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Wormholes

2022-08-30T13:10:11Z

Sevoris: /* Wormholes and time travel */

Wormholes are hypothetical structures in space-time allowed by the general theory of relativity.
They provide short cuts through space-time that connect one region of space and time to another; potentially, these regions can be very distant from each other.
The symmetries in space-time of our universe mean that wormholes move like physical objects, and acquire the conserved (or in some cases approximately conserved) quantities of things that go into them and lose those quantities of things that come out – properties like energy, mass, momentum, angular momentum, and electric charge.

==What is a wormhole? The geometry of space-time and how to twist it into a pretzel==

We're going to have to start out with some pretty heady stuff. Like the very nature of existence. And what is reality, anyway?
Usually, space and time are so much an integral part of our existence that we don't even think of them at all. They simply form a backdrop on which all the interesting stuff plays out. But when we do think about them, we're so accustomed to living in a world where they exist and where distances are well defined and events can be uniquely identified based on <i>where</i> and <i>when</i> they happened (or are happening or will happen) that a lot of stuff seems so obvious that we don't stop to think about if it really has to be this way or if there are other possible ways of doing things.

===A toy model===

So lets consider a hypothetical scenario, where an ant can be in any of several states, represented by the squares in the pictures below. The ant can transition to any other state that has a numbered side with the same number as the state it is in. We will start on the left. If the ant chooses the "2" transition, we end up with the picture in the middle. If it then chooses the "10" transition, we end up with the state on the right.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Space_elements_1.png|frameless]]
<td width=400>[[File:Space_elements_2.png|frameless]]
<td width=400>[[File:Space_elements_3.png|frameless]]
</table>
But now notice that the behavior of this system is exactly the same as if we re-arrange the states into a grid, as shown below on the left. The disconnected states now become a space; an expanse of coordinates on which the ant can exist. Now, the ant's transition between states can be described as a trajectory as shown in the right picture below.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Space_elements_grid.png|frameless]]
<td width=255>[[File:Space_elements_grid_path.png|frameless]]
</table>

This gives us an inkling of how something like space-time might end up being assembled from individual events. But what if we don't have things match up into quite such a perfect grid? We
ll take our space elements and tile them into an infinite grid. Two sections of the grid are shown below. But now we remove two of those space-tiles, and we connect the edges adjacent to the removed tiles with a corresponding edge of the other removed tile (although, as shown in the figure, we don't have to keep the same orientation). Now the ant, entering side 1 (which we have color-coded red for convenience) comes out of side 1 of the other missing square. This is perhaps the most basic wormhole.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Space_elements_wormhole_grid_1.png|frameless]]
<td width=400>[[File:Space_elements_wormhole_grid_2.png|frameless]]
</table>

Of course, this simple assembly only stitches together a space. But the world we live in is one of space-time. As the ant transitions from one square to another, it is also transitioning further ahead in time. But when the ant passes through the wormhole, while it locally experiences no discontinuity that wormhole might connect to a different time as well as a different region of space. So the ant might emerge far in the past or future of when it stepped through.

===Manifolds and coordinate patches===

The toy model used above hopefully prepares you for some of the ideas to follow - one of how we can piece together a complete space and time by connecting together different pieces. But before we go further, lets define an important term: a <i>manifold</i> is a "shape" of a given number of dimensions where, if you zoom in close enough, you can always get to a scale where it looks like the shape is flat. We'll use this definition to help us describe how we can put space and time together to get a wormhole. But first, let's look at a few illustrative examples.

====One dimension====

The simplest one dimensional shape is a line. And a line is everywhere flat. So a line is a one dimensional manifold.

A one dimensional shape might seem to be a wiggly path if inscribed in two or more dimensions, but it has no <i>intrinsic</i> curvature. That is, if everything about your world is confined to that path, you can't tell that it is curved – all you can do is measure the distance you go along the path. This greatly limits the kinds of shapes that we can consider. So any curved path of infinite extent looks from the inside like a line, and so any infinite length path is also a manifold.

But there's another class of one dimensional shapes that are manifolds – loops. A loop is a path of finite length that connects back with itself. You can make one by drawing a closed curve in a higher dimensional space. Or you can take a straight line but then say that a given point <i>here</i> maps on to another point <i>there</i>, and all the points between <i>here</i> and <i>there</i> form a loop. When you are marching along the line starting at <i>there</i> and you reach <i>here</i> you are also at <i>there</i>, so if you go further you just end up back where you were. This last bit may seem like a pointless bit of pedantry that just confuses a simple problem, but it will be a very useful way of looking at things as we go on.

====Two dimensions====

The simplest two dimensional manifold is a plane. A plane is everywhere flat, and extends off to infinity. A small region around any point on a plane (naturally) locally looks like a plane, so it meets our definition of a manifold.

=====Curved manifolds=====

But in two dimensions, we can have something that we don't have in one dimension – intrinsic curvature. Consider a sphere. If you get close enough to the sphere, it will look like you are on a plane; we live on a world that is (approximately) spherical (if you neglect mountains and ocean basins and all that other stuff), but when you stand up and look around you it mostly looks flat. So a sphere is a manifold. But take a line on the surface of your sphere. Extend it out along the surface, drawing it as straight as possible. Eventually, it will go all the way around the sphere, dividing it in two, and meet itself. This path is called a great circle. The Earth's equator is (approximately) a great circle, as are the lines of longitude (but not lines of latitude, because they do not divide the Earth into two equal-sized halves). The straightest-possible path on a curved surface (like a great circle on a sphere) is called a <i>geodesic</i>. But to keep the jargon down, we'll just call them lines for this non-technical presentation, and hopefully this will keep things more intuitive and understandable to the layman.

In flat space, two lines that are initially parallel remain parallel, always remaining the same distance apart. But in a curved space, initially parallel lines do not remain the same distance apart. On a sphere, for example, any two great circles will intersect at two points. A sphere thus has intrinsic curvature.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Sphere_and_great_circles.png|frameless]]
<td>[[File:Hyperbolic_surface_and_lines.png|frameless]]
<td>[[File:Cylinder_and_lines.png|frameless]]
<tr>
<td width=400>Two great circles going around a sphere. Although initially parallel, they meet at two points.
<td width=400>Two initially parallel lines on a hyperbolic surface. As you get farther from the place where they are parallel, the lines diverge.
<td width=400>Lines that are initially parallel on a cylinder remain parallel everywhere.
</table>

On a sphere, parallel lines always end up converging. One example of a curved surface where parallel lines diverge is a hyperbolic surface. Like a sphere, a hyperbolic surface is a manifold. But it has the opposite sense of curvature.

Now consider a cylinder. Again, a cylinder is a manifold. A cylinder can be made by rolling up a plane. Like our construction of a loop, this can be done by simply taking a long rectangular section of a plane and saying that two of the parallel edges are actually the same (this is essentially what you do if you roll up a sheet of paper so that two of its parallel edges touch, and then pretend that they are merged together where they meet). This "rolling up" preserves distances, so any two parallel lines on the plane remain the same distance apart on the cylinder (think about that sheet of paper, and if you drew two parallel lines on it - they will stay the same distance apart when the paper is rolled up). A cylinder has <i>extrinsic</i> curvature – you can tell that it is curved when looking at it from three dimensions. But it has no <i>intrinsic</i> curvature – if you are confined to the surface of the cylinder it behaves locally as if it is flat.

A manifold that is flat obeys all the rules of Euclidean geometry (although you'll need to go to the three dimensional case for dealing with volumes), and the standard Cartesian coordinate system (representing position by two orthogonal coordinates, commonly called <math>x</math> and <math>y</math>) can be used to specify any point. A manifold that is not flat violates Euclid's fifth postulate; the geometry of the manifold will be <i>non-Euclidean</i> (somewhere, H. P. Lovecraft is quietly having a mental breakdown). It cannot be represented by Cartesian coordinates, but different sets of coordinates – or perhaps patches of multiple coordinate systems that apply in different places – can be defined instead to locate points.

=====Topology: simply and multiply connected manifolds=====
If you take a plane or a sphere, any closed loop on the manifold can be shrunk in a continuous manner until it vanishes. This is called <i>simply connected</i>. But not all manifolds are simply connected. Those which are not simply connected are called <i>multiply connected</i>. The most basic multiply connected manifold is a torus, like a bagel or doughnut or bicycle inner tube. While there are closed loops on a torus that can be shrunk to a point and made to vanish, those that go all the way around the torus tube cannot – they'll get caught up circling the tube. Likewise, loops that go all the way along the tube will get caught up on the central hole if they are shrunk and again cannot be made to go to a point.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:torus_and_loops.png|frameless]]
<td width=400>[[File:Topological_torus_and_loops.png|frameless]]
<tr>
<td colspan=2>Three classes of loops on a torus. On the left is a standard geometrical torus; on the right is a topological torus that is flat but has opposite edges identified so that they are connected. The blue loop doesn't go around the torus, it can be continuously deformed to shrink to a point and vanish. The red line goes around the torus the long way. If you shrink it, it will get caught up going around the hole in the middle and get stuck there; it cannot be continuously deformed to make it any smaller. The green line loops around the torus from the outside to the inside and back. Again, if you shrink it the line gets caught up circling the torus tube and can't be made any smaller.
</table>

And this brings us to another interesting kind of manifold. You can get a manifold that is flat but has the topology of a torus. You can do this by taking a rectangle and saying that the opposite sides are actually the same. This gives you the effect of a "wrap-around screen" like on that old arcade game Asteroids – if you go off the edge on one side, you just come back on the opposite side at the same height. in physics, this is known as "periodic boundary conditions". Closed loops that go all the way around one way or the other cannot be shrunk to a point. But this kind of surface <i>cannot</i> be embedded in three-dimensional space! We might have grown used to surfaces that we can envision existing in the space we live in, but there are manifolds, even fairly simple two-dimensional manifolds, that cannot be embedded in higher-dimensional spaces like that. Remember this when we get to actual wormholes.

You can have multiply connected manifolds that are not toruses, even though they might have the same topology; any manifold with one or more "loops" or "handles" is multiply connected.

=====Non-orientable manifolds=====

And there is one other interesting kind of manifold that is relevant to wormholes – manifolds that are <i>non-orientable</i>. An orientable manifold is one where you can define a consistent sense of clockwise and counter-clockwise everywhere on the manifold. If you start with a clock spinning clockwise, or a right-handed glove, or a copy of Lewis Carrol's "Through the Looking-Glass, and What Alice Found There", you can move it anywhere on the surface and if it meets back up with itself it will still be spinning clockwise, or right handed, or normally readable. But in a non-orientable surface, there are ways you can move the object so that when you get it back to where it originally was the clock is spinning counter-clockwise, or the glove is left-handed, or the book is written in mirror writing. The simplest example of such a surface is a Mobius strip. As shown below, if you move an oriented circle around the strip, it comes back with the opposite orientation.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Topological_Mobius_strip.png|frameless]]
<td width=400>[[File:Mobius_twist.png|frameless]]
<td width=400>[[File:Mobius_strip.png|frameless]]
<tr>
<td colspan=3>A Mobius strip … a strip of surface with a twist so the edges where it connects meet up "the wrong way around". On the right is a 2D strip where you connect along the short edge matching the 1's together and the 2's together. The center shows the same strip with a twist in it so you can line the edges up the way you are supposed to, demonstrating how the twist reverses the orientation of the spinning circles. An embedded image in 3D space with the edges connected together is shown on the left. You cannot define a consistent orientation to this surface – starting at the red counter-clockwise circle, if you move the circle continuously along the surface when you get back where you started you will find it is going clockwise!
</table>

====Three dimensions====

The most basic three-dimensional manifold is a flat space. This is, well, just a normal space. You can go off in any combination of three independent directions (left-right, up-down, front-back), and because it is flat, initially parallel lines stay parallel. It all behaves much as you would expect three dimensional spaces to behave.

But you can get variations on this. They are harder to visualize than two-dimensional cases, because we don't have the brainspace for processing curved spaces inside of a four-dimensional hyperspace. But you can do things like identifying two planes to be the same, such that if you go far enough in one direction you can end back up where you started again (akin to a cylinder), or make them curved such that distances are distorted and lines don't remain parallel and maybe you can put larger things inside of smaller surfaces. There are 3D equivalents of spherical and hyperbolic surfaces (called, naturally enough, spherical and hyperbolic spaces). There are even non-orientable spaces.

One way of trying to envision a curved space, and to gain some intuition about them, is to take a two-dimensional slice of that curved space and embed that in a flat three-dimensional space. This is called an embedding diagram. Remember that this is a reduced representation that throws away a lot of the information (unless there are special symmetries you can exploit) and is not actually the thing being described.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Space_with_embedded_plane.png|frameless]]
<td width=400>[[File:Embedded_plane.png|frameless]]
<td width=400>[[File:Embedding_diagram_example.png|frameless]]
<td width=400>[[File:Embedding_diagram_example_very_curved.png|frameless]]
<tr>
<td>A space with an embedded plane. We show a sphere in this space, with the plane bisecting the sphere.
<td>Just the embedded plane. Note how in a spherically symmetric manifold (like this space), a circle on the embedding diagram represents a complete 2-dimensional spherical surface.
<td>An embedding diagram of a "dimpled" space. Just as before, the closed circles represent closed spherical surfaces. It is just that in this case, the distance between two concentric spheres of radius <math>R_1</math> and <math>R_2</math> is longer than the usual <math>R_1 - R_2</math> because of the spatial curvature, as shown by the increasing distance along the dimple's surface.
<td>An embedding diagram of an extremely curved section of space. In this example, there are several places where we have a bigger sphere inside of a smaller one! Curved space is weird like that.
</table>

Remember that not every manifold can be embedded, so not every curved space can be represented by an embedding diagram. But you can take patches of the curved space and show them by embedding diagrams, and then describe how these patches stitch together to create a contiguous surface.

====Space-time: three space dimensions and one time dimension====

This is the world we live in. So this is the class of manifolds that we really need to use when making a wormhole. Sometimes it's abbreviated as 3+1D (as opposed to 1D, 2D, 3D, etc.).

The basic, infinite, flat space-time manifold is often called Minkowski space. I know, it's a big word using scary foreign-sounding names. But all it means is that its a 3+1D manifold that goes on forever and is flat. In flat space-time, everything obeys the rules of special relativity. Which means it really doesn't work when masses become significant enough to make gravity.

Our best model of how gravity works is general relativity. In this theory, the curvature of space-time is created by mass. At least, that's what they always say in the over-simplified pop-sci summaries of general relativity. Actually, space-time curvature is created by all the components of the stress-energy tensor. It's just that in almost all cases, mass is the most significant part of this. But momentum, stresses, pressures, and shear forces all have contributions as well. In general relativity, physical objects move along the straightest possible lines in this curved space-time. The way the curvature causes the lines to bend and converge is what leads to the appearance of gravity as a force if the curvature is sufficiently weak.<ref name="Misner_Thorne_and_Wheeler">Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler, "Gravitation", W. H. Freemann and Company, New York (1973)</ref>

The most basic curved space-time manifold is the Schwarzschild geometry, which is what you get when you have an uncharged, non-spinning spherical mass doing its distortion thing on space-time. The Schwarzschild geometry is <i>static</i> meaning it does not change in time, and is <i>spherically symmetric</i>, meaning that it has a definite center and it doesn't matter which direction you are relative to that center – it is independent of angle. If the mass becomes dense enough, space-time gets so distorted that an event horizon forms. Nothing that goes into the event horizon can ever come back out, and the inevitability of the geometry crushes the mass down to an infinitely dense point at the center called a singularity. Nothing on the inside of the horizon can ever escape. It forms a <i>black hole</i>, that forever traps any light or matter that enters it. For masses that don't exceed this critical density, the gravitational distortion outside of the object is the same as for a black hole of equal mass, but there is no event horizon. The object can still be seen and interacted with.<ref name="Misner_Thorne_and_Wheeler"></ref>

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Schwarzschild_star.png|frameless]]
<td width=400>[[File:Schwarzschild_black_hole.png|frameless]]
<tr>
<td>An embedding diagram of the Schwarzschild geometry of a dense, massive star. The physical extent of the star is shown in blue.
<td>An embedding diagram of the Schwarzschild geometry of a black hole of the same mass as the star to the left. The event horizon is shown in red.
</table>

There is one distinction between space-time manifolds that will become important later. Some manifolds have an isolated region of high curvature (corresponding to a concentration of mass, or at least of stress-energy) but as you go farther away from the strongly curved region the space-time becomes increasingly flat. If you can always go out far enough away from a curved region that space-time becomes flatter that whatever flatness criterion you choose, no matter what direction you choose to leave the curved region, the manifold is called <i>asymptotically flat</i>. For our purposes, if you can go far enough away that the Newtonian approximation to general relativity is accurate, and gravity can be described as a force rather than requiring a geometric description, then the geometry can be considered asymptotically flat. You can also allow gravitational waves in your asymptotically flat space-time – their very small space-time curvature is not enough to cause problems and they can still be described by a linearized version of gravity. Both the Schwarzschild and the Minkowski geometries are asymptotically flat. Spherical or hyperbolic geometries are not asymptotically flat. Planets, stars, galaxies, neutron stars, black holes, and galactic clusters are all approximately asymptotically flat; but the universe as a whole is <i>not</i>. As long as we confine our attention to merely galactic clusters, we can make the approximation of asymptotic flatness.

===Making a wormhole===

Now that we have covered some of the introductory material, we can get down to the main course. To make a wormhole, take two regions of space-time, usually far apart and usually asymptotically flat (although neither of these is strictly required) and connect them together. That's it. You're done. Sounds easy, doesn't it?

Note that if you are connecting two regions in the same universe, your new geometry will be multiply connected. A loop that goes through the wormhole and then connects back up outside the wormhole cannot be shrunk to a point. There will now be two equivalent ways to get between any two points – the path that goes through your usual space-time, and the path that goes through the wormhole. If you use a wormhole to connect two separate universes, the combination of the two universes and the wormhole can still be simply connected.

In general relativity, you can distort the geometry of space-time with stress-energy, but there is no way to change its topology; that is, no way to turn a simply connected universe into one that is multiply connected, or to add or remove more "loops" or "handles" to an already multiply connected universe. Such changes, if they are possible at all, would only be possible in the realm of quantum gravity and we do not know enough about quantum gravity to know if it would be possible or not. So, it might not be possible to make intra-universal wormholes at all. Or maybe quantum gravity does allow topology changes in space-time and there are ways to connect different parts of our universe together with wormholes. On the plus side, if you can't change the topology of space-time, you can always bud out a new universe from our own – perhaps by triggering the inflationary field that started our universe. It will be connected to ours by a wormhole. And because the topology can't change, that wormhole would never be able to break. We would have an eternal bridge to the new universe. So a system of physics that does not allow topology changes still allows wormholes to other universes.

==Common kinds of wormholes==

===Spherically symmetric wormholes===

General relativity involves a lot of non-trivial math. You need to deal with four-dimensional curvature tensors with up to 20 independent components, and figure out how they relate to a given distribution of matter, fields, energy, stresses, and material flows. If you can reduce the number of dimensions you need to keep track of, it can simplify things a lot. One way of doing this is assuming your space-time geometry does not change with time (it is static), and looks the same in all directions when viewed from a central point (it is spherically symmetric). This makes things a lot easier to analyze.

Astute readers will have noticed that these conditions are satisfied by the Schwarzschild geometry. In fact, if you play around with the Schwarzschild geometry enough, you will find that you can even get a wormhole-like solution. At the event horizon, you can connect your black hole solution to a time-reversed structure called a white hole. This is called an Einstein-Rosen bridge. Unfortunately, it has some issues. For one, it can't form naturally from infalling matter creating a black hole. For another, going through an Einstein-Rosen bridge involves passing through an event horizon, so the trip is necessarily one way. And finally, you can show that even the smallest amount of matter or radiation falling into the Einstein-Rosen bridge leads to it collapsing before the matter or radiation can pass through, trapping it inside the black hole. So this is not looking particularly useful.

So what are we going to do? Well, the Einstein-Rosen bridge, like a lot of early work in general relativity, was made by assuming a distribution of stress-energy (in this case, matter) that was assumed to be physically plausible, and finding out what the resulting geometry would be. But we want a wormhole, dangit! So what happens if we just demand a static tunnel connecting two asymptotically flat regions of space-time? If you do this, you get a Morris-Thorne wormhole<ref name="MorrisThorne">Michael S. Morris and Kip S. Thorne, "Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity", American Journal of Physics <b>56</b>(5), 395-412, May 1988</ref>. You still have the relation between stress-energy and geometry, but now it is your assumed geometry dictating the stress-energy rather than the other way around. One difficulty with this method is that the distribution of stress-energy you end up with might not end up seeming physically plausible. In particular, it can be shown that any wormhole must have regions in it with negative energy density<ref name="Visser_Lorentzian_wormholes">Matt Visser, "Lorentzian Wormholes: From Einstein to Hawking", Springer/AIP Press (1996)</ref>, and these spherically symmetric wormholes are no different. We will discuss the issues around and solutions to the negative energy density problem later.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Morris_Thorne_wormhole.png|frameless]]
<tr>
<td>An embedding diagram of a Morris-Thorne wormhole. For this wormhole, we just took two copies of the Schwarzschild geometry from above and connected the middle "dimples" with a cylindrical patch of space-time.
</table>

The places where you enter a wormhole are called it's <i>mouths</i>, and most wormholes have two. The space-time tunnel between the mouths is called the wormhole <i>throat</i>.

====Some common problems with spherically symmetric wormholes====

if you don't have your distribution of matter and energy going off to infinity, then eventually if you go far enough away your geometry will reduce to the Schwarzschild geometry. A good many published examples of such wormholes are actually Schwarzschild-like up until pretty close to where the event horizon would be. The first problem you might encounter, then, is the formation of a horizon. A single horizon makes any trip one-way. Going into it on the side with the horizon would be indistinguishable from entering a black hole - except that you pop out somewhere else instead of getting removed from existence at the singularity. And because of the horizon, you couldn't ever go back. If both sides have a horizon, you simply can't get out of it. Why do you even have this thing, anyway?

The strong curvature as you get close to the throat can also produce very strong tidal forces. Due to the gravitational forces, if you go in feet-first your feet could be pulled much harder than your head as you go in and you could end up getting pulled apart – spaghettified is the technical term. In addition to these pure spatial tides, it turns out that you also have velocity-dependent tides. These tides are transverse to the radial direction - if you dive in really fast, you will be squeezed even harder from side to size.

Early attempts to limit the tidal forces while still keeping a throat large enough to fit a person through resulted in very large masses. Like planetary scale masses or even solar scale masses. This would mean you would need to spend a lot of propellant de-orbiting into the wormhole and then rocketing your way back out. Some of these early wormholes also had issues like the extreme gravity blue-shifting the cosmic microwave background to blow-torch intensity x-rays. Sometimes, attempts to keep the tides down also lead to really large throat sizes. Like the distance from the Sun to the Earth sizes. Or even light year sizes. These gigantic wormholes would be rather inconvenient.

None of these are necessarily intrinsic limitations to spherical wormholes. But it does indicate that a perfectly reasonable assumption for the wormhole tech in your science fiction universe could involve wormholes with Jupiter-level masses, extreme tides, and other fun details.

===Thin shell wormholes===

Take a shape located somewhere in flat (Minkowski) space-time. The important part of this shape is that it must be a closed surface that completely encloses a given volume. Now go someplace else in space-time and take an identical shape. Remove all of the space-time from inside both shapes, and then say that the corresponding parts of the two surfaces of the shapes are actually the same points. Now you have a thin shell wormhole. The throat region is infinitesimally short – the section that corresponds to the wormhole is just that surface of your shape, hence the name "thin shell".

The first thin shell wormholes proposed were simple polyhedra <ref name="Polyhedral_wormholes">Matt Visser, "Traversable wormholes: Some simple examples", Physical Review D vol. 29 no. 10 pages 3182-3184 (1989)</ref>. These are attractive choices for a potential wormhole-user because the flat faces of a polyhedral thin-shell wormhole have no tidal effects whatsoever. Further, the flat faces do not have any stress-energy either. It is simply flat space-time, no different from any other area of flat space-time. Going through the face of a polyhedral wormhole is no different from going through the flat space-time of the doorway separating your hallway from your bedroom. The negative energy stuff is concentrated entirely in the edges and corners of the polyhedron. And hoo boy, do you need a lot of it. For a cubic wormhole, each edge requires approximately <math>-1.52 \times 10^{43}</math> J/m of length. This is <math>-1.69 \times 10^{26}</math> kg/m, or close to a tenth of a Jupiter mass per meter of edge length. It has been proposed that perhaps cosmic strings – defects of the primordial universe where things couldn't line up the right way and left a linear flaw in reality – with negative tension might be able to meet this requirement and hold open a polyhedral wormhole.

But there's a strange thing when you consider the mass that these wormholes require. They are embedded in flat space-time. Which means that outside of that thin shell they have, they induce no curvature on the space-time around them. Remember that mass causes space-time to curve, dimpling up as in the Schwarzschild geometry shown earlier. Because these wormholes induce no far-flung curvature to imprint on the surrounding space-time, they have zero mass (technically ADM mass). And because mass is equivalent to energy, it should technically be possible to make one of these for no energy input at all from those of us in flat space-time. Forces are the rate of change of energy with distance, pressures are the rate of change of energy with volume. But because the energy is always zero no matter what size they are (as measured from out in flat space-time), there would not be any force or pressure needed to hold them open. The extreme conditions of their interior contrasts with the relatively benign conditions once away from the thin shell (or, indeed, within any flat area within the shell).

A curious feature of polyhedral wormholes is that each face of the polyhedron might connect to a <i>different</i> polyhedron. You might, for example, have a collection of seven cubical wormhole mouths, and each of the six faces of the wormhole you are facing connect to a different one of the six remaining mouths.

For more general thin-shell wormholes, you just need to smear that negative mass out a bit. As long as you keep a flat area to go through, travelers will not experience any strange tides or encounter space-warping exotic energies. You could, for example, have your standard circular portal beloved of fiction. Or you could have any other arbitrary shape you wanted.

Thin shell wormholes are what you get when you take a mathematical limit; they would be an approximation of a more physical distribution of stress and energy. Any realizable version would have a finite, although possible small, thickness to its shell and hence a finite, although possible small, throat length. As we will discuss later, they would likely also have a positive, although possible small, mass. But as an approximation, and as a valid solution to Einstein's field equations in general relativity, they allow us to explore many aspects of wormhole physics using a relatively simple model that is easy to calculate, and can lead to a lot of interesting inspiration for fictional ideas of wormholes.

====Tidal distortion during transit====

Consider a thin shell wormhole whose shell-shape is everywhere convex - no flat areas this time. For convenience, let's make it a sphere. As a traveler passes through the shell, the sections of his body that pass the shell must suddenly "pop" from curving one way to curving the other way, as illustrated below:

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Velocity_dependent_tides_pre_transit.png|frameless]]
<td width=400>[[File:Velocity_dependent_tides_in_transit.png|frameless]]
<tr>
<td>A traveler (left rectangular shape) about to go through a spherical thin-shell wormhole (here represented as circles, with the interior volume blacked out to indicate that there is no space-time there and nothing can exist in that region).
<td>The traveler in the process of passing through the wormhole. Note that on the wormhole shell, the portion of the traveler to the left has a leftward-curvature, while the portion on the right has a rightward curvature. As the traveler passes through, each section that goes through the shell immediately jumps from left-curving to right-curving. This will induce stresses in the traveler as his interior is deformed.
</table>

it is clear that as the curvature of the wormhole becomes large compared to the size of the traveler, the distortion will decrease. In the limit of a flat section of the surface, no distortion is experienced at all.

A consequence of this is that something going through a curved section of a thin shell wormhole will experience stresses (distortions) and strains (forces). The elastic energy stored by a strained object is proportional to the product of the stress and the strain, added up over all of its volume. Assuming that the object going in to the wormhole is initially in its relaxed state, it will require energy to push it into the wormhole shell. If it enters with insufficient initial kinetic energy to get it through or external forces pushing on it, it will just bounce off.

===Cylindrical and toroidal wormholes===

In principle, you can have a wormhole with cylindrical symmetry
<ref>Kirill A. Bronnikov and José P. S. Lemos, "Cylindrical wormholes", arXiv:0902.2360 [gr-qc] 24 Feb 2009</ref>
. This forms a valid solution to Einsteins field equations in general relativity. However, a true cylindrical wormhole would be infinite in length. So they probably don't exist naturally, and you won't be able to build one. Unlike all the other wormholes discussed here, they are not asymptotically flat. Unlike wormholes of finite extent, not all cylindrical wormholes require negative energy regions. However, it has been shown that negative energy stuff is needed if the geometry radially far enough away from the wormhole approaches the flat space-time of our usual universe, so a universe with a cylindrical wormhole without negative energy would not look like our own.

So the thought is that maybe you can get close enough to a cylindrical wormhole by making a really long wormhole that bends ever so slightly and comes back and meets itself, forming an extremely stringy torus. The general case for this has not yet been worked out, but a thin shell approximation of a toroidal wormhole has been developed
<ref>Vladimir Dzhunushaliev, Vladimir Folomeev, Burkhard Kleihaus, and Jutta Kunz, "Thin-shell toroidal wormhole", arXiv:1901.07545v2 [gr-qc] 28 Jan 2019</ref>
.
These thin shell toriodal wormholes have been shown to be stable to perturbations, one nice benefit that has not been established for a lot of other kinds of wormholes. They do require some negative energy regions on their surface, but there will also be some regions that do not contain negative energy.

===Non-orientable wormholes===

In the discussion on non-orientable manifolds, we showed how in these manifolds you can't properly determine the clockwise - counter-clockwise orientation of objects or motions, or the orientation of objects with respect to their mirror reflections. If you move an object along certain paths, when it comes back it might come back the wrong way around. So it is probably no surprise that a non-orientable wormhole turns the universe into a non-orientable manifold. Something going through a non-orientable wormhole comes out looking like it has been mirror reflected.

Normally this seems like it would be rather boring and useless, but for an obscure constraint from quantum physics called the CPT theorem. This theorem, which holds for all physical phenomena, means that the product of the discrete symmetries of charge conjugation (C), parity transformation (P), and time reversal (T) always returns you to your original state. Huh? What does that mean? Well, parity transformation is equivalent to a mirror reflection; so passing through a non-orientable wormhole means parity is inverted. This means that exactly one of the other two symmetries must also be inverted for the CPT theorem to hold. The thing coming out on the other end is still manifestly going forward in time, so it must be charge conjugated, whatever that is.

Charge conjugation means you turn all particles into their antiparticles. So anything passing through a non-orientable wormhole emerges on the other end made entirely out of antimatter.

While this particular turn of events might discourage people from passing through, it does offer obvious application for energy generation, space propulsion, and weaponry.

==Exotic energy conditions==

It is fairly easy to see than any wormhole requires negative energy densities (or more generally, regions where the tension is higher than the energy in natural units) to exist. Take a look at any of the wormhole diagrams above – the spherically symmetric ones will probably make it the most clear. If you send a parallel ray of stuff – maybe particles or light or golf balls – through the wormhole, and trace out the straightest possible path of all of those parallel lines, you will see that after they go through the wormhole all those initially parallel trajectories will now be diverging. Positive energy (such as mass) causes attractive forces, drawing trajectories toward it and making the paths converging. To get diverging paths you need the opposite – repulsive gravity – which you can get from negative energy densities or very high tensions. Having the tension be higher than the energy density turns out to be equivalent to a negative energy density – in some frames of reference observers will measure a negative energy density there.

A lot of physicists used to be reflexively dismissive of negative energy densities. After all, they can't really exist; all energy in the real world is positive.

Except that it is not. There are certain odd cases that allow for negative energy densities. One of these is the Casimir vacuum. A conducting boundary cannot support an electric field parallel to that boundary (if there was such a field, charge would flow under the force of the field until the parallel component of the field is screened out). So if you have two parallel conductive plates, the only electromagnetic waves that you can get between them require their fields to vanish at the plates. This restricts the number of ways the electromagnetic field can vibrate, called modes. In quantum mechanics, merely the potential for a vibration is associated with a certain energy (called <i>zero point energy</i>) even if no quanta of vibration are actually present. By making these modes impossible, they cannot contribute their zero point energy to the region between the plates. This lowers the total energy of the space between the plates compared to the region outside the plates (or with the plates absent). Because the energy of empty space is zero, this means that the region between these Casimir plates has negative energy. The Casimir effect has been measured in the laboratory. It is fairly weak, but it is present. In fact, not long after traversable wormholes were first described, one proposal suggested holding open the wormhole using the Casimir effect<ref name=MirrosThorneYurtsever">Michael S. Morris, Kip S. Thorne, and Ulvi Turtsever, "Wormholes, Time Machines, and the Weak Energy Condition", Physical Review Letters, Volume 61, Number 13, 26 September 1988, pages 1446-1449</ref>

Another situation that gives rise to a negative energy density is a <i>squeezed vacuum</i>. A squeezed state is a way of manipulating a quantum oscillator (like a pulse of light) to get around the uncertainty relations of quantum mechanics to increase the precision in one variable by increasing the uncertainty in its conjugate variable. Applying this operator reduces the energy of the oscillator, and if this is applied to the vacuum state, which already has zero energy, you can end up with regions of negative energy.

It is worth noting that not just any old negative energy region of exotic space-time will work. In particular, dark energy is a condition of the universe we live in where even bare vacuum itself has a small amount of negative energy. The current best model we have for this is as a cosmological constant - where every part of the universe has an equal density of negative energy and a related amount of pressure. But the wormhole allowing contributions to space-time curvature of that negative energy are exactly cancelled by the wormhole-denying contributions of the pressure. So according to our best current models, dark energy will not support a wormhole.

Also, the Casimir vacuum described above is also problematic. The exotic vacuum between the plates does have the necessary properties (if not magnitudes) to support a wormhole, but this is generally far overwhelmed by the immediately adjacent positive mass matter making up the conductive plates. The technical term for the criterion that determines whether you can have a wormhole or not is the Averaged Null Energy Condition (ANEC). If you project a ray through space-time that follows the path that light (in a vacuum) could take, and do some math along that ray and add up all the contributions from all the places along that ray, then if your result is zero or positive then the ANEC is said to be satisfied. If this is the case, you can't get a wormhole. If the ANEC is less than zero, it is said to be violated and in principle you could use that weird exotic energy stuff to prop open a wormhole's throat. There are other so-called energy conditions, and if you have regions of space-time that also violate those energy conditions they can be used to hold open wormholes as well - but all of these also violate the ANEC, so the ANEC is all you need to consider.

===So just how do I get the ANEC-violating stuff I need?===

The short answer is, we don't know.

As we already said, a cosmological constant form of dark energy won't do it (although different ideas about dark energy, such as phantom energy, could conceivably work). The inflationary vacuum thought to have existed very briefly during the first instances of our universe's existence is also usually thought to be a cosmological constant (although one that stopped existing after a very brief period of time, so I guess it is less than constant), so the inflationary vacuum probably won't work either.

If you can get the Casimir vacuum without the massive conducting plates that bound it, you might be able to do something with that. For example, the twisted geometry of space-time itself in a wormhole throat imposes boundary conditions that restrict the modes of vibration inside of it similar to the Casimir effect. This is called the <i>topological Casimir effect</i>, potentially it could be used to help support a wormhole. Also, although the ANEC is satisfied if the probing space-time ray goes through the Casimir plates, a ray that goes between the plates but never touches them will violate the ANEC. Can you use this to hold open a wormhole? Maybe a sufficiently clever person will eventually figure out how to do so.

Even more serendipitous, it turns out that gravity itself naturally squeezes the vacuum
<ref name="gravitationally_squeezed_vacuum">David Hochberg and Thomas W. Kephart, "Lorentzian wormholes from the gravitationally squeezed vacuum", Physics Letters B 268, 377-383 (1991)</ref>
. If you have the curved space-time of gravity, all of the modes of oscillation in it become squeezed states. So it might be possible that the curved space-time of a wormhole can squeeze the vacuum enough to support itself. Or a combination of gravitational vacuum squeezing and the topological Casimir vacuum from the wormhole's geometry constraints might be all that is needed. At least one proposal has suggested a class of wormholes that can be self supporting in such a way
<ref>S. V. Krasnikov, "Toward a Traversable Wormhole", arXiv:gr-qc/0003092v1 22 Mar 2000</ref>
<ref>S. Krasnikov, "Traversible Wormhole", Physical Review D, volume 62, article 084028 (2000)</ref>,
although the throat of these wormholes is very "wrinkly" or "crumpled up" and it is unclear what the effects of that would be on anyone passing through.

====Quantum energy inequalities====

When you apply quantum field theory to what happens when you have a region where the quantum fields produce local negative energy densities, you end up with a limit of how much negative energy stuff you can have before it has to be balanced out by an even greater amount of nearby positive energy stuff
<ref>[https://web.archive.org/web/20060206055950/http://maths.york.ac.uk/www/PhysicsQIneq.htm|Quantum Energy Inequalities]</ref>
<ref>[http://www.phys.lsu.edu/mog/mog20/node16.html|Quantum field theory on curved spacetime at the Erwin Schrödinger Institute]</ref>
<ref name="Fewster_Roman_2005">Christopher J. Fewster and Thomas A. Roman, "On wormholes with arbitrarily small quantities of exotic matter", Physical Review D <b>72</b>, 044023 (2005)</ref>
.

====Matter with negative mass====

It might be tempting to try to solve the negative energy density requirement for wormholes by introducing some sort of matter that naturally has negative mass.
However, negative mass would introduce all kinds of problems to the smooth operation of how the world works. For one thing, if negative mass (or equivalently energy) can exist, why doesn't it just spontaneously pop out of empty space accompanied by an equal magnitude of positive mass (or energy)? For another, negative masses lead to all kinds of runaway instabilities. For example, imagine what happens if you push on an object with a negative mass. You exert a force on it with your hand. But by Newton's second law of motion, an object's acceleration is proportional to its mass. So the negative mass thing will accelerate backwards, against the direction of the force, and into your hand. Which means you will be pushing even harder on it, which makes it push back even harder than before. You have a runaway process that blows up until your hand breaks, and any fragments of flesh and bone and sinew that were in the way of the negative mass thing have been flung off.

Or consider a negative mass thing with an electric charge. If you put something with an electric charge in an electric field the field will exert a force on the charge. For a normal positive mass charged thing, it will move in the direction of the force. The work done is the force in the direction of the displacement in position, so because the force and motion of the object are in the same direction, the field does work on the charged object and energy flows from the field into the kinetic energy of the charged object. But if the charged object has negative mass it will begin to move opposite to the direction of the force. This will do work on the field instead of the object, increasing the energy of the field. The energy comes from the kinetic energy of the negative mass charged object – but because kinetic energy is proportional to mass and increases with increasing speed, as the negative mass object loses kinetic energy it gains speed, going ever faster. Both the field strength and the speed of the object increase without bound. If the electric field originates from the electric part of an electromagnetic wave, it will vary sinusoidally with time and space. As the negative mass object is forced into a sinusoidal trajectory opposing the electric force on it, it will amplify that wave making it ever more intense. A single, isolated charge exposed to an electromagnetic wave will eventually have its inertia keep it moving from the previously imposed force even after the force has switched direction, so if the negative mass is free with no other forces acting on it there will be no net amplification. But if there are any other damping forces, or internal arrangements of bound charge that can resonate with the wave, then instead of being absorbed (as would happen with a positive mass), the wave will be amplified indefinitely and the ever-increasing energy of the wave will come from the ever increasing temperature of the negative mass thing (because the mass is negative, the thermal motion of temperature gives it a negative energy whose magnitude only increases as the temperature rises).

And, of course, the quantum energy inequalities indicate that you can't just have isolated lumps of negative mass stuff floating around.

So just having lumps of stuff that have a negative mass is probably not realistic, and it were possible it would lead to all kinds of universe-shattering cataclysms. Hopefully, we can get wormholes to stay open just using the exotic vacuum states from the Casimir effect and squeezed vacuums.

===Reducing or eliminating ANEC violating stuff===

As soon as people figured out that the wormholes they want needed stuff they couldn't get, they set about trying to find ways to use as little of the stuff they couldn't get as possible. A number of attempts have been made to hammer out a geometry for spherical wormholes that are stable, meet the quantum energy inequalities, and allow a person to pass through them in a reasonable length of time without being shredded by tides
<ref name="Fewster_Roman_2005"></ref>
<ref name="Visser_Kar_Dadhich">Matt Visser, Sayan Kar, and Naresh Dadhich, "Traversable Wormholes with Arbitrarily Small Energy Condition Violations", Physical Review Letters Vol. 90 No. 20 article 201102 (2003)</ref>
<ref name="Kuhfittig">Peter K. F. Kuhfittig, "More on wormholes supported by small amounts of exotic matter", Physical Review D <b>73</b>, 084014 (2006); Peter K. F. Kuhfittig, "Wormholes supported by small amounts of exotic matter: some corrections", arXiv:gr-qc/0508060v1 15 Aug 2005</ref>
.
Often, these resulting structures end up having extreme dimensions, such as throat sizes on the order of light years
<ref name="Fewster_Roman_2005"></ref>
<ref>Remo Garattini, "Generalized Absurdly Benign Traversable Wormholes powered by Casimir Energy", arXic:2008.05901v1 [gr-qc] 12 Aug 2020</ref>

Thin shell wormholes have also been the subject of investigation. One study <ref name="Mazharimousavi_Halilsoy">S. Habib Mazharimousavi, M. Halilsoy, "3 + 1-dimensional thin shell wormhole with deformed throat can be supported by normal matter", Eur. Phys. J. C (2015) 75:271 DOI 10.1140/epjc/s10052-015-3506-6</ref> found a number of thin shell wormholes with sharp-edged shapes could be supported entirely without any exotic energy. However, this only applies if the edges are infinitely sharp. A finite radius of curvature at the edges would require negative energy at those edges.

If the wormhole is taken have the properties of a class of subatomic particles called a fermion, then if the ratio of the electric charge to the mass is sufficiently large you can get a wormhole that can remain open without any negative energy regions at all
<ref name="Blazquez-Salcedo_Knoll_Radu">Jose Luis Blázquez-Salcedo, Christian Knoll, and Eugen Radu, “Transversable wormholes in Einstein-Dirac-Maxwell theory”, Physical Review Letters <b>126</b>, 101102 (2021); ArXiv: 2010.07317v2 [gr-qc].</ref>
.

==Fields and conserved properties==

Our best theories of how the world works – quantum mechanics and general relativity – predict that certain things are [[Conservation_Laws:_Limits_to_Cheating|conserved]]. For asymptotically flat space-times, these quantities are energy, linear momentum (or just momentum), angular momentum, and electric charge. Furthermore, in the non-relativistic limit mass is conserved independently of non-mass energy.

And not only are these things conserved, but they are always conserved <i>locally</i>. This means you can't just get rid of, say, some energy in one place and expect things to balance out by having that energy appear somewhere else.

What does it mean for something to be conserved? Mathematically, it means that the thing obeys the continuity equation. There are various ways of writing the continuity equation, all equivalent, but basically it boils down to the following: if you have a region of space surrounded by a closed surface, then the amount of conserved stuff inside that surface can only increase if some of the stuff enters through the surface and it can only decrease if some of the stuff leaves through the surface. Makes sense, right? If you have three loaves of bread in a room, then if the deliveryman brings in another two loaves you will have five loaves. But unlike loaves of bread, which can be baked or eaten or grow moldy or burn up or be broken into crumbs, the only way to get energy or electric charge or the other conserved things into or out of the room is to have them come into or go out of the room.

Each of these conserved quantities has an associated field. Electric charges create an electric field. Energy and momentum and angular momentum each create a distribution of the curvature tensor of space-time. In the non-relativistic limit, this is the mass part of the energy creating the gravitational field.

There is a relationship between the amount of a conserved quantity in our hypothetical closed surface and the net amount of field that penetrates that surface. This is called [https://en.wikipedia.org/wiki/Gauss%27s_law <i>Gauss's law</i>]. Basically, it says that if you add up all of the amount of field that goes out of the surface, and subtract off all of the field that goes back into the surface, the resulting amount will be directly proportional to the amount of the conserved stuff in there. Do you start off with a kilogram of mass inside your room? Then you have a certain amount of gravitational field from that mass that leaks out of the room. Add another kilo for two kilos total, and the amount of gravitational field leaving is also doubled. The distribution of where that field comes out may change, but as long as you don't take that mass out of the room or add more mass in the room, the total amount of field leaking out won't change. And any gravitational field from masses outside of the room that leaks into the room must also leak back out again <i>somewhere</i>. Those masses outside the room will not affect the net amount of field leaking through the surface. What this means is that you can determine the amount of conserved stuff inside a surface if you know the amount of field going through that surface.

This can be visualized in the field line approximation. The field around a positive charge can be approximated as a series of directed lines that radiate away from the charge. The field around a negative charge in this approximation is a series of lines that converge on the charge (for non-relativistic gravitation, you only have positive charges because mass is always positive. We'll just ignore the fact that wormholes require negative energy density stuff in order to exist which will lead to negative mass – we're just trying to get across the basic idea here). Field lines can only ever start on a positive charge and only ever end on a negative charge. The number of lines radiating away or converging on the charge is proportional to the size of that charge (so if you double the charge, you double the number of lines that connect to it). Field lines propagating through space bend away from a positive charge and bend toward a negative charge.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=322>[[File:charges_positive.png|frameless]]
<td width=322>[[File:charges_negative.png|frameless]]
<td width=322>[[File:charges_dipole.png|frameless]]
<tr>
<td>The field lines radiating away from a positive electric charge.
<td>The field lines converging on a negative electric charge.
<td>The field lines emerging from a positive charge and curving around to go into a negative charge of the same magnitude. This configuration, with a positive and negative charge of the same magnitude, is called a <i>dipole</i>.
</table>

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=322>[[File:charges_positive_Gauss_law.png|frameless]]
<td width=322>[[File:charges_negative_Gauss_law.png|frameless]]
<td width=322>[[File:charges_dipole_Gauss_law.png|frameless]]
<td width=322>[[File:charges_exterior_dipole_Gauss_law.png|frameless]]
<tr>
<td>A closed surface encloses a positive charge. A net amount of field lines exit the surface.
<td>A closed surface encloses a negative charge. A net amount of field lines enter the surface.
<td>A closed surface encloses a dipole. The total amount of charge inside the surface is zero because the negative charge cancels the positive charge. No net amount of field goes through the surface - just as many field lines enter as leave.
<td>A closed surface that has no charge inside of it, but various charges outside of it (in this case a dipole). Again, no net amount of field goes through the surface - every field line that enters comes back out somewhere else.
</table>

Now consider the case of an isolated positive charge, like one one shown above, that is far away from anything else. We will bring up one mouth of a wormhole close to the charge. Because the field lines going out of the charge cannot break, they cannot go through the wormhole. Instead, they will curve around the wormhole mouth. As the charge enters the wormhole mouth and travels into its throat, it drags its field lines along with it. With all of its field lines entering the mouth, that mouth looks like it has a net positive charge. As the charge leaves the other mouth, it continues to drag ints field line out of the wormhole - the field lines that start on the charge now curve around to enter the wormhole mouth. The mouth that the charge exited from now looks like it has a negative charge.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Wormhole_with_charges_initial.png|frameless]]
<td width=400>[[File:Wormhole_with_charges_inside.png|frameless]]
<td width=400>[[File:Wormhole_with_charges_passed_through.png|frameless]]
<td width=400>[[File:Wormhole_with_charges_long_gone.png|frameless]]
<tr>
<td>The charge approaches the wormhole.
<td>The charge is inside the wormhole. The mouth of the wormhole that the charge entered looks like it has a positive charge.
<td>The charge has passed through the wormhole. The mouth that the charge entered still looks like it has a positive charge, but now the mouth that the charge came out of looks like it has a negative charge.
<td>The positive charge has now moved very far away. Both mouths of the wormhole still appear to be charged.
</table>

This can also be understood by the continuity equation. Picture an imaginary closed surface around the mouth of the wormhole. This is like the case of the room, before, except that <i>now there is a way out of the enclosed volume that does not pass through the surface</i> – by going through the wormhole! Any conserved quantity that goes into the surface adds its quantity to the stuff in the surface. If it goes through the wormhole, it does not come out through the surface. The conserved quantity inside that surface stays the same! To an outside observer, it looks like the wormhole mouth has that quantity.

Either way you look at it, the results are the same. A wormhole mouth acquires any conserved quantity of the things that enter it, and loses any conserved quantity of the stuff that exits from it.
<ul>
<li> A wormhole mouth acquires the <i>energy</i> of anything that goes into it. Its energy decreases by the energy of anything that goes out of it.
<li> In the non-relativistic limit, this means that a wormhole mouth acquires the <i>mass</i> of anything that goes into it. Its mass decreases by the mass of anything that goes out of it.
<li> A wormhole mouth acquires the <i>electric charge</i> of anything that goes into it. It gains negative the electric charge of anything that goes out of it.
<li> A wormhole mouth acquires the <i>momentum</i> vector of anything that goes into it. It gains the negative of the momentum vector of anything that goes out of it.
<li> A wormhole mouth acquires the <i>angular momentum</i> vector of anything that goes into it. It gains the negative of the angular momentum vector of anything that goes out of it.
</ul>

This perhaps should not be a surprising result – other general relativistic space-time distortions also exhibit this behavior. A black hole, for example, gains the mass, charge, momentum, and angular momentum of anything that goes into it.

You can use these same arguments to find that the same results apply to any asymptotically flat manifolds of space-time. They do not necessarily hold for space-time geometries that are not asymptotically flat. If the gravitational distortion of a wormhole extends infinitely far away, you will not be able to even define its total energy or momentum or angular momentum, so it will be impossible to say if they are conserved. But do note that all of the wormhole examples that we have discussed so far (except for the cylindrical wormhole of infinite extent) are asymptotically flat, and will obey the local conservation of all of these quantities.

===Wormhole dynamics===

A wormhole mouth has the gravitational field and conserved properties of mass, momentum, and angular momentum. By the equivalence principle of general relativity, this must mean that it behaves as if it had those properties. So a wormhole mouth will follow ballistic trajectories in vacuum through space-time as if it were an object in free fall. If it is around a concentrated mass, like a planet or sun, it will have a Keplerian orbit around that object (or, if the wormhole has more mass than the planet or sun, perhaps it would be more reasonable to say that the planet or sun would orbit it). The mass of the wormhole mouth will resist acceleration like any other mass, according to Newton's laws of motion. Force is the time rate of change of momentum, so as the wormhole mouth's momentum changes it will experience forces, and these will change its trajectory in the same way as if that momentum (or forces) acted on a material object with the same mass. And if the wormhole mouth has an electric charge, it will experience forces from electric and magnetic fields; again, these can change its trajectory.

In other words, a wormhole mouth acts like a physical object of the same mass, charge, energy, momentum, and angular momentum.

The motion of the wormhole mouths through our normal space does not affect their relative separation through the wormhole's throat. That can remain the same length regardless of the behavior of the mouths in our normal universe.

As we covered, if an object enters a wormhole mouth that mouth gains the object's momentum. This will give the wormhole a "kick" changing its motion in the direction that the entering object was originally moving. The dynamics work out the same as if the wormhole were a sticky blob and the object collided and stuck to it. In both cases, you get a final object with the combined masses of both initial things (the mouth and the entering object) that has been knocked off course a bit by the momentum of the entering object hitting it.

If that object comes out of the other mouth, then as that mouth gains the opposite of the objects momentum it gets a recoil kick in the opposite direction. Again, this results in dynamics that are similar to things we already know about – like a gun with a bullet loaded firing the bullet. As the bullet leaves, the gun + bullet system loses the mass of the bullet, and as the bullet shoots away the gun recoils in the opposite direction.

So now consider what happens if you direct a constant stream of material through a wormhole. As the jet of material exits its mouth, that mouth loses mass at the same rate as the mass flow rate of the jet. It also continually gains momentum by the recoil kick of the gas. These dynamics are identical to that of a rocket - as the propellant is ejected, the rocket loses the propellant's mass and the recoil momentum from the escaping gas jet pushes the rocket in the opposite direction. So by shooting stuff through a wormhole you can turn it into a rocket, with dynamics identical to a rocket, obeying the [https://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation Tsiolkovsky rocket equation], and everything else that happens with rockets.

===Does gravity go through a wormhole?===

The above diagrams of the electric field of a charge as it approaches a wormhole mouth also hold for the gravitational field of a mass that is near a wormhole mouth. In the same way that the electric field cannot enter the wormhole's interior or leak through to the other side unless the charge enters the wormhole or goes through to the other side, so to will the gravitational field of a nearby mass curve around the wormhole. The gravity of a nearby planet will fall to zero inside the wormhole throat and will not affect those near the opposite mouth.

===Can waves go through a wormhole?===

So if gravity won't go through a wormhole, and static electric and magnetic fields can't go through a wormhole, then what about light? That's made up of electricity and magnetism, right?

So, turns out that light can go through a wormhole. So can other electromagnetic waves like radio and x-rays. So can gravitational waves. The rules for drawing field lines only really work for static charges; for radiating waves you also need to include the parts about how the changing field generates more field. In the end, waves that are smaller in wavelength than the size of the wormhole can go through just fine. You'll get some [[Diffraction|diffraction]] around the mouths, but for wavelengths much smaller than the mouth size even that might not be noticable.

===Can negative mass wormholes exist?===

So a wormhole mouth loses the mass of anything that comes out of it. What happens if you send something through a wormhole with more mass than the mouth which it exits from? This rule says that the mass must become negative, but as we saw earlier this leads to all kinds of problems (especially because the wormhole can also be charged) and is likely to be forbidden by the laws of nature.

If a wormhole's mouth can reach negative mass, you can also get other interesting instabilities. The positive mass mouth will attract matter into it, which will pass through, increasing the mass of the positive mass mouth and making the mass of the negative mass mouth even more negative. Meanwhile the gravity of the negative mass mouth will repel matter, preventing any return flow. Both the negative and positive masses will gradually increase over time.

So what's an aspiring science fiction writer to do? If you have a particular desire for negative mass things in the fiction you create, go ahead and allow it. Just be aware of the consequences. Otherwise, it is probably easier to say that negative mass wormhole mouths can't exist, and justify that with the quantum energy inequalities. Which, of course, brings up the question of what happens when you try to push an object through a wormhole that would make the mass of one of its mouths go negative? Two obvious possibilities are that either you get a back-reaction force that makes it so you can't ever make something go through the mouth to make its mass negative (perhaps by the formation of a horizon as the mass approaches zero), or the wormhole collapses before the mass can ever reach zero. Perhaps inventive science fiction authors and fans can come up with other plausible options as well.

==Wormholes and time travel==

Wormholes connect across both space and time. So it is natural to consider the possibility of using wormholes for time travel.

Now maybe you don't want to have to deal with time travel in your fiction. It introduces all kinds of opportunities for paradoxes, and you have to keep track of plot lines in a self-consistent way that go back and forth through time, influencing themselves in complicated ways. Okay, no problem. Lets just have our wormholes always connect to the same time. Easy right? In fact, it is natural to justify this. Physics is local, so if you can make wormholes you probably have to create both ends right next to each other at the same time. You can then move the wormhole ends away from each other – maybe putting one end on an interstellar spacecraft or something – but they stay at the same time. Right?

Right?

Except ...

In relativity, motion not only affects how fast you go through space, but also how fast you go through time. When things move, they experience time dilation, so if you move one wormhole rapidly far away and then bring it back, the time dilation it experienced will naturally allow it to form a time machine so you can send things and people back to interfere with their own pasts. Even worse, time can also be slowed down by gravity, so if your wormholes are in areas affected differently by your galaxy's own gravity, or around stars of different mass, or just at different elevations on the same planet, the rate they experience time will be different. Temporal complications ahoy!

Let's look at how this works in more detail. We'll take our standard science fiction empire, called The Empire. Like all empires, it has a central area of authority, the metropole (which we will place on a planet called Metropole), which extracts wealth from various colonies and subjugated client states for its own enrichment in the form of taxes, tribute, and forced favorable trading opportunities. Let's give our Empire two colony worlds which we'll call Colony A and Colony B (these Empire folks sure aren't very creative in their naming, are they?). Just for convenience, we'll put Metropole, Colony A, and Colony B all 100 light years away from each other, in an equilateral triangle.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:WormholesTimeTravel graphic1 wb.svg|600px|frameless]]
<td>[[File:WormholesTimeTravel graphic2 wb.svg|600px|frameless]]
<tr>
<td width=400>Metropole and its two colonies A and B
<td width=400>Metropole sending a wormhole to Colony A.
</table>

So Metropole launches a wormhole mouth out to Colony A. They want to get to Colony A quickly, so they'll make the wormhole go really fast. Let's say they fling it out at 99.9999% the speed of light. At this speed, it takes 100.0001 years for the wormhole mouth to reach its destination in the reference frame of Metropole and Colony A. But due to relativistic time dilation, in the reference frame of the projected wormhole mouth it only takes 0.1414 years to go from Metropole to Colony A. Because the techs at Metropole can look through the wormhole, they will see that 0.1414 years after launch it arrives at its destination. At that point, brave explorers from Metropole can go through and set foot on Colony A – they only need wait about a month and a half to feel alien soil under their feet, rather than a century. In Metropole's reference frame, going through the wormhole takes you 100 light years away, and 99.8596 years into the future. Going the other way, from Colony A to Metropole, takes you 100 light years away and 99.8596 years into the past.

[[File:WormholesTimeTravel graphic3 wb.svg|800px|frameless]]

One might think that these time warps would let you engage in all kinds of time travel. It is easy to see that the Metropole—Colony A situation described here doesn't allow these kinds of shenanigans. For practical purposes, you only have a time machine when you can go back to the place you left at a time before you left. And you can't do that here. Go from Colony A to Metropole and you go back in time 99.8596 years. Go back to Colony A through the wormhole, and you go forward in time the same amount, plus any time you spent on Metropole, so you get back after you left. If you go back through flat space-time, it will always take at least 100 years since you can't go faster than the speed of light so you also get back after you left. No paradoxes for you!

However, it is easy to imagine situations where a wormhole, or a configuration of wormholes, does make a time machine. For example, what happens if we immediately turn around and send the wormhole mouth from Colony A back to Metropole? In the reference frame of Metropole, the wormhole comes back with its "Return to sender" sticker 200.0002 years after it was launched. But going through from the left-behind mouth to the round trip mouth will take you 199.7192 years into the future. And if you step into the round trip mouth, you will go back in time by 199.7192 years into your own past. You now have a real honest to goodness time machine, complete with paradoxes.

[[File:WormholesTimeTravel graphic4 wb.svg|800px|frameless]]

Or maybe the wormhole mouth stays at Colony A. And Metropole sends another one to Colony B, that also goes 99.8596 years into the future. Now Colony A sends a wormhole to Colony B. This wormhole also goes 99.8564 years into the future as a consequence of its trip. This means if a traveler at Colony B went through the Colony A wormhole he would go back in time 99.8564 years. Then going from Colony A to Metropole he would go back in time another 99.8564 years. Then he could go from Metropole to Colony B and go forward in time 99.8564 years. The net result is that he ended up back where he started nearly a century before he left.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:WormholesTimeTravel graphic5 wb.svg|600px|frameless]]
<td>[[File:WormholesTimeTravel graphic6 wb.svg|600px|frameless]]
</table>

And it is not just motion that causes time dilation. Being different depths into a gravitational potential does this as well. If Colony A is deeper into the gravity well of the galaxy, or orbits a heavier star, it will experience more gravitational time dilation than Metropole. So eventually it will build up more than 100 years of time slip between the two end of the wormhole and you can have a time machine.

So is time travel inevitable if you have wormholes? What a nuisance!

Well, maybe not. Here's why. Think about what happens when the Colony A – Colony B wormhole has gone just far enough that a light signal going through the wormholes can get back to where it left just as it is leaving. Now, since the propagating signal and the newly transmitted signal are both leaving at the same time, you have double the intensity. So this doubled intensity signal goes around and meets itself again, quadrupling its intensity. And so on. At this point, just as the configuration is on the verge of becoming a time machine, it becomes a perfect resonator for light signals, which then build up to arbitrarily high intensities until something breaks and you don't have an incipient time machine any more.

Now, this won't always work if you just consider light signals. Light can get defocused or sent off in odd directions or something so that you don't have a good path back until after you end up with a time machine. But if you look at what happens in semiclassical quantum gravity, will get a buildup in the amplitude of quantum fluctuations as soon as you start to get a time machine, called <i>vacuum polarization</i>. These fluctuations are expected to build up to sufficient amplitude as to destroy the time machine to be, at least for the few simple cases studied. Or perhaps it just causes the wormholes to bounce away from each other so as to prevent time machine formation. This has not yet been shown to occur in general. But to does seem to happen for all possible configurations of one or two wormholes about to become time machines
<ref name="Visser_Lorentzian_wormholes"></ref>
.

But let us assume for the moment that wormholes cannot form time machines in any configuration due to this vacuum polarization mechanism. Now there are limits on just how you can place and move your wormholes. You don't want to accidentally almost form a time machine and break your transportation network. There are several ways to do this:
<ul>
<li> Move your wormholes slowly, so as not to build up much time dilation (or <i>time lag</i>, if you will).
<li> Don't form closed loops in your wormhole transportation network. The less distance you need to travel to go all the way around a loop to get back where you started, the less time lag is needed to make a time machine.
<li> "Discharge" the time lag by periodically rapidly moving the mouth of a wormhole that is "behind" in time in such a way that it comes back to where it was. Perhaps you can put a charge on it by sending a [[Particle_Beam_Weapons|particle beam]] through it, and then accelerate it up to relativistic speeds in a synchrotron for a while.
<li> Send new wormholes that you are going to place through the existing wormhole network. This way they acquire the same time lag across their mouths as all of the wormholes they traverse. Until they build up some additional time lag of their own, you provide a perfect return trip without making a time machine.
</ul>

In conclusion, if you want a story or setting that involves time travel, you can plausibly still use wormholes somehow to allow that (although it might take loops of several of them). If you want a story that avoids the complications of time travel, you can still plausibly have wormholes in that setting (although you now need to beware of the limitations on your wormhole transportation network imposed by the no time machine rule).

==Practical implications==

===Getting places===

Usually in science fiction, people want wormholes because they want to be able to get to cool places far away without taking forever. Very often, this involves the technology (or the protagonist, with some kind of psychic power) "opening" a wormhole from the current location to the desired location when there wasn't one before.

Sadly, the ability to just create a wormhole to anywhere you want to seems unlikely. Physics tends to be local, which means that if you can make a wormhole both ends will emerge right next to each other. You will need to move the ends to where you want them to be using other methods (mailing them by post, putting them on a spacecraft, etc.). As was mentioned earlier, if you shoot a stream of stuff through a wormhole, the end that the stuff comes out of becomes a rocket. So one way to move an end of a wormhole is to shoot stuff through the wormhole. Because you can leave all the big heavy equipment at home (generators, rocket motors, cooling systems, and everything else), if you can make your wormhole light enough this could be a low cost and efficient method of getting it places. Remember that you can exploit time dilation to make it seem like you can get someplace in much less time than you would expect light to take going to that place (at the expense of taking you some time forward in the future as well when you step through). If you can get your wormhole mass down to a few grams, or even several kilograms, using powerful lasers to shoot through the wormhole to make a photon rocket could be a highly efficient way to go places at relativistic speeds. Perhaps it could even fuel itself by collecting mass from the interstellar medium (ISM) as it flies along, and it might be able to brake against the ISM or solar wind of the destination star system to slow down, reducing mass loss even further.

===Keeping time===

We have already shown that wormholes connect across both space and time, so that a trip between star systems could take you hundreds of years into the future, and the return trip takes you hundreds of years back in time. And this is even before we throw in how time slips between planets when considering relativistic time dilation due to different speeds and gravitational potentials.

Fortunately, all the weirdness of different time rates and going backward and forward in time can be ignored by the average person. This is because you never need to go from one world to another, or back, across the vast gulfs of interstellar space. You just take the wormhole between them. All you ever need to worry about is the coordinate frame that goes across the wormhole. When considering this reference frame, you're not hopping all over the place in time. If it takes ten minutes to cross the wormhole between the two planets, when you get to your destination world the clocks will read ten minutes later than they did when you left your departure world. By coordinating their time-keeping across the wormhole network, all of the worlds of the network can agree on a common time to coordinate their activities. This is all travelers ever need to worry about, and they can then ignore all the relativistic weirdness. Your network engineers will still need to keep track of relative time drift and how close a given configuration is getting to a time loop. But unless your protagonist is a network engineer, they can just ignore all that stuff. And, as an author, so can you! Assume your engineers are competent, you have good regulatory bodies and standards institutions, and don't worry about any of this "time travel" that doesn't actually let you cause paradoxes.

===Keeping wormholes away from planets===

By now, people expect science fiction to have spaceships. Wormholes offer a convenient excuse to have spaceships, because it provides a way for those spaceships to get to other worlds that are not in our solar system. But if you have wormholes, sometimes you need to ask why you still have spaceships? Why not just put one end on Earth and the other on an alien planet and step through to get where you are going?

Sometimes there are good reasons for this. If your wormhole has as much mass as Jupiter, you are not just going to keep it in Topeka, Kansas. That would kind of end the Earth. So you would put it in orbit around the Sun, and send spaceships from Earth out to the wormhole and through it to wherever it goes.

But all too often we see authors using special pleading to pigeonhole in their spaceships without thinking about good physics. Things like "it can't work in atmosphere" (use an airlock), "it can't work within a certain number of planetary radii" (why? There is no physical plausibility to such a restriction), "It can't work in a gravitational field of more than X" (violates the equivalence principle), and so on. But if you insist on having spaceships, and you also insist on wormholes that mass less than a planet, there are a few plausible ways to justify your preconceived ideas that at least plausibly align with physics.

For example, the closest you can plausibly get to "X number of planetary radii" or "gravity field less than X" is a criterion that relies on local measurements of space-time curvature only. And it turns out that this quantity is the tidal forces (curvature is a direct measurement of tides). Tidal forces are going to be proportional to the mass of the nearby thing causing the tides, and inversely proportional to the cube of the distance. If <math>R</math> is a measure of your space-time curvature, <math>M</math> is your object's mass, and <math>r</math> is the distance from the object to the wormhole, then

<div align=center><math>R \propto \frac{M}{r^3}</math>.</div>

So if you want your wormholes to be at least ten planetary radii from Earth (<math>r = 64 \times 10^6</math> m, <math>M = 6 \times 10^{24}</math> kg), then the curvature limit is <math>R = 23</math> kg/m³ in some convenient system of units where we don't have to include constants of proportionality. But now suppose you want to send a <math>26 \times 10^6</math> kg ore freighter through that wormhole. You'll only be able to get your freighter within about 100 meters of the wormhole before the freighter's tides collapse the wormhole. This <i>might</i> work if the freighter were really long and skinny, much longer than 100 meters, to reduce its transverse tides; or if the wormhole was much larger than 100 meters in diameter. But in general, note that the curvature and tides have units of density, so if the average density in any volume around the wormhole exceeds the critical density – oops, there it goes.

Also, having a limit on the geometric curvature of space-time similar to what you get around mere planets doesn't end up being very plausible – the curvatures in the wormhole itself exceed that by so very many orders of magnitude that having so stringent of a curvature tolerance seems very implausible, and just dancing on the edge of disaster. For example, if your wormhole is about 100 meters across, its curvature in these same units will be approximately 500 billion trillion kg/m³. Keeping the tides to within 23 kg/m³ is the equivalent of an engineering tolerance of one part in 25 billion trillion.

So if you don't really have a good way to limit where you can place a wormhole based on the nearby "gravity", are there other ways you can exclude them from the vicinity of planets?

Safety might be one good reason. We have already discussed various ways a wormhole might collapse (forming a time machine, having its mass go negative). What happens when a wormhole collapses? If there are any parts of the wormhole or things in the wormhole that are charged or made of charged particles (like ordinary matter) you could justify a considerable fraction of the wormhole's mass-energy being converted to electromagnetic radiation. In addition, if the wormhole collapses into a black hole it will evaporate away its mass-energy primarily in the form of electromagnetic waves as it undergoes Hawking radiation. Large black holes will last a long time, but if the black hole left behind has a mass of 500 tons or less it will last less than 10 seconds. And 500 tons worth of energy delivered in 10 seconds would do quite a number on a planet. These sorts of considerations could lead to a regulatory environment that require wormholes to be located several light seconds away from inhabited places (like planets).

===Wormholes on planets===

But sometimes you might want to have the wormhole on a planet. Perhaps you are envisioning one of those circular or oval fantasy portals, looking like a window to an exotic foreign land, and when you step through you are actually there. We already know that thin-shell wormholes can be very low mass, and can have a circular shape. So that should work, right?

There are at least two issues to consider here. One is that even if both wormhole ends are on the same planet, natural variations in atmospheric pressure due to weather will lead to very significant differences in air pressure from one side of the wormhole to another (unless the wormholes are so close that they are within a small distance of the extent of the same weather system – say, within a few km of each other). This pressure difference will drive severe winds through the wormhole, usually comparable to the winds of a hurricane (although there will be periods of relative calm when the pressure at both wormholes just happens to be at the same pressure). This effect will only be exacerbated if the wormhole mouths are at different altitudes, due to the decrease in air pressure with altitude. And if the wormhole connects between different worlds, the effect will be much worse. In addition to making travel inconvenient and possibly damaging the equipment, the winds will produce large uncontrollable mass flows across the wormhole (air on earth has a density of about 1 kg/m³, so if you have 50 m/s winds blowing through a 2 meter diameter portal you have a mass flow of about 160 kg/s through your wormhole). This will make it difficult for mass balance, and keeping both ends having high enough masses that neither gets so close to zero that the wormhole breaks or forms a horizon or goes negative mass and causes a runaway planetary catastrophe or whatever wormholes do when one end gets driven to near negative mass by the stuff leaving it.

As a consequence, planetary wormhole mouths will probably be kept in airlocks.

The other issue is that different latitudes on a planet will be rotating at different speeds; at the poles they won't be moving at all while at the equator they will be going a full planetary circumference every day. This means wormhole mouths at different latitudes will be building up different time dilations with respect to each other. In addition, wormholes at different altitudes will experience different gravitational time dilation rates. If you let this go on for long enough, the differences in time across the wormhole mouths will form a time machine. Whether this collapses the wormholes, makes them "bounce" apart from each other (likely ruining a lot of property and infrastructure in the process), or just forms a time loop that leads to various inconvenient paradoxes depends on the assumptions you made for the physics of your world. But it will probably be something you will want to avoid. Re-balancing the time differences between wormhole ends will probably need to be done every few decades.

===Wormholes on spaceships===

The necessity for radiators on spacecraft is really annoying. Dealing with all that waste heat is not fun, and these issues are not awesome kewl stuff like gigawatts and megatons and hundreds of g's of thrust and other things that let you show everyone how much your spaceships totally rule and everyone else's spacecraft totally drool. Instead you need these big fragile stupid-looking things hanging off the side of your ship, ruining your aesthetics and efficiency and being vulnerable to your enemies blowing them up, and lighting you up so bright that you're easily visible from the other side of the solar system. Hey! I know! Let's put a tiny wormhole on our spaceships so we can just send all the heat someplace else. Great idea! That'll work, right?

Actually, yes. If you have small enough wormholes, it will work great. But there are consequences.

So you're running your coolant loops from your reactor through the wormhole to some cooling towers on your home planet. Much better than radiators. But – why do you still have that reactor on your spaceship? It's big and heavy and needs lots of big and heavy shielding and poses nasty radiation hazards. Surely you could leave it behind on your home planet as well, next to those cooling towers, and just run a power line through the wormhole? You'll be able to run a bigger, more powerful reactor that way anyway, and that extra weight won't be bogging your ship down.

And why does your spaceship have its lasers on board? You can have a much bigger laser on the planet that you don't need to lug around. Just shine the laser beam through the wormhole and re-direct it with mirrors.

And all of your missiles? Might as well just feed them through the wormhole when you need to fire them.

And then do you really need all that life support equipment? Just run an HVAC and plumbing that sends fresh air and water to the spacecraft and takes used air and water off. And remember to send the occasional snack, too.

Oh, but wait. Why is all your crew even on the spacecraft again? They have a wormhole to the spaceship. You can just have all your crew run the spacecraft remotely from a mission control station. Most of your equipment is on the planet anyway, so most of the engineering staff is already here. Now you don't actually need to put people in harm's way. If someone does need to fix something just put them in a space suit and send them through, let them do their job, then bring them back. Most of that can be done by a tele-operated robot, in any event. And being able to go back to their families when their shift is over is great for morale.

Hmmm, your sensors could just look through the wormhole instead of having all the vulnerable and heavy and expensive equipment actually on the spacecraft. And as long as you are doing that, you can just leave your laser beam pointer at home too, and aim the laser by pointing it through the wormhole.

So now what's actually left in your actual spacecraft? Oh, the rocket itself. But remember how we already talked about how we could just keep the rocket engine and propellant on the planet and shoot the rocket jet through the wormhole, and it will move just like a rocket? Yeah, so no rocket either.

Congratulations, you've successfully gotten rid of your spaceship. The wormhole itself is your spaceship. It's all you need, it's safer, it's more capable, it's stealthier, and just better all around than some contraption of steel and Mylar and fissioning uranium fuel rods and fragile people going through space.

===Adding mass to wormholes===

So you just made a pair of wormholes. Chances are, you pulled them apart as near Planck mass objects, with about 20 micrograms of mass. Well, that's not very useful! What if you want to send something through that's bigger than 20 micrograms? You're going to need to add mass to your wormhole mouths.

A similar issue comes up if you project a wormhole at some far away planet or star system using the wormhole-rocket trick already discussed, maybe by using powerful lasers to make a photon rocket. To be practical, your projected wormhole mouth will need to be very low mass. But once you get it to where you want to go, you'll need to give the mouth enough mass to let your explorers and colonists and equipment through.

A simple way to do this is to simply suck stuff into the wormhole. This adds mass to the wormhole mouth the stuff comes through. Of course, if you take it out from the other side, that other mouth will lose that much mass, and if it didn't have than much mass to begin with you'll likely have problems. So just leave that stuff in there! Shove it off to the side. Let it clutter up the unused areas out of the main traffic route in the way that basements and attics and the unused spaces of garages accumulate all that old junk that you never use any more but don't want to throw away. Build a wall inside of your wormhole, so travelers don't have to see all the asteroidal rock or super salty water or very surprised natives or whatever it was you scooped up on your alien world. Shove it into the undesirable high curvature places with nasty tides. This wall can also help keep the legitimate passengers away from the inner working of the wormhole, the strange energies that support it, the cables and pipes and crawlspaces that only your maintenance workers should access anyway, and the extreme space-time curvatures that would rip the passengers to shreds if they ever were directly exposed to them.

===Tiny wormholes===

Wormholes that are too small to fit people through could still be useful for sending information. You could use them to rapidly communicate over long distances. If the wormholes were light enough and cheap enough, they might even replace cell phones and laggy land lines with super high bit rate connections.

You could also send energy through tiny wormholes, in the form of light or electric current. Storms that knock down power lines no longer need to cut off your home's electricity. You could also use them to power machinery that need light weight and high power output – electric transportation like airplanes, shipping, or trucking; directed energy weapons; or high specific impulse rockets.

===Causality denial, causality attacks===

We have shown that moving wormholes around can lead to time travel, and that there are plausible physical mechanisms that can prevent time-travel allowing wormhole configurations from happening. People being people, you just know that if this is possible, that it will not be long before these effects are used in politics and warfare.

In a network of wormholes that is about to create a time machine, one likely resolution is that things break until a time machine is no longer possible. In any chain stressed to its point of failure, it is the weakest link that breaks. So in a path through wormholes and normal space-time that is about to form a time machine, it is the weakest wormhole that will collapse (or otherwise have something bad happen to it). So if your rival polity has a wormhole that allows then easy trade with a rich world, and you want to edge in on that trade and simultaneously deny that trade to your rival, you can just send a bigger, stronger wormhole in such a way that forces a time loop. Now your rival's wimpy wormhole breaks, and you get access to the trade opportunities with the world! This is the basics of a causality attack – use the causality-enforcing properties of wormholes to attack your enemy's wormhole infrastructure.

But what if your rival polity anticipated this? What if they have strong wormholes, with an arrangement in space-time that blocks off easy access of your wormholes to a profitable region of space? This is a causality denial action, or a causality fortress. By controlling the time displacement of the wormholes in their network, a polity can control what additional connections can be made. This can be used to enforce borders, prevent infiltration, or restrict civilian wormholes to legitimate uses.

===Moving one wormhole through another wormhole===

If you have a wormhole mouth, you can move it through another wormhole as long as the mouth (and any supporting structures) is small enough to fit through the throat of the wormhole it is going through. The geometry all works out. In fact, this has a rather nice side-effect. If you are looking through the wormhole being moved from back home, you only see it take as long to go through the other wormhole as a wormhole transit usually takes even if it got projected far into the future or past. So the moved wormhole acquires the same time differences as the wormhole it just went through (added on to any time differences it already had, it doesn't get reset or anything). This can let you easily build wormhole networks that connect to themselves in loops. It also makes the network more susceptible to breaking, because there is much less leeway for error with a closed loop as far as making a time machine and breaking the loop. So it can be done, but there are consequences.

===Dropping a wormhole into a black hole===

In principle, if you have a wormhole connection to the inside of a black hole, you can get information out from inside the event horizon. The theorem that prevents things from escaping a black hole was made under the assumption that the energy can't go negative. And as we've already seen, wormholes require regions of negative energy to work. You still wouldn't be able to see further toward the singularity at the center than the wormhole's location. And the wormhole would be inevitably dragged into the singularity in a finite amount of time. But, in principle, during that time you could see what is happening in the black hole.

But there's a big catch with this. You can't actually get the wormhole into the black hole. Not if you assume that time machines can't form, anyway. From the point of view of the wormhole mouth falling into the black hole, it will pass through the event horizon in a finite amount of time. From the point of view of someone watching from outside, the infalling mouth gets more and more time dilated to the point that it never crosses the horizon. This time dilation sets up the conditions for a time machine, and any time machine preventing physics happens on the wormhole before it can ever get in.

== Possible Networks ==
=== Acyclic wormhole networks ===

Recall that wormholes are space-time connections. They bridge space as well as time. Assuming we want to build this network in a stable manner, we want to avoid Closed Timelike Curves from appearing. Let us start with some simple anaylsis of consequences on this. Assume projected wormholes have one mouth sent at relativistic speeds to a distant target and the other mouth is kept at rest at home. From the perspective of the dispatching party, this will lead to the new connection opening very quickly. Done again and again, large volumes of space can be accessed in a short time, a powerful advantage.<br />

[[File:Wormholes PartialOrder.svg|400px|thumb|The partial order relationship of the wormhole connection]]

Every projected connection thus extends onto the future. Then we must avoid forming loops, or more generally "backtracking" back towards a dispatched node. This nets us a graph with no closed loops - an acyclical graph, also called a tree. Since connections may not return to previously established nodes if they are time-shifting from the past into the future, we also get a partial order: nodes <math display="inline">u</math> that dispatch wormholes have a higher order over the nodes <math display="inline">v</math> they establish: <math display="inline">u < v</math>. This lets us establish a direction on the whole graph. For a particular node <math display="inline">n</math>, its '''parent node''' established it, <math display="inline">parent(n) < n</math>, and it in turn establishes '''children nodes''', <math display="inline">n < child(n)</math>. The node that only has children nodes is the tree's '''root node'''. Nodes that have no children are '''leafs''' of the tree. The trees '''height''' is the longest outward path one can take from the root to a leaf.<br />

In terms of traversal, such nodes have some interesting and useful properties. Since wormhole traversals may be assumed to be effectively instantaneously bridging large distances, travel time is mostly defined by the travel distance between wormhole mouths. And as such, we want to minimize the amount of transits we have to make to get from a node <math display="inline">A</math> to a node <math display="inline">B</math> the fastest. It turns out, in order to achieve this, we want to have as large a choice of connections at each node. In the ideal case, any node except root can be reached in two transits: one transit to root, and from there to the other node. In the less ideal case we have to make one further stop-over, choosing from a number of connections at a second node to reach our destinations. The less ideal case from that sees us make two transits through such "switch yards". Assuming each node except the leafs in such a wormhole network has exactly <math display="inline">k</math> connections and there are <math display="inline">n</math> nodes in the whole network, we can reach any destination from root in <math display="inline">\log_k (n)</math> choices of wormhole transit. This is a powerful way to get to connections quickly through centralization. (In computer science, such k-nary trees are of great interest for quickly search-able data structures for this reason.) Configurations that are less "balanced" net appropriately lesser advantages and longer transit times.<br />

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Wormholes AcyclicGraphExample.svg |600px|frameless]]
<td>[[File:Wormholes Trees.svg|600px|frameless]]
<tr>
<td width=400>An example of an acyclic wormhole network in space-time.
<td width=400>The connected tree graph of the network exampke, and a k-nary tree example.
</table>

Also, traffic to most other nodes ''has'' to go through a root-wards node. This gives these nodes significant presence in the larger network. Routing through them is unavoidable if you want to reach most of the rest of the network. Root nodes can profit from taxation, services to travelers, control of routing infrastructure for information, and so on. But there is also a catch. Assuming leaf nodes each send the same amount of traffic inward. The amount of traffic load increases polynomial as you go towards root. Your wormhole infrastructure (and what transports cargo between wormholes, in space or on planets) has to cope with this scaling. Congestion might become an issue.<br />

=== Mesh wormhole networks ===

The acyclic network may have some bothersome problems. The central routing focuses power and can keep transit times down quite significantly depending on the specific network infrastructure, but it’s also congestion-prone, and some close locations may be widely separated on the network. Or maybe you want to avoid centralized routing dependencies for political reasons. Root might like being root; everyone else being subject to root’s whims, not so much. So, can we build networks where there are two routes between the same two nodes? We can, but it comes with some necessary adjustment.<br />

Recall why the acyclic network is acyclic: to avoid forming CTCs due to the time connection between future and past from the projected wormholes. But if we let both wormhole mouths experience the same time dilation, the problem that ultimately leads to CTCs - one connection routing displacing further into the future than the other connection, and thus allowing one to arrive at a node before ones departure - is avoided entirely, since there is no time displacement to be accumulated. Thus we get meshes, where there may be many connections and cycles between nodes. An observer at one node watching for his transition from another node 10 light years away would have to wait for ten years to watch his light arrive. Closed timelike loops are not possible so long as this network is kept balanced.<br />

[[File:Wormholes MeshNetwork.svg|thumb|center|700px|An example of such a mesh network]]

The disadvantage - this network must take great care to ensure that the time dilation that inenvitably occurs due to different star velocities and the galactic gravitational field don’t accidentally create a time shift and CTC in the network. And also we cannot exploit time dilation to open connections very quickly, from the home node’s perspective. One has to wait out the full years it takes the wormhole’s far mouth to get to a remote destination, before the connection can be used.<br />

=== Semi-privileged wormhole networks ===

The interesting thing is, it is very much possible to build networks that make use both of relativistic projection to seemingly accelerate the network construction time, ''and'' contain cycles, with alternate routes to get to another node. The acyclic construction is a strategy to avoid CTC formation, but we should remember the ultimate constraint reason: we avoid cycles so that there are no connections which, through a certain time shift on them, a CTC could be formed. Acyclic construction is one strategy. Only building cyclic connections with careful time shift balancing is another!<br />

The foundation for such networks is an acyclic "base graph" that may be isolated from the larger network graph. We also call this the '''red graph'''. The base graph maintains all the partially ordered properties of the simple acyclic graph. We can call this the "privileged" graph, since its nodes and their connections define the principle positions in space-time of the network, and we must avoid generating CTCs on this definition. How do we avoid forming CTCs but still create alternative paths and thus loops?<br />

Let’s assume we have two nodes <math display="inline">A</math> and <math display="inline">B</math> ten light years apart. They both had connections established from a parent node <math display="inline">P</math>. Those connections are time-shifted over 99 years and part of the privileged base graph.

We couldn’t build a time-shifted connection between <math display="inline">A</math> and <math display="inline">B</math> without causing a CTC to be formed. But we are entirely fine sending signals '''Fast-as-Light''' from <math display="inline">A</math> to <math display="inline">B</math> and back. This is not a violation. Now consider: how does a wormhole with no time shift across it behave? In the mesh network case, such connections don’t cause CTCs to appear. And it doesn’t cause one to appear here either. How can we build such connections? We can of course use projection with relativistic speeds at both ends, like how we would build Mesh networks. But there is an even faster solution. If we can move wormholes through wormholes, we can shift the new mouths through the existing network to create this connection!<br />

The main catch is that we have to tightly watch out for temporal drift on such new connections. Through adding such edges that overall form a "blue graph" on top of the red graph, we are creating loops on the red graph underpinning the network. If one end of a wormhole mouth drifts more in time than the other (which will inevitably happen because stars won’t have the same mass, experience the same galactic gravity, or have the same velocity relative to each other) the conditions for a CTC will start to appear. The only way to pre-empt this is with continuous maintenance, taking connections offline and adjusting their temporal position using synchrotons, and having "safety space-time" spacing between wormholes. In this way, there is some "room" where signals can arrive faster and faster, but not before they were sent, allowing the connection to be used most of the time with only short intervals of adjustment.<br />

This works even when both ends connected don’t have the same time distance from the root node - the important thing is that the new shortcut accomodates this difference with a time shift of its own. We can calculate this difference easily. For two non-root nodes <math display="inline">A,B</math>, for each node we sum the temporal difference over the path from root to the nodes along the red graph, then subtract the difference from each other. The resulting remaining temporal difference is that the new connection on the blue graphh must have to keep the network stable. This means we can even do things like connect systems far out directly to core systems, or bridge from one edge of the graph to the other - arbitrary routings are possible. These routings can allow faster signal transmission and mass transport than through root, mitigating bandwith issues. The main problem is to watch out for network stability. As nodes accumulate, CTC avoidance becomes more and more delicate, and safety spacing must be maintained diligently. This can add to the effective travel time through such alternate connections, and thus maintains some of the "privilege" of the acyclic base graph.<br />

[[File:Wormholes BuildingCyclics.svg|800px|center]]

Arguably this network configuration is the most powerful, marrying many advantages and removing some disadvantages. It is however also bound to the need for safety spacing. Depending on your infrastructure assumptions, this can pose issues. If you want to implement serious stretches of safety space, wormhole links not part of the acyclic base graph would likely have to be located at interplanetary distances away from the main links, with light hours to light days of safety spacing. If your main network runs on planetary trains, this is rather a bother. Adjusting the internal throat length of wormholes can help in this regard, but of course now poses the issues of how to safely navigate such a long wormhole. Maybe you could extend something like space elevator tethers through such stretchy throats. With no gravity inside the wormhole itself, such cables could probably stretch quite significant distances. If your culture is handling wormholes mostly in space, you still have to account for added travel times through the safety spacing. Communications links would probably be easier connected between arbitrary nodes than macroscopic, craft-traversable wormholes.

[[File:Wormholes ComplexExample.svg|800px|thumb|center|An example of a complex semi-privileged graph wormhole network]]

==Credit==
Authors: Luke Campbell and Sevoris

==References==

[[Category:Physics]][[Category:Infrastructure]][[Category:Physics & Math & Engineering‏‎]][[Category:Spacecraft & Infrastructure‏‎]]

Intelligent Aliens

2022-08-26T15:50:45Z

Sevoris:

==Summary Quick Facts==
Alien species creation is the entry point for "soft sciences" like anthropology and sociology. As fiction is ultimately about emotional resonance, alien species are the canvas a writer (or exo-archeologist or exo-sociologist) uses to contrast with human cultures, and give characters interesting people to talk to. There are several "tiers" of alien species, ranging from Exaggerated Human Cultural Trope Aliens, all the way to fully realized alien societies and cultures.

==Historical Context==

The first alien species in SF that protagonists communicated with were thinly veiled "savages and sages" tropes, dating back to Edgar Rice Burroughs's novels of Pellucidar and Barsoom. E.E. "Doc" Smith carried many of these tropes forward in the Skylark series, and eventually the Lensman series. In the earliest tradition, aliens are nearly always "people in need of a Western White Man" to lead them. They might be amazingly technologically advanced, they might be some decadent elder civilization overcome with sloth and ennui, but they were always in need of the Bold Dashing Earthman to get them out of their cultural decline and to aid the hero in things. In this, they are direct inheritors of the Penny Dreadful pulp tradition and stories of Darkest Africa from the 19th Century.

John W. Campbell, the original editor of Astounding Science Fiction, and the editor who discovered many of the Golden Age authors (Heinlein, Asimov, van Vogt, and several others), reputedly challenged his authors to "Make something that thinks as well as a man, but different." Some authors (Stanley G. Weinbaum, and to a lesser extent, van Vogt) made their reputations from doing this during the span from the 1930s through the 1950s. Later authors, like Larry Niven, refined the techniques of their predecessors by making one or two aspects of biology determine what makes an alien species different from humans, and altering how they think, often with a "motivational difference" that catches a human point-of-view character by surprise. While the "New Wave" of science fiction in the late 1960s and through the 1970s de-emphasized engineering as a driver for stories in many markets, it made focusing on sociology, psychology, and gender role examination as legitimate areas of exploration in "serious" science fiction. David Brin's '''Uplift''' series may be the high point of "John W. Campbell"-style aliens.

A sea change in alien portrayals came with C.J. Cherryh, whose earliest books covered alien species in contrast with human cultures, then had a diversion through her all-human Alliance-Union setting, and then the Novels of the Compact, starting with the Pride of Chanur. A high school Latin teacher who was well read in history, mythology, anthropology and sociology, and a determined researcher, Cherryh's aliens were well realized enough on a sociological and psychological level that she could write entire series of novels totally from an alien's point of view. Other writers have built on the foundation that Cherryh laid.

===Types of Fictional Aliens===

In modern SF, intelligent aliens tend to fit one of three "tiers" of complexity: Television aliens, biologically deterministic aliens, and sociologically complex aliens.

Aliens for television are generally constrained to humanoid bipeds with lines of spoken dialog that miraculously translate to English. These limitations are driven by audience accessibility, and make it easier to hire actors. Motion capture technology may greatly expand what can be portrayed on screen. Television aliens are sometimes called "bumpy forehead" aliens or "latex aliens", depending on how much prosthetic costuming is needed. If much thought has been put into what makes them "alien," it's a human characteristic carried to an extreme: Klingons are the Honorable Warrior Race, Vulcans are Stoics, Narns are Oppressed Rebels about to re-enact a cycle of vengeance, the Centauri are stand-ins for decadent Romans, and so on. While some details are Clearly Different (most often hinted at with sexual reproductions, with Vulcan ''pon farr'', Narns being marsupials, and Centauri males having six penes...), the television alien culture has obvious human analogs, usually to highlight the Virtues of Human Culture by contrast. Many television aliens (and quite a few Bad Guy Aliens in novels) are obvious ports of human cultures. The original Klingons in Star Trek were stand-ins for an authoritarian empire. They became pseudo-Samurai by the TNG era, and one of Worf's character arcs over the series involved the reconciliation of what his childhood painted as Klingon culture, and the actual reality of dealing with Klingons.

Writers should be careful of their own biases and cultural tropes when using "ported human culture dialed up" as a template for aliens. For example, the original portrayal of Star Trek's Ferengi were exaggerated capitalists who contrasted with the peaceful utopian post-replicator technology of the Federation; between that and several unfortunate choices in prosthetics, the Ferengi started out as an overt port of anti-Semitic tropes; they were eventually rehabilitated in later series. Similarly, most Honorable Warrior Race aliens have dark skin in Western media and in anime, because it's a way to quickly code the aliens as "savage barbarians" or "formidable opponents" with minimal explanation to the audience, and without granting them cultural parity or equivalency.

Biologically derived aliens range from the aliens of the Lensman series (where aliens range from "humans with odd skin colors" to creatures that live at liquid helium temperatures), the Slan of A.E. van Vogt, and most of Larry Niven's Known Space aliens. Nearly all of these aliens have different behaviors driven by some biological trait, that grew out of the environment they evolved in. This can be a (Lensman series) Rigellian sense-of-perception arising from their homeworld with an opaque atmosphere, to Niven's Kzinti being obligate carnivores (and aggressive) and his Puppeteers being herbivores and obligate cowards. Niven's later alien species, like the Moties and the Fthip through to Niven's later alien species like the Moties and the Fithp.

While "bumpy forehead aliens" are largely limited to television and film, biologically derived aliens show up in games and written SF. Both are used to highlight and contrast aspects of human societies. Even with biologically driven aliens, there is often an unspoken bias, using the cultural changes driven by biology as evidence that the aliens are hampered by their biology. The process of making a biologically derived alien starts by looking into some of the odder corners of terrestrial biology, and templating off their reproductive strategies, what they eat, or how they contest for territory. The advantage (for the SF writer) is that it's a lot easier to research an alien species based off of a translation of a terrestrial animal type than it is to come up with an entire alien culture! Again, Brin's '''Uplift''' books, along with Vernor Vinge's '''Zones of Thought''' books are good examples of this, and most of Gannon's '''Caine Riordan''' universe aliens are from this tradition. (Because house cats are common pets, and clearly interface with humans on something approaching their own terms, there are more cat-like aliens in SF than you can shake a stick at!)

Sociologically complex intelligent aliens are made by an iterative process, and if your book is going to be '''about''' aliens, putting more work into your aliens at the beginning will pay off later.

==Sociologically Complex Aliens==

Making sociologically complex intelligent aliens requires asking questions and taking notes. There are three techniques that work well in tandem to make well-realized aliens, ones that are more versatile and interesting than The Honorable Warrior Race aliens, or the Declining Elder Species Aliens and similar. These techniques are the Seven Sociological Questions (which also brush onto biology) and the Three Ripple Rule, and social adaptations to overcome things that weren't limits from the environment they involved in, but are when society becomes more complex.

===The Seven Sociological Questions===

Ask yourself what things are true about the aliens you're creating, working from this list:

# The physical environment the species evolved in, and how that shapes the environments they seek out.
# How they bear their young, how the young are raised, and how instinct and nurturing interact in raising the young, and on cultural transmission to younger generations.
# The way the species conducts spatial and living space planning, including how personal and family dwellings are arranged, with distances between buildings, how family groups do (or don't) live together, hunting territories, agricultural areas, and animal husbandry if practiced. This also covers using architecture for ceremonial purposes and showing status.
# What the species eats, the relative difficulty of getting enough calories per day, what methods are used to obtain and prepare food, how food is stored, and cultural practices and taboos for food preparation and consumption.
# The mechanisms and processes used for recording new discoveries, transmitting and sharing new discoveries, and how this is used to reinforce prejudices, tribal identity and culture.
# How the species views death, mourning, dying and handling of family members who won't live out a season, including beliefs (or non-belief!) in an afterlife.
# How does the species think about thinking, self-definition and identity, and conceptualizations of the universe they inhabit.

There are no 'right' answers to these questions, but the more thoroughly you interrogate these concepts, the more different you can make your aliens. Having two or three of these questions give "different from human" answers will make aliens distinct.

Many of these questions have had different answers for humans in different cultures, and at different points in time. For example, many of Earth's late Neolithic sites, such as Gobeki Tepli and Stonehenge, were not permanent residences. They appear to have been pilgrimage sites used only at certain times, much like the Kaaba of the Grand Mosque of Medina is today.

===The Three Ripple Rule===

The Three Ripple Rule holds that any technology that makes transportation, commerce, entertainment or communications easier will have consequences unforseen by its inventors or the people who promulgate it. The typical example from Western cultures is that anyone could've predicted the automobile, and several people did. The combination of the automobile and movies led to drive-in theaters. The combination of automobiles, movies and drive in theaters led to a ''sweeping'' change in sexual mores, which terrified local political structures and sparked the sexual revolution a decade later.

This makes the automobile the change, the first ripple was drive through theaters, the second ripple was more privacy for teenagers and young adults out of the home, and the third ripple is the dawn of female birth control and the sexual revolution. A good resource for thinking about the Three Ripple Rule is the television series by James Burke from the late 1970s called '''Connections'''. Burke mostly focuses on technology, but covers a bit of sociology here. He did a similar series for the BBC called '''The Day The Universe Changed'''.

When you answer the Seven Sociological Questions, look for ripple effects that follow through from the answers you make.

For example, human females are constantly in estrous, and much of the daunting history of maternal risks in childbirth comes from the fact that the human pelvis can ''just barely'' accommodate the diameter of a newborn baby's skull as it's squeezed out the birth canal. Remaining in estrous constantly makes the risks inherent in a "just barely works" birthing mechanism less likely to result in the collapse of the species at a genetic bottleneck or climate disaster. Much of human society's historically awful treatment of women as chattel derives from this, as does the peculiar obsessions humans have with sex.

Let's contrast this with a species that lays and broods eggs, with females only going into estrous at certain times of the year. In a wide range of species, estrous can be triggered by diet, or from environmental cues like the length of day. The first ripple is that a birth control or artificially triggering estrous might happen much earlier in a society's development. The second ripple to consider is that unlike humans, who think regularly about sex from puberty to late middle age, sex and reproduction might be a seasonal. They may completely lack the concept of romance novels and might lack erotic fiction entirely as a culture. The third ripple is more speculative: In terrestrial species with mating seasons, the competition for mates is intense, which might be disruptive in a larger, more interconnected society. There may be cultural behaviors put in place to act as escape valves, or there may be constant competition for status that leads to proximity to females, or rituals for joining families or clans. They would certainly consider the human near constant obsession with sexual signaling to be odd by comparison.

Similarly, an alien species with different childhood development patterns (and a larger load of instincts) may have parenting styles that humans find appalling, with resulting differences in spiritual beliefs.

===Social Adaptation To Compensate For Instincts===

Instincts are a shorthand imposed by evolutionary adaptation to keep members of a species alive long enough to raise children. In human societies, we have a solid instinct to seek out sugar, salt and fats in foods, because in the environment we evolved in, sugars and fats were signs of high caloric density foods, and eating a lot of them when they were available would allow you to put on the weight needed to get through the next famine. Now that we've made foods that fit those cravings *exceptionally* cheap and greatly reduced how often famines occur, the Western diet has well documented health effects. When looking at your alien species, look for instincts that don't quite fit a technological civilization.

Modern opposition to family planning in Western cultures is an example of counter-reactions to social adaptations. Many of the arguments against family planning boil down to control of women, derived from unquestioned assumptions about the need for larger families and the role of women in raising children. Cherryh's Chanur novels have similar unquestioned assumptions about the role of males in an alien society, and are particularly subtle because it's shown with an unquestioned leap of logic without intervening steps being explained, shown through the thoughts a point of view character who encounters a male in an unexpected place and sees his presence only as an unwelcome complication.

A useful point of differentiation when making alien species is a disputed sociological concept called ''Dunbar's Number''. On average, a human being can keep track of 100 to 150 distinct relationships, in varying degrees of closeness (about 20 close friends, about 70 to 100 intermediate friends, and 30 to 50 second order relationships, based on "friends of friends." Humans have remarkably high Dunbar's numbers; our closest genetic cousins, bonobos, have Dunbar's numbers that seem to cap out at 50 to 70 members. Even so, a lot of human society can be looked at in terms of 'how do we circumvent the limits of Dunbar's Number?" In parts of the world where tribal and clan loyalties run paramount, you can see how Dunbar's Number shapes society, as the mechanisms that Westerners assume always work...don't. For example, it's not that Afghans are materially more corrupt than Westerners; it's that their cultural incentives don't give them many reasons to think beyond the group of 100 to 150 people that they have relationships with, many of whom are tied by familial bonds. This makes perfect sense in situations of periodic scarcity; it also makes trusting in the law and police departments to do their job rather than favor family members seem like incredible naivete...

What other mechanisms are there to compensate for lower Dunbar's numbers? What does a society where individuals with a higher Dunbar's number look like? In most Earthly societies, people who can maintain larger social networks gain wealth and status. How does that work for your alien species? What does that look like? What do higher status members of your society do that others cannot, or are not permitted to do?

[[Category:Life]]

Structural Biological Materials

2022-08-26T15:50:19Z

Sevoris:

{{PageConstructionNotice}}

Real organisms use a wide variety of structural materials to give toughness, strength, rigidity, and other desirable material properties to their bodies, both on the cellular (as in cell walls) and tissue (as in bones or exoskeletons) level. Moreover, these materials are rarely homogenous - biological structural materials are almost always composites of more than one material, becoming something greater than the sum of its parts.

Rather than simply take the approach of listing options, this page will aim to explore the general categories of structural biological materials - both real and theoretical - and explain how their particular properties play into how they're used in living creatures.

==Polysaccharides==

Polysaccharides are perhaps the most widespread - and simplest to evolve - structural materials we find in living creatures. Saccharides, especially glucose, are universally used as energy storage in Earth life, and it's likely that alien life - at least, that which shares our basic biochemistry - will do the same. It therefore isn't much of a step to derive your structural materials from the stuff you already use for energy storage!

Even though monosaccharides are typically hydrophilic, and readily dissolve in water, most polysaccharides are, in fact, quite hydrophobic and insoluble. This makes them quite well-suited for assembly-on-the-spot - the component monosaccharides can be dissolved in the cellular medium for transport, and will precipate out of the water as they are assembled into polysaccharides.

One defining element of polysaccharides, on the chemical level, is the multiple free hydroxyl groups on each monomer of the chain. Through a variety of processes, these can be replaced by other functional groups, altering the properties of the polysaccharide; they may also serve as useful bonding sites for other structural molecules, such as peptides and polyphenols. This chemical versatility is another reason why they are so ubuiquitous in Earth life - they excel at a wide variety of structural purposes, and make a very good "foundation" for other structural biological materials to build on in composite materials.

===Cellulose, and other simple linear polysaccharides===

[https://en.wikipedia.org/wiki/Cellulose Cellulose] is the simplest and most widespread polysaccharide that sees structural use on Earth, and is likely to be so on alien worlds as well. It's a simple linear chain of β(1→4) linked units of d-glucose - the same molecule nearly all life on Earth uses to store energy. Aliens might use ʟ-glucose instead, but this should have identical mechanical properties.

Contrary to popular belief, cellulose is actually quite flexible, when alone. The rigidity of plant cells - famously with cell walls made out of cellulose - is provided by other materials bonded with cellulose, such as [[Structural Biological Materials#Polyphenols (lignin, tannin, etc.)|lignin]], along with the hydrostatic pressure provided by the water within the cell.

A number of Earth organisms are demonstrated to use mannose, or other simple sugars, in the place of glucose, forming structural polysaccharides such as [https://en.wikipedia.org/wiki/Mannan_(polysaccharide) mannan]. Their properties are similar to cellulose - though, how they may bond to other polymers might differ, as we'll discuss later with glucomannans.

Some other organisms, such as oomycetes, use other β-glucans - that is, linear chains of glucose - which have different linkages (such as the β(1→3) linkages of chrysolaminarin) as structural materials. In bulk, their mechanical properties are fairly similar.

[[File:Cellulose.png|thumb|none|The structural formula of cellulose.]]

===Hemicelluloses - heterogenous simple polysaccharides===

In plants, cellulose is not found alone - it is typically accompanied by filaments of [https://en.wikipedia.org/wiki/Hemicellulose hemicellulose], otherwise known as '''polyose''', a group of highly heterogenous polysaccharides composed of a semi-random mix of miscellaneous saccharide molecules - predominately arabinose and xylose. The heterogeneity of these fibers inhibits the formation of crystalline structures the way cellulose might form, rendering the result relatively amorphous and weak: if isolated, hemicelluloses would have a gum- or gel-like texture. While that may seem undesirable in a structural material, hemicelluloses perform important functions in plants - in particular, cross-linking cellulose fibers together, adding additional strength and toughness to the resulting matrix.

[[File:Xylan_hardwood.png|thumb|none|Structural formula of a xylan hemicellulose found in a hardwood species.]]

===Chitin, and other poly-amino-sugars===

[https://en.wikipedia.org/wiki/Chitin Chitin] is the second most widespread structural polysaccharide on Earth, and for good reason. Like cellulose, it is a linear chain of single monomers. Rather than being made of simple glucose, however, it is composed of [https://en.wikipedia.org/wiki/N-Acetylglucosamine ''N''-acetylglucosamine] - an amino-sugar of glucose where one of the hydroxyl groups is replaced by an acetylamino moiety. This acetylamino moiety forms stronger hydrogen bonds with adjacent polymers than the hydroxyl group would, rendering chitin much stronger and slightly more rigid than cellulose - though, it still remains plenty flexible, without other components to rigidify it.

''N''-acetylmannosamine and ''N''-acetylgalactosamine, derivatives of mannose and galactose, respectively, also exist in nature - to my knowledge, they are not used to form polymers like chitin, but an alien organism may do so!

[[File:Chitin Highlighted.png|thumb|none|Structural formula of chitin. Notice the acetylamino moieties, here highlighted in red.]]

===Hydrophilic polysaccharides===

Though most polysaccharides are hydrophobic, this does not apply to all. Many are possessed of an overall negative charge and/or are internally polar, allowing them to become quite hydrophilic. As hydrophilic molecules, they will readily dissolve in water - a characteristic exploited in certain biological lubricants, such as the fluid that fills the space between joints in vertebrates, which consists of relatively short chains of hydrophilic polysaccharides dissolved in water (when you pop your knuckles - or other joints - what you're doing is popping bubbles in this fluid).

However, though these polysaccharides dissolve in water, this ''does not break the bonds of the polymers''. This is a property of water-soluble polymers in general - though they may "dissolve", they do not break apart. This means relatively long polymers, especially those with extensive cross-linkages between them, will maintain their structural form when hydrated - instead, the water infiltrates the polymer structure, causing it to swell into a sort of hydrogel or gum, which may be stiffer or softer depending on how closely interlinked the polymer chains within it are. Water can also pass through the polysaccharide gel, at a rate which is, again, dependent on the exact structure of the gel - which may assist organisms, particularly unicellular or colonial ones, in osmoregulation. This structural usage of hydrophilic polysaccharides is obviously better suited for aquatic biota, which do not have to worry about their polysaccharide hydrogels drying out - but can still be quite useful in the interior structures of terrestrial biota.

====Algin, and other polyuronic acids====
[https://en.wikipedia.org/wiki/Alginic_acid Algin], or alginic acid, a polyuronic acid, is a primary structural component of the cell walls of brown algae, such as kelp. It is a polymer of two uronic acids in alternating sequence - β-D-mannuronate and α-L-guluronate. Uronic acids are derivatives of saccharides where the hydroxyl group opposite the carbonyl has been oxidized to a carboxyl group, making them a kind of carboxylic acid. This makes polyuronic acids acidic (as implied by the name) - which in turn makes them hydrophilic and water-soluble.

The hydrophilic nature of algin is responsible for the gummy, gel-like texture/structure of the brown algae.

The carboxyl group replacement that defines the uronic acids very rarely co-occurs with replacements of other hydroxyls on the same monomer with other functional groups, for reasons I don't personally understand. For example, I know of only one amino uronic acid - [https://en.wikipedia.org/wiki/N-Acetyltalosaminuronic_acid ''N''-Acetyltalosaminuronic acid].

[[File:AlginHighlighted.png|thumb|none|Structural formula of algin. Notice the carboxyl groups, highlighted in blue.]]

====Sulfated polysaccharides====
Sulfated polysaccharides are common throughout Earth life, and should be expected to be common among alien organisms with broadly similar biochemistry. In sulfated polysaccharides, one or more free hydroxyl groups are replaced by a sulfate ion; in some cases, sulfates occur only once every several monomers, while in others, nearly every single hydroxyl group in the entire chain is replaced by a sulfate.

Like polyuronic acids, sulfated polysaccharides are hydrophilic, due to the negative charge of the sulfate ion. However, they have a range of properties which distinguishes them profoundly from their uronic acid equivalents.

First, whereas the carboxyl groups that define the uronic acids rarely co-occur with other hydroxyl-replacements on the same monomer, sulfation ''frequently'' occurs with other functional group replacements - particularly, sulfated amino sugars, where one or more of the remaining hydroxyl groups (after the addition of the amine) has been sulfated, are quite common in nature, allowing a single monomer to bear the characteristics brought by both sulfation and other derivative processes. This extends to, as previously mentioned, molecules where each monomer has had ''all'' free hydroxyl groups sulfated, resulting in a much greater negative charge (and thus, greater hydrophilicity) than in uronic acids.

Second, sulfated polysaccharides are quite good at binding and precipitating metal ions, such as calcium or sodium, out of solution. Relatively short sulfated polysaccharides, therefore, often play a non-structural role in regulating the transport of ions throughout an organism; and sulfated polysaccharides in the outer surfaces of aquatic organisms, such as in the cell walls of many algae, may assist in the capture of necessary ions from the water. However, this also lends itself to a structural purpose - sulfated polysaccharides appear to play a role in biomineralization in some clades, such as in the coralline algaes, precipitating metal ions out of solution to form hard, mineralized coatings on the organism's exterior.

===Disaccharide polysaccharides===

As demonstrated by algin, above, many structural polysaccharides, rather than consisting of a homogenous chain of a single monosaccharide molecule, are instead often composed of two distinct monosaccharides in alternating sequence. This allows the copolymer to have some of the properties offered by both monosaccharides.

====Glycosaminoglycans====
[https://en.wikipedia.org/wiki/Glycosaminoglycan Glycosaminoglycans], also known as '''mucopolysaccharides''', are a common form of alternating polysaccharide, consisting of alternating units of a uronic acid and an amino sugar, the latter of which may be sulfated (as in [https://en.wikipedia.org/wiki/Heparin heparin]) or not (as in [https://en.wikipedia.org/wiki/Hyaluronic_acid hyaluronan]). These polysaccharides are ''very'' polar, and attract water very strongly as a result, while also self-adhering quite well.

Glycosaminoglycans fulfill a variety of important functions, particularly in animals. Since diffusion through them is possible, but quite slowly, they help slow and regulate the intercellular diffusion of substances like ions and proteins; since they can adhere quite well, they serve as an adhesive binding different cells together; and since they attract water so well, they are used to assist in the retention of moisture where watery surfaces must be exposed, such as in the eyes of vertebrates.

====Peptidoglycan====
[https://en.wikipedia.org/wiki/Peptidoglycan Peptidoglycan], also known as murein, is the primary structural component of bacterial cell walls. It is an alternating copolymer of ''N''-acetylglucosamine (as found in chitin) and ''N''-acetylmuramic acid - a complex monosaccharide where a lactic acid moiety has replaced yet another hydroxyl group on ''N''-acetylglucosamine. Like in chitin, the acetylamino moieties serve to assist in the formation of crystalline structures through strong hydrogen bonds; the addition of the lactic acid to every other monomer in the chain, however, serves as a strong binding site for short oligopeptides (consisting of three to five amino acids) which bind to a different polysaccharide chain at each end, forming a regular mesh that resembles a chain-link fence.

The resulting structure is very hard to break apart chemically, and strong enough to maintain the regular shape of cells observed in most bacteria. There is no particular reason, in fact, why it is not found in eukaryotes or archaeans, other than these two groups simply not having evolutionarily stumbled upon it - peptidoglycan only evolved once. So, complex alien life might use it!

===Branching polysaccharides===

As seen above, almost all polysaccharides used for structural purposes in living organisms are linear chains, though branching polysaccharides are commonly used for energy storage. However, some branching polysaccharides do, in fact, see structural use - such as the '''glucomannans''' found in some yeasts and plants. Glucomannans consist of a linear chain of mannose units (called a mannan), with short glucose branches. Unlike linear polysaccharides, branched polysaccharides resist forming neat, crystalline structures (as the branches interfere with hydrogen-bond formation between polymers), and thus are able to slide past each other, forming a rubbery, hydrophilic solid, or even a viscous liquid (when hydrated).

==Polyphenols (lignin, tannin, etc.)==

Polyphenols, such as lignin and tannin, are a broad category of biopolymer found almost exclusively in plants on Earth, consisting of many phenol units oxidatively coupled together. Unlike other common structural biopolymers, polyphenols are typically non-linear - instead forming complex 3-dimensional structures. They are also often highly heterogenous, made of many different monomers. This makes it difficult to talk about sub-types of polyphenol, as we have done for polysaccharides - instead, this section will talk about the properties different polyphenols often possess.

===Color===

Polyphenols' complex 3d structures leads to a variety of interactions with light waves that result in them being typically opaque and strongly colored. While most polyphenols are dark brown - lignin being responsible for the dark brown color of most tree bark - some polyphenols have brilliant colors, such as the anthocyanidins that provide flower petals with vivid reds, purples, and blues.

While structural polyphenols are typically quite separate from pigment polyphenols, an organism that is able to synthesize one is likely capable of synthesizing the other!

===Reactivity===

Whereas most structural biopolymers are only weakly reactive, polyphenols are quite reactive, specifically to oxidation. This might sound like a vulnerability for a structural material, but polyphenols' reactivity is usually localized to a few specific sites on the molecule. This makes them very good at '''binding to other structural polymers,''' like polysaccharides and proteins, at multiple sites, via oxidative coupling. Other structural biopolymers bound by polyphenols in this way tend to become exceptionally hard and rigid - like tree bark.

===Resistance to degradation===

Some - not all - polyphenols, such as lignins, are exceptionally resistant to bio-degradation, making them very difficult for other living things to break down and digest - and this extends to the other polymers they bind to. This makes them very good at protecting from penetration by parasites or pathogens, and limits the possible predators of organisms that use polyphenol cross-linking in their structures.

===Affinity for saccharides===

Polyphenols have something of a special affinity for saccharides, and will not only readily bond with them, but also often include one or more saccharide molecules in their structure - such as in tannic acid, a polyphenol composed of phenol branches stemming from a glucose molecule at its center.

{|style="margin: 0 auto;"
| [[File:Phenol2.png|thumb|upright=0.59|Structural formula of phenol, the simplest phenol.]]
| [[File:TannicAcid.png|thumb|Structural formula of tannic acid, a polyphenol. Notice that it is built with a monosaccharide - glucose, specifically - at its center.]]
| [[File:Lignin.png|thumb|none|Idealized structure of lignin from a softwood. In reality, lignin molecules are highly heterogenous, and do not have a consistent structure.]]
|}

==Cross-linking proteins (collagens, keratins, sclerotins, etc.)==

Proteins - perhaps unsurprisingly, given their ubiquity - are common components of structural biological materials, specifically in the form of cross-linking proteins like keratins or sclerotins, which bond to each other at many different sites, allowing for not only toughness and rigidity but a ''tunable'' degree of both - via controlling the frequency of cross-linkages, materials can smoothly vary in rigidity and hardness, often forming distinct gradients in biological structures (such as arthropod cuticle becoming more flexible around the joints).

Most cross-linking structural proteins tend to be brown to black in color, though are not particularly opaque, and some can be outright colorless. Their versatility also lends themselves to the creation of complex microstructures resulting in vivid structural colorations, and tissue that includes them often also plays host to a variety of pigments that add their own colors to the material.

===Collagens===

Collagens are the primary family of structural proteins found in all metazoans, though they are most prominent in the deuterostomes - in protostomes, many of the functions typically performed by collagens are instead performed by chitin. In mammals, collagens make up 25-35% of all proteins in the body! They make up the connective tissue, while also adding strength to other tissues (such as blood vessels) throughout the body, and also providing the matrix on which hydroxyapatite is deposited to form bone and cartilage in vertebrates.

Collagen consists of a left-handed triple helix of elongated peptide chains. About half of its amino acid content is glycine and proline alone, which appear to play a role in binding the three components of the helix together. In addition, sites throughout the chain are saturated in hydroxylysine - a hydroxylated derivative of the amino acid lysine. It is by these hydroxylysine sites that collagen helixes are cross-linked - via aldol condensation, strong covalent bonds are formed through conjugated enone moieties between the hydroxylysine sites on adjacent collagen helixes. This binds the adjacent helixes together very strongly, while still allowing hydrolysis of the linkages when they must be split apart, unlike certain other structural proteins.

Unlike most structural proteins, collagen appears colorless, or sometimes white. It is also extremely flexible - collagenous tissue must rely on other components to add hardness and rigidity, if necessary.

===Keratins===

Keratins are the primary family of structural proteins found in vertebrate integument - composing hair, nails, feathers, scales, etc, while also reinforcing the epithelial tissue (the outermost layer of the organs and blood vessels). There are two primary categories of keratin - α-keratins, which are found in all vertebrates, and β-keratins, which are found only in sauropsids. The primary difference between these is that α-keratins form double-helix filaments and are more flexible, whereas β-keratins form β-pleated sheets and are significantly harder and more rigid.

Both types of keratin crosslink by means of disulfide bridges - they contain large amounts of the amino-acid cysteine, which bonds to cysteines on adjacent keratin polymers via the sulfur. These bonds are very strong and rigid, and, as mentioned before, the precise degree of hardness and rigidity can be precisely controlled by determining the frequency of disulfide bridges. The presence of so much sulfur-containing cysteine is why heavily-keratinized tissues - like human hair - smell pungent when burned.

===Sclerotins===

Sclerotins are another widespread family of structural proteins, found primarily in arthropods but also some annelids. Sclerotins are a much broader family of proteins than keratins, and do not have a particular identifiable geometric structure, unlike the α-helixes and β-pleated sheets of keratins. They are also not bonded by disulfide bridges - instead, they are cross-linked by a process, called sclerotization, similar to what is done to tan animal hide into leather.

Sclerotin proteins contain large numbers of free amine and thiol groups, which tend towards oxidative reactions with other molecules. In the process of sclerotization, quinones are enzymatically introduced to tissue containing large numbers of sclerotins, typically by enzymatic action on dopamine-derivatives such as ''N''-acetyldopamine. These quinones then react, forming strong covalent bonds, with the amine and thiol groups on the sclerotins, cross-linking them together into an exceptionally tough, hard, and rigid material. Like keratins, the exact degree of hardness and rigidity can be controlled, this time via the amount of quinones introduced into the tissue.

Sclerotization is effectively non-reversible - cross-linked sclerotins being extremely hard to break apart - and thus organisms must shed sclerotinized parts if they need to change them, hence the necessity of molting in arthropods.

Some annelids, such as the intimidatingly-named bloodworm, have special sclerotins which bind metal ions, such as copper and iron, prior to undergoing sclerotization. This results in unique metal-composite structures which are exceptionally hard and strong.<ref name="BloodwormJaw">[https://www.cell.com/matter/fulltext/S2590-2385(22)00153-9]</ref>

===Other cross-linking proteins===

Other cross-linking proteins are found in other clades, though none as archetypical and widespread as keratins and sclerotins.

For example, cephalopod mollusks possess their own particular cross-linking proteins that add hardness and rigidity to their beaks, operating in a two-tiered system - they are, unfortunately, absent a catchy name like keratins and sclerotins, and are instead simply referred to as '''chitin-binding beak proteins (CBPs)''' and '''histidine-rich beak proteins (HBPs)'''. CBPs simply bind to multiple chitin chains at both ends, forming cross-links between them. HBPs add a second tier to this matrix, and bond to both CBPs and each other, forming an additional set of cross-links to add even more strength, hardness, and rigidity to the result.<ref name="CephalopodBeak">[https://www.nature.com/articles/nchembio.1833]</ref>

Alien organisms may be expected to exhibit yet more variations on the common motif of proteins cross-linked together, unseen on Earth - instead of directly using one found on Earth, it may be better to simply specify a, say, "keratin-like protein"!

==Biominerals==

While all the structural materials discussed previously - and those which will be discussed after - are polymers, mineral crystals also play a major role in structural biological materials. Biominerals excel at adding hardness, strength, and rigidity to biological structures; only a handful of organic polymers are capable of matching even the softest biominerals in hardness and rigidity.

For obvious reasons, tissue that is heavily biomineralized cannot be flexible - although, semi-flexible partially-biomineralized tissue, such as cartilage in vertebrate, does exist in nature.

In addition to performing structural functions, biomineralized tissue also often functions as a reservoir of biologically-essential chemicals, such as metal ions, phosphate, and sulfate. In fact, many biomineralized structures in modern Earth organisms are believed to have evolved from what were initially purely storage deposits - for example, the bones of vertebrates are believed to be derived from what was initially the mere deposition of calcium and phosphate for storage.

All biomineralization processes found in nature (at least, that I am aware of) involve the creation of ''insoluble'' crystals from ''soluble'' components. The necessity of this particular combination of characteristics - soluble components, insoluble products - is obvious when you consider the challenges involved in creating them: chemicals must be soluble in water (or whatever alternative solvent) to be acquired from the environment and transported through an organism's tissues; and upon forming their insoluble products, they immediately precipitate out of solution, forming hard, insoluble crystals, and thus useful structural materials. This means that '''components that are insoluble cannot be acquired and transported, and products that are soluble cannot be made into useful structural materials.''' This renders certain sci-fi go-tos - such as giving an alien organism bones of titanium - implausible (as titanium is insoluble in water in all forms). However, there is still plenty of room to explore hypothetical biominerals, as will soon be demonstrated.

While most biominerals are salts - specifically, salts of metal cations and oxyanions - we will first discuss the exceptions to that rule.

===Silicates===

[[File:Diatoms through the microscope.jpg|thumb|right|200px|An assortment of unicellular diatoms under light microscopy.]]
[[File:Diatoms Egge and Aksnes 1992 plot.png|thumb|right|200px|A chart showing diatoms as a proportion of algal populations, relative to silicate concentration. It shows the extreme dependency of silicate-biomineralizing organisms on silicic acid concentrations.]]

'''Silicates''', otherwise known as common glasses (literally, like the kind in your window), are perhaps the most common and widespread kind of structural biominerals on Earth, having evolved independently countless times throughout Earth's tree of life. This may seem surprising, given that you probably know of very few living things that are made out of glass - and that's because almost all organisms that use silica as a structural biomineral are unicellular. While clades that use silica as a ''minor'' structural component are common among multicellular life here on Earth - such as the horsetail ferns - there is only one that is known to use it as the primary mineral component of their structural materials: the Hexactinellid sponges, otherwise known as the '''glass sponges.'''

Silicates are also one of the few biominerals thare are not typically found in the form of a salt - instead being typically formed of silica (SiO₂) in its various crystalline forms. Silica is extremely chemically inert, and therefore virtually useless to any living organism for biochemical processes - therefore, unlike many other biominerals, it is ''solely'' a structural material, and not a reservoir of essential nutrients. On the other hand, this non-reactivity makes it exceptional for structural purposes, and it is one of the hardest and strongest biominerals found in nature!

One may find the apparent ubiquity of biogenic silica confusing, in context of previous remarks on the necessity that biominerals be the insoluble products of soluble components, as silicates are not usually thought to be soluble - the truth, however, is that silicates are ''very weakly'' soluble, depending on their crystalline structure. While, ordinarily, such weak solubility would be insufficient to supply enough of the compound to a planet's oceans, the key lies in that ''silica makes up the vast majority of the Earth's crust'', and can be expected to do the same on alien worlds of similar composition to Earth; thus, the relatively small ''portion'' of silica that dissolves is actually quite gigantic, in absolute terms! And once this dissolved silica has been re-formed into crystals, it does not readily dissolve again - thus fulfilling that same criterion.

Silicate biomineralization is typically performed by precipitating silicic acid - the dissolved form of silica - out of water, onto a matrix of specialized biopolymer filaments. This not only results in a very strong structural material, it is also, energetically, exceptionally cheap for the organism in question - organisms that use silica as a structural material spend only a fraction of the energy forming silicate structures as other biominerals require.

This, however, does mean that an organism's ability to build biological structures out of silica is strictly limited by the availability of silicic acid in its environment - and given how silicic acid enters the environment in the first place, this quantity is quite limiting indeed. In fact, competition over silicic acid in the biosphere of Earth is quite fierce: Hexactinellid sponges once were one of the dominant clades of filter-feeder throughout the water column, forming reefs comparable to today's corals, but are today limited entirely to deep water environments - by the Cenozoic period, the diatoms had become ecologically ascendant (though they are believed to have first appeared in the Triassic, they did not become ecologically dominant until much later), and nearly monopolized the supply of silicic acid in the ocean's photic zone, making it impossible for the shallow-water glass sponges to obtain enough to build their spicules. The implications of this for alien ecosystems are profound - in order for silica to be more widely-used in an alien ecosystem than it is on Earth, something will have to dramatically increase the availability of silicic acid in the planet's waters. As an example - adding ammonia or methanol to water substantially increases the solubility of silica, but not enough to make silica less useful as a biomineral; thus, the subsurface oceans of many ice worlds (provided the lower surface is silica, and not another layer of ice), thought to consist of eutectic mixtures of water and ammonia, may be replete with silica-biomineralizing life!

As a biomineral, silica is essentially always found in the form of [https://en.wikipedia.org/wiki/Opal opal], the hydrated, amorphous form of silica. This is, perhaps unfortunately, ''not'' typically what we would identify as ''precious'' opal - the gemstone known for its vibrant plays of color - but common opal, weakly translucent and milky-gray in color. Given that precious opal is merely a ''particular arrangement'' of hydrated silica, however, it is entirely plausible that an alien organism might evolve surface structures of precious opal as a form of display!

===Metal Oxides and Sulfides===

A few organisms, in need of exceptionally hard and tough biominerals, have evolved the use of transition metal oxides and sulfides - particularly the oxides and sulfides of iron. Unlike metal salts, they do not make particularly good reservoirs of essential metals for other biological purposes - they are quite strongly bound, and thus, like silicates, are almost solely used as structural materials. However, they do also, potentially, have one other interesting application - many of these minerals (particularly the iron-oxide [https://en.wikipedia.org/wiki/Magnetite magnetite] and the iron-sulfide [https://en.wikipedia.org/wiki/Greigite greigite]) are ferromagnetic, and thus many organisms use them as components of [https://en.wikipedia.org/wiki/Magnetosome magnetosomes] - organelles for sensing geomagnetic fields, and a kind of biological compass.

It is not particularly common for these biominerals to occur as major structural components throughout an organism's body - rather, most cases involve, quite specifically, ''teeth''. Metal oxides and sulfides are generally energetically and nutritionally expensive to form; therefore, they tend to be used for specialized purposes, with more common biominerals being used elsewhere. As an example, [https://en.wikipedia.org/wiki/Chiton chitons] and [https://en.wikipedia.org/wiki/Limpet limpets] both have independently-evolved iron oxide teeth ([https://en.wikipedia.org/wiki/Magnetite magnetite] in the former case; [https://en.wikipedia.org/wiki/Goethite goethite] in the latter), which they use to scrape algae and other organisms from hard rocks on the seabed (their teeth being so exceptional at this that they are significant contributors to erosion) - however, both possess defensive shells of the rather more ordinary calcium carbonate, instead.

The exception, of course, occurs where metals to oxidize - and oxygen or sulfur to react them with - are easily found, such as around hydrothermal vents, which spew large amounts of sulfur and iron into the surrounding ocean. It is here that the snail [https://en.wikipedia.org/wiki/Scaly-foot_gastropod Chrysomalion squamiferum], the scaly-foot snail, is found - with its shell and protective scales made of iron sulfides.

These oxides and sulfides of metals are typically formed by reacting oxygen or sulfur with metal ions inside of tissues, which then preciptates out as the corresponding metal oxide/sulfide.

===Salts===

As previously mentioned, the most common biominerals on Earth are mineral salts - particularly, salts of a transition or alkaline metal cation (such as calcium, magnesium, or iron) with an oxyanion (such as carbonate, phosphate, or sulfate).

As with other biominerals, the anion and the cation must be soluble in water; however, the resulting salt crystal must be insoluble, and thus able to precipitate out of solution. Unlike with silicates, which are always very weakly soluble, and with metal oxides and sulfides, which go from very soluble to completely insoluble, the gap between the solubilities of salts and their component ions is typically rather small; changing the pH of the solution may render insoluble salts soluble, or soluble salts insoluble, changing which make for workable biominerals.

====Cations====

While there are quite a large number of metals, both alkaline and transitional, that can potentially form salts with oxyanions, only a handful make for plausible components of biominerals. Most are simply too rare to be used in bulk for structural materials in biology. Many of those that remain are biochemically inert - particularly those that are completely insoluble in water, such as aluminum and titanium (sorry, SpecEvo writers who like giving their creatures titanium bones - it doesn't work! Titanium is biochemically inert). And some that are both common and very biochemically active, such as sodium and potassium, only seem to form salts that are soluble.

While we typically characterize biominerals as being composed of one cation in particular (e.g. we speak of our bones as being made from calcium), in reality, virtually all biomineral salts actually contain a mix of cations, though one might predominate - the bones of vertebrates actually contain significant amounts of magnesium and manganese in addition to calcium, for instance.

Notably, it is typically the cations, not the anions, that typically determines the color of mineral salts - in the absence of other pigments. In minerals that contain a mix of cations, the color will, naturally, be a mix of the cations' particular colors.

=====Calcium=====

Calcium is the most common cation in biomineral salts here on Earth - as found in the bones of vertebrates and the shells of many sea creatures. This is unsurprising, given that, while it is less biogically abundant than sodium and potassium, which don't form structural biominerals, it is much more abundant in biology than any other metal cation that ''does'', playing vital roles in an enormous variety of biochemical processes, from the nerve activations of metazoans to the opening and closing of stomata in plants. Even alien organisms that primarily use some other cation in their biominerals are likely to use calcium as a substantial component as well.

Calcium salts do have a particular weakness, however, although it is not one that is ''typically'' relevant on Earth - they tend to be highly soluble in even relatively weakly acidic solutions. Many alien worlds may have waters too acidic for calcium salts to be viable as biominerals - indeed, Earth's own oceans were likely once too acidic for calcium carbonate to exist, and human pollution of the Earth's waters is making it more difficult for organisms to maintain calcium carbonate shells. Such cases offer opportunities to allow other metal cations to shine (as primary components of biomineral salts) - though, there is no particular reason biominerals primarily using other metals couldn't just appear anyways on occasion.

Calcium minerals tend to be very opaque whites - as popular knowledge might already tell you.

=====Magnesium=====

Magnesium is about half as abundant in biochemical systems as calcium, here on Earth, and also plays a variety of incredibly important roles therein - including being a component of ATP/ADP, and the chlorophyll plants use for photosynthesis.

Magnesium salts are substantially less soluble in weak acids than calcium salts, meaning they are potentially favored as biominerals over calcium salts on more acidic planets. They also tend to be slightly harder and stronger than calcium salts.

Magnesium minerals tend towards light, greenish-grays.

=====Manganese=====

Manganese is yet another abundant cation in biochemical systems, though it is significantly rarer in Earth's crust than either calcium or magnesium (being about as common as phosphorous - which, given phosphorous is a common component of biominerals, is easily enough for structural usage). It is one of the most common metal cofactors found in enzymes, and thus plays an enormous role in biochemistry.

Unlike calcium and magnesium, which are found diffused throughout the world's waters, manganese tends to be concentrated in particular locations - particularly, around hydrothermal vents, which carry manganese from deep in the Earth to the ocean, or near the mouths of rivers, which dissolve manganese from sediment as they run to the sea. In the open ocean, manganese content is essentially nil - meaning organisms that live there must obtain it from the food chain, from organisms that were able to obtain it in coastal waters or those around hydrothermal vents.

Like magnesium salts, manganese salts tend to be less soluble in weak acids than calcium salts, and are also slightly harder and stronger. They also tend to be notably ''heavier and denser'' than either calcium or magnesium salts - rhodochrosite, for instance, has a specific gravity of about 3.7, compared to calcite's ~2.7.

Manganese minerals tend strongly towards bright, vibrant pinks - even in minerals where manganese makes up less than half of the cations: some kutnohorite specimens that are only ~40% manganese, and nearly ~60% calcium, are still vibrant pink colors.

=====Iron=====

Iron is abundant in both biochemical systems and in Earth's crust (in the latter case, far more so than any other metal, save silicon and aluminum, which are biochemically inert). Yet, there is very little of it in Earth's oceans, and it is only occasionally used in biominerals. Why?

The answer has less to do with the commonality of iron, or its solubility in water, than with how it reacts with oxygen. Prior to the Great Oxygenation Event, Earth's own oceans were replete with dissolved iron; it is believed it was far more commonly used in biochemistry than even today, as well, with organisms performing photosynthesis using iron and sulfur instead of carbon dioxide and water. However, as oxygen was added into Earth's atmosphere - and thus dissolved into the oceans as well - it began to react with the dissolved iron, forming iron oxides which precipated out, sunk to the seafloor, and became buried in sediment.

This means the bioavailability of iron in a planets waters has less to do with the sheer amount of iron on the planet - and more to do with how long it is able to last before becoming oxidized in the planet's atmosphere. In heavily oxidizing atmospheres, iron's bioavailability will be drastically curtailed - and in atmospheres less oxidizing than Earth's, it will be increased.

Iron minerals tend to range from black to orange-red.

=====Copper=====

Copper, while less abundant than iron, also plays enormous roles in biochemical systems throughout Earth's biosphere. It is especially important in electron and oxygen transport; all eukaryotes depend on copper-containing proteins for aerobic respiration, and the copper-containing hemocyanin is a common alternative to hemoglobin for carrying oxygen in blood among Earth life (and should be prevalent among alien life, too - especially on cold worlds with relatively low oxygen pressures, where it is more efficient than hemoglobin).

Similarly to iron, dissolved copper in water will, in an oxidizing atmosphere, slowly convert to copper oxide, which tends to precipitate out of solution; its bioavailability, therefore, is again limited more by its half-life in the world's waters than by its raw abundance. It is, however, slightly more soluble than iron oxide - notably, it dissolves in aqueous ammonia, and, as has already been noted, many alien worlds can be expected to have oceans of aqueous ammonia.

Copper biominerals are found in a number of clades on Earth, usually playing specialized roles similar to those played by iron; they are found in the jaws of bloodworms and the mycelia of certain fungi.

Copper minerals tend to be very vivid, cool colors, ranging from blue-green to bright blue.

=====Strontium=====

The presence of strontium in this list may be surprising, given its high atomic number and relatively low level of biological activity, but strontium is actually found as a primary component of biominerals on Earth - particularly, in the [https://en.wikipedia.org/wiki/Acantharea Acantharea], which possess skeletons of strontium sulfate. It is surprisingly common as an ion in Earth's waters, owing to how easily it tends to be absorbed out of rock and soil, similarly to calcium.

Overall, strontium minerals tend to have near-identical physical and chemical properties to their calcium equivalents - save for the fact that strontium minerals tend to be much, much heavier and denser. This is believed to be part of the reason why the Acantharea evince skeletons of strontium sulfate - the heavy weight of the strontium helps them sink to the ocean floor, their preferred habitat.

Strontium minerals tend to be either white, like their calcium equivalents, or light, ethereal blues.

=====Barium=====

The presence of barium here is likely even more surprising than strontium, given it is even less biologically active - but, again, its biomineralization is demonstrated in Earth life, typically as an accessory component to strontium biominerals. Biomineralization of barium sulfate is notably found in the Desmidiales, a clade of unicellular algae closely related to modern plants.

Not only is barium not particularly biochemically useful, its high reactivity renders it highly toxic to virtually all biological systems. One might think this is contrary to its use as a component of biominerals, but, in fact, barium occurs as a component of biominerals not despite, but ''because of'' its toxicity. In the form of mineral salts, barium's reactivity is neutralized; it is no longer a danger to the organism, once bound into salt crystals. As a result, many organisms have evolved the ability to secrete crystals of barium salts, in order to remove the toxin from the environment - and some then further evolved the ability to incorporate it into their structural materials.

Barium minerals are generally white or colorless.

====Anions====

Salts always involve a cation and an anion, and in the case of biominerals, the anion is always, specifically, an oxyanion - the common monatomic anions are too reactive, and the polyatomic anions which use nitrogen in the place of oxygen (such as cyanide and cyanate) are too unstable in oxidizing atmospheres (but may see more use in exotic biochemistries, such as in alkane solvent life).

The physical properties of mineral salts are typically more dependent upon the anion than the cation, perhaps surprisingly - for instance, the vast majority of phosphate minerals are harder and stronger than the vast majority of carbonates.

Unlike with the cations, mineral salts typically include only a single anion; this applies to both biogenic and inorganic minerals.
=====Carbonate=====

Carbonate (CO₃²⁻) is by far the most common anion found in biomineral salts in life on Earth - and is likely to be so on alien worlds as well. It is utterly ubiquitous throughout the world's waters (and thus, carbonate biominerals will be far more limited by the availability of their cations than by the availability of carbonate), and the same should apply to virtually any alien ocean - carbonate readily forms wherever carbon dioxide and water interact, and on a world with carbon dioxide in its atmosphere (which should be any alien world capable of sustaining life-as-we-know-it), that becomes "anywhere there is water".

Carbonates tend to be slightly more soluble in weak acids than the other common biomineral salts (though it depends on the cation as well - magnesium carbonate is less soluble than calcium carbonate, for instance).

<gallery mode=nolines heights=300px>
File:Foraminifères de Ngapali.jpg|thumb|Calcite tests of foraminifers.
File:NautilusCutawayLogarithmicSpiral.jpg|thumb|Cutaway of a nautilus shell, made of nacre - tiny aragonite plates layered on top of each other.
</gallery>

=====Oxalate=====

Oxalate (C₂O₄²⁻), another carbon-containing anion (structurally, essentially two carbonates joined together by replacing the double-bonded oxygen with a bond between the two carbons), is also found in biomineral salts on Earth, though typically only in minor roles - though, it may easily be far more important in alien life!

Oxalate, unlike carbonate, does not appear to be generated by inorganic processes; it is instead ubiquitously produced by plants (and a single genus of mold - [https://en.wikipedia.org/wiki/Aspergillus ''Aspergillus'']), through incomplete oxidation of carbohydrates. It is, in fact, in human health, most notable for the form of biomineralization that occurs when consuming too much of foods high in oxalate/oxalic acid - the formation of kidney stones, which largely consist of calcium oxalate, as the oxalate bonds to calcium ions in your gut and forms solid crystal precipitates.

Oxalates tend to be somewhat harder and stronger than their carbonate equivalents, though not as much so as phosphates.

=====Phosphate=====

Phosphate (PO₄³⁻) is, perhaps surprisingly, not actually that common on earth, despite the fact that your very own bones are primarily composed of hydroxyapatite (Ca₁₀(PO₄)₆(OH)₂), and phosphate minerals of inorganic origin are quite common. Only vertebrates and the mantis shrimp (very specifically the mantis shrimp - no other arthropods) have been shown to have independently evolved phophate biomineralization; this leaves it having evolved only twice, compared to the dozens of times carbonate biomineralization has evolved.

The reason why phosphate biomineralization is so (evolutionarily) rare is not particularly well understood; it may be that the more complex crystalline structures of phosphates (relative to carbonates) do not occur to biological systems so easily, or that phosphate (itself a very essential component for life) is too precious to waste on structural material for most species, or some other reason. Regardless - this implies it is far from likely that alien organisms will have bones (or other mineralized components) made out of the same stuff as our own!

It is perhaps better to address the evolution of phosphate biomineralization not from the persepctive of ''why'' rather than ''why not?'' - it is, after all, far from the default, even if we tend to (in this case, erroneously) assume our own biochemistry's way of doing things is "normal". While there are multiple theories, one I've personally found convincing for how vertebrates evolved phosphate biomineralization is that it was deeply tied to the origins of vertebrate bone as a form of essential nutrient storage - not only is phosphate itself an essential nutrient, and storing it along with calcium a potentially useful adaptation in and of itself, primitive vertebrates would undergo periodic changes in the pH of their blood based on level of activity - changes in pH that might damage carbonate biomineral deposits, but which phosphate biominerals would be more resistant to.<ref name="VertebrateBone">[https://www.jstor.org/stable/2409087]</ref>

Phosphate biominerals tend to be dramatically harder and stronger than carbonates, and slightly denser and heavier.

=====Sulfate=====

Sulfate (SO₄²⁻) is yet another common anion in biomineral salts; substantially more common than phosphate, but much less so than carbonate.

Sulfate biominerals are, in fact, generally marginally softer and weaker than their carbonate equivalents, which one might imagine would make them worse as structural materials; one must wonder why organisms bother, when carbonate is so ubiquitous. The answer has to do with some of the particular chemical properties of sulfate.

For one, sulfates tend to be hygroscopic without being hydrophilic - they tend to absorb water from the environment, but don't tend to dissolve easily. This already makes them more viable in environments where carbonates are not, due to the relatively high solubility of carbonates; but it also allows a slightly odd capacity to tweak the properties of sulfate minerals via controlling their level of hydration. While sulfate is, itself, heavier than carbonate, and sulfate minerals therefore tend to be heavier as well (again, as shown in the case of strontium sulfate), highly hydrated sulfates will actually be lighter and less dense than carbonates; this means organisms can effectively "tune" the density of their sulfate biominerals over the course of their evolution, via the level of absorbed water present in the crystal structure, evolving them to be lighter or heavier as the organism adapts to different environments.

Second, sulfate tends to much more easily bind metal ions and precipitate out of solution than carbonate does; this makes sulfate biominerals generally cheaper, energetically, to construct, and makes them preferred for capturing relatively rare cations (such as the aforementioned strontium or barium).

==Polyesters==

Given that polyesters are typically thought of as synthetic polymers, it may be surprising to hear that polyesters do, in fact, occur and play structural roles in Earth life - although, in only two known cases, both of which are relatively marginal. In particular, the cutin that makes up the waxy outer layer covering the leaves of plants; and a special polymer secreted by bees of the genus ''Colletes'' to protect their brood cells. Both cases are, unfortunately, poorly understood - but they show, without a doubt, that polyesters are viable as a structural biological material, and there's no reason why alien organisms might not make much wider use of them! Some bacteria also make use of polyhydroxylalkanoates as an energy and carbon storage mechanism, but it seems to serve no structural purpose.

Polyesters tend to be divided into two categories, based on the motifs present in their chemical structure, which are, incidentally, also the two classes of hydrocarbon: '''aliphatic''' polyesters, where carbon atoms form open chains; and '''aromatic''' polyesters, where the carbon atoms form closed, six-member rings. Aromatic polyesters tend to be much more crystalline in structure (and therefore much harder and more rigid), while also being more chemically and thermally stable. Most are, however, rather difficult for organisms to synthesize.

While the most well-known synthetic polyester - and an obvious thought when hearing that polyesters are viable structural biological materials - is polyethylene terephthalate, otherwise known as PET, the conditions under which this is produced are not suitable for biological processes. There are, however, a number of other interesting polyesters to look at.

===Polyesters from hydroxy acids===

All polyesters that appear in nature here on Earth - although, this is a rather limited selection, given their relative rarity - are, it seems, derived from hydroxy acids (that is, carboxylic acids with at least one hydroxyl group), particularly short-chain fatty acids. The exact chemical mechanisms by which biochemical processes produce these polyesters is generally poorly understood, although polyhydroxylalkanoates are known to be produced via action of an enzyme on the hydroxy acids.

====Cutin====
[https://en.wikipedia.org/wiki/Cutin Cutin], along with the related polymer '''suberin''', is what provides plants with the tough, waxy outer coating of their leaves, assisting them in retaining their structure and making them hydrophobic so that rain slides off their surfaces. While it is not very well understood, it is thought to be a branched polymer of C16 and C18 hydroxy fatty acids, such as hydroxy palmitic acid. These very-long-chain acids, and its branching structure, result in an amorphous, soft, waxy structure, that is nonetheless fairly tough, completely insoluble in water, and highly hydrophobic.

====Polyhydroxylalkanoates====
[https://en.wikipedia.org/wiki/Polyhydroxyalkanoates Polyhydroxylalkanoates], such as polyhydroxylbutyrate, are another polyester that appears in nature - albeit not for structural purposes. It is used by a number of species of bacteria for energy and carbon storage. Nevertheless, it still provides an option to us for a structural biopolymer, as, on Earth, polyhydroxylalkanoates are routinely harvested from the bacteria that produce them and used to make strong plastics, particularly for biomedical use (as they are non-toxic).

As a structural material, polyhydroxylalkanoates are generally quite ductile, and may or may not be quite elastic depending on exact composition. They are tough and strong, with similar tensile strength to polypropylene. They are generally permeable to oxygen, but ''not'' to water. Compared to other polyester plastics, they also tend to have quite good UV resistance.

====Phenolic acid polyesters====
[https://en.wikipedia.org/wiki/Phenolic_acid Phenolic acids] - that is, a phenol with a carboxyl group, such as salicylic acid<ref name=Polysalicylate>https://pubs.acs.org/doi/10.1021/acsmacrolett.9b00890</ref> - which are widely found in nature, have been demonstrated to be capable of forming polyesters under a variety of conditions. In the case that these are homogenous, linear polyesters (i.e. have no chain branches), their properties should generally be similar to other aromatic polyesters - that is, highly crystalline, and therefore hard and rigid.

More complex polyphenolic acids may instead more closely resemble polyphenols - indeed, phenolic-acid polyesters are both polyphenols and polyesters!

===Polyesters from lactones===
[https://en.wikipedia.org/wiki/Lactone Lactones] can be readily converted into polyesters via a variety of catalytic ring-opening polymerization reactions. While there are, as far as I am aware, no natural polyesters that are derived this way, lactones do occur in nature, and catalytic processes are well-suited for biochemistry.

====Polylactic acid====
[https://en.wikipedia.org/wiki/Polylactic_acid Polylactic acid], most often known by its acronym "PLA", is a plastic most known, here on Earth, for its use as a 3D-printing filament. While it is not found in nature, it is catalytically synthesized from lactide (the lactone of lactic acid), which very much is, meaning alien organisms that synthesize it are highly plausible.

Unlike most other aliphatic polyesters, PLA ranges from semi-crystalline to highly crystalline in structure depending on exact composition, and is thus quite hard and rigid - it is somewhere in between polystyrene and PET in mechanical properties.

PLA is also sensitive to UV degradation, but this simply breaks off monomers, which could be biologically recuperated and the PLA repaired.

====Polycaprolactone====
WIP - needs more research

====Polybutyrolactone====
WIP - needs more research

[[Category:Life]]

Category:Life

2022-08-26T15:49:47Z

Sevoris: Created page with "Life, biology, neurology, and everything else."

Life, biology, neurology, and everything else.

Active Structures

2022-08-26T15:49:04Z

Sevoris:

Active structures rely on constant power input, in addition to the material and mechanical properties of their construction materials (active support). This is in contrast to passive structures, which solely rely on the aforementioned properties (passive support). An example of an active structure is the force of a jet of water holding up a tethered lid of a trashcan in the air, versus the passive structure of a concrete pillar.

Nearly everything, from skyscrapers to houses are passive structures. Low-power active structures are in use now, for things like roof support.

The advantage of active structures is that they can be much more massive than passive structures <ref group="Footnote">Passive structures can attain extremely tall heights, however, they require pyramid-like tapering with a significant base area to support the weight.</ref>, enabling structures many kilometers tall without requiring significant tapering. Some proposals for [[non-rocket launch infrastructure]] rely on active support, with the advantage of the option for being built by modern, existing materials.

Most known designs of active structures rely on the force of a stream of mass to support them, using an accelerator to drive the mass stream.

<references group="Footnote"/>
=Active Support Principles=
As gravity<ref> https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation
<br/>Wikipedia article about gravity in classical mechanics.</ref>is what pulls down objects, active support must counteract gravity. Since it is the acceleration that causes objects to be pulled down, it follows that active support should accelerate in the opposite direction; the acceleration must be equal to gravity to support the structure.

The gravitational acceleration of the planet is given by:

<math alt=>A_g = G(M_p/r_p^2)</math>
* Where <math alt=>A_g</math> is the gravitational acceleration.
* Where <math alt=>G</math> is the [https://en.wikipedia.org/wiki/Gravitational_constant universal gravitational constant], defined to be 6.6743e11 m<sup>3</sup>/kg/s<sup>2</sup>.
* Where <math alt=>M_p</math> is the mass of the planet.
* Where <math alt=>r_p</math> is the radius of the planet.

On Earth, <math alt=>A_g</math> equals 9.80665 m/s<sup>2</sup>, a constant known as <math alt=>g</math>.

==Mass Streams==
Mass streams generally use particle accelerators or similar technology to create the streams. They use a deflector, usually magnetic, to receive the force from the stream and redirect it back towards the ground to create a loop.

Before we can calculate the kinetic energy required for each particle/pellet in the stream, important in determining energy input, the force, or the weight of the active structure must be calculated:

<math alt=> F=MA_g</math>

* Where F is the weight of the active structure.
* Where M is the mass of the active structure.

With that information in mind, the kinetic energy of the stream is given by:

<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="font-weight:bold;line-height:1.6;">Long form</div>
<div class="mw-collapsible-content">
<math alt=> KE_s = (v-\sqrt{v^2-2A_gh} )/A_g \cdot F/(2 \sqrt{v^2-2A_gh}) v^2 </math>
</div></div>

<div class="toccolours mw-collapsible mw-collapsed" style="width:400px; overflow:auto;">
<div style="font-weight:bold;line-height:1.6;">Simple form</div>
<div class="mw-collapsible-content">
<math alt=> KE_s = hF(b(\frac{1}{\sqrt{1-1/b }}-1)) </math>
<br/><math alt=> b = v^2/2A_gh</math>
</div></div>

* Where <math alt=>KE_s</math> is the kinetic energy of the stream.
* Where <math alt=>v</math> is the velocity of the stream.
* Where <math alt=>h</math> is the altitude of the deflector.

Notes:
# As a rule of thumb, whenever <math alt=>v>\sqrt{10A_gh}</math> energy costs are almost optimal. Provided that the stream velocity is above that threshold, you can forego the complex equations involved and go with <math alt=> KE_s \approx 0.6hF</math> instead. The choice of pellet characteristics can then fall under engineering constraints.
# The functions overestimate the energy content due to the presence of a constant gravitational field - in reality, it would not be constant.

Finally, the kinetic energy of the stream is divided by the number of pellets/particles in the stream to give the kinetic energy of each pellet/particle:

<math alt=>KE_p=KE_s/N</math>

* Where <math alt=>KE_p</math> is the kinetic energy of the particle/pellet in the stream.
* Where <math alt=>N</math> is the number of pellets/particles in the stream.

Additionally, the mass of the pellet/particle can be calculated with:

<math alt=>m=2KE_p/v^2</math>

* Where <math alt=>m</math> is mass of the particle/pellet.

A calculated example:

: Given a 1,000 ton structure on Earth (implying a <math alt=>A_g</math> of 9.8066 m/s<sup>2</sup>), a height of 10 km, 1,000 pellets in the stream and pellet velocity of 10 km/s, the force is 9,806,650 newtons and the kinetic energy of the pellet is 49,105.495 megajoules. Secondarily calculating the mass with the KE divided by the number of pellets and given velocity we get 0.982 kg.

=Active Structures=
==Existing==
* [https://www.youtube.com/watch?v=i4kTF0xw3y8 Small-scale demonstration of incomplete mass stream technology using water].

* The air-supported fabric roofs of the Tokyo Dome, Japan, and the Silverdome, USA use (and for the latter, used) constant fan pressure to keep the roofs aloft.</code>

==Proposed==
* The [[Lofstrom Launch Loop]] is a thin 2000+ km long and 80 km tall active structure, and uses its own mass stream to help launch payloads to orbit. It uses attractive magnetic levitation for the mass stream; the mass stream is a solid continuous iron rotor. The loop suffers from some unaddressed instability concerns.

* The [[Space Cable]] is a similar concept to the launch loop. It differs from the launch loop in that it uses magnetically interacting bolts instead of a continuous rotor, is smaller in length, and has addressed instability concerns.

* The [[Bolonkin Kinetic Space Tower / Anti-Gravitator]], using cables and rollers instead of using rotors or bolts or pellets to support various things. It has completely unaddressed instability concerns.

* The [[Space Fountain / Space Tower]], which aims to make a [[Space Elevator|space elevator]] using active support. Its technology can also extend to shorter buildings.

* The [[Orbital Ring]], which uses a mass stream travelling faster than the [[Orbits|orbital velocity]] to support a ring above a planet, as the stream keeps it from falling through momentum, and is tethered to the earth for stability.

* The [[Pneumatic Freestanding Tower]], which uses pressurized gas to support large structures such as a space tower. It utilizes compressors to provide pressurized gas and alleviate leaks. The main concerns are buckling due to the height of the tower, though it has mechanisms in place to prevent this.

=Control Systems=
[[Control Systems]]
<br/>Active structures can suffer from stability issues as mentioned before, such as for example in the launch loop unstable attractive magnetic levitation of the mass-stream in the launch-loop requiring active control of the deflector magnet. The unpredictable winds in the atmosphere are also a concern. Control systems are also needed in even just skyscrapers, with devices like tuned mass dampers to deal with vibration<ref>https://en.wikipedia.org/wiki/Tuned_mass_damper <br/>https://en.wikipedia.org/wiki/Active_structure <br/> https://www.youtube.com/watch?v=f1U4SAgy60c<br/>Wikipedia articles about control systems and the aforementioned tuned mass damper, as well a Practical Engineering video on it.</ref>.

=Safety Engineering=
Active structures are subject to the problem of how to ensure that they don't fail, or a bit worse, only fail gracefully, when something in their active systems breaks down. This is not a question of if; entropy breaks everything. All electrical and mechanical systems have a mean time between failure. If an active structure is only supported by a single active support "string", the failure of that "string" will cause a catastrophic failure.

By adding redundancy to our structure, we can ensure it can tolerate the failure of some of its components, and possible "fail gracefully" with a time period allowing for evacuation and response measures to be taken, and/or a "controlled failure" of the structure in which terminal velocity of the falling structure and the production of energetic debris is reduced.

Accepting an increase in mass, we split the support power required between some <math>N</math> "strings" operating in parallel. Each string only operates at a partial power, with an oversize factor of <math>1/N</math> added on top. If one or a few of the strings in the parallel system fail, the other strings are ramped to full power, generating sufficient support power to ensure the active structure remains standing despite the failure of some of its "strings". We can also use this to shut down strings intentionally for inspections, maintenance, overhauls, or other work, overall allowing us also to keep the active structure alive over time by incrementally replacing and upgrading its parts.

Usually safety redundancies have at least three systems operating in parallel. You may consider having a larger number of systems.

Note: in terms of safety engineering, no degree of redundancy reduces the chance of a total, catastrophic failure to zero. There is some chance that even a very redundant system may experience the failure of all its critical components at once. But this is given for any system, and you could consider pushing the safety factor of an active structure to the same point (or beyond) any passive structure.

=Additional Reading=

=Additional References=
<references/>

==Derivation of the Gravitational Acceleration Equation==
# The gravitational force equation<ref group="GAE">https://en.wikipedia.org/wiki/Newton%27s_law_of_universal_gravitation <br/> Reference for the gravitational force equation.</ref> is <math alt=>F = G(m_1m_1/r^2)</math>, where <math alt=>F</math> is the force, <math alt=>G</math> is the universal gravitational constant, <math alt=>m_1</math> is the mass of the first object, <math alt=>m_2</math> is the mass of the second object and <math alt=>r</math> is the distance between their centers of mass.
# <math alt=>r</math> becomes the radius of the planet from the frame of reference of a planet.
# The equation <math alt=>F = MA</math>, which gives the force needed to accelerate an object is rearranged to give acceleration, thus <math alt=>A=F/M</math>
# Since <math alt=>A =F/M</math>, where <math alt=>A</math> is the acceleration, divide by the mass of the second object and therefore cancel out its inclusion in the expression, giving you the gravitational acceleration equation.

'''Reference for the Derivation of the Gravitational Acceleration Equation'''
<references group="GAE"/>

==Derivation of the Stream Kinetic Energy Equations==
===Long Form===
# Given a classical particle accelerating upwards, it will achieve a final velocity of <math alt=>v_f=\sqrt{v^2-2A_gh}</math> where v is the initial speed.<ref group="SKE">https://www.school-for-champions.com/science/energy_potential.htm <br/> The expression is derived from the expression for the velocity of falling objects: <math alt=>V=\sqrt{2A_gh}</math>. This can be thought as of the amount of velocity that gravity subtracts from the initial velocity.</ref>.
# The necessary force for deflecting the particle down is given by <math alt=>F=2\dot{m}v_f</math><ref group="SKE"> <math alt=>\dot{m}</math> is the mass of the particle <math alt=>\bullet</math> the repetition rate, the repetition rate being analogous to <math alt=>N</math>, and <math alt=>\dot{m}</math> analogous to <math alt=>KE_s/N</math>.</ref>. The total energy content is then <math alt=>mv^2/2</math>.
# The stream must be long enough for one full circulation up and down, so any particle going up the height of the stream to the deflector takes a time <math alt=>T</math> where <math alt=>v_f = v - A_gT \Rightarrow \color{green}{T = (v-v_f)/A_g}</math>, for a full circulation it takes <math alt=>2T</math>.
# Therefore <math alt=>m = 2T\dot{m}</math>, so <math alt=> KE_s = T\dot{m}v^2</math>. To get rid of <math alt=>\dot{m}</math> we divide <math alt=>F</math> by <math alt=>2v_f</math>.
# This gives our stream kinetic energy equation, taking the limit as <math alt=>v</math> monotonically approaches infinity <math alt=>hF/2</math> from above.

===Simple Form===
# The simple form is just a factorized version of the long form, where <math alt=>k</math> is defined as <math alt=>2A_gh/v^2</math> or <math alt=>1/b</math>.

===Regarding the notes===
# The simple form shows that whenever <math alt=>v^2/2A_gh > 5</math>, the energy content in the stream is less than <math alt=>0.6hF</math>. Rearranging the equation for <math alt=>v</math> and accounting for 5 gives the rule of thumb.
# The upper bound nature of the equations means that:
:: For <math alt=>v > \sqrt{10A_gh}</math>, <math alt=>0.5hF < E < 0.6hF</math> for any <math alt=>h</math> and any gravitational field that is decreasing.

----
'''Reference for the Derivation of the Stream Kinetic Energy Equations'''
<references group="SKE"/>

=Credit=
To Tshhmon for writing the article
* To SOPHONT SIMP and pMXoTJFu for sweeping the article.
* To AdAstraGames for contributing useful information and sweeping the article.
* To Sevoris for writing the safety engineering section.
* To MatterbeamToughSF, Kerr in particular and Favalli for help with the math.

[[Category:Infrastructure]][[Category:Engineering]]

Category:Infrastructure

2022-08-26T15:48:28Z

Sevoris: Created page with "Category for all infrastructure articles and discussions."

Category for all infrastructure articles and discussions.

Wormholes

2022-08-26T15:47:23Z

Sevoris:

Wormholes are hypothetical structures in space-time allowed by the general theory of relativity.
They provide short cuts through space-time that connect one region of space and time to another; potentially, these regions can be very distant from each other.
The symmetries in space-time of our universe mean that wormholes move like physical objects, and acquire the conserved (or in some cases approximately conserved) quantities of things that go into them and lose those quantities of things that come out – properties like energy, mass, momentum, angular momentum, and electric charge.

==What is a wormhole? The geometry of space-time and how to twist it into a pretzel==

We're going to have to start out with some pretty heady stuff. Like the very nature of existence. And what is reality, anyway?
Usually, space and time are so much an integral part of our existence that we don't even think of them at all. They simply form a backdrop on which all the interesting stuff plays out. But when we do think about them, we're so accustomed to living in a world where they exist and where distances are well defined and events can be uniquely identified based on <i>where</i> and <i>when</i> they happened (or are happening or will happen) that a lot of stuff seems so obvious that we don't stop to think about if it really has to be this way or if there are other possible ways of doing things.

===A toy model===

So lets consider a hypothetical scenario, where an ant can be in any of several states, represented by the squares in the pictures below. The ant can transition to any other state that has a numbered side with the same number as the state it is in. We will start on the left. If the ant chooses the "2" transition, we end up with the picture in the middle. If it then chooses the "10" transition, we end up with the state on the right.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Space_elements_1.png|frameless]]
<td width=400>[[File:Space_elements_2.png|frameless]]
<td width=400>[[File:Space_elements_3.png|frameless]]
</table>
But now notice that the behavior of this system is exactly the same as if we re-arrange the states into a grid, as shown below on the left. The disconnected states now become a space; an expanse of coordinates on which the ant can exist. Now, the ant's transition between states can be described as a trajectory as shown in the right picture below.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Space_elements_grid.png|frameless]]
<td width=255>[[File:Space_elements_grid_path.png|frameless]]
</table>

This gives us an inkling of how something like space-time might end up being assembled from individual events. But what if we don't have things match up into quite such a perfect grid? We
ll take our space elements and tile them into an infinite grid. Two sections of the grid are shown below. But now we remove two of those space-tiles, and we connect the edges adjacent to the removed tiles with a corresponding edge of the other removed tile (although, as shown in the figure, we don't have to keep the same orientation). Now the ant, entering side 1 (which we have color-coded red for convenience) comes out of side 1 of the other missing square. This is perhaps the most basic wormhole.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Space_elements_wormhole_grid_1.png|frameless]]
<td width=400>[[File:Space_elements_wormhole_grid_2.png|frameless]]
</table>

Of course, this simple assembly only stitches together a space. But the world we live in is one of space-time. As the ant transitions from one square to another, it is also transitioning further ahead in time. But when the ant passes through the wormhole, while it locally experiences no discontinuity that wormhole might connect to a different time as well as a different region of space. So the ant might emerge far in the past or future of when it stepped through.

===Manifolds and coordinate patches===

The toy model used above hopefully prepares you for some of the ideas to follow - one of how we can piece together a complete space and time by connecting together different pieces. But before we go further, lets define an important term: a <i>manifold</i> is a "shape" of a given number of dimensions where, if you zoom in close enough, you can always get to a scale where it looks like the shape is flat. We'll use this definition to help us describe how we can put space and time together to get a wormhole. But first, let's look at a few illustrative examples.

====One dimension====

The simplest one dimensional shape is a line. And a line is everywhere flat. So a line is a one dimensional manifold.

A one dimensional shape might seem to be a wiggly path if inscribed in two or more dimensions, but it has no <i>intrinsic</i> curvature. That is, if everything about your world is confined to that path, you can't tell that it is curved – all you can do is measure the distance you go along the path. This greatly limits the kinds of shapes that we can consider. So any curved path of infinite extent looks from the inside like a line, and so any infinite length path is also a manifold.

But there's another class of one dimensional shapes that are manifolds – loops. A loop is a path of finite length that connects back with itself. You can make one by drawing a closed curve in a higher dimensional space. Or you can take a straight line but then say that a given point <i>here</i> maps on to another point <i>there</i>, and all the points between <i>here</i> and <i>there</i> form a loop. When you are marching along the line starting at <i>there</i> and you reach <i>here</i> you are also at <i>there</i>, so if you go further you just end up back where you were. This last bit may seem like a pointless bit of pedantry that just confuses a simple problem, but it will be a very useful way of looking at things as we go on.

====Two dimensions====

The simplest two dimensional manifold is a plane. A plane is everywhere flat, and extends off to infinity. A small region around any point on a plane (naturally) locally looks like a plane, so it meets our definition of a manifold.

=====Curved manifolds=====

But in two dimensions, we can have something that we don't have in one dimension – intrinsic curvature. Consider a sphere. If you get close enough to the sphere, it will look like you are on a plane; we live on a world that is (approximately) spherical (if you neglect mountains and ocean basins and all that other stuff), but when you stand up and look around you it mostly looks flat. So a sphere is a manifold. But take a line on the surface of your sphere. Extend it out along the surface, drawing it as straight as possible. Eventually, it will go all the way around the sphere, dividing it in two, and meet itself. This path is called a great circle. The Earth's equator is (approximately) a great circle, as are the lines of longitude (but not lines of latitude, because they do not divide the Earth into two equal-sized halves). The straightest-possible path on a curved surface (like a great circle on a sphere) is called a <i>geodesic</i>. But to keep the jargon down, we'll just call them lines for this non-technical presentation, and hopefully this will keep things more intuitive and understandable to the layman.

In flat space, two lines that are initially parallel remain parallel, always remaining the same distance apart. But in a curved space, initially parallel lines do not remain the same distance apart. On a sphere, for example, any two great circles will intersect at two points. A sphere thus has intrinsic curvature.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Sphere_and_great_circles.png|frameless]]
<td>[[File:Hyperbolic_surface_and_lines.png|frameless]]
<td>[[File:Cylinder_and_lines.png|frameless]]
<tr>
<td width=400>Two great circles going around a sphere. Although initially parallel, they meet at two points.
<td width=400>Two initially parallel lines on a hyperbolic surface. As you get farther from the place where they are parallel, the lines diverge.
<td width=400>Lines that are initially parallel on a cylinder remain parallel everywhere.
</table>

On a sphere, parallel lines always end up converging. One example of a curved surface where parallel lines diverge is a hyperbolic surface. Like a sphere, a hyperbolic surface is a manifold. But it has the opposite sense of curvature.

Now consider a cylinder. Again, a cylinder is a manifold. A cylinder can be made by rolling up a plane. Like our construction of a loop, this can be done by simply taking a long rectangular section of a plane and saying that two of the parallel edges are actually the same (this is essentially what you do if you roll up a sheet of paper so that two of its parallel edges touch, and then pretend that they are merged together where they meet). This "rolling up" preserves distances, so any two parallel lines on the plane remain the same distance apart on the cylinder (think about that sheet of paper, and if you drew two parallel lines on it - they will stay the same distance apart when the paper is rolled up). A cylinder has <i>extrinsic</i> curvature – you can tell that it is curved when looking at it from three dimensions. But it has no <i>intrinsic</i> curvature – if you are confined to the surface of the cylinder it behaves locally as if it is flat.

A manifold that is flat obeys all the rules of Euclidean geometry (although you'll need to go to the three dimensional case for dealing with volumes), and the standard Cartesian coordinate system (representing position by two orthogonal coordinates, commonly called <math>x</math> and <math>y</math>) can be used to specify any point. A manifold that is not flat violates Euclid's fifth postulate; the geometry of the manifold will be <i>non-Euclidean</i> (somewhere, H. P. Lovecraft is quietly having a mental breakdown). It cannot be represented by Cartesian coordinates, but different sets of coordinates – or perhaps patches of multiple coordinate systems that apply in different places – can be defined instead to locate points.

=====Topology: simply and multiply connected manifolds=====
If you take a plane or a sphere, any closed loop on the manifold can be shrunk in a continuous manner until it vanishes. This is called <i>simply connected</i>. But not all manifolds are simply connected. Those which are not simply connected are called <i>multiply connected</i>. The most basic multiply connected manifold is a torus, like a bagel or doughnut or bicycle inner tube. While there are closed loops on a torus that can be shrunk to a point and made to vanish, those that go all the way around the torus tube cannot – they'll get caught up circling the tube. Likewise, loops that go all the way along the tube will get caught up on the central hole if they are shrunk and again cannot be made to go to a point.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:torus_and_loops.png|frameless]]
<td width=400>[[File:Topological_torus_and_loops.png|frameless]]
<tr>
<td colspan=2>Three classes of loops on a torus. On the left is a standard geometrical torus; on the right is a topological torus that is flat but has opposite edges identified so that they are connected. The blue loop doesn't go around the torus, it can be continuously deformed to shrink to a point and vanish. The red line goes around the torus the long way. If you shrink it, it will get caught up going around the hole in the middle and get stuck there; it cannot be continuously deformed to make it any smaller. The green line loops around the torus from the outside to the inside and back. Again, if you shrink it the line gets caught up circling the torus tube and can't be made any smaller.
</table>

And this brings us to another interesting kind of manifold. You can get a manifold that is flat but has the topology of a torus. You can do this by taking a rectangle and saying that the opposite sides are actually the same. This gives you the effect of a "wrap-around screen" like on that old arcade game Asteroids – if you go off the edge on one side, you just come back on the opposite side at the same height. in physics, this is known as "periodic boundary conditions". Closed loops that go all the way around one way or the other cannot be shrunk to a point. But this kind of surface <i>cannot</i> be embedded in three-dimensional space! We might have grown used to surfaces that we can envision existing in the space we live in, but there are manifolds, even fairly simple two-dimensional manifolds, that cannot be embedded in higher-dimensional spaces like that. Remember this when we get to actual wormholes.

You can have multiply connected manifolds that are not toruses, even though they might have the same topology; any manifold with one or more "loops" or "handles" is multiply connected.

=====Non-orientable manifolds=====

And there is one other interesting kind of manifold that is relevant to wormholes – manifolds that are <i>non-orientable</i>. An orientable manifold is one where you can define a consistent sense of clockwise and counter-clockwise everywhere on the manifold. If you start with a clock spinning clockwise, or a right-handed glove, or a copy of Lewis Carrol's "Through the Looking-Glass, and What Alice Found There", you can move it anywhere on the surface and if it meets back up with itself it will still be spinning clockwise, or right handed, or normally readable. But in a non-orientable surface, there are ways you can move the object so that when you get it back to where it originally was the clock is spinning counter-clockwise, or the glove is left-handed, or the book is written in mirror writing. The simplest example of such a surface is a Mobius strip. As shown below, if you move an oriented circle around the strip, it comes back with the opposite orientation.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Topological_Mobius_strip.png|frameless]]
<td width=400>[[File:Mobius_twist.png|frameless]]
<td width=400>[[File:Mobius_strip.png|frameless]]
<tr>
<td colspan=3>A Mobius strip … a strip of surface with a twist so the edges where it connects meet up "the wrong way around". On the right is a 2D strip where you connect along the short edge matching the 1's together and the 2's together. The center shows the same strip with a twist in it so you can line the edges up the way you are supposed to, demonstrating how the twist reverses the orientation of the spinning circles. An embedded image in 3D space with the edges connected together is shown on the left. You cannot define a consistent orientation to this surface – starting at the red counter-clockwise circle, if you move the circle continuously along the surface when you get back where you started you will find it is going clockwise!
</table>

====Three dimensions====

The most basic three-dimensional manifold is a flat space. This is, well, just a normal space. You can go off in any combination of three independent directions (left-right, up-down, front-back), and because it is flat, initially parallel lines stay parallel. It all behaves much as you would expect three dimensional spaces to behave.

But you can get variations on this. They are harder to visualize than two-dimensional cases, because we don't have the brainspace for processing curved spaces inside of a four-dimensional hyperspace. But you can do things like identifying two planes to be the same, such that if you go far enough in one direction you can end back up where you started again (akin to a cylinder), or make them curved such that distances are distorted and lines don't remain parallel and maybe you can put larger things inside of smaller surfaces. There are 3D equivalents of spherical and hyperbolic surfaces (called, naturally enough, spherical and hyperbolic spaces). There are even non-orientable spaces.

One way of trying to envision a curved space, and to gain some intuition about them, is to take a two-dimensional slice of that curved space and embed that in a flat three-dimensional space. This is called an embedding diagram. Remember that this is a reduced representation that throws away a lot of the information (unless there are special symmetries you can exploit) and is not actually the thing being described.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Space_with_embedded_plane.png|frameless]]
<td width=400>[[File:Embedded_plane.png|frameless]]
<td width=400>[[File:Embedding_diagram_example.png|frameless]]
<td width=400>[[File:Embedding_diagram_example_very_curved.png|frameless]]
<tr>
<td>A space with an embedded plane. We show a sphere in this space, with the plane bisecting the sphere.
<td>Just the embedded plane. Note how in a spherically symmetric manifold (like this space), a circle on the embedding diagram represents a complete 2-dimensional spherical surface.
<td>An embedding diagram of a "dimpled" space. Just as before, the closed circles represent closed spherical surfaces. It is just that in this case, the distance between two concentric spheres of radius <math>R_1</math> and <math>R_2</math> is longer than the usual <math>R_1 - R_2</math> because of the spatial curvature, as shown by the increasing distance along the dimple's surface.
<td>An embedding diagram of an extremely curved section of space. In this example, there are several places where we have a bigger sphere inside of a smaller one! Curved space is weird like that.
</table>

Remember that not every manifold can be embedded, so not every curved space can be represented by an embedding diagram. But you can take patches of the curved space and show them by embedding diagrams, and then describe how these patches stitch together to create a contiguous surface.

====Space-time: three space dimensions and one time dimension====

This is the world we live in. So this is the class of manifolds that we really need to use when making a wormhole. Sometimes it's abbreviated as 3+1D (as opposed to 1D, 2D, 3D, etc.).

The basic, infinite, flat space-time manifold is often called Minkowski space. I know, it's a big word using scary foreign-sounding names. But all it means is that its a 3+1D manifold that goes on forever and is flat. In flat space-time, everything obeys the rules of special relativity. Which means it really doesn't work when masses become significant enough to make gravity.

Our best model of how gravity works is general relativity. In this theory, the curvature of space-time is created by mass. At least, that's what they always say in the over-simplified pop-sci summaries of general relativity. Actually, space-time curvature is created by all the components of the stress-energy tensor. It's just that in almost all cases, mass is the most significant part of this. But momentum, stresses, pressures, and shear forces all have contributions as well. In general relativity, physical objects move along the straightest possible lines in this curved space-time. The way the curvature causes the lines to bend and converge is what leads to the appearance of gravity as a force if the curvature is sufficiently weak.<ref name="Misner_Thorne_and_Wheeler">Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler, "Gravitation", W. H. Freemann and Company, New York (1973)</ref>

The most basic curved space-time manifold is the Schwarzschild geometry, which is what you get when you have an uncharged, non-spinning spherical mass doing its distortion thing on space-time. The Schwarzschild geometry is <i>static</i> meaning it does not change in time, and is <i>spherically symmetric</i>, meaning that it has a definite center and it doesn't matter which direction you are relative to that center – it is independent of angle. If the mass becomes dense enough, space-time gets so distorted that an event horizon forms. Nothing that goes into the event horizon can ever come back out, and the inevitability of the geometry crushes the mass down to an infinitely dense point at the center called a singularity. Nothing on the inside of the horizon can ever escape. It forms a <i>black hole</i>, that forever traps any light or matter that enters it. For masses that don't exceed this critical density, the gravitational distortion outside of the object is the same as for a black hole of equal mass, but there is no event horizon. The object can still be seen and interacted with.<ref name="Misner_Thorne_and_Wheeler"></ref>

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Schwarzschild_star.png|frameless]]
<td width=400>[[File:Schwarzschild_black_hole.png|frameless]]
<tr>
<td>An embedding diagram of the Schwarzschild geometry of a dense, massive star. The physical extent of the star is shown in blue.
<td>An embedding diagram of the Schwarzschild geometry of a black hole of the same mass as the star to the left. The event horizon is shown in red.
</table>

There is one distinction between space-time manifolds that will become important later. Some manifolds have an isolated region of high curvature (corresponding to a concentration of mass, or at least of stress-energy) but as you go farther away from the strongly curved region the space-time becomes increasingly flat. If you can always go out far enough away from a curved region that space-time becomes flatter that whatever flatness criterion you choose, no matter what direction you choose to leave the curved region, the manifold is called <i>asymptotically flat</i>. For our purposes, if you can go far enough away that the Newtonian approximation to general relativity is accurate, and gravity can be described as a force rather than requiring a geometric description, then the geometry can be considered asymptotically flat. You can also allow gravitational waves in your asymptotically flat space-time – their very small space-time curvature is not enough to cause problems and they can still be described by a linearized version of gravity. Both the Schwarzschild and the Minkowski geometries are asymptotically flat. Spherical or hyperbolic geometries are not asymptotically flat. Planets, stars, galaxies, neutron stars, black holes, and galactic clusters are all approximately asymptotically flat; but the universe as a whole is <i>not</i>. As long as we confine our attention to merely galactic clusters, we can make the approximation of asymptotic flatness.

===Making a wormhole===

Now that we have covered some of the introductory material, we can get down to the main course. To make a wormhole, take two regions of space-time, usually far apart and usually asymptotically flat (although neither of these is strictly required) and connect them together. That's it. You're done. Sounds easy, doesn't it?

Note that if you are connecting two regions in the same universe, your new geometry will be multiply connected. A loop that goes through the wormhole and then connects back up outside the wormhole cannot be shrunk to a point. There will now be two equivalent ways to get between any two points – the path that goes through your usual space-time, and the path that goes through the wormhole. If you use a wormhole to connect two separate universes, the combination of the two universes and the wormhole can still be simply connected.

In general relativity, you can distort the geometry of space-time with stress-energy, but there is no way to change its topology; that is, no way to turn a simply connected universe into one that is multiply connected, or to add or remove more "loops" or "handles" to an already multiply connected universe. Such changes, if they are possible at all, would only be possible in the realm of quantum gravity and we do not know enough about quantum gravity to know if it would be possible or not. So, it might not be possible to make intra-universal wormholes at all. Or maybe quantum gravity does allow topology changes in space-time and there are ways to connect different parts of our universe together with wormholes. On the plus side, if you can't change the topology of space-time, you can always bud out a new universe from our own – perhaps by triggering the inflationary field that started our universe. It will be connected to ours by a wormhole. And because the topology can't change, that wormhole would never be able to break. We would have an eternal bridge to the new universe. So a system of physics that does not allow topology changes still allows wormholes to other universes.

==Common kinds of wormholes==

===Spherically symmetric wormholes===

General relativity involves a lot of non-trivial math. You need to deal with four-dimensional curvature tensors with up to 20 independent components, and figure out how they relate to a given distribution of matter, fields, energy, stresses, and material flows. If you can reduce the number of dimensions you need to keep track of, it can simplify things a lot. One way of doing this is assuming your space-time geometry does not change with time (it is static), and looks the same in all directions when viewed from a central point (it is spherically symmetric). This makes things a lot easier to analyze.

Astute readers will have noticed that these conditions are satisfied by the Schwarzschild geometry. In fact, if you play around with the Schwarzschild geometry enough, you will find that you can even get a wormhole-like solution. At the event horizon, you can connect your black hole solution to a time-reversed structure called a white hole. This is called an Einstein-Rosen bridge. Unfortunately, it has some issues. For one, it can't form naturally from infalling matter creating a black hole. For another, going through an Einstein-Rosen bridge involves passing through an event horizon, so the trip is necessarily one way. And finally, you can show that even the smallest amount of matter or radiation falling into the Einstein-Rosen bridge leads to it collapsing before the matter or radiation can pass through, trapping it inside the black hole. So this is not looking particularly useful.

So what are we going to do? Well, the Einstein-Rosen bridge, like a lot of early work in general relativity, was made by assuming a distribution of stress-energy (in this case, matter) that was assumed to be physically plausible, and finding out what the resulting geometry would be. But we want a wormhole, dangit! So what happens if we just demand a static tunnel connecting two asymptotically flat regions of space-time? If you do this, you get a Morris-Thorne wormhole<ref name="MorrisThorne">Michael S. Morris and Kip S. Thorne, "Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity", American Journal of Physics <b>56</b>(5), 395-412, May 1988</ref>. You still have the relation between stress-energy and geometry, but now it is your assumed geometry dictating the stress-energy rather than the other way around. One difficulty with this method is that the distribution of stress-energy you end up with might not end up seeming physically plausible. In particular, it can be shown that any wormhole must have regions in it with negative energy density<ref name="Visser_Lorentzian_wormholes">Matt Visser, "Lorentzian Wormholes: From Einstein to Hawking", Springer/AIP Press (1996)</ref>, and these spherically symmetric wormholes are no different. We will discuss the issues around and solutions to the negative energy density problem later.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Morris_Thorne_wormhole.png|frameless]]
<tr>
<td>An embedding diagram of a Morris-Thorne wormhole. For this wormhole, we just took two copies of the Schwarzschild geometry from above and connected the middle "dimples" with a cylindrical patch of space-time.
</table>

The places where you enter a wormhole are called it's <i>mouths</i>, and most wormholes have two. The space-time tunnel between the mouths is called the wormhole <i>throat</i>.

====Some common problems with spherically symmetric wormholes====

if you don't have your distribution of matter and energy going off to infinity, then eventually if you go far enough away your geometry will reduce to the Schwarzschild geometry. A good many published examples of such wormholes are actually Schwarzschild-like up until pretty close to where the event horizon would be. The first problem you might encounter, then, is the formation of a horizon. A single horizon makes any trip one-way. Going into it on the side with the horizon would be indistinguishable from entering a black hole - except that you pop out somewhere else instead of getting removed from existence at the singularity. And because of the horizon, you couldn't ever go back. If both sides have a horizon, you simply can't get out of it. Why do you even have this thing, anyway?

The strong curvature as you get close to the throat can also produce very strong tidal forces. Due to the gravitational forces, if you go in feet-first your feet could be pulled much harder than your head as you go in and you could end up getting pulled apart – spaghettified is the technical term. In addition to these pure spatial tides, it turns out that you also have velocity-dependent tides. These tides are transverse to the radial direction - if you dive in really fast, you will be squeezed even harder from side to size.

Early attempts to limit the tidal forces while still keeping a throat large enough to fit a person through resulted in very large masses. Like planetary scale masses or even solar scale masses. This would mean you would need to spend a lot of propellant de-orbiting into the wormhole and then rocketing your way back out. Some of these early wormholes also had issues like the extreme gravity blue-shifting the cosmic microwave background to blow-torch intensity x-rays. Sometimes, attempts to keep the tides down also lead to really large throat sizes. Like the distance from the Sun to the Earth sizes. Or even light year sizes. These gigantic wormholes would be rather inconvenient.

None of these are necessarily intrinsic limitations to spherical wormholes. But it does indicate that a perfectly reasonable assumption for the wormhole tech in your science fiction universe could involve wormholes with Jupiter-level masses, extreme tides, and other fun details.

===Thin shell wormholes===

Take a shape located somewhere in flat (Minkowski) space-time. The important part of this shape is that it must be a closed surface that completely encloses a given volume. Now go someplace else in space-time and take an identical shape. Remove all of the space-time from inside both shapes, and then say that the corresponding parts of the two surfaces of the shapes are actually the same points. Now you have a thin shell wormhole. The throat region is infinitesimally short – the section that corresponds to the wormhole is just that surface of your shape, hence the name "thin shell".

The first thin shell wormholes proposed were simple polyhedra <ref name="Polyhedral_wormholes">Matt Visser, "Traversable wormholes: Some simple examples", Physical Review D vol. 29 no. 10 pages 3182-3184 (1989)</ref>. These are attractive choices for a potential wormhole-user because the flat faces of a polyhedral thin-shell wormhole have no tidal effects whatsoever. Further, the flat faces do not have any stress-energy either. It is simply flat space-time, no different from any other area of flat space-time. Going through the face of a polyhedral wormhole is no different from going through the flat space-time of the doorway separating your hallway from your bedroom. The negative energy stuff is concentrated entirely in the edges and corners of the polyhedron. And hoo boy, do you need a lot of it. For a cubic wormhole, each edge requires approximately <math>-1.52 \times 10^{43}</math> J/m of length. This is <math>-1.69 \times 10^{26}</math> kg/m, or close to a tenth of a Jupiter mass per meter of edge length. It has been proposed that perhaps cosmic strings – defects of the primordial universe where things couldn't line up the right way and left a linear flaw in reality – with negative tension might be able to meet this requirement and hold open a polyhedral wormhole.

But there's a strange thing when you consider the mass that these wormholes require. They are embedded in flat space-time. Which means that outside of that thin shell they have, they induce no curvature on the space-time around them. Remember that mass causes space-time to curve, dimpling up as in the Schwarzschild geometry shown earlier. Because these wormholes induce no far-flung curvature to imprint on the surrounding space-time, they have zero mass (technically ADM mass). And because mass is equivalent to energy, it should technically be possible to make one of these for no energy input at all from those of us in flat space-time. Forces are the rate of change of energy with distance, pressures are the rate of change of energy with volume. But because the energy is always zero no matter what size they are (as measured from out in flat space-time), there would not be any force or pressure needed to hold them open. The extreme conditions of their interior contrasts with the relatively benign conditions once away from the thin shell (or, indeed, within any flat area within the shell).

A curious feature of polyhedral wormholes is that each face of the polyhedron might connect to a <i>different</i> polyhedron. You might, for example, have a collection of seven cubical wormhole mouths, and each of the six faces of the wormhole you are facing connect to a different one of the six remaining mouths.

For more general thin-shell wormholes, you just need to smear that negative mass out a bit. As long as you keep a flat area to go through, travelers will not experience any strange tides or encounter space-warping exotic energies. You could, for example, have your standard circular portal beloved of fiction. Or you could have any other arbitrary shape you wanted.

Thin shell wormholes are what you get when you take a mathematical limit; they would be an approximation of a more physical distribution of stress and energy. Any realizable version would have a finite, although possible small, thickness to its shell and hence a finite, although possible small, throat length. As we will discuss later, they would likely also have a positive, although possible small, mass. But as an approximation, and as a valid solution to Einstein's field equations in general relativity, they allow us to explore many aspects of wormhole physics using a relatively simple model that is easy to calculate, and can lead to a lot of interesting inspiration for fictional ideas of wormholes.

====Tidal distortion during transit====

Consider a thin shell wormhole whose shell-shape is everywhere convex - no flat areas this time. For convenience, let's make it a sphere. As a traveler passes through the shell, the sections of his body that pass the shell must suddenly "pop" from curving one way to curving the other way, as illustrated below:

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Velocity_dependent_tides_pre_transit.png|frameless]]
<td width=400>[[File:Velocity_dependent_tides_in_transit.png|frameless]]
<tr>
<td>A traveler (left rectangular shape) about to go through a spherical thin-shell wormhole (here represented as circles, with the interior volume blacked out to indicate that there is no space-time there and nothing can exist in that region).
<td>The traveler in the process of passing through the wormhole. Note that on the wormhole shell, the portion of the traveler to the left has a leftward-curvature, while the portion on the right has a rightward curvature. As the traveler passes through, each section that goes through the shell immediately jumps from left-curving to right-curving. This will induce stresses in the traveler as his interior is deformed.
</table>

it is clear that as the curvature of the wormhole becomes large compared to the size of the traveler, the distortion will decrease. In the limit of a flat section of the surface, no distortion is experienced at all.

A consequence of this is that something going through a curved section of a thin shell wormhole will experience stresses (distortions) and strains (forces). The elastic energy stored by a strained object is proportional to the product of the stress and the strain, added up over all of its volume. Assuming that the object going in to the wormhole is initially in its relaxed state, it will require energy to push it into the wormhole shell. If it enters with insufficient initial kinetic energy to get it through or external forces pushing on it, it will just bounce off.

===Cylindrical and toroidal wormholes===

In principle, you can have a wormhole with cylindrical symmetry
<ref>Kirill A. Bronnikov and José P. S. Lemos, "Cylindrical wormholes", arXiv:0902.2360 [gr-qc] 24 Feb 2009</ref>
. This forms a valid solution to Einsteins field equations in general relativity. However, a true cylindrical wormhole would be infinite in length. So they probably don't exist naturally, and you won't be able to build one. Unlike all the other wormholes discussed here, they are not asymptotically flat. Unlike wormholes of finite extent, not all cylindrical wormholes require negative energy regions. However, it has been shown that negative energy stuff is needed if the geometry radially far enough away from the wormhole approaches the flat space-time of our usual universe, so a universe with a cylindrical wormhole without negative energy would not look like our own.

So the thought is that maybe you can get close enough to a cylindrical wormhole by making a really long wormhole that bends ever so slightly and comes back and meets itself, forming an extremely stringy torus. The general case for this has not yet been worked out, but a thin shell approximation of a toroidal wormhole has been developed
<ref>Vladimir Dzhunushaliev, Vladimir Folomeev, Burkhard Kleihaus, and Jutta Kunz, "Thin-shell toroidal wormhole", arXiv:1901.07545v2 [gr-qc] 28 Jan 2019</ref>
.
These thin shell toriodal wormholes have been shown to be stable to perturbations, one nice benefit that has not been established for a lot of other kinds of wormholes. They do require some negative energy regions on their surface, but there will also be some regions that do not contain negative energy.

===Non-orientable wormholes===

In the discussion on non-orientable manifolds, we showed how in these manifolds you can't properly determine the clockwise - counter-clockwise orientation of objects or motions, or the orientation of objects with respect to their mirror reflections. If you move an object along certain paths, when it comes back it might come back the wrong way around. So it is probably no surprise that a non-orientable wormhole turns the universe into a non-orientable manifold. Something going through a non-orientable wormhole comes out looking like it has been mirror reflected.

Normally this seems like it would be rather boring and useless, but for an obscure constraint from quantum physics called the CPT theorem. This theorem, which holds for all physical phenomena, means that the product of the discrete symmetries of charge conjugation (C), parity transformation (P), and time reversal (T) always returns you to your original state. Huh? What does that mean? Well, parity transformation is equivalent to a mirror reflection; so passing through a non-orientable wormhole means parity is inverted. This means that exactly one of the other two symmetries must also be inverted for the CPT theorem to hold. The thing coming out on the other end is still manifestly going forward in time, so it must be charge conjugated, whatever that is.

Charge conjugation means you turn all particles into their antiparticles. So anything passing through a non-orientable wormhole emerges on the other end made entirely out of antimatter.

While this particular turn of events might discourage people from passing through, it does offer obvious application for energy generation, space propulsion, and weaponry.

==Exotic energy conditions==

It is fairly easy to see than any wormhole requires negative energy densities (or more generally, regions where the tension is higher than the energy in natural units) to exist. Take a look at any of the wormhole diagrams above – the spherically symmetric ones will probably make it the most clear. If you send a parallel ray of stuff – maybe particles or light or golf balls – through the wormhole, and trace out the straightest possible path of all of those parallel lines, you will see that after they go through the wormhole all those initially parallel trajectories will now be diverging. Positive energy (such as mass) causes attractive forces, drawing trajectories toward it and making the paths converging. To get diverging paths you need the opposite – repulsive gravity – which you can get from negative energy densities or very high tensions. Having the tension be higher than the energy density turns out to be equivalent to a negative energy density – in some frames of reference observers will measure a negative energy density there.

A lot of physicists used to be reflexively dismissive of negative energy densities. After all, they can't really exist; all energy in the real world is positive.

Except that it is not. There are certain odd cases that allow for negative energy densities. One of these is the Casimir vacuum. A conducting boundary cannot support an electric field parallel to that boundary (if there was such a field, charge would flow under the force of the field until the parallel component of the field is screened out). So if you have two parallel conductive plates, the only electromagnetic waves that you can get between them require their fields to vanish at the plates. This restricts the number of ways the electromagnetic field can vibrate, called modes. In quantum mechanics, merely the potential for a vibration is associated with a certain energy (called <i>zero point energy</i>) even if no quanta of vibration are actually present. By making these modes impossible, they cannot contribute their zero point energy to the region between the plates. This lowers the total energy of the space between the plates compared to the region outside the plates (or with the plates absent). Because the energy of empty space is zero, this means that the region between these Casimir plates has negative energy. The Casimir effect has been measured in the laboratory. It is fairly weak, but it is present. In fact, not long after traversable wormholes were first described, one proposal suggested holding open the wormhole using the Casimir effect<ref name=MirrosThorneYurtsever">Michael S. Morris, Kip S. Thorne, and Ulvi Turtsever, "Wormholes, Time Machines, and the Weak Energy Condition", Physical Review Letters, Volume 61, Number 13, 26 September 1988, pages 1446-1449</ref>

Another situation that gives rise to a negative energy density is a <i>squeezed vacuum</i>. A squeezed state is a way of manipulating a quantum oscillator (like a pulse of light) to get around the uncertainty relations of quantum mechanics to increase the precision in one variable by increasing the uncertainty in its conjugate variable. Applying this operator reduces the energy of the oscillator, and if this is applied to the vacuum state, which already has zero energy, you can end up with regions of negative energy.

It is worth noting that not just any old negative energy region of exotic space-time will work. In particular, dark energy is a condition of the universe we live in where even bare vacuum itself has a small amount of negative energy. The current best model we have for this is as a cosmological constant - where every part of the universe has an equal density of negative energy and a related amount of pressure. But the wormhole allowing contributions to space-time curvature of that negative energy are exactly cancelled by the wormhole-denying contributions of the pressure. So according to our best current models, dark energy will not support a wormhole.

Also, the Casimir vacuum described above is also problematic. The exotic vacuum between the plates does have the necessary properties (if not magnitudes) to support a wormhole, but this is generally far overwhelmed by the immediately adjacent positive mass matter making up the conductive plates. The technical term for the criterion that determines whether you can have a wormhole or not is the Averaged Null Energy Condition (ANEC). If you project a ray through space-time that follows the path that light (in a vacuum) could take, and do some math along that ray and add up all the contributions from all the places along that ray, then if your result is zero or positive then the ANEC is said to be satisfied. If this is the case, you can't get a wormhole. If the ANEC is less than zero, it is said to be violated and in principle you could use that weird exotic energy stuff to prop open a wormhole's throat. There are other so-called energy conditions, and if you have regions of space-time that also violate those energy conditions they can be used to hold open wormholes as well - but all of these also violate the ANEC, so the ANEC is all you need to consider.

===So just how do I get the ANEC-violating stuff I need?===

The short answer is, we don't know.

As we already said, a cosmological constant form of dark energy won't do it (although different ideas about dark energy, such as phantom energy, could conceivably work). The inflationary vacuum thought to have existed very briefly during the first instances of our universe's existence is also usually thought to be a cosmological constant (although one that stopped existing after a very brief period of time, so I guess it is less than constant), so the inflationary vacuum probably won't work either.

If you can get the Casimir vacuum without the massive conducting plates that bound it, you might be able to do something with that. For example, the twisted geometry of space-time itself in a wormhole throat imposes boundary conditions that restrict the modes of vibration inside of it similar to the Casimir effect. This is called the <i>topological Casimir effect</i>, potentially it could be used to help support a wormhole. Also, although the ANEC is satisfied if the probing space-time ray goes through the Casimir plates, a ray that goes between the plates but never touches them will violate the ANEC. Can you use this to hold open a wormhole? Maybe a sufficiently clever person will eventually figure out how to do so.

Even more serendipitous, it turns out that gravity itself naturally squeezes the vacuum
<ref name="gravitationally_squeezed_vacuum">David Hochberg and Thomas W. Kephart, "Lorentzian wormholes from the gravitationally squeezed vacuum", Physics Letters B 268, 377-383 (1991)</ref>
. If you have the curved space-time of gravity, all of the modes of oscillation in it become squeezed states. So it might be possible that the curved space-time of a wormhole can squeeze the vacuum enough to support itself. Or a combination of gravitational vacuum squeezing and the topological Casimir vacuum from the wormhole's geometry constraints might be all that is needed. At least one proposal has suggested a class of wormholes that can be self supporting in such a way
<ref>S. V. Krasnikov, "Toward a Traversable Wormhole", arXiv:gr-qc/0003092v1 22 Mar 2000</ref>
<ref>S. Krasnikov, "Traversible Wormhole", Physical Review D, volume 62, article 084028 (2000)</ref>,
although the throat of these wormholes is very "wrinkly" or "crumpled up" and it is unclear what the effects of that would be on anyone passing through.

====Quantum energy inequalities====

When you apply quantum field theory to what happens when you have a region where the quantum fields produce local negative energy densities, you end up with a limit of how much negative energy stuff you can have before it has to be balanced out by an even greater amount of nearby positive energy stuff
<ref>[https://web.archive.org/web/20060206055950/http://maths.york.ac.uk/www/PhysicsQIneq.htm|Quantum Energy Inequalities]</ref>
<ref>[http://www.phys.lsu.edu/mog/mog20/node16.html|Quantum field theory on curved spacetime at the Erwin Schrödinger Institute]</ref>
<ref name="Fewster_Roman_2005">Christopher J. Fewster and Thomas A. Roman, "On wormholes with arbitrarily small quantities of exotic matter", Physical Review D <b>72</b>, 044023 (2005)</ref>
.

====Matter with negative mass====

It might be tempting to try to solve the negative energy density requirement for wormholes by introducing some sort of matter that naturally has negative mass.
However, negative mass would introduce all kinds of problems to the smooth operation of how the world works. For one thing, if negative mass (or equivalently energy) can exist, why doesn't it just spontaneously pop out of empty space accompanied by an equal magnitude of positive mass (or energy)? For another, negative masses lead to all kinds of runaway instabilities. For example, imagine what happens if you push on an object with a negative mass. You exert a force on it with your hand. But by Newton's second law of motion, an object's acceleration is proportional to its mass. So the negative mass thing will accelerate backwards, against the direction of the force, and into your hand. Which means you will be pushing even harder on it, which makes it push back even harder than before. You have a runaway process that blows up until your hand breaks, and any fragments of flesh and bone and sinew that were in the way of the negative mass thing have been flung off.

Or consider a negative mass thing with an electric charge. If you put something with an electric charge in an electric field the field will exert a force on the charge. For a normal positive mass charged thing, it will move in the direction of the force. The work done is the force in the direction of the displacement in position, so because the force and motion of the object are in the same direction, the field does work on the charged object and energy flows from the field into the kinetic energy of the charged object. But if the charged object has negative mass it will begin to move opposite to the direction of the force. This will do work on the field instead of the object, increasing the energy of the field. The energy comes from the kinetic energy of the negative mass charged object – but because kinetic energy is proportional to mass and increases with increasing speed, as the negative mass object loses kinetic energy it gains speed, going ever faster. Both the field strength and the speed of the object increase without bound. If the electric field originates from the electric part of an electromagnetic wave, it will vary sinusoidally with time and space. As the negative mass object is forced into a sinusoidal trajectory opposing the electric force on it, it will amplify that wave making it ever more intense. A single, isolated charge exposed to an electromagnetic wave will eventually have its inertia keep it moving from the previously imposed force even after the force has switched direction, so if the negative mass is free with no other forces acting on it there will be no net amplification. But if there are any other damping forces, or internal arrangements of bound charge that can resonate with the wave, then instead of being absorbed (as would happen with a positive mass), the wave will be amplified indefinitely and the ever-increasing energy of the wave will come from the ever increasing temperature of the negative mass thing (because the mass is negative, the thermal motion of temperature gives it a negative energy whose magnitude only increases as the temperature rises).

And, of course, the quantum energy inequalities indicate that you can't just have isolated lumps of negative mass stuff floating around.

So just having lumps of stuff that have a negative mass is probably not realistic, and it were possible it would lead to all kinds of universe-shattering cataclysms. Hopefully, we can get wormholes to stay open just using the exotic vacuum states from the Casimir effect and squeezed vacuums.

===Reducing or eliminating ANEC violating stuff===

As soon as people figured out that the wormholes they want needed stuff they couldn't get, they set about trying to find ways to use as little of the stuff they couldn't get as possible. A number of attempts have been made to hammer out a geometry for spherical wormholes that are stable, meet the quantum energy inequalities, and allow a person to pass through them in a reasonable length of time without being shredded by tides
<ref name="Fewster_Roman_2005"></ref>
<ref name="Visser_Kar_Dadhich">Matt Visser, Sayan Kar, and Naresh Dadhich, "Traversable Wormholes with Arbitrarily Small Energy Condition Violations", Physical Review Letters Vol. 90 No. 20 article 201102 (2003)</ref>
<ref name="Kuhfittig">Peter K. F. Kuhfittig, "More on wormholes supported by small amounts of exotic matter", Physical Review D <b>73</b>, 084014 (2006); Peter K. F. Kuhfittig, "Wormholes supported by small amounts of exotic matter: some corrections", arXiv:gr-qc/0508060v1 15 Aug 2005</ref>
.
Often, these resulting structures end up having extreme dimensions, such as throat sizes on the order of light years
<ref name="Fewster_Roman_2005"></ref>
<ref>Remo Garattini, "Generalized Absurdly Benign Traversable Wormholes powered by Casimir Energy", arXic:2008.05901v1 [gr-qc] 12 Aug 2020</ref>

Thin shell wormholes have also been the subject of investigation. One study <ref name="Mazharimousavi_Halilsoy">S. Habib Mazharimousavi, M. Halilsoy, "3 + 1-dimensional thin shell wormhole with deformed throat can be supported by normal matter", Eur. Phys. J. C (2015) 75:271 DOI 10.1140/epjc/s10052-015-3506-6</ref> found a number of thin shell wormholes with sharp-edged shapes could be supported entirely without any exotic energy. However, this only applies if the edges are infinitely sharp. A finite radius of curvature at the edges would require negative energy at those edges.

If the wormhole is taken have the properties of a class of subatomic particles called a fermion, then if the ratio of the electric charge to the mass is sufficiently large you can get a wormhole that can remain open without any negative energy regions at all
<ref name="Blazquez-Salcedo_Knoll_Radu">Jose Luis Blázquez-Salcedo, Christian Knoll, and Eugen Radu, “Transversable wormholes in Einstein-Dirac-Maxwell theory”, Physical Review Letters <b>126</b>, 101102 (2021); ArXiv: 2010.07317v2 [gr-qc].</ref>
.

==Fields and conserved properties==

Our best theories of how the world works – quantum mechanics and general relativity – predict that certain things are [[Conservation_Laws:_Limits_to_Cheating|conserved]]. For asymptotically flat space-times, these quantities are energy, linear momentum (or just momentum), angular momentum, and electric charge. Furthermore, in the non-relativistic limit mass is conserved independently of non-mass energy.

And not only are these things conserved, but they are always conserved <i>locally</i>. This means you can't just get rid of, say, some energy in one place and expect things to balance out by having that energy appear somewhere else.

What does it mean for something to be conserved? Mathematically, it means that the thing obeys the continuity equation. There are various ways of writing the continuity equation, all equivalent, but basically it boils down to the following: if you have a region of space surrounded by a closed surface, then the amount of conserved stuff inside that surface can only increase if some of the stuff enters through the surface and it can only decrease if some of the stuff leaves through the surface. Makes sense, right? If you have three loaves of bread in a room, then if the deliveryman brings in another two loaves you will have five loaves. But unlike loaves of bread, which can be baked or eaten or grow moldy or burn up or be broken into crumbs, the only way to get energy or electric charge or the other conserved things into or out of the room is to have them come into or go out of the room.

Each of these conserved quantities has an associated field. Electric charges create an electric field. Energy and momentum and angular momentum each create a distribution of the curvature tensor of space-time. In the non-relativistic limit, this is the mass part of the energy creating the gravitational field.

There is a relationship between the amount of a conserved quantity in our hypothetical closed surface and the net amount of field that penetrates that surface. This is called [https://en.wikipedia.org/wiki/Gauss%27s_law <i>Gauss's law</i>]. Basically, it says that if you add up all of the amount of field that goes out of the surface, and subtract off all of the field that goes back into the surface, the resulting amount will be directly proportional to the amount of the conserved stuff in there. Do you start off with a kilogram of mass inside your room? Then you have a certain amount of gravitational field from that mass that leaks out of the room. Add another kilo for two kilos total, and the amount of gravitational field leaving is also doubled. The distribution of where that field comes out may change, but as long as you don't take that mass out of the room or add more mass in the room, the total amount of field leaking out won't change. And any gravitational field from masses outside of the room that leaks into the room must also leak back out again <i>somewhere</i>. Those masses outside the room will not affect the net amount of field leaking through the surface. What this means is that you can determine the amount of conserved stuff inside a surface if you know the amount of field going through that surface.

This can be visualized in the field line approximation. The field around a positive charge can be approximated as a series of directed lines that radiate away from the charge. The field around a negative charge in this approximation is a series of lines that converge on the charge (for non-relativistic gravitation, you only have positive charges because mass is always positive. We'll just ignore the fact that wormholes require negative energy density stuff in order to exist which will lead to negative mass – we're just trying to get across the basic idea here). Field lines can only ever start on a positive charge and only ever end on a negative charge. The number of lines radiating away or converging on the charge is proportional to the size of that charge (so if you double the charge, you double the number of lines that connect to it). Field lines propagating through space bend away from a positive charge and bend toward a negative charge.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=322>[[File:charges_positive.png|frameless]]
<td width=322>[[File:charges_negative.png|frameless]]
<td width=322>[[File:charges_dipole.png|frameless]]
<tr>
<td>The field lines radiating away from a positive electric charge.
<td>The field lines converging on a negative electric charge.
<td>The field lines emerging from a positive charge and curving around to go into a negative charge of the same magnitude. This configuration, with a positive and negative charge of the same magnitude, is called a <i>dipole</i>.
</table>

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=322>[[File:charges_positive_Gauss_law.png|frameless]]
<td width=322>[[File:charges_negative_Gauss_law.png|frameless]]
<td width=322>[[File:charges_dipole_Gauss_law.png|frameless]]
<td width=322>[[File:charges_exterior_dipole_Gauss_law.png|frameless]]
<tr>
<td>A closed surface encloses a positive charge. A net amount of field lines exit the surface.
<td>A closed surface encloses a negative charge. A net amount of field lines enter the surface.
<td>A closed surface encloses a dipole. The total amount of charge inside the surface is zero because the negative charge cancels the positive charge. No net amount of field goes through the surface - just as many field lines enter as leave.
<td>A closed surface that has no charge inside of it, but various charges outside of it (in this case a dipole). Again, no net amount of field goes through the surface - every field line that enters comes back out somewhere else.
</table>

Now consider the case of an isolated positive charge, like one one shown above, that is far away from anything else. We will bring up one mouth of a wormhole close to the charge. Because the field lines going out of the charge cannot break, they cannot go through the wormhole. Instead, they will curve around the wormhole mouth. As the charge enters the wormhole mouth and travels into its throat, it drags its field lines along with it. With all of its field lines entering the mouth, that mouth looks like it has a net positive charge. As the charge leaves the other mouth, it continues to drag ints field line out of the wormhole - the field lines that start on the charge now curve around to enter the wormhole mouth. The mouth that the charge exited from now looks like it has a negative charge.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Wormhole_with_charges_initial.png|frameless]]
<td width=400>[[File:Wormhole_with_charges_inside.png|frameless]]
<td width=400>[[File:Wormhole_with_charges_passed_through.png|frameless]]
<td width=400>[[File:Wormhole_with_charges_long_gone.png|frameless]]
<tr>
<td>The charge approaches the wormhole.
<td>The charge is inside the wormhole. The mouth of the wormhole that the charge entered looks like it has a positive charge.
<td>The charge has passed through the wormhole. The mouth that the charge entered still looks like it has a positive charge, but now the mouth that the charge came out of looks like it has a negative charge.
<td>The positive charge has now moved very far away. Both mouths of the wormhole still appear to be charged.
</table>

This can also be understood by the continuity equation. Picture an imaginary closed surface around the mouth of the wormhole. This is like the case of the room, before, except that <i>now there is a way out of the enclosed volume that does not pass through the surface</i> – by going through the wormhole! Any conserved quantity that goes into the surface adds its quantity to the stuff in the surface. If it goes through the wormhole, it does not come out through the surface. The conserved quantity inside that surface stays the same! To an outside observer, it looks like the wormhole mouth has that quantity.

Either way you look at it, the results are the same. A wormhole mouth acquires any conserved quantity of the things that enter it, and loses any conserved quantity of the stuff that exits from it.
<ul>
<li> A wormhole mouth acquires the <i>energy</i> of anything that goes into it. Its energy decreases by the energy of anything that goes out of it.
<li> In the non-relativistic limit, this means that a wormhole mouth acquires the <i>mass</i> of anything that goes into it. Its mass decreases by the mass of anything that goes out of it.
<li> A wormhole mouth acquires the <i>electric charge</i> of anything that goes into it. It gains negative the electric charge of anything that goes out of it.
<li> A wormhole mouth acquires the <i>momentum</i> vector of anything that goes into it. It gains the negative of the momentum vector of anything that goes out of it.
<li> A wormhole mouth acquires the <i>angular momentum</i> vector of anything that goes into it. It gains the negative of the angular momentum vector of anything that goes out of it.
</ul>

This perhaps should not be a surprising result – other general relativistic space-time distortions also exhibit this behavior. A black hole, for example, gains the mass, charge, momentum, and angular momentum of anything that goes into it.

You can use these same arguments to find that the same results apply to any asymptotically flat manifolds of space-time. They do not necessarily hold for space-time geometries that are not asymptotically flat. If the gravitational distortion of a wormhole extends infinitely far away, you will not be able to even define its total energy or momentum or angular momentum, so it will be impossible to say if they are conserved. But do note that all of the wormhole examples that we have discussed so far (except for the cylindrical wormhole of infinite extent) are asymptotically flat, and will obey the local conservation of all of these quantities.

===Wormhole dynamics===

A wormhole mouth has the gravitational field and conserved properties of mass, momentum, and angular momentum. By the equivalence principle of general relativity, this must mean that it behaves as if it had those properties. So a wormhole mouth will follow ballistic trajectories in vacuum through space-time as if it were an object in free fall. If it is around a concentrated mass, like a planet or sun, it will have a Keplerian orbit around that object (or, if the wormhole has more mass than the planet or sun, perhaps it would be more reasonable to say that the planet or sun would orbit it). The mass of the wormhole mouth will resist acceleration like any other mass, according to Newton's laws of motion. Force is the time rate of change of momentum, so as the wormhole mouth's momentum changes it will experience forces, and these will change its trajectory in the same way as if that momentum (or forces) acted on a material object with the same mass. And if the wormhole mouth has an electric charge, it will experience forces from electric and magnetic fields; again, these can change its trajectory.

In other words, a wormhole mouth acts like a physical object of the same mass, charge, energy, momentum, and angular momentum.

The motion of the wormhole mouths through our normal space does not affect their relative separation through the wormhole's throat. That can remain the same length regardless of the behavior of the mouths in our normal universe.

As we covered, if an object enters a wormhole mouth that mouth gains the object's momentum. This will give the wormhole a "kick" changing its motion in the direction that the entering object was originally moving. The dynamics work out the same as if the wormhole were a sticky blob and the object collided and stuck to it. In both cases, you get a final object with the combined masses of both initial things (the mouth and the entering object) that has been knocked off course a bit by the momentum of the entering object hitting it.

If that object comes out of the other mouth, then as that mouth gains the opposite of the objects momentum it gets a recoil kick in the opposite direction. Again, this results in dynamics that are similar to things we already know about – like a gun with a bullet loaded firing the bullet. As the bullet leaves, the gun + bullet system loses the mass of the bullet, and as the bullet shoots away the gun recoils in the opposite direction.

So now consider what happens if you direct a constant stream of material through a wormhole. As the jet of material exits its mouth, that mouth loses mass at the same rate as the mass flow rate of the jet. It also continually gains momentum by the recoil kick of the gas. These dynamics are identical to that of a rocket - as the propellant is ejected, the rocket loses the propellant's mass and the recoil momentum from the escaping gas jet pushes the rocket in the opposite direction. So by shooting stuff through a wormhole you can turn it into a rocket, with dynamics identical to a rocket, obeying the [https://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation Tsiolkovsky rocket equation], and everything else that happens with rockets.

===Does gravity go through a wormhole?===

The above diagrams of the electric field of a charge as it approaches a wormhole mouth also hold for the gravitational field of a mass that is near a wormhole mouth. In the same way that the electric field cannot enter the wormhole's interior or leak through to the other side unless the charge enters the wormhole or goes through to the other side, so to will the gravitational field of a nearby mass curve around the wormhole. The gravity of a nearby planet will fall to zero inside the wormhole throat and will not affect those near the opposite mouth.

===Can waves go through a wormhole?===

So if gravity won't go through a wormhole, and static electric and magnetic fields can't go through a wormhole, then what about light? That's made up of electricity and magnetism, right?

So, turns out that light can go through a wormhole. So can other electromagnetic waves like radio and x-rays. So can gravitational waves. The rules for drawing field lines only really work for static charges; for radiating waves you also need to include the parts about how the changing field generates more field. In the end, waves that are smaller in wavelength than the size of the wormhole can go through just fine. You'll get some [[Diffraction|diffraction]] around the mouths, but for wavelengths much smaller than the mouth size even that might not be noticable.

===Can negative mass wormholes exist?===

So a wormhole mouth loses the mass of anything that comes out of it. What happens if you send something through a wormhole with more mass than the mouth which it exits from? This rule says that the mass must become negative, but as we saw earlier this leads to all kinds of problems (especially because the wormhole can also be charged) and is likely to be forbidden by the laws of nature.

If a wormhole's mouth can reach negative mass, you can also get other interesting instabilities. The positive mass mouth will attract matter into it, which will pass through, increasing the mass of the positive mass mouth and making the mass of the negative mass mouth even more negative. Meanwhile the gravity of the negative mass mouth will repel matter, preventing any return flow. Both the negative and positive masses will gradually increase over time.

So what's an aspiring science fiction writer to do? If you have a particular desire for negative mass things in the fiction you create, go ahead and allow it. Just be aware of the consequences. Otherwise, it is probably easier to say that negative mass wormhole mouths can't exist, and justify that with the quantum energy inequalities. Which, of course, brings up the question of what happens when you try to push an object through a wormhole that would make the mass of one of its mouths go negative? Two obvious possibilities are that either you get a back-reaction force that makes it so you can't ever make something go through the mouth to make its mass negative (perhaps by the formation of a horizon as the mass approaches zero), or the wormhole collapses before the mass can ever reach zero. Perhaps inventive science fiction authors and fans can come up with other plausible options as well.

==Wormholes and time travel==

Wormholes connect across both space and time. So it is natural to consider the possibility of using wormholes for time travel.

Now maybe you don't want to have to deal with time travel in your fiction. It introduces all kinds of opportunities for paradoxes, and you have to keep track of plot lines in a self-consistent way that go back and forth through time, influencing themselves in complicated ways. Okay, no problem. Lets just have our wormholes always connect to the same time. Easy right? In fact, it is natural to justify this. Physics is local, so if you can make wormholes you probably have to create both ends right next to each other at the same time. You can then move the wormhole ends away from each other – maybe putting one end on an interstellar spacecraft or something – but they stay at the same time. Right?

Right?

Except ...

In relativity, motion not only affects how fast you go through space, but also how fast you go through time. When things move, they experience time dilation, so if you move one wormhole rapidly far away and then bring it back, the time dilation it experienced will naturally allow it to form a time machine so you can send things and people back to interfere with their own pasts. Even worse, time can also be slowed down by gravity, so if your wormholes are in areas affected differently by your galaxy's own gravity, or around stars of different mass, or just at different elevations on the same planet, the rate they experience time will be different. Temporal complications ahoy!

Let's look at how this works in more detail. We'll take our standard science fiction empire, called The Empire. Like all empires, it has a central area of authority, the metropole (which we will place on a planet called Metropole), which extracts wealth from various colonies and subjugated client states for its own enrichment in the form of taxes, tribute, and forced favorable trading opportunities. Let's give our Empire two colony worlds which we'll call Colony A and Colony B (these Empire folks sure aren't very creative in their naming, are they?). Just for convenience, we'll put Metropole, Colony A, and Colony B all 100 light years away from each other, in an equilateral triangle.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:WormholesTimeTravel graphic1 wb.svg|600px|frameless]]
<td>[[File:WormholesTimeTravel graphic2 wb.svg|600px|frameless]]
<tr>
<td width=400>Metropole and its two colonies A and B
<td width=400>Metropole sending a wormhole to Colony A.
</table>

So Metropole launches a wormhole mouth out to Colony A. They want to get to Colony A quickly, so they'll make the wormhole go really fast. Let's say they fling it out at 99.9999% the speed of light. At this speed, it takes 100.0001 years for the wormhole mouth to reach its destination in the reference frame of Metropole and Colony A. But due to relativistic time dilation, in the reference frame of the projected wormhole mouth it only takes 0.1414 years to go from Metropole to Colony A. Because the techs at Metropole can look through the wormhole, they will see that 0.1414 years after launch it arrives at its destination. At that point, brave explorers from Metropole can go through and set foot on Colony A – they only need wait about a month and a half to feel alien soil under their feet, rather than a century. In Metropole's reference frame, going through the wormhole takes you 100 light years away, and 99.8596 years into the future. Going the other way, from Colony A to Metropole, takes you 100 light years away and 99.8596 years into the past.

[[File:WormholesTimeTravel graphic3 wb.svg|800px|frameless]]

One might think that these time warps would let you engage in all kinds of time travel. It is easy to see that the Metropole—Colony A situation described here doesn't allow these kinds of shenanigans. For practical purposes, you only have a time machine when you can go back to the place you left at a time before you left. And you can't do that here. Go from Colony A to Metropole and you go back in time 99.8596 years. Go back to Colony A through the wormhole, and you go forward in time the same amount, plus any time you spent on Metropole, so you get back after you left. If you go back through flat space-time, it will always take at least 100 years since you can't go faster than the speed of light so you also get back after you left. No paradoxes for you!

However, it is easy to imagine situations where a wormhole, or a configuration of wormholes, does make a time machine. For example, what happens if we immediately turn around and send the wormhole mouth from Colony A back to Metropole? In the reference frame of Metropole, the wormhole comes back with its "Return to sender" sticker 200.0002 years after it was launched. But going through from the left-behind mouth to the round trip mouth will take you 199.7192 years into the future. And if you step into the round trip mouth, you will go back in time by 199.7192 years into your own past. You now have a real honest to goodness time machine, complete with paradoxes.

[[File:WormholesTimeTravel graphic6´4 wb.svg|800px|frameless]]

Or maybe the wormhole mouth stays at Colony A. And Metropole sends another one to Colony B, that also goes 99.8596 years into the future. Now Colony A sends a wormhole to Colony B. This wormhole also goes 99.8564 years into the future as a consequence of its trip. This means if a traveler at Colony B went through the Colony A wormhole he would go back in time 99.8564 years. Then going from Colony A to Metropole he would go back in time another 99.8564 years. Then he could go from Metropole to Colony B and go forward in time 99.8564 years. The net result is that he ended up back where he started nearly a century before he left.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:WormholesTimeTravel graphic5 wb.svg|600px|frameless]]
<td>[[File:WormholesTimeTravel graphic6 wb.svg|600px|frameless]]
</table>

And it is not just motion that causes time dilation. Being different depths into a gravitational potential does this as well. If Colony A is deeper into the gravity well of the galaxy, or orbits a heavier star, it will experience more gravitational time dilation than Metropole. So eventually it will build up more than 100 years of time slip between the two end of the wormhole and you can have a time machine.

So is time travel inevitable if you have wormholes? What a nuisance!

Well, maybe not. Here's why. Think about what happens when the Colony A – Colony B wormhole has gone just far enough that a light signal going through the wormholes can get back to where it left just as it is leaving. Now, since the propagating signal and the newly transmitted signal are both leaving at the same time, you have double the intensity. So this doubled intensity signal goes around and meets itself again, quadrupling its intensity. And so on. At this point, just as the configuration is on the verge of becoming a time machine, it becomes a perfect resonator for light signals, which then build up to arbitrarily high intensities until something breaks and you don't have an incipient time machine any more.

Now, this won't always work if you just consider light signals. Light can get defocused or sent off in odd directions or something so that you don't have a good path back until after you end up with a time machine. But if you look at what happens in semiclassical quantum gravity, will get a buildup in the amplitude of quantum fluctuations as soon as you start to get a time machine, called <i>vacuum polarization</i>. These fluctuations are expected to build up to sufficient amplitude as to destroy the time machine to be, at least for the few simple cases studied. Or perhaps it just causes the wormholes to bounce away from each other so as to prevent time machine formation. This has not yet been shown to occur in general. But to does seem to happen for all possible configurations of one or two wormholes about to become time machines
<ref name="Visser_Lorentzian_wormholes"></ref>
.

But let us assume for the moment that wormholes cannot form time machines in any configuration due to this vacuum polarization mechanism. Now there are limits on just how you can place and move your wormholes. You don't want to accidentally almost form a time machine and break your transportation network. There are several ways to do this:
<ul>
<li> Move your wormholes slowly, so as not to build up much time dilation (or <i>time lag</i>, if you will).
<li> Don't form closed loops in your wormhole transportation network. The less distance you need to travel to go all the way around a loop to get back where you started, the less time lag is needed to make a time machine.
<li> "Discharge" the time lag by periodically rapidly moving the mouth of a wormhole that is "behind" in time in such a way that it comes back to where it was. Perhaps you can put a charge on it by sending a [[Particle_Beam_Weapons|particle beam]] through it, and then accelerate it up to relativistic speeds in a synchrotron for a while.
<li> Send new wormholes that you are going to place through the existing wormhole network. This way they acquire the same time lag across their mouths as all of the wormholes they traverse. Until they build up some additional time lag of their own, you provide a perfect return trip without making a time machine.
</ul>

In conclusion, if you want a story or setting that involves time travel, you can plausibly still use wormholes somehow to allow that (although it might take loops of several of them). If you want a story that avoids the complications of time travel, you can still plausibly have wormholes in that setting (although you now need to beware of the limitations on your wormhole transportation network imposed by the no time machine rule).

==Practical implications==

===Getting places===

Usually in science fiction, people want wormholes because they want to be able to get to cool places far away without taking forever. Very often, this involves the technology (or the protagonist, with some kind of psychic power) "opening" a wormhole from the current location to the desired location when there wasn't one before.

Sadly, the ability to just create a wormhole to anywhere you want to seems unlikely. Physics tends to be local, which means that if you can make a wormhole both ends will emerge right next to each other. You will need to move the ends to where you want them to be using other methods (mailing them by post, putting them on a spacecraft, etc.). As was mentioned earlier, if you shoot a stream of stuff through a wormhole, the end that the stuff comes out of becomes a rocket. So one way to move an end of a wormhole is to shoot stuff through the wormhole. Because you can leave all the big heavy equipment at home (generators, rocket motors, cooling systems, and everything else), if you can make your wormhole light enough this could be a low cost and efficient method of getting it places. Remember that you can exploit time dilation to make it seem like you can get someplace in much less time than you would expect light to take going to that place (at the expense of taking you some time forward in the future as well when you step through). If you can get your wormhole mass down to a few grams, or even several kilograms, using powerful lasers to shoot through the wormhole to make a photon rocket could be a highly efficient way to go places at relativistic speeds. Perhaps it could even fuel itself by collecting mass from the interstellar medium (ISM) as it flies along, and it might be able to brake against the ISM or solar wind of the destination star system to slow down, reducing mass loss even further.

===Keeping time===

We have already shown that wormholes connect across both space and time, so that a trip between star systems could take you hundreds of years into the future, and the return trip takes you hundreds of years back in time. And this is even before we throw in how time slips between planets when considering relativistic time dilation due to different speeds and gravitational potentials.

Fortunately, all the weirdness of different time rates and going backward and forward in time can be ignored by the average person. This is because you never need to go from one world to another, or back, across the vast gulfs of interstellar space. You just take the wormhole between them. All you ever need to worry about is the coordinate frame that goes across the wormhole. When considering this reference frame, you're not hopping all over the place in time. If it takes ten minutes to cross the wormhole between the two planets, when you get to your destination world the clocks will read ten minutes later than they did when you left your departure world. By coordinating their time-keeping across the wormhole network, all of the worlds of the network can agree on a common time to coordinate their activities. This is all travelers ever need to worry about, and they can then ignore all the relativistic weirdness. Your network engineers will still need to keep track of relative time drift and how close a given configuration is getting to a time loop. But unless your protagonist is a network engineer, they can just ignore all that stuff. And, as an author, so can you! Assume your engineers are competent, you have good regulatory bodies and standards institutions, and don't worry about any of this "time travel" that doesn't actually let you cause paradoxes.

===Keeping wormholes away from planets===

By now, people expect science fiction to have spaceships. Wormholes offer a convenient excuse to have spaceships, because it provides a way for those spaceships to get to other worlds that are not in our solar system. But if you have wormholes, sometimes you need to ask why you still have spaceships? Why not just put one end on Earth and the other on an alien planet and step through to get where you are going?

Sometimes there are good reasons for this. If your wormhole has as much mass as Jupiter, you are not just going to keep it in Topeka, Kansas. That would kind of end the Earth. So you would put it in orbit around the Sun, and send spaceships from Earth out to the wormhole and through it to wherever it goes.

But all too often we see authors using special pleading to pigeonhole in their spaceships without thinking about good physics. Things like "it can't work in atmosphere" (use an airlock), "it can't work within a certain number of planetary radii" (why? There is no physical plausibility to such a restriction), "It can't work in a gravitational field of more than X" (violates the equivalence principle), and so on. But if you insist on having spaceships, and you also insist on wormholes that mass less than a planet, there are a few plausible ways to justify your preconceived ideas that at least plausibly align with physics.

For example, the closest you can plausibly get to "X number of planetary radii" or "gravity field less than X" is a criterion that relies on local measurements of space-time curvature only. And it turns out that this quantity is the tidal forces (curvature is a direct measurement of tides). Tidal forces are going to be proportional to the mass of the nearby thing causing the tides, and inversely proportional to the cube of the distance. If <math>R</math> is a measure of your space-time curvature, <math>M</math> is your object's mass, and <math>r</math> is the distance from the object to the wormhole, then

<div align=center><math>R \propto \frac{M}{r^3}</math>.</div>

So if you want your wormholes to be at least ten planetary radii from Earth (<math>r = 64 \times 10^6</math> m, <math>M = 6 \times 10^{24}</math> kg), then the curvature limit is <math>R = 23</math> kg/m³ in some convenient system of units where we don't have to include constants of proportionality. But now suppose you want to send a <math>26 \times 10^6</math> kg ore freighter through that wormhole. You'll only be able to get your freighter within about 100 meters of the wormhole before the freighter's tides collapse the wormhole. This <i>might</i> work if the freighter were really long and skinny, much longer than 100 meters, to reduce its transverse tides; or if the wormhole was much larger than 100 meters in diameter. But in general, note that the curvature and tides have units of density, so if the average density in any volume around the wormhole exceeds the critical density – oops, there it goes.

Also, having a limit on the geometric curvature of space-time similar to what you get around mere planets doesn't end up being very plausible – the curvatures in the wormhole itself exceed that by so very many orders of magnitude that having so stringent of a curvature tolerance seems very implausible, and just dancing on the edge of disaster. For example, if your wormhole is about 100 meters across, its curvature in these same units will be approximately 500 billion trillion kg/m³. Keeping the tides to within 23 kg/m³ is the equivalent of an engineering tolerance of one part in 25 billion trillion.

So if you don't really have a good way to limit where you can place a wormhole based on the nearby "gravity", are there other ways you can exclude them from the vicinity of planets?

Safety might be one good reason. We have already discussed various ways a wormhole might collapse (forming a time machine, having its mass go negative). What happens when a wormhole collapses? If there are any parts of the wormhole or things in the wormhole that are charged or made of charged particles (like ordinary matter) you could justify a considerable fraction of the wormhole's mass-energy being converted to electromagnetic radiation. In addition, if the wormhole collapses into a black hole it will evaporate away its mass-energy primarily in the form of electromagnetic waves as it undergoes Hawking radiation. Large black holes will last a long time, but if the black hole left behind has a mass of 500 tons or less it will last less than 10 seconds. And 500 tons worth of energy delivered in 10 seconds would do quite a number on a planet. These sorts of considerations could lead to a regulatory environment that require wormholes to be located several light seconds away from inhabited places (like planets).

===Wormholes on planets===

But sometimes you might want to have the wormhole on a planet. Perhaps you are envisioning one of those circular or oval fantasy portals, looking like a window to an exotic foreign land, and when you step through you are actually there. We already know that thin-shell wormholes can be very low mass, and can have a circular shape. So that should work, right?

There are at least two issues to consider here. One is that even if both wormhole ends are on the same planet, natural variations in atmospheric pressure due to weather will lead to very significant differences in air pressure from one side of the wormhole to another (unless the wormholes are so close that they are within a small distance of the extent of the same weather system – say, within a few km of each other). This pressure difference will drive severe winds through the wormhole, usually comparable to the winds of a hurricane (although there will be periods of relative calm when the pressure at both wormholes just happens to be at the same pressure). This effect will only be exacerbated if the wormhole mouths are at different altitudes, due to the decrease in air pressure with altitude. And if the wormhole connects between different worlds, the effect will be much worse. In addition to making travel inconvenient and possibly damaging the equipment, the winds will produce large uncontrollable mass flows across the wormhole (air on earth has a density of about 1 kg/m³, so if you have 50 m/s winds blowing through a 2 meter diameter portal you have a mass flow of about 160 kg/s through your wormhole). This will make it difficult for mass balance, and keeping both ends having high enough masses that neither gets so close to zero that the wormhole breaks or forms a horizon or goes negative mass and causes a runaway planetary catastrophe or whatever wormholes do when one end gets driven to near negative mass by the stuff leaving it.

As a consequence, planetary wormhole mouths will probably be kept in airlocks.

The other issue is that different latitudes on a planet will be rotating at different speeds; at the poles they won't be moving at all while at the equator they will be going a full planetary circumference every day. This means wormhole mouths at different latitudes will be building up different time dilations with respect to each other. In addition, wormholes at different altitudes will experience different gravitational time dilation rates. If you let this go on for long enough, the differences in time across the wormhole mouths will form a time machine. Whether this collapses the wormholes, makes them "bounce" apart from each other (likely ruining a lot of property and infrastructure in the process), or just forms a time loop that leads to various inconvenient paradoxes depends on the assumptions you made for the physics of your world. But it will probably be something you will want to avoid. Re-balancing the time differences between wormhole ends will probably need to be done every few decades.

===Wormholes on spaceships===

The necessity for radiators on spacecraft is really annoying. Dealing with all that waste heat is not fun, and these issues are not awesome kewl stuff like gigawatts and megatons and hundreds of g's of thrust and other things that let you show everyone how much your spaceships totally rule and everyone else's spacecraft totally drool. Instead you need these big fragile stupid-looking things hanging off the side of your ship, ruining your aesthetics and efficiency and being vulnerable to your enemies blowing them up, and lighting you up so bright that you're easily visible from the other side of the solar system. Hey! I know! Let's put a tiny wormhole on our spaceships so we can just send all the heat someplace else. Great idea! That'll work, right?

Actually, yes. If you have small enough wormholes, it will work great. But there are consequences.

So you're running your coolant loops from your reactor through the wormhole to some cooling towers on your home planet. Much better than radiators. But – why do you still have that reactor on your spaceship? It's big and heavy and needs lots of big and heavy shielding and poses nasty radiation hazards. Surely you could leave it behind on your home planet as well, next to those cooling towers, and just run a power line through the wormhole? You'll be able to run a bigger, more powerful reactor that way anyway, and that extra weight won't be bogging your ship down.

And why does your spaceship have its lasers on board? You can have a much bigger laser on the planet that you don't need to lug around. Just shine the laser beam through the wormhole and re-direct it with mirrors.

And all of your missiles? Might as well just feed them through the wormhole when you need to fire them.

And then do you really need all that life support equipment? Just run an HVAC and plumbing that sends fresh air and water to the spacecraft and takes used air and water off. And remember to send the occasional snack, too.

Oh, but wait. Why is all your crew even on the spacecraft again? They have a wormhole to the spaceship. You can just have all your crew run the spacecraft remotely from a mission control station. Most of your equipment is on the planet anyway, so most of the engineering staff is already here. Now you don't actually need to put people in harm's way. If someone does need to fix something just put them in a space suit and send them through, let them do their job, then bring them back. Most of that can be done by a tele-operated robot, in any event. And being able to go back to their families when their shift is over is great for morale.

Hmmm, your sensors could just look through the wormhole instead of having all the vulnerable and heavy and expensive equipment actually on the spacecraft. And as long as you are doing that, you can just leave your laser beam pointer at home too, and aim the laser by pointing it through the wormhole.

So now what's actually left in your actual spacecraft? Oh, the rocket itself. But remember how we already talked about how we could just keep the rocket engine and propellant on the planet and shoot the rocket jet through the wormhole, and it will move just like a rocket? Yeah, so no rocket either.

Congratulations, you've successfully gotten rid of your spaceship. The wormhole itself is your spaceship. It's all you need, it's safer, it's more capable, it's stealthier, and just better all around than some contraption of steel and Mylar and fissioning uranium fuel rods and fragile people going through space.

===Adding mass to wormholes===

So you just made a pair of wormholes. Chances are, you pulled them apart as near Planck mass objects, with about 20 micrograms of mass. Well, that's not very useful! What if you want to send something through that's bigger than 20 micrograms? You're going to need to add mass to your wormhole mouths.

A similar issue comes up if you project a wormhole at some far away planet or star system using the wormhole-rocket trick already discussed, maybe by using powerful lasers to make a photon rocket. To be practical, your projected wormhole mouth will need to be very low mass. But once you get it to where you want to go, you'll need to give the mouth enough mass to let your explorers and colonists and equipment through.

A simple way to do this is to simply suck stuff into the wormhole. This adds mass to the wormhole mouth the stuff comes through. Of course, if you take it out from the other side, that other mouth will lose that much mass, and if it didn't have than much mass to begin with you'll likely have problems. So just leave that stuff in there! Shove it off to the side. Let it clutter up the unused areas out of the main traffic route in the way that basements and attics and the unused spaces of garages accumulate all that old junk that you never use any more but don't want to throw away. Build a wall inside of your wormhole, so travelers don't have to see all the asteroidal rock or super salty water or very surprised natives or whatever it was you scooped up on your alien world. Shove it into the undesirable high curvature places with nasty tides. This wall can also help keep the legitimate passengers away from the inner working of the wormhole, the strange energies that support it, the cables and pipes and crawlspaces that only your maintenance workers should access anyway, and the extreme space-time curvatures that would rip the passengers to shreds if they ever were directly exposed to them.

===Tiny wormholes===

Wormholes that are too small to fit people through could still be useful for sending information. You could use them to rapidly communicate over long distances. If the wormholes were light enough and cheap enough, they might even replace cell phones and laggy land lines with super high bit rate connections.

You could also send energy through tiny wormholes, in the form of light or electric current. Storms that knock down power lines no longer need to cut off your home's electricity. You could also use them to power machinery that need light weight and high power output – electric transportation like airplanes, shipping, or trucking; directed energy weapons; or high specific impulse rockets.

===Causality denial, causality attacks===

We have shown that moving wormholes around can lead to time travel, and that there are plausible physical mechanisms that can prevent time-travel allowing wormhole configurations from happening. People being people, you just know that if this is possible, that it will not be long before these effects are used in politics and warfare.

In a network of wormholes that is about to create a time machine, one likely resolution is that things break until a time machine is no longer possible. In any chain stressed to its point of failure, it is the weakest link that breaks. So in a path through wormholes and normal space-time that is about to form a time machine, it is the weakest wormhole that will collapse (or otherwise have something bad happen to it). So if your rival polity has a wormhole that allows then easy trade with a rich world, and you want to edge in on that trade and simultaneously deny that trade to your rival, you can just send a bigger, stronger wormhole in such a way that forces a time loop. Now your rival's wimpy wormhole breaks, and you get access to the trade opportunities with the world! This is the basics of a causality attack – use the causality-enforcing properties of wormholes to attack your enemy's wormhole infrastructure.

But what if your rival polity anticipated this? What if they have strong wormholes, with an arrangement in space-time that blocks off easy access of your wormholes to a profitable region of space? This is a causality denial action, or a causality fortress. By controlling the time displacement of the wormholes in their network, a polity can control what additional connections can be made. This can be used to enforce borders, prevent infiltration, or restrict civilian wormholes to legitimate uses.

===Moving one wormhole through another wormhole===

If you have a wormhole mouth, you can move it through another wormhole as long as the mouth (and any supporting structures) is small enough to fit through the throat of the wormhole it is going through. The geometry all works out. In fact, this has a rather nice side-effect. If you are looking through the wormhole being moved from back home, you only see it take as long to go through the other wormhole as a wormhole transit usually takes even if it got projected far into the future or past. So the moved wormhole acquires the same time differences as the wormhole it just went through (added on to any time differences it already had, it doesn't get reset or anything). This can let you easily build wormhole networks that connect to themselves in loops. It also makes the network more susceptible to breaking, because there is much less leeway for error with a closed loop as far as making a time machine and breaking the loop. So it can be done, but there are consequences.

===Dropping a wormhole into a black hole===

In principle, if you have a wormhole connection to the inside of a black hole, you can get information out from inside the event horizon. The theorem that prevents things from escaping a black hole was made under the assumption that the energy can't go negative. And as we've already seen, wormholes require regions of negative energy to work. You still wouldn't be able to see further toward the singularity at the center than the wormhole's location. And the wormhole would be inevitably dragged into the singularity in a finite amount of time. But, in principle, during that time you could see what is happening in the black hole.

But there's a big catch with this. You can't actually get the wormhole into the black hole. Not if you assume that time machines can't form, anyway. From the point of view of the wormhole mouth falling into the black hole, it will pass through the event horizon in a finite amount of time. From the point of view of someone watching from outside, the infalling mouth gets more and more time dilated to the point that it never crosses the horizon. This time dilation sets up the conditions for a time machine, and any time machine preventing physics happens on the wormhole before it can ever get in.

== Possible Networks ==
=== Acyclic wormhole networks ===

Recall that wormholes are space-time connections. They bridge space as well as time. Assuming we want to build this network in a stable manner, we want to avoid Closed Timelike Curves from appearing. Let us start with some simple anaylsis of consequences on this. Assume projected wormholes have one mouth sent at relativistic speeds to a distant target and the other mouth is kept at rest at home. From the perspective of the dispatching party, this will lead to the new connection opening very quickly. Done again and again, large volumes of space can be accessed in a short time, a powerful advantage.<br />

[[File:Wormholes PartialOrder.svg|400px|thumb|The partial order relationship of the wormhole connection]]

Every projected connection thus extends onto the future. Then we must avoid forming loops, or more generally "backtracking" back towards a dispatched node. This nets us a graph with no closed loops - an acyclical graph, also called a tree. Since connections may not return to previously established nodes if they are time-shifting from the past into the future, we also get a partial order: nodes <math display="inline">u</math> that dispatch wormholes have a higher order over the nodes <math display="inline">v</math> they establish: <math display="inline">u < v</math>. This lets us establish a direction on the whole graph. For a particular node <math display="inline">n</math>, its '''parent node''' established it, <math display="inline">parent(n) < n</math>, and it in turn establishes '''children nodes''', <math display="inline">n < child(n)</math>. The node that only has children nodes is the tree's '''root node'''. Nodes that have no children are '''leafs''' of the tree. The trees '''height''' is the longest outward path one can take from the root to a leaf.<br />

In terms of traversal, such nodes have some interesting and useful properties. Since wormhole traversals may be assumed to be effectively instantaneously bridging large distances, travel time is mostly defined by the travel distance between wormhole mouths. And as such, we want to minimize the amount of transits we have to make to get from a node <math display="inline">A</math> to a node <math display="inline">B</math> the fastest. It turns out, in order to achieve this, we want to have as large a choice of connections at each node. In the ideal case, any node except root can be reached in two transits: one transit to root, and from there to the other node. In the less ideal case we have to make one further stop-over, choosing from a number of connections at a second node to reach our destinations. The less ideal case from that sees us make two transits through such "switch yards". Assuming each node except the leafs in such a wormhole network has exactly <math display="inline">k</math> connections and there are <math display="inline">n</math> nodes in the whole network, we can reach any destination from root in <math display="inline">\log_k (n)</math> choices of wormhole transit. This is a powerful way to get to connections quickly through centralization. (In computer science, such k-nary trees are of great interest for quickly search-able data structures for this reason.) Configurations that are less "balanced" net appropriately lesser advantages and longer transit times.<br />

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Wormholes AcyclicGraphExample.svg |600px|frameless]]
<td>[[File:Wormholes Trees.svg|600px|frameless]]
<tr>
<td width=400>An example of an acyclic wormhole network in space-time.
<td width=400>The connected tree graph of the network exampke, and a k-nary tree example.
</table>

Also, traffic to most other nodes ''has'' to go through a root-wards node. This gives these nodes significant presence in the larger network. Routing through them is unavoidable if you want to reach most of the rest of the network. Root nodes can profit from taxation, services to travelers, control of routing infrastructure for information, and so on. But there is also a catch. Assuming leaf nodes each send the same amount of traffic inward. The amount of traffic load increases polynomial as you go towards root. Your wormhole infrastructure (and what transports cargo between wormholes, in space or on planets) has to cope with this scaling. Congestion might become an issue.<br />

=== Mesh wormhole networks ===

The acyclic network may have some bothersome problems. The central routing focuses power and can keep transit times down quite significantly depending on the specific network infrastructure, but it’s also congestion-prone, and some close locations may be widely separated on the network. Or maybe you want to avoid centralized routing dependencies for political reasons. Root might like being root; everyone else being subject to root’s whims, not so much. So, can we build networks where there are two routes between the same two nodes? We can, but it comes with some necessary adjustment.<br />

Recall why the acyclic network is acyclic: to avoid forming CTCs due to the time connection between future and past from the projected wormholes. But if we let both wormhole mouths experience the same time dilation, the problem that ultimately leads to CTCs - one connection routing displacing further into the future than the other connection, and thus allowing one to arrive at a node before ones departure - is avoided entirely, since there is no time displacement to be accumulated. Thus we get meshes, where there may be many connections and cycles between nodes. An observer at one node watching for his transition from another node 10 light years away would have to wait for ten years to watch his light arrive. Closed timelike loops are not possible so long as this network is kept balanced.<br />

[[File:Wormholes MeshNetwork.svg|thumb|center|700px|An example of such a mesh network]]

The disadvantage - this network must take great care to ensure that the time dilation that inenvitably occurs due to different star velocities and the galactic gravitational field don’t accidentally create a time shift and CTC in the network. And also we cannot exploit time dilation to open connections very quickly, from the home node’s perspective. One has to wait out the full years it takes the wormhole’s far mouth to get to a remote destination, before the connection can be used.<br />

=== Semi-privileged wormhole networks ===

The interesting thing is, it is very much possible to build networks that make use both of relativistic projection to seemingly accelerate the network construction time, ''and'' contain cycles, with alternate routes to get to another node. The acyclic construction is a strategy to avoid CTC formation, but we should remember the ultimate constraint reason: we avoid cycles so that there are no connections which, through a certain time shift on them, a CTC could be formed. Acyclic construction is one strategy. Only building cyclic connections with careful time shift balancing is another!<br />

The foundation for such networks is an acyclic "base graph" that may be isolated from the larger network graph. We also call this the '''red graph'''. The base graph maintains all the partially ordered properties of the simple acyclic graph. We can call this the "privileged" graph, since its nodes and their connections define the principle positions in space-time of the network, and we must avoid generating CTCs on this definition. How do we avoid forming CTCs but still create alternative paths and thus loops?<br />

Let’s assume we have two nodes <math display="inline">A</math> and <math display="inline">B</math> ten light years apart. They both had connections established from a parent node <math display="inline">P</math>. Those connections are time-shifted over 99 years and part of the privileged base graph.

We couldn’t build a time-shifted connection between <math display="inline">A</math> and <math display="inline">B</math> without causing a CTC to be formed. But we are entirely fine sending signals '''Fast-as-Light''' from <math display="inline">A</math> to <math display="inline">B</math> and back. This is not a violation. Now consider: how does a wormhole with no time shift across it behave? In the mesh network case, such connections don’t cause CTCs to appear. And it doesn’t cause one to appear here either. How can we build such connections? We can of course use projection with relativistic speeds at both ends, like how we would build Mesh networks. But there is an even faster solution. If we can move wormholes through wormholes, we can shift the new mouths through the existing network to create this connection!<br />

The main catch is that we have to tightly watch out for temporal drift on such new connections. Through adding such edges that overall form a "blue graph" on top of the red graph, we are creating loops on the red graph underpinning the network. If one end of a wormhole mouth drifts more in time than the other (which will inevitably happen because stars won’t have the same mass, experience the same galactic gravity, or have the same velocity relative to each other) the conditions for a CTC will start to appear. The only way to pre-empt this is with continuous maintenance, taking connections offline and adjusting their temporal position using synchrotons, and having "safety space-time" spacing between wormholes. In this way, there is some "room" where signals can arrive faster and faster, but not before they were sent, allowing the connection to be used most of the time with only short intervals of adjustment.<br />

This works even when both ends connected don’t have the same time distance from the root node - the important thing is that the new shortcut accomodates this difference with a time shift of its own. We can calculate this difference easily. For two non-root nodes <math display="inline">A,B</math>, for each node we sum the temporal difference over the path from root to the nodes along the red graph, then subtract the difference from each other. The resulting remaining temporal difference is that the new connection on the blue graphh must have to keep the network stable. This means we can even do things like connect systems far out directly to core systems, or bridge from one edge of the graph to the other - arbitrary routings are possible. These routings can allow faster signal transmission and mass transport than through root, mitigating bandwith issues. The main problem is to watch out for network stability. As nodes accumulate, CTC avoidance becomes more and more delicate, and safety spacing must be maintained diligently. This can add to the effective travel time through such alternate connections, and thus maintains some of the "privilege" of the acyclic base graph.<br />

[[File:Wormholes BuildingCyclics.svg|800px|center]]

Arguably this network configuration is the most powerful, marrying many advantages and removing some disadvantages. It is however also bound to the need for safety spacing. Depending on your infrastructure assumptions, this can pose issues. If you want to implement serious stretches of safety space, wormhole links not part of the acyclic base graph would likely have to be located at interplanetary distances away from the main links, with light hours to light days of safety spacing. If your main network runs on planetary trains, this is rather a bother. Adjusting the internal throat length of wormholes can help in this regard, but of course now poses the issues of how to safely navigate such a long wormhole. Maybe you could extend something like space elevator tethers through such stretchy throats. With no gravity inside the wormhole itself, such cables could probably stretch quite significant distances. If your culture is handling wormholes mostly in space, you still have to account for added travel times through the safety spacing. Communications links would probably be easier connected between arbitrary nodes than macroscopic, craft-traversable wormholes.

[[File:Wormholes ComplexExample.svg|800px|thumb|center|An example of a complex semi-privileged graph wormhole network]]

==Credit==
Authors: Luke Campbell and Sevoris

==References==

[[Category:Physics]][[Category:Infrastructure]]

Particle Beam Weapons

2022-08-26T15:46:22Z

Sevoris:

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Like [[Laser Weapons]], particle beams are a popular concept in science fiction. Streams of charged or neutralized particles are accelerated to awesome velocities and projected at a remote target. Like a hail of trillions of atom-sized (or smaller) bullets, the particle beam smashes into the target, releasing thermal energy and creating radiation. The unlucky target is irradiated, cored through, or exploded into chunks!

Particle beams combine some properties of larger-projectile kinetic energy weapons and the laser directed energy weapon. Particle beams can travel at high-relativistic velocities close to the speed of light, able to reach targets only a tiny bit slower than a beam of photons. But their working medium are particles with mass, which gives them different propagation and interaction behavior. Often high-energy particles can penetrate into a target or release showers of secondary radiation that penetrate deeply. This irradiation also offers a soft, insidious means to make technology fail and kill biological beings (or anything dependent on nanometer structures and molecular systems) without having to physically melt or blow apart the target. Yet a particle beam's mass flow is generally so tiny that they are not limited by physical ammunition but the power required to drive this little mass up to high speeds and thus energy.

Depending on configuration, particle beams can be highly utilitarian systems with a wide range of not just weapon but also sensing and propulsion applications. They can deal significant radiation damage at low power or focus, or transfer enough energy for explosive vaporization at high energies. Their accuracy is exceptional and their range can be significant, providing direct-fire engagment capability out to the limits of the weapon mounts and supporting sensors.

Defending against particle beams is possible with appropiate material composition and in some cases, magnetic fields.

== The limitation of bloom ==
There is one fundamental problem with weaponizing particle beams. To accelerate a particle beam efficiently, we want to use electromagnetic fields - gravity is way too weak and dissipates very quickly to accelerate particle beams without completely absurd assumptions (consider: the entire mass of the Earth amounts to 1G of acceleration. The forces in a particle accelerator can be in the hundreds of millions of Gs), and neither the weak nor the strong nuclear force reach sufficiently far. Electromagnetic fields are wonderful! To accelerate particles with an electromagnetic field, they must be charged positively or negatively. But a cloud of positively or negatively charged particles will experience charge repulsion, giving rise to '''electrostatic bloom''' - the tightly bunched-together group will disperse into a weak, wide-area cloud. Too unfocused, the fearsome particle beam will not do much damage. Maybe it will irradiate the target, but the beam may disperse so widely that it becomes indistinguishable from the cosmic background radiation you find everywhere in deep space.

In older, popularized Sci-fi analysis, this issue of was generally considered the critical issue with charged particle beam weapons at ranges beyond a few hundred kilometers.

{{Quote|
They have a disadvantage of possessing a much shorter range. The beam tends to expand the further it travels, reducing the damage density ("electrostatic bloom"). This is because all the particles in the beam have the same charge, and like charges repel, remember? Self-repulsion severely limits the density of the beam, and thus its power.

::: - Atomic Rockets, [http://www.projectrho.com/public_html/rocket/spacegunconvent2.php#id--Particle_Beams Particle Beams page]}}

{{Quote|
Particle beams disperse for a lot more reasons than laser beams, unfortunately, so it's harder to give a simple formula. It will depend on things like magnetic and electric fields in the region between the source and the target (if the particles have spin, for example, they will couple to the magnetic field gradient even if they are neutral).
However, for a neutral particle beam traversing empty, field-free space, the dispersion is proportional to the temperature of the beam. Using, for the sake of a simple example, a mercury ion beam (dispersion decreases proportional to square root of atomic mass, and mercury is a convenient high-mass atom that ionizes easily), the lateral (spreading rate) velocity of the beam is:

<math> V = 1.4 \sqrt{T} \, m/sec </math> for <math>T</math> in Kelvin

To calculate the actual angular spread of the beam, you need to know the beam velocity. For a quick calculation, you could say it's no more than the speed of light, <math>300,000,000 m/sec</math>. So the dispersion in nano-radians is <math>5 \sqrt{T}</math>.

So, for a beam with an effective temperature of, say, <math>1000K</math>, dispersion for mercury is <math>150\, nR</math>, or <math>0.15</math> micro-radians. Dispersion at a distance of <math>100,000 \, km</math> would be <math>0.015 \, km</math>, or 15 meters. A hydrogen beam would disperse <math>\sqrt{80}= 9</math> times more.
[note that if the beam is actually relativistic, you have to apply a relativistic correction, which I'll ignore here.]

::: - Dr. Geoffrey A. Landis, via [http://www.projectrho.com/public_html/rocket/spacegunconvent2.php#id--Particle_Beams Atomic Rockets]}}

{{Quote|
Particle beams come in three varieties: Charged particle beams (electrons or protons), neutral particle beams (neutrons) or neutralized atomic beams (high velocity hydrogen or helium atoms).

Each presents a number of technical problems. Electron beams are very easy to produce; indeed, anyone who’s sat in front of a CRT-based television has sat in front of one. However, electron beams are also very easy to redirect via running a modest charge through the surface of the target, and Coulomb repulsion between the electrons in the beam would cause it to disperse rapidly in roughly 40 to 80 km in a vacuum. The other variety of charged particle beam just replaces the electron with protons, which have the advantage of greater mass per particle (by a factor of a bit over 1,836). This is an advantage, in that the beams will avoid spreading quite as quickly, but also requires considerably more energy to get them up to speed in the first place.

Einsteinian time dilation from the frame of reference of the particles in the beam mean that beam spread happens more slowly from the frame of reference outside the beam, but higher speed beams have less time to interact with the atomic nuclei of their targets, making the optimum velocity of the beam somewhat hard to determine even for a theoretical model for a game. In round numbers, a proton beam would have ranges of roughly triple an electron beam for about the same energy inputs...which is still shorter than the lasers currently in the game.

The second type of particle beam is the neutralized hydrogen or helium atom beam. You impart a charge on the beam and accelerate it with a cyclotron or linear accelerator, then when the beam has enough energy stored in it, you run it through a filter that strips the electrons off, and you get a beam of neutrally charged particles. In theory, this beam of particles could penetrate armor or ignore armor within a given distance of the skin of the ship before random collisions cause it to lose energy. In practice, most filters that will de-ionize the hydrogen or helium atoms will also impart enough random momentum among the atoms in the beam that it will rapidly disperse within a few kilometers.
The third type of beam is the most problematic. If you presume that there is a way to generate neutron sources, capture neutrons and accelerate them, you could, in theory, make a beam entirely out of neutrons. Neutron beams, as beams, have some very powerful advantages. They don’t have an electrical charge, so they don’t spread from mutual repulsion of like charges. This allows them to be diffraction limited, but without the inherent limitations of photons. Depending on the tolerances of the system, and the final velocity of the neutrons, they could be very long ranged indeed; a reasonable ball park figure is around 800 to 1200 km for the first range bracket. At 20 km per hex, a 40 hex ‘short range bracket’ mean they would quickly render every other weapon in the game obsolete.

Even worse, neutron beams launching particles the speeds needed to make them weaponizable wouldn’t damage the ships. They’d sweep through the ships and turn everything they crossed into a radioactive hell, killing the crew in seconds. A game of “Both ships shoot, both ships become unmanned wrecks” wouldn’t be terribly fun to play.

Now, it is also possible, that like charged particle beam, the same forces that can be used to accelerate these neutrons into a beam could also be used to shield the target; from a game play perspective, this probably results in a perfect defense that, if it fails, results in instant death for the crew, which, will somewhat better than the “two ships fire, two ships die” model above, is much less compelling than the game that’s already present.

::: - ''Designers Note: Where are the Particle beams?'' Ken Burnside, Attack Vector Tactical 2nd Edition Rulebook, '''p. 104''', Ad Astra Games, Pelican Rapids, MN 56572}}

Expected ranges were generally poor because the charged particle beam appeared to disperse too fast. The use of neutralized ion particle beams is discussed more favorably in multiple analyses - [http://toughsf.blogspot.com/2018/12/particle-beams-in-space.html including in a longer blog post on ToughSF] - but neutralization of a charged particle beam requires the use of heavier ions as a base. At the least, single protons must be used. The neutralization of the beam removes the problem of mutual electrostatic repulsion, but the recombination of ions and electrons is not without leftover energy. This leftover energy adds random motion to what is now a relativistic gas. Once again the particle beam begins to disperse!

To get an effective weapon, we want the exact opposite! We want to fight that bloom! Only by keeping our beams focused on target, can we achieve beam intensities that do fun things like melt the target, or combust or, or explode parts of it - or just irradiate it so thoroughly that anything delicate won’t work any longer. And in deep space bloom is an even bigger problem if we want to push the range up to that of powerful lasers. Or maybe we even want to take the beams beyond the range of pesky lasers and ram a rod of particles up that smug laserstar nose to rocket nozzle!

=== Solving electrostatic bloom one way ===
The wonderful news is that we can solve the problem of (electrostatic) bloom, and it’s surprisingly easy with the humble electron. Electrons are wonderful subatomic particles to weaponize. The electron has a mass 1840 times less that of the proton and neutron, while having the same magnitude of charge as a proton. This makes it much easier to accelerate, since the charge to mass ratio is much higher. Easier to accelerate particles can attain more energy in a shorter length of accelerator. And for this magic trick we‘ll need a lot of acceleration to happen in our high-powered hat! The fact that we can most easily accelerate electrons to very, very high energies in a short distance allows us to exploit relativity to our advantage. More specifically, relativistic time dilation.

Time dilation happens as some object moves closer to the speed of light. From the perspective of the object at relativistic speed, time slows down relative to the surrounding universe. A journey that from the perspective of an observer at rest (not at relativistic speed) takes years, can happen in seconds from the perspective of the object.

We can use the same thing in a particle beam: that the electrons will repulse from each other and disperse cannot effectively matter when from the electrons point of view, the journey to the target has happened before they had significant time to repulse each other. The beam arrives before it has had time to experience electrostatic bloom from its massively dilated point of observation. This concept has been known about for decades and has been considered for real world weapons, but accelerator technology was nowhere near up to the task to do so in a compact format. Even now our best accelerators are a bit big for current military applications and we don’t really have any need for space battleships… yet!

This is the foundation of the '''Ultra-relativistic electron beam (UREB)'''. By driving electrons with energies of, minimum viable, 10 Giga-electronvolt (a measure of particle energy) at the target, the time dilation experienced by the beam checks its electrostatic repulsion. We may consider particle energies of up to 1 Tera-electronvolts and more! The equation for the doubling distance of a charged, ultra-relativistic beam is shown below just to provide an idea of what we are working with.

::: <math alt=> Z = 1.32 \cdot 10^2(\gamma^2-1)^{3/4}r/\sqrt{I} </math><ref name=USPAS2016>Martin Reiser, "Theory and Design of Charged Particle Beams", 2008, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim</ref>

<math>Z</math> is the distance for the beam radius <math>r</math> to double in size due to electrostatic bloom. The term is the Lorentz gamma <math>\gamma</math> factor (which we will talk about later) and <math>I</math> is the beam current in amperes. For an example of the power of this equation, if we have a <math>500 \, GeV</math> electron beam (<math>\gamma = 978474</math> and pretty high end for electron beams) and an average beam current of <math>2</math> milliamps (pretty normal and even low for particle accelerators) and a beam radius of <math>1 \, cm</math>, we get a doubling distance of nearly <math>30</math> million km! Although this is an elementary first approximation, the potential is clear!

A UREB will still eventually disperse. However, significant defocusing at the discussed particle energy will only occur well after other limitations to weapons range. There is the constant spectre of space plasmas and electromagnetic fields complicating this basic analysis, but that is multiple PhD thesis levels of work and most likely only an issue at really extreme ranges.

In theory, we could consider using ultra-relativistic speeds to fight bloom on particle beams of heavier particle beams as well. Practically, this gets prohibitively complicated due to the much higher mass of ions: with masses thousands to tens of thousands or more times heavier than a single electron, the speed gained per invested energy is much lower, and getting up to ultra-relativistic speeds requires enormous amounts of energy and gigantic accelerators. However, arguably this is also not necessarily required. On ion beams, we can fight bloom in other ways.

=== Solving electrostatic bloom the other way ===
In the introduction, we mentioned using neutral ion beams. This is actually a quite powerful technique, though the devil is in the details! Instead of using relativity to suppress the problems of charge repulsion, we can remove the problem of charge repulsion from the equation by using neutral particles.

The catch is, we need electric charge for all of our acceleration tricks to work. (Since gravity has the gall to lack in easily applicable fields and is a horrendously weak force anyway - look at how much mass we need for a paltry 1G of acceleration compared to the trillions of G’s in a typical particle accelerator!) So during the acceleration process, we require ions - atoms with fewer or more electrons in their orbits than protons in their cores.

We also need the aforementioned much heavier particles! Our mass per particle rises by at least three orders of magnitude in using protons. Using light ions, we are now working with particles four orders of magnitude heavier, and for heavy atoms, we are at five orders of magnitude and more! This causes us a bunch of problems in accelerating these heavy beasts quickly, and pretty much leaves neutralization as the only way to work with them at significant ranges.

For comparison, if we were to use the ion beam without neutralization: let us take a hydrogen beam fired at 100 MeV. This beam doubles its radius in just 1.9 km! The quotes from the introduction come to mind. A heavy ion beam can end up much, much worse due to the higher charge per particle!

With neutralization, our particle beam is accelerated and handled internally as ions and then neutralized upon firing. There are two primary ways to do this, either strip an extra electron that was added to make a negative ion or add the needed electrons to a positive ion. Each method has its own advantages and disadvantages to consider. For the extra electron version you get some accelerator physics advantages for hydrogen beams, but beams of any heavier ions are just harder to accelerate properly due to the poor charge to mass ratio (charge of one electron vs the mass of some sort of large ion). Stripping that extra electron also can be very difficult with the most popular method being gas or foil strippers that make the beam spot bigger and can take huge (up to 50%) cuts out of the beam power. This won’t do for a weapon beam at! There are also optical electron stripping methods available, although those can potentially require large amounts of laser power to pull off. The real advantage of this method though, is that you end up with a truly neutral beam of relativistic atoms that will laugh at anything but a solid chunk of matter.

The second method of ionization involves accelerating a beam of electrons up to the same speed as the ions and then injecting them into the final beam optics with the ions. Luckily this is very easy with a 100 MeV proton beam only needing electrons at 51 keV to keep up with it. That can be generated with an old CRT TV tube! These electrons will naturally mix with the ions and form a cold plasma that travels to the target. Yes, you heard me right A PLASMA! Now you would think that the plasma would neutralize out into a gas, and given enough time you would be right! However, the time scale for this to happen at any large scale is typically too long to matter to a beam traveling a good chunk of the speed of light. If you end up shooting at other solar systems with one of these, do be sure to write and tell us how that works out! The neat thing with plasma is that it typically ends up screening out external fields as well, so attempts at electric or magnetic shields will struggle to work for a well designed weapon beam even if the needed field strengths can be reached. That is not to say it can’t be done, but it sure won’t be easy or consistent if you make your weapon right! On the other hand, this also means that a beam could be made to be easily deflected and that is wonderful for propelling spacecraft!

The catch with either of these neutralization methods is that gas or plasma has a temperature, and random brownian motion. The hotter the gas, the higher the average energy and thus velocity of the brownian motion: the quicker the beam disperses. Instead of being forced apart by electrostatic repulsion, the beam drifts apart. Due to high mass and difficulty of accelerating ions we can’t always rely on relativity to save us here. A 100 MeV proton beam is “only” doing 41% of the speed of light and carbon beam of equivalent energy is even slower! Thus relativistic time contraction is not something that will always play a role in neutral ion beams. As a consequence, controlling the temperature of the beam during generation, acceleration and neutralization is key. The stripped extra electron beams will have to be careful with how the electron is stripped (although that effort always adds “heat” to the beam) and the injected electron neutralized beams will need to be extremely careful with electron injection as well as temperature. If you know the beam temperature, the doubling distance can be found with the following equation.

::: <math>Z = (r / \sqrt{T/m}) \cdot \beta \cdot c \cdot \gamma</math>

Where <math>Z</math> is the doubling distance like before, <math>r</math> is the initial beam radius, <math>T</math> is the beam temperature in eV, <math>m</math> is the ion mass and <math>\beta</math>, <math>c</math>, and <math>\gamma</math> are the typical relativistic terms for the beam and light speed respectively.

Compared to UREBs, the analysis of neutralized beam performance is more complicated, as it depends on more separate factors. While depending on analysis, neutral particle beams typically have a higher effective emittance, ions are much easier to deflect than electrons without generating synchrotron radiation. This allows for the ion beams to be re-circulated in “storage rings”, effectively creating a kind of storage device for particle beams quite akin to a weird sort of momentum wheel. This allows for the buildup of far larger particle bunches with much higher energies, which can then be projected onto targets in an instant, on demand, separated from burst power or continuous power supply capabilities of the firing vessel. One could consider a barrage of bunches built-up and stored over hours, and fired in milliseconds- each pulse significantly more powerful than what any electron beam weapon could easily provide.

=== Solving electrostatic bloom the third way ===
If we aren’t fighting in the depths of space, but are on good ole Terra Firma (or anywhere else with an atmosphere) there is a third way to fight beam bloom that has received a LOT of R&D. In this case rather than use relativity or neutralization to fight the beam bloom, we will use the atmosphere itself! When a high ''current'' (i.e. lots of charges per second) beam is injected into the atmosphere it will turn that atmosphere into plasma (consuming some beam energy). Now a neat thing about plasma is that it always acts to neutralize any electric or magnetic fields that are around and our high current beam has a lot of both! The air will then act to neutralize and (with the right beam parameters and air pressure) further focus the beam! Sadly the physics of this phenomenon doesn’t lend itself to easy analytic equations (damn plasma physics!), but we can say that kiloamp class and MeV particle energy beam pulses are needed for this to work. Your standard CRT tube won’t cut the mustard here!

This technique has been shown to allow for low energy (single to tens of MeV) electron or ion beams to propagate much further through the atmosphere than a simple beam scattering analysis would lead one to believe. This is all thanks to a neat phenomena known as the Bennett pinch that continues to get open and classified research to this very day! During the SDI days high current, 300 MeV electron beams were investigated for killing incoming ballistic missiles and found to be one of the most promising interception methods out there. There have been a variety of experiments conducted at US national labs to show that these beams can fly kilometers through the air while still bringing sufficient energy to the target to cause an obscene amount of damage. In the SDI days the beams were expected to reach 80 km up right to the edge of space! This whole scheme works even better with heavier particles (less loss of beam to scatter in air) so if you can make a really intense proton beam instead of an electron beam it could be worth it.

== Building Weapons ==
That is the introduction out of the way! We know some basic things about the fundamental limitation to particle beams. The good news is: particle beams are alive, menacing and ''they can reach out''. They are not just short-range weapons that work at hundreds of kilometers only. We can build particle beams that keep together in deep space, near and inside an atmosphere, and even pass through an atmospheric interface!

Building a weaponized particle accelerator requires understanding various bits about [[Particle_Accelerators]] in general.

Of some specific interest here are [[Particle_Accelerators#Particles_for_acceleration|the particles we can use]], the two main accelerator types of interest for us, [[Particle_Accelerators#Radiofrequency_Linear_Accelerator|Radiofrequency accelerators]], their [[Particle_Accelerators#Dielectric_Wakefield_Accelerator|advanced derivative]] and the [[Particle_Accelerators#Laser-Plasma_Accelerator|plasma accelerators]] required to implement UREBs, various [[Particle_Accelerators#Other_parts||secondary facilities]] and the two main design shapes of [[Particle_Accelerators#Accelerator_Shapes|LINAC and synchrotron facilities]].

Weaponized particle beams in many ways follow the design considerations and general technologies of particle accelerators. The main divergence is in output parameters and support considerations. Scientific accelerators are aimed at either providing specific imaging of materials through the use of particle interactions or synchrotron radiation on one hand; or the creation of exotic conditions allowing for the probing of fundamental physics. Commercial accelerators are used to image materials, cut and solder, implant material, or other such uses.

The main goal of a weaponized particle beam is the delivery of focused energy into the target material, with at least the goal of releasing ionizing radiation that degrades or destroys sensitive parts. On top, it may deliver sufficient energy for thermal effects. If this energy is delivered rapidly, the thermal effects become explosive.

== Design It Yourself ==

=== Example Builds ===

==Further Reading==
* N. Bloembergen, C. K. N. Patel, P. Avizonis, R. G. Clem, et al. "Report to The American Physical Society of the study group on science and technology of directed energy weapons", Reviews of Modern Physics, Vol 59, Issue 3, 1 July 1987, pp1-201, DOI https://doi.org/10.1103/RevModPhys.59.S1
* Andre Gsponder, "The Physics of high-intensity high-energy Particle Beam Propagation in open Air and outer-space Plasmas", Independent Scientific Research Institute, Oxford, OX4 4YS, England, 11 January 2009, http://arxiv.org/abs/physics/0409157

==Credits==
Sevoris and Gerrit Bruhaug for writing

Gerrit Bruhaug for scientific research, math wrangling and advise

ATriffleMind and Luke W. Campbell for proof reading.

==References==

[[Category:Beams]][[Category:Engineering]][[Category:Military]]

Category:Military

2022-08-26T15:45:41Z

Sevoris: Created page with "Military articles, military technologies, everything else concerning the use of sharpened rocks or angry electrons."

Military articles, military technologies, everything else concerning the use of sharpened rocks or angry electrons.

Laser Weapons

2022-08-26T15:45:05Z

Sevoris:

[[File:Lasers_at_Starfire_Optical_Range.png|thumb|Lasers at the Starfire Optical Range.]]
Lasers are common in science fiction as futuristic weapons. Lasers project light at damaging intensities onto a target. To be effective, this light must be focused through optics (laser beams do not travel as perfect columns! Focusing makes the beam converge to a tight spot on the target). The ability to focus is limited by [[Diffraction|diffraction]] and, in an atmosphere, [[Thermal Blooming|thermal blooming]]; and energy can be lost from the beam due to atmospheric [[Attenuation#Absorption|absorption]], [[Attenuation#Scattering|scattering]], and other phenomena. [[Lasers_and_the_electromagnetic_spectrum|The color (wavelength, frequency, or per-photon energy) of the light you choose can be very important for your ability to get the light to the target]].

Lasers can be expected to be exceptionally accurate. Depending on the power, energy, and intensity you can deliver to the target, they could be limited to relatively shallow surface burns or could drill deeply enough to reach vital equipment or organs deep inside the target. You can defend against lasers with armor or smoke, but mirroring will not work against higher intensity lasers.

Certain colors of lasers can cause unintended blindness to bystanders or friendlies, including all visible colors of light. Use with caution.

==Introduction==

Lasers are remarkable tools, with a wide range of applications across all realms of technology - communications, medicine, remote sensing, computing, data storage - but of course you don’t care about any of that. No, what draws us to places like this is using lasers to <i>blow stuff up</i>! So what is a laser, anyway? What makes it good at blowing stuff up, and why can’t it blow stuff up even better?

When we think about some new technology, we often default to thinking about it the same way as some analogous technology we are already familiar with. Early writers imagined airplanes as giant cruising air battleships rather than jet fighters. Early robots were very often metal humans that could talk and sense and interact with their environment rather than mechanical arms carrying out rote motions in factories or disk-shaped vacuum cleaners. In the same way, we often default to thinking about laser weapons as just like bullet-shooting guns. But they are not. Laser weapons, even laser guns, will be shaped by their own physics and design constraints into machines that optimize their potential and very often this leads to appearance and behavior that will be very different from the bullet guns that we expect.

If you look at the usual definitions of a laser, they will say that a laser is a device that uses stimulated emission to produce monochromatic, unidirectional, coherent light. All of this is mostly correct, but there are exceptions to each of these cases. Real life gets complicated!
<ul>
<li>Stimulated emission is a process where an atom, molecule, or other quantum system in an excited (energized) state can be made to transition to a lower energy state by being hit by a quantum unit (or particle) of light (a photon), and in the process produce a second photon that is identical in every way to the first. Stimulated emission is a useful method to make directed beams of light, but there are machines called lasers that produce laser-like light without stimulated emission.
<li>Monochromatic means the light the laser produces is only one color (or wavelength, frequency, or photon energy … any one of these terms suffices to specify the color). In fact, monochromatic means that the laser is exactly one color - but this is unphysical. Every real phenomena has some finite distribution of colors, even if this distribution is so narrow that it can be hard to detect and can be neglected for many purposes. Most lasers are very close to only one color. But a number of lasers, especially those made for extremely high powered pulses, produce beams with a wide spectral range of colors.
<li>Unidirectional means all the light is going in one direction. As we will see later, this is ultimately not possible. The wave nature of light makes light that seems to originally be going in the same direction eventually diverge. In fact, it is often helpful to have the light coming out of a laser be slightly diverging or converging, and adjust that later with lenses or mirrors.
<li>Coherent (or, for our purposes, transversely coherent - there’s also longitudinal coherence but we don’t need to worry about that here) means that if you look across the beam at any given distance along the beam, all the light will be pointing or wiggling the same way. It is this property that allows lasers to be focused and directed so well. Loss of coherence means your beam can’t focus as well as it potentially could. Even incoherent light can be focused to some extent by forming an image - witness using a magnifying lens to focus sunlight - but coherent light can be focused much better.
</ul>

Fundamentally, a laser is a tool that amplifies light. But even if we could, we wouldn’t want to amplify just any old light. No, we want to amplify light that has only those properties we want. For laser combat, that usually means light that will be focused on the target after going through the appropriate optics.

Technically, what happens is that to make it efficient to amplify light, we put the light we are trying to amplify into a cavity, a structure that allows light of a certain kind to resonate (or bounce back and forth a very large number of times with little loss). The material, substance, or device that amplifies the light is called the gain medium. This gain medium is placed so that the resonating cavity light goes through it, getting amplified with each pass.

The ways that the light can bounce around inside of a cavity are called its modes. A given mode will have only one color, and will have a specific relative intensity and direction profile that depends on the geometry of the cavity. The light in a given mode might be converging (focused) or diverging (defocused), but that can be altered later with optics. Any single mode will be fully coherent. The gain medium will usually only allow certain colors to be amplified, and thus only certain cavity modes can be excited. What we want to do is arrange the cavity so that only the modes that are most useful to us are amplified. When we do this, we can make beams with a minimum divergence that we can focus to destructive power on our enemies.

==The Nature of Light==

Lasers emit light. So to understand lasers we need to understand light. [[Lasers and the electromagnetic spectrum]] goes over some fundamental properties of light, and how the different colors of light impact what your laser can do.

==Focus==

[[File:Airbornelaserturret.jpg|thumb|Boeing YAL-1 Airborne laser beam pointer turret.]]
[[File:1920px-THEL-ACTD.jpg|thumb|Tactical High Energy Laser (THEL) beam pointer.]]
[[File:1920px-Laser_Weapon_System_aboard_USS_Ponce_(AFSB(I)-15)_in_November_2014_(05).jpg|thumb|The laser beam pointer of the AN/SEQ-3 LaWS (Laser Weapons System) aboard the USS <i>Ponce</i>.]]
So, we have a beam of really bright light that’s mostly one color and going mostly in one direction. That’s great. Now what? If you are fighting someone, you probably want to shine that light on them. And you probably want the light to be as intense as possible when it gets to them. However, if the intensity of the laser is too high while it is in your laser generating machinery, it will melt all that expensive machinery that is making the laser. You need the laser to be spread out enough in your laser gun that your laser gun can survive, but intense enough at your target that you can do bad things to it.

The trick is to use a lens or mirror to focus the laser to a tight spot on your target. This lets you use a large, spread out beam in your laser gun, but still blast your target with concentrated energy!

To do this, you need to be able to focus the laser at different distances. Also, you need to know the distance at which the beam needs to be focused. If you get either of these wrong, your target will just get heated up a bit, or get a painful but mostly harmless surface burn.
The first issue can be solved by using motors and moveable or deformable lenses or mirrors. This is similar to the auto-focus technology of modern cameras. The second issue also uses the same technology that cameras use, either by bouncing a ranging laser pulse off whatever the laser is aimed at and seeing how long it takes to return, or by adjusting the focus until whatever is aimed at displays crisp lines.
A consequence of all this is that laser weapons will see their beams directed to some large focal assembly that looks like a large camera lens or telescope mount, rather than just taking the beam that comes directly out of the laser generator.

But you want a big focusing aperture for more than just preventing damage to your beam optics. A wide aperture also lets you better overcome [[Diffraction|diffraction]] to focus better at longer ranges.
In addition, in the air your ability to focus will be limited by [[Thermal Blooming|thermal blooming]], [[Adaptive Optics|twinkle]], and [[Jitter|jitter]]. If your beam is too high powered and you focus too much, you will run into problems with [[Thermal Blooming|thermal blooming]], [[Two-Photon Absorption|two-photon absorption]], [[Stimulated Scattering|stimulated scattering]], and [[Dielectric Breakdown|cascade breakdown]].

==Diffraction==

Diffraction gives the ultimate limit to how well you can focus your laser light. [[Diffraction|This page]] covers the effect in more detail, but the end result is that the larger your focal aperture, the shorter the [[Lasers_and_the_electromagnetic_spectrum|wavelength of the light]] you are using, or the shorter the distance to your target the tighter your focus will be.

==Beam-Atmosphere Interactions==

Getting light through air can be tricky. Sure, we can see through air just fine, so maybe you’d think it wouldn’t be too much of an issue. And sometimes it isn’t! But when the intensity of the light gets really high all sorts of crazy non-linear things can happen. We’ll try to go through the more important of these things so you can get some idea of what the limitations are for your laser and how you might get around them.

If your only concern is blowing up enemy spacecraft with your own spacecraft, you don’t need to worry about atmosphere because there isn’t any. Have fun floating around up there in orbit, though, wistfully gazing down on all those nice juicy targets that you can’t blast with your vacuum-only rated lasers.

<ul>
<li>[[Attenuation]] will remove energy from your laser beam, giving less juice to blow up your target. There are two main forms of attenuation:
<ul>
<li>[[Attenuation#Absorption|Absorption]].
<li>[[Attenuation#Scatter|Scatter]].
</ul>
<li>[[Adaptive_Optics|Twinkle]] can make your beam wander around randomly and lose focus if you don't correct for it.
<li>[[Thermal_Blooming|Thermal blooming]] will make your beam lose focus. It is a non-linear phenomenon so you need to be careful that the steps you take to correct it don't just end up making it worse.
<li>[[Two-Photon_Absorption|Two-photon absorption]] can absorb high power pulses of ultraviolet light when they are tightly focused.
<li>[[Stimulated_Scattering|Stimulated scattering]] turns your laser into a laser with the result that energy is removed from your beam. It is mostly important for high power pulses.
<li> [[Dielectric_Breakdown|Cascade breakdown]] can occur when a beam of longer wavelengths is focused too tightly. It turns the air into a plasma, which absorbs the beam and prevents it from getting to your target.
<li> [[Atmospheric_Hole_Burning|Atmospheric hole burning]] might allow the really high frequency beams that are rapidly absorbed by air a way to get through the air anyway.
<li> [[Filamentation]] is a strange phenomenon where the light in your beam self-focuses. Laser filaments have unexpected behavior – they might cause problems but they might end up being useful.
</ul>

==Accuracy==

Before your beam can do anything, you need to shine it on your target. Fortunately for you (and unfortunately for your target) lasers can be exceptionally accurate.

===Deflection===
Imagine shooting out a ray of light. It goes through an optical obstacle course, consisting of any variety of lenses, prisms, and mirrors until it finally lands somewhere on a screen and makes a bright spot. So here’s a neat trick - if you take the light at the spot and reverse its direction (or perhaps more practically, put a light source at the spot that shoots out a ray backwards along the direction that the original ray came in at), the resulting ray moves back exactly along the path it originally took, retracing the same route through the mirrors and prisms and lenses, so that it ends up at the original source.

Now let’s see what that means for lasers. Suppose you have a laser gun. You see your target in your scope and you line up your scope reticle with the image of the target, then slowly squeeze your trigger. Zap! You send a beam of collimated photonic death at your enemy. But wait! It is a hot day, full of mirages, and the heat distortion is bending the light from your target so that it takes a round-about path to get to you. Your laser was never actually pointing at your target. Surely you must miss, right? Well, no. Because the laser beam takes the same path back to the target as the light from the target took to get to you. As long as you aim at the target’s image, no matter how it was displaced by optical trickery, the beam will arrive at the target.

In this way a laser can be much more accurate at long range than bullet guns, which have to contend with drop due to gravity and deflection by the wind.

===Leading the target===
Laser beams also travel much faster than bullets. Any waterfowl hunter knows they need to lead their target by a bit if they want duck dinner, because it takes the shot a slight amount of time to get there. Laser beams travel so fast that over planetary scales you don’t need to worry about leading your target.

Over vast distances in space, though, even the incredible speed of light gives a slight lag between when the information you are using for targeting is given off by your target, and when the beam reaches your target. If your target is moving on a known trajectory, you can correct for this and most spacecraft gunnery control systems should have this functionality. But this lag does give the target a small amount of time to dodge your beam.

This light lag might never be a problem, even in space, for some kinds of lasers. A visible or near visible light wavelength would need to be focused through an impractically large mirror to cause damage out to even a light second with any reasonably achievable beam power, due to the focusing limits of diffraction. So for lasers like this, lag and dodging will not be a problem. But to get very long range lasers by going farther and farther into the ultraviolet or x-rays, or by making enormous focusing mirrors, may give you beams that can reach many light seconds or even light minutes. At these ranges dodging becomes an issue. Very long range lasers might end up being tools that force an enemy to expend propellant to avoid being hit, eventually resulting in insufficient propellant left to complete their mission.

===Parallax===
The scope on top of a rifle is set maybe 6 cm above the barrel. So if the scope is perfectly aligned with the barrel (it isn’t) and the bullet goes exactly straight (it doesn’t) , when you shoot a rifle you will hit about 6 cm below where you shot at. This is called parallax. Usually, this being within 6 cm is adequate - it’s still close enough for a hunter to put a bullet through a deer’s vitals, or a sniper to drop a terrorist with a shot to the center torso. (As an aside, scopes are usually set up to point somewhat down compared to the barrel, so that the initial rise of the bullet is pulled back down by gravity, and the bullet first rises a bit above the line of aim before coming back down - thus nicely countering one effect by the other and extending the distance over which you don’t have to adjust for bullet drop considerably.)

But what if we could do better? A laser uses optics to focus the beam. The shooter uses optics to aim the beam. So let’s use the same optics for both purposes. This idea is common in cameras, where it is called a single lens reflex camera. A flippable mirror directs light from the lens to an eyepiece, showing exactly what the camera will record when you click the shutter and flip the mirror around so the light goes to the sensor (or film) instead. You can do the same thing with your laser gun, except that now instead of letting the light in to a sensor you let the light out from the laser beam generator. Either way, the laser beam goes directly to where the reticle was pointing, without any parallax at all.

===Jitter===

Sometimes it's not so much about how much the beam itself moves around, but how steady <i>you</i> can hold the beam. [[Jitter|This page]] covers much of the issues with jitter on fixed mounts. Hand-held laser weapons will mostly be limited by the jitter imposed by the person aiming the gun – their breathing, pulse, tiny muscle tremors – all these can throw off a gunner's aim. Fortunately, modern camera makers have come up with ways to correct for that. Electronic auto-stabilization that works on consumer camera optics will also work on laser optics, making it easier to hold your aim rock-steady.

===Recoil===

A gun that shoots bullets will kick back as the bullet is launched. This might be from the pressure of the gas inside the barrel pushing back on the gun. For a science-fictional gauss gun, it might be from the interaction of the induced currents in the bullet with the magnetic field and the currents in the gun’s barrel. But no matter how you do it, Newton’s laws of motion and the conservation of momentum mean that when you shoot the bullet forward, something else has to go backward and almost always that thing that goes backward is the gun.

Recoil has a number of drawbacks. If you don’t have experience shooting and you are holding the gun wrong, it can knock you off balance. If you are shooting really big bullets going really fast, the gun recoiling can hurt or even leave bruises, especially if you are a small framed person. Many shooters start to subconsciously anticipate the kick, and flinch as they pull the trigger which will throw off aim. Most seriously, perhaps, is that when you are shooting rapid fire the recoil makes it hard to control the gun. On fully automatic fire, after the first two or three bullets the rapid recoil usually torques the barrel of the gun up into the air so all you are shooting at is the sky. And for really big guns, like cannons, you need to engineer in shock absorbers so the recoil doesn’t damage the cannon’s mount or throw off the aim of subsequent shots.

Lasers don’t have this problem. Or rather, if they do have this problem you are dealing with such massive overkill that your battlefield will look like nothing we have ever experienced. For any practical laser power, the recoil will be so minuscule that you can’t notice it. Lasers do produce a tiny amount of recoil. The force they give is the power divided by the speed of light; the total momentum given to the gun is the energy divided by the speed of light<ref>Halliday, Resnick, and Walker, “Fundamentals of Physics”, John Wiley & Sons (2005)</ref>. Because light is so incredibly fast unless the power and energy is extraordinarily high the recoil will never be felt. If your laser kicks with the recoil typical of a normal gun, you are throwing energies typical of the detonation of tons of TNT downrange at your enemies with each shot.

==Effect on target==
So now you are shining an intense beam of light on your target. What does it do? Does it just smoulder? Does it burst into flames? Is it drilled through in a shower of sparks? Does it explode messily? [[Beam-Target_Interactions|This page]] will help you find out.

==Defenses==
So we’ve been having fun pretending to blow up bad guys with lasers. But what happens when it’s the bad guy that has the laser? How are we going to protect ourselves from becoming the extra crispy variety of barbeque?

===Smoke===
The first step in not getting shot is not being seen. While stealth methods have long been known and used to prevent the enemy from shooting at you, for lasers not being seen has an additional meaning. Lasers operate by emitting light. If light can’t get between you and your attacker, his beam can’t reach you. It doesn’t matter if the evil emperor's goons have a precise lock on you with their radar. If you are behind a kilometer thick fog bank, their laser guns are not going to reach you.

Now clearly, mere darkness is not going to stop a laser. Even if your foe can’t see you because the lights are out, his beam can still get to you. But if you can put something in the way that will absorb or scatter the light out of the beam, then his beam can’t reach you and you will be safe. This might involve deploying a smoke screen. Or flying above the clouds - or below the clouds if the enemy is in orbit and you are not.

A rough rule of thumb for visible and near-visible colors is that if they can see you through clouds or smoke, they can shoot you with the laser. This is not a strict rule. Human eyes are good at picking out low-contrast or faint patterns, so the smoke might let someone see faintly through it but still attenuate laser beams to near uselessness. But there are factors that also swing the other way, too. A lot of the reason that we have trouble seeing in fog is not just because light is being scattered out of the path between us and the thing we are looking at, making it dimmer; but also because other light is getting scattered into the light path from the sides, which reduces the signal to noise (or contrast, if you prefer). A laser beam will be affected by the first of these, but won’t care one whit about the second (although the second effect will make the whole fog bank light up like a colored flash bomb when the beam turns on). Taken together, this usually means that if the bad guy can’t see you (which, due to the reversibility of light paths means you can’t see the bad guy) he can’t shoot you. If he can barely see you, he can shoot you but with reduced effectiveness. And if he can see you clearly you’ve got no protection from his laser beam.

Very powerful beams might be able to burn through smoke or fog. This is more difficult when the beam is wide (near the shooter) than when it is focused and narrow (near the target) because it will be of lower intensity and will also have to burn more junk out of the way. The upshot is that if you can burn aerosols out of the air, it is harder to shoot out of your own smokescreen than it is to shoot into it.

On the other hand, smoke that absorbs light will make thermal blooming much worse. You might be able to get your beam to your target, but the beam may be so smeared out by thermal distortion that it doesn’t do much.

Of course, you don’t need a cloud of aerosols to block a light beam. A solid barrier can do so as well. A sheet of paper could, in principle, stop the beam. If the beam is still spread out and not focused, it might not have enough intensity to burn through. Even if it can burn through, it might spend enough energy doing so that the effect of the beam is reduced. If the beam is focused on the paper sheet, it will blast through no problem. But then the beam is focused on the paper - not you. When a beam focuses to a point, it then starts to spread out again once it gets past that point. In photography, this is called depth of field. When you focus on something, both the stuff in front of and behind the thing you are focusing on can be blurry. The same effect in lasers is depth of focus, where if a beam focuses on something and shoots through it, it will spread out and become less intense once it passes through its focus point. So if you are hiding in bushes, the laser that your enemy shoots at you might put holes in a few leaves and sticks but if you are far enough into the bushes the beam might not have much of an effect on you by the time it reaches you.

One complication of all this “not being seen” stuff is that it is assuming visible or near-visible light beams. A glass window might let you see through it and even shoot a visible light beam through it, but glass is good at blocking ultraviolet light. If you tried to shoot through the glass with an ultraviolet-B beam, the glass would act like an opaque barrier. Likewise, fog will affect infrared light differently from visible or ultraviolet light. To make things even more complicated, “infrared” is a very broad spectral range, and light at one color of infrared will go through fog quite differently than light at another color of infrared. So when you start getting to off-visible colors, a more useful comparison might be “if your enemy can see you with a sensor that detects the same color of light as the beam he is using, he can shoot you.”

===And mirrors===

So we can defend against laser beams by absorbing or scattering the beam out of the way. What about reflecting it? It seems like everybody’s favorite laser defense is covering themselves with mirrors. But does it work?

The answer is “it depends, but mostly no.” The problem is that mirrors are not perfect. And even if you could somehow make a perfect mirror in the clean room of your multi-million dollar laboratory facility, it will not long remain perfect once it is out in the environment collecting dust and corrosion and micro-cracks and who knows what. The mirrors used for your beam guides and beam pointer telescopes can be sealed away. Mirrors used for armor - not so much.

That can be okay for heat ray type lasers. If you have an outer surface that reflects away 80% of the incident light, that can be enough to keep you from getting a hole melted through your innards, or catching on fire, or something equally bad.

But once the intensity at the point of focus of the laser starts getting more intense than heat ray scales, bad things start happening to the mirror. The small amount of light the mirror absorbs starts to damage the mirror. And if the mirror gets damaged, it doesn’t work as well so it absorbs more light. So it gets even more degraded. And then you don’t really have a mirror any more. This will happen very quickly, once the mirror starts to degrade it is all over. Some very intense pulses can just bypass the reflectivity of a mirror altogether, with electric field strengths high enough to simply rip electrons right off the atoms so that mere linear optical properties like reflectivity mean nothing. The mirrors used to direct and focus the beam work with the beam dispersed over a larger area and are thus protected. If the beam is focused onto a tiny spot, it could damage even the same mirrors used to direct it.

But there is one way that mirrors could potentially help against lasers. Remember that the laser needs to be focused on its target. If the laser beam pointer can’t get the right range to the thing it's pointing at, the laser will focus someplace else. A mirrored surface can potentially confuse most methods of range finding. Send out a ranging laser pulse and see how long it takes to come back? Oops, the beam pointer measures the range to the image reflected in the mirror, not the mirror. Use passive methods of adjusting focus until the lines are sharp? Same story. So if you wear a really shiny suit, your enemy’s lasers just might focus on whatever they see reflected in your mirrored suit instead of you. And then the laser is not tightly focused when it hits you and your mirror works just fine to bounce the beam away to whatever it thinks it should be focusing on.

===Armor===

The time-honored way of protecting yourself from something bad is to put a bunch of stuff between you and it. Usually something that is strong and tough. With the hopes that the bad thing will not be able to get through the strong and tough stuff, give up, and go someplace else.

This works for lasers, too. Armoring is a perfectly reasonable way to keep a laser from blasting your innards into outards.

Against a laser of heat ray intensity, mirroring has already been mentioned. In addition, it will help to use materials with a high melting temperature, high thermal conductivity (to wick heat away from the laser-heated spot), high specific heat capacity (so it can absorb a lot of heat without raising its temperature too much), and preferably something that won’t thermally decompose or catch fire. Active cooling can also help - send a flow of water or other coolant through capillary channels to absorb the heat and transport it somewhere else.

When the laser gets intense enough to start melt ejection, you probably don’t have much time to engage in active cooling. But a high specific heat of fusion (the amount of energy it takes to turn something into a vapor) will help.

For lasers that are so intense that they are blasting out craters or channels in things through the sheer force of their vapor pressure, thermal properties no longer really matter. All the damage is mechanical, so you want the same things that will protect you against bullets and explosive blasts. A high strength-to-weight ratio is good for making it difficult for the pulses to blast very far into the armor. And a low brittleness will help you take multiple hits without your armor shattering.

As a laser gets closer to its target, its spot size gets more tightly focused. And a very small spot size makes the laser more effective at penetrating armor. More so than bullets, a laser’s ability to penetrate armor will fall off with range, and will also increase faster as range decreases. A laser gun shooting a train of blast-pulses with the same overall energy of a bullet will punch a significantly deeper hole through armor at close enough range where the focus can be kept tight.

The main limitation on how much you can increase your armor penetration just by getting closer so you have a smaller spot size and more concentrated beam is that lasers have trouble drilling out holes with too high of an aspect ratio. Plausibly, a laser could drill a 1:20 or even 1:30 aspect ratio hole in a target’s armor, but is likely to stall out around that depth. If you get even closer, you won’t drill deeper. So get to within the range where your laser is giving you its maximum achievable aspect ratio hole, and shoot from there.

===Deflector shield force screens===

Science fiction likes to imagine fantastical technology that projects invisible barriers that stop incoming attacks. Note the "invisible" part. These technological force screens are almost invariably portrayed as letting light go right through them. Logically, this means they won't do much to a laser beam; at least not a visible light laser beam. If the force screen is a completely reflective perfect mirrored surface or totally black sphere of inky onyx midnight or something, protecting from laser beams becomes a lot easier to justify. But then of course the people inside can't see out. So choose your poison.

==Logistics and Energy Supply==

You've supplied all your troops with laser guns. Your tanks have laser cannons, and your orbital spacecraft have giant laser mirrors. But how do you make sure your lasers have enough ammo to keep working?

This is one area where lasers are really expected to stand out from the crowd of other sci-fi weapons, because most of them run on electricity. You're going to be using electricity anyway, to run your computers and lights and communications gear and maybe even your hover tanks and VTOL jets and space-jeeps or whatever else you're using to get around. So you won't need an entirely different logistics leg to keep your lasers in ammo. In addition, electricity is cheap. Even if you have to generate it out in the field using Diesel fuel, it is still much cheaper than bullets. And if you can plug into the grid, it gets cheaper still.

There are sometimes ideas for laser ammo that doesn't just involve plugging their batteries in to recharge. Chemical lasers are one of the biggest culprits. They would require shipments of exotic, reactive, flammable, toxic, and corrosive chemicals with special handling needs. So needless to say, they were abandoned pretty quickly when electrical lasers came along.

Another idea occasionally floated is using an explosively pumped flux compression generator. These generators are a way to make a very high power pulse of electricity, presumably to energize a very high power laser pulse. The way they do this is by detonating a length of high explosive cord inside of an electromagnet. Needless to say, neither the explosive nor the electromagnet survive the process. So you end up with laser "bullets", where one cartridge supplies one shot. Probably accompanied by a very loud bang. Flux compression generators don't have a great specific energy, but they are one way to get very high specific powers. They'll also end up with expensive ammo and extra logistical headaches, so militaries will avoid them if they can get away with rechargeable energy storage.

So what is that electrical energy storage? A number of examples are covered [[Energy_Storage | here]].

==Safety==

===Eye safety===
The cornea, lens, and vitreous humor of the eye are transparent to wavelengths between roughly 0.350 μm and 1.4 μm (the entire visible range, as well as near infrared and part of the ultraviolet-A band). Light that scatters off of your target can be focused by your very own eyes into an intense spot on your retina that can cause burns and permanent injury. Even the light that remains after multiple reflections can be hazardous. High power lasers in this wavelength range can cause instant total and permanent blindness. Anyone working with laser weapons in this range will need to take precautions to protect their eyesight. Not every eye injury will be permanent. Bright lasers can also dazzle, leave you seeing spots, or temporarily blind you. And some injuries may be permanent but only cause partial loss of vision. But this is a major concern anywhere that lasers are used<ref>Proceedings of the U. S. Army Natick Laboratories Flash Blindness Symposium, 8-9 November 1967, John M. Davies and David T. Randolph, Ed.</ref>.

Note that part of the wavelength range of eye-hazardous light is invisible. You might never see the beam that blinds you. But every visible light laser weapon represents a threat to your eyesight.

The incandescent plasma produced by an incident laser beam can also be bright enough to cause eye damage. This is less hazardous than the directly scattered light from the beam, however - similar to the danger involved with directly viewing an electric arc.

Eye damage from indirect exposure to the beam can be avoided by wearing protective glasses that filter out the wavelength of the laser being used. If you want to have a laser gun that can be used without worrying about blinding people without eye protection, you can use short-wave infrared or longer wavelengths (wavelength > 1.4 μm) or the higher registers of the UV-A band or shorter wavelengths (wavelength < 0.35 μm).

===Ionizing radiation===
Some lasers emit beams of ionizing radiation. Ionizing radiation is radiation whose energy-per-photon is high enough to knock electrons off of atoms, creating ions. Free ions in tissue can attack cells, causing cell death, mutations, or cancer. This is the same stuff people worry about when exposed to radioactive elements or nuclear technology. The main wavelengths to worry about are hard x-rays and gamma rays. Light with these wavelengths can penetrate deeply through matter and deposit its energy inside the body. At high enough doses this can lead to increased cancer risk, sickness, or death.

Technically, extreme ultraviolet and soft x-rays are also ionizing radiation. But since they can’t get through air, clothes, or environment suits they are not much to worry about. Even if you do reach you, they will primarily cause radiation burns similar to sunburn on your skin, which is definitely unpleasant and will increase your risk of skin cancer, but not nearly as bad as having all your bone marrow killed so you die slowly of anemia and infection like the deeply penetrating forms of radiation.

But what if you don’t have a laser that emits ionizing radiation? Do you have to miss all the fun? Not necessarily. Absurdly high intensities of non-ionizing radiation can do crazy physics shenanigans that result in the production of some ionizing radiation. These kinds of laser intensities will probably be confined to laboratories, because you don’t actually need situations that extreme to blow a hole in somebody and that kind of intensity in air can actually impede the propagation of your beam in many cases. But who knows? Maybe future research will show that the best way to blast bug eyed aliens is to make uber-powered beams that kinda sorta make some ionizing radiation as a minor side effect. Nothing to worry about, citizen. Move along. Our experts have it all under control.

===Gun safety===
There are certain basic precautions that should be used with any ranged weapon, including laser weapons.
<ul>
<li> Treat every laser weapon as if it were charged and ready to fire. Always keep the aperture pointed in a safe direction.
<li> Keep your finger off the trigger until you are ready to fire. Use your safety, but remember that safeties sometimes fail.
<li> Be sure of your target and what lies behind it before firing.
<li> Be certain nobody around you is at risk of being blinded or irradiated by the laser.
<li> Never use a laser weapon unless you are familiar with how it works.
<li> Be sure the aperture is clear of obstructions.
<li> Never point a laser weapon at anything you do not want to shoot.
<li> Remove the power source of a laser weapon when not in use. Store power supplies and laser weapons separately.
<li> Never use alcohol or drugs before or during shooting.
</ul>

If you are designing a hand-held laser weapon for your science fiction setting, there are a few common-sense safety designs to include in your weapon. Trigger guards, for example. You don't want some stray branch or obstruction to catch on the trigger and shoot a hole in your foot. Safety switches are also a Really Good Idea, to help prevent negligent discharges.

==What does it look like?==

So you're making science fiction, and you want to know what a person at the scene would experience when lasers are around or in use. We'll go over that a bit here. In this discussion, we assume that the lasers under discussion are laser weapons, and intense enough to cause some damaging effect to the target.

===The beam in vacuum===

In vacuum, there is nothing to scatter the light out of the beam and into your eye. So unless the laser is pointed into your eye, you won't see anything. If the laser is pointed into your eye, then in a very short time you won't be seeing anything either.

This might seem disappointing to people who want flashy beams of light flickering between their spacecraft during cinematic space battles, but sadly that's the way it is.

===The beam in air===

If your laser weapon is operating at visible light wavelengths in an atmosphere, it will produce a visually obvious beam due to [[Attenuation#Scatter|scatter]]. Even lasers too weak to be effective weapons (in the few watt range) produce quite visible beams in daylight for Earth-typical conditions. This also holds true for lasers going through other transparent media, like water.

An invisible light beam will, usually, be invisible. If there are a lot of particulates in the air like dust, smoke, or pollen, you might get a glow as the particulates heat up and combust or shine from their heat. Of course, if you are looking at the beam with optics that let you see light of that wavelength, it will be as if the beam was using visible light.

The same atmospheric heating that produces [[Thermal_Blooming|thermal blooming]] will also produce pressure waves that you can hear. Any pulsed beam will make a snap, pop, or crack sound from this heated air. A continuous beam that turns on suddenly will also make a sound like this.

Long wavelength beams that trigger [[Dielectric_Breakdown|cascade breakdown]] will make a glowing ball of plasma where the beam is being absorbed – but if this is happening you are doing it wrong and your laser will not be very effective.

If your beam is powerful enough to cause [[Filamentation|filamentation]], it will appear as a white glow once filametation has been initiated.

A beam of ionizing radiation – x-rays or gamma rays – will produce a blue glow from the ionized air molecules (and Cherenkov radiation in water or other transparent condensed media). If the beam is being used for [[Atmospheric_Hole_Burning|hole burning]], the air plasma in the beam will be blazingly bright like a straight lightning stroke.

And remember – any light that you can see is energy that is not being deposited in your target. To make a beam that efficiently blows up your enemy you incidentally want to make your beam as invisible as possible even disregarding considerations like not giving away your position.

===The point of incidence===

The place where the beam strikes its target (or, if it misses, the point where the beam hits <i>something</i>) is called the point of incidence.

For heat ray type lasers, the point of incidence is likely to get heated to glowing incandescence. If it [[Beam-Target_Interactions#Combustion|combusts]], you will get flames and smoke. If the target is merely being [[Beam-Target_Interactions#Cooking|cooked]], you will still probably get steam and vapors and some smoke coming off. You will hear the crackling of the flame, the sizzling of the flesh, and smell the smoke or odor of cooking meat.

At higher powers, you will see a brilliant flare of plasma and incandescent vapors. [[Beam-Target_Interactions#Melt_ejection|Melt ejection]] will produce a spray of glowing sparks. You will get a roar for a continuous beam cutting through an object. A high powered pulse beam will produce a brilliant plasma flash and explosion; as with any explosion there will be a very loud bang.

Most obvious of all, however, will be visible light beams. A reasonable fraction of the light will be scattered away, this makes a dazzling flare the color of the beam. The intense glare from the point of incidence will make it hard to look at anything in that direction.

===The laser itself===

The actual device that produces the beam will require a large aperture for its focal array (unless you are trying to go for the atmospheric hole burning route). For hand-held models, this will result in something that looks like a cross between a firearm and a camera. There will not be any barrel. Instead, you'll have a big lens. Because it will probably use [[Laser_Weapons#Parallax|single lens reflex sights]], you won't have a separate scope (although you might include the notches of iron sights for when you don't want to use the in-built scope for some reason). But the ergonomics of a pistol grip or rifle-style stock mean that you will still get these design elements on your weapon (for weapons designed for human use, anyway. Aliens might want other ways to grab on to their lasers).
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:BattleLaser.png|frameless]]
<td>[[File:BattleLaser2.png|frameless]]
<td>[[File:raygun4.png|frameless]]
<tr>
<td>[[File:HeavyLaserPistol2.jpg|frameless]]
<td>[[File:MediumLaserPistol.jpg|frameless]]
<tr>
<td colspan=3>Various possible designs of portable laser weapons.
</table>

Larger lasers are likely to have a beam generator that's in a box, probably below decks or deep under the armor of your AFV or buried underground or something. It will send its laser light up through beam pipes using mirror arrays or fiber optics to a beam pointer. The beam pointer will be a turreted telescope with the afore-mentioned large aperture.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Ratel_tank.png|frameless]]
<tr>
<td>This spider tank robot shows a laser beam pointer, as the rotating spotlight-like turret on top of the main turret. The actual laser generator is in the middle of the robot's body.
</table>

==Laser Technologies==
People have invented many different kinds of lasers. [[Kinds_of_lasers|This page]] summarizes some of the more important ones from a weapons perspective.

==Worked Examples==

===Vehicular heat ray===

This is an example of a potential near term laser weapon. It uses a fiber laser to put out 100 kW of 1 μm near infrared light through a 30 cm aperture beam pointer telescope-turret. The entire device has a mass of 400 kg, and is mounted in an all-terrain hybrid Diesel-electric truck. It gets its power from the same generator and battery pack that powers the truck.

The truck and heat ray are meant to provide zone defense for the army units they are deployed with. The laser is used for shooting down incoming missiles, mortar shells, artillery shells, drones, and thin-skinned aircraft, although it can certainly be used to fry personnel, light ground vehicles and sensitive points on light armored vehicles as well - or anything the gunner can see, really. The truck has a really big radar array for detecting airborne threats.
Let’s consider this weapon being used to burn down some incoming mortar shells. The shells are flying toward us at 200 m/s from 1 km away. The shell’s transverse speed is 50 m/s at this moment. The windspeed is 3 m/s, but with the weather, dust kicked up by the vehicles, and battlefield smoke the attenuation and absorption length in the air is about 3 km.

Looking just at the diffraction limit, we might naively expect the laser can focus all 100 kW of its beam down to a mere 4 mm in diameter, delivering intense photonic drilling death in mere moments. But it has all this air to go through. The range is still far less than the attenuation length, so that won’t cause us to lose too much power from our beam. We’re at far too low of a frequency for two photon absorption to happen, our adaptive optics are correcting the twinkle, cascade breakdown and stimulated scattering are not a worry, and our jitter is less than our diffraction limit. But thermal blooming is going to be an issue. This early model heat ray doesn’t have fancy algorithms for predicting and correcting the non-linear effects of thermal blooming. In order to keep runaway blooming from happening, the spot size at the target can only be focused down to 15 cm at best. At this intensity, the beam is burning through the steel casing at about a quarter millimeter per second. It takes four to five seconds for the beam to burn through the case and deflagrate the explosives inside. Because it only takes five seconds for the mortar shells to reach their target, this seems to be a problem. Fortunately, as the shells get closer the beam can be focused tighter. At 800 m, the beam spot focuses down to 10 cm, and now it can burn down a shell in just over a second. With the damage done from the first second of burning, the first of the mortar shells is incinerated after two second. Now at 600 m, the beam switches to the second shell with a 6 cm wide beam. The blistering radiant heat flashes through the second, and third shell in about half a second each. At 400 m, the beam spot is down to a mere 3 cm. The laser is blasting down the shells nearly as soon as the beam touches them. It is limited more by its rate of slew between shells than its dwell time on each shell.

===Man-portable heat ray pack===

Now let’s step a few decades into the future. Laser technology has advanced enough to allow a 10 kW laser to be fit into a 10 kg backpack with a fiber optic cable to a 4 kg beam pointer “rifle” with a 20 cm aperture. The pack is fueled by another 10 kg of high-capacity, high discharge rate lithium-sulfur batteries. With 50% electricity-to-laser-light efficiency, the batteries supply enough juice for 80 seconds of continuous lasing. This laser pack is designed to emit light at 1.5 μm, to reduce the blinding hazard to bystanders.

The heat ray pack is a weapon of the Space Marines, for use in vacuum or when boarding enemy vessels. The low recoil is considered to outweigh the limited range in atmosphere due to thermal blooming - you can keep a 1 mm spot size focus at 70 m in air even with poor air quality, and very few spacecraft corridors are anywhere near that long. In vacuum or clearer air, the heat ray can maintain its 1 mm focus up to 110 m away.

Within that range, this weapon is murderous. In 1/10th of a second, it can cut an unarmored person in half with a slash across their torso, blood and viscera splattering from the wound as explosive vaporization of tissue blasts out a 25 cm deep, 1.3 cm wide swath of destruction. Against hardened steel, sparks of molten metal fly out in an incandescent jet as the narrow beam cuts a hole up to 2 cm deep at up to 2 cm/s. Thinner steel can be cut proportionally faster. Spectra<sup>TM</sup> woven armor, effective against bullets, withers and burns under the beam as it vaporizes a 2 cm deep hole at up to 13 cm/s. Anywhere the beam penetrates leaves grievous wounds. Only hard armor provides much protection - it takes 1/10th of a second to burn a 2 cm deep hole into a rifle plate, and the beam can only cut to 2 cm depth at a rate of 1 cm/s - again, thinner plates can be cut proportionally faster. It is hard to keep a beam focused on a maneuvering target to within 1 cm for a second, so heat pack users usually aim for less armored parts.

===Anti-spacecraft laser===

While the Space Marines are storming contested space habitat units, what is the Space Force doing when facing enemy spacecraft? They use a deep ultraviolet free electron laser. This beast of a weapon emits a time average of 100 MW of 50 nm wavelength light. Because FELs naturally operate in pulse mode, this beam is made up of a rapid-fire train of 10 J, 100 fs pulses emitted at a rate of 10 MHz. With efficient superconducting linacs, energy recovery of the linac pulses, and good resonators and amplifiers, the laser gets 66% efficiency - always useful when heat rejection is such a problem in the vacuum of space. The linacs and laser light generators and other associated equipment have a total mass of 40 metric tons. The 2.5 meter diameter beam pointer telescope and turret mount have a mass of another 10 tons, although the Vanguard-class attack beamstar spacecraft generally mount four beam pointers to give full coverage of all firing arcs with overlapping fields of fire.

As a bonus, FELs are frequency-agile. This laser can tune the wavelength of its beam down into the near ultraviolet, visible or near infrared for through-atmosphere bombardment.

As an example of this laser in action, we’ll look at the Excelsior, a Vanguard-class beamstar in medium orbit around the colony world of Clementine. She’s patrolling with the Wyvern and Desperado, both Excalibur-class missile slingers. The task force has been tracking several sets of inbound craft, presumably alien hostiles first noted by their drive plumes weeks ago.

The aliens are coming in on a hyperbolic trajectory, plotted to pass within a few tens of kilometers of the planet’s surface. Anticipating an Oberth burn at periapsis, the two Excalibur-class vessels spend precious propellant to adjust their orbits to planet-skimming retrograde to the incoming craft. The Excelsior puts herself in a higher overwatch position. As the hostiles approach to within the orbits of Clementine’s two small moons, Wyvern and Desperado loose their first flight of missiles, boosted by the Oberth effect for extra closing speed. At around the same time, multiple high thrust burns are detected coming from the invaders as they release their own swarms of kinetics.

Coming up over the limb of the planet, the enemy missiles come into view of the Excelsior, at a range of 25,000 km or almost two planetary radii. She opens up, sending searing beams of ultraviolet at the incoming swarms. At this distance, the beam has expanded from diffraction to a 60 cm spot. The pulses flash a thin layer of the armored graphite nosecones to plasma, ten million times per second. Gradually, the alien missiles exposed to this hellish radiation erode away, their fronts being ground off at a rate of a few millimeters per second. But seconds are what the Excelsior has - at a closing speed of 20 km/s, it will take ⅔ of an hour for the missiles to cross the gap. Initially, it takes over a minute and a half for the thickly armored nosecones to be ablated away. One by one the missiles lose their armor, and take a flood of hard UV radiation through their innards. The Excelsior’s own beam pointer scopes act as telescopes to assess the damage, watching for the change in spectral lines to indicate a change from carbon plasma to iron and silicon and aluminum and oxygen to tell her she got through the outer shell of the missile to the chewy center. As the missiles get closer, the beam gets more intense. At 12,000 km, the missiles are falling at a rate of one every half minute and at 6,000 km the last missile is burned through in less than eight seconds.

But by this time the enemy spacecraft are behind the limb of the planet. They had waged their own battle with the missiles from the two missile craft, using some kind of radiation beam to fry the missiles’ electronics. Now it was time to wait. The Excelsior pivoted to bring her beam pointers to bear on where she expected the craft to re-appear. The crew wait, fidgeting, watching the telemetry from the Wyvern and Desperado. They release the last of their missiles to skim around the top of the planet, then burn hard for a higher orbit. And sure enough, the missiles meet the enemy coming from the opposite direction. With little time to engage with beams, the enemy runs straight into the incoming munitions. Flashes of plasma and the sparkle of tumbling debris light up the planet’s limb. Only three of the targets made it through the barrage. They turn their radiation beams on the two missile carriers. At this range, the sleeting radiation burns holes in the armor while delivering a lethal dose to the crew.

But as the enemy climb into view, the Excelsior is waiting. At a range of only 300 km, her beams focus to less than a centimeter across. With a beam pointer dedicated to each enemy vessel, beams flicker from one to the next nearly instantly. Her targets flare brightly, the beams first coring their particle emitters, then piercing the reactor coolant lines, and finally cutting gashes in the hull at over 12 m/s. The alien spacecraft come apart, their pieces tumbling away into the void. Over the next several weeks, the people on the colony world below occasionally see meteor trails of infalling debris, a reminder of the heroic Space Force men and women who risked and gave their lives to defend them.

===Infantry pulse laser===
As the Galactic Federation expands ever further into the galaxy, their laser tech increases. Instead of cumbersome heat packs, the glorious Space Marines are now equipped with powerful new pulse lasers. A pulse laser emits a 2.5 kJ pulsed beam with a 1 millisecond duration, for a peak power of 2.5 MW. The beam is tuneable from 1.5 μm short-wave infrared across the entire visible spectrum up to 0.3 μm UV-B. For eye safety, 1.5 μm is the default setting unless longer range is needed.

This laser has a 7 cm aperture. At up to 50 m, it can put its pulse into a 1.25 mm diffraction-limited spot - any tighter and the laser just wastes its energy trying to drill a hole with too high of an aspect ratio. This range can be extended by decreasing the wavelength, in the usual fashion for diffraction, at up to 180 m for a violet beam. Without compensation, the beam would be moderately impacted by thermal blooming in dirty air, but improvements in non-linear control systems implemented in the laser’s adaptive optics allow it to overcome this level of beam disruption. At longer ranges, the beam loses focus, but can still cause lethal damage to an unarmored target at up to 400 m in the short-wave infrared, or 1.5 km in violet. Two photon absorption pretty much kills any ultraviolet beam in air, but in space ultraviolet frequencies can extend the range of maximum effect to 250 m and potentially lethal range to 2 km.

At close focus, the beam can instantly punch a 2.5 cm deep hole through the hardest armor steel. It can flash-drill through 1.2 cm of the nano-diamondoid armor the Space Marine’s foes can be expected to wear. And it can literally explode out a cavity that can punch through up to three people in a row (if they’re all close enough to be within the depth of focus of about 6 m for short-wave infrared or 25 m for violet). At the maximum effective kill range, it will blast out an even wider and messier hole with enough penetration to reach vital organs.

A pulse laser is equipped with a 1 kg ring shaped superconductive magnetic energy storage (SMES) magazine, roughly doughnut shaped if you like doughnuts large enough that you can fit your hand through the hole in the middle. This SMES stores up to 20 MJ of circulating electrical energy, enough for 4000 shots at the 50% efficiency of the laser generator. Space Marines usually don’t need to carry extra magazines. The entire laser gun, magazine included, has a mass of 5 kg.

===Blaster gun===
The venerable pulse laser has seen the Federation Marines through numerous wars. It saw the downfall of the K’mrugh Alliance, the Ghrell Hegemony, and other foes of the Feds. But now a new weapon has come along to replace it.

The Mk. 28 Infantry Micropulse Laser, or blaster as the gunts affectionately call it, emits a train of 100 pulses within a millisecond duration. Spaced 10 μs apart, each pulse has an energy of 35 J and lasts for only 10 ns. At close focus, it has a spot size of 1.5 mm. The pulses, falling one on top of the other, can blast out a sequence of stacked craters up to 7.5 cm deep in armored steel, 3 cm deep into the toughest and strongest nano-carbon armor, and blows right through any reasonable number of people that can fit in its depth of focus. With a similar 7 cm focal aperture to the pulse laser, a blaster has only about a 20% longer range for its close focus. But its effective lethal range extends out to 4 km for short-wave infrared and 15 km for violet light and 20 km for ultraviolet.

In order to keep the beams from dispersing due to stimulated Raman scattering, each shot contains a number of weak “seed” beams tuned specifically to stimulate Raman scattering themselves. But these beams stimulate the scattering in a direction that is still focused, preemptively removing available photons that would otherwise be scattered at random and allowing the beam to keep tight focus much further. The number of additional parallel seed wavelengths ranges from just one for short-wave infrared up to 25 for violet light.

Each shot has a combined energy for the full pulse train of 3.5 kJ. But despite the increased energy, improved efficiency and better SMES energy rings allow the blaster to get up to 6000 shots from a single fully energized magazine. An infantry blaster has a mass of 4.5 kg fully loaded.

==Further Reading==
<ul>
<li>Philip E. Nielsen, “Effects of Directed Energy Weapons”, (2012)
<li>Bob Preston, Dana Johnson, Sean J.A. Edwards, Michael Miller, Calvin Shipbaugh, “Space Weapons Earth Wars”, RAND 2002.
<li>N. Bloembergen et al., “Report to The American Phsyical Society of the study group on science and technology of directed energy weapons”, Reviews of Modern Physics, Vol. 59, No. 3, Part ll, July 1987
</ul>

==Credit==
Author: Luke Campbell
<br>
Helpful comments by Sevoris Doe and Gerrit Bruhaug

==References==

[[Category:Lasers]] [[Category:Beams]][[Category:Engineering]][[Category:Military]]

Category:Engineering

2022-08-26T15:44:18Z

Sevoris: Created page with "All kinds of engineering discussions - how you build things, and what to consider for them."

All kinds of engineering discussions - how you build things, and what to consider for them.

Two-Photon Absorption

2022-08-26T15:43:27Z

Sevoris:

It takes about 6.2 eV of energy to ionize an air molecule. So any laser beam that has an energy-per-photon of 6.2 eV or more will almost immediately be absorbed by air. This corresponds to a wavelength of 0.2 μm.

But if you have a beam of light with a wavelength between 0.2 μm and 0.4 μm, then two photons acting together will have enough energy to ionize a molecule in the air. This is much harder than just ionizing with one photon, but if the beam is very intense it can happen. 0.4 μm happens to be just on the threshold between violet and ultraviolet light, and the 0.2 μm to 0.4 μm range spans the near ultraviolet between vacuum ultraviolet and the visible part of the spectrum, If you have an ultraviolet laser and are trying to focus it to a very intense spot at your target, two photon absorption can end up removing most of the power from your beam before it gets to your target!

If you start with a power of <math>P_0</math> in your beam, emitted from a focusing aperture of diameter <math>D</math> and focused to a spot of size <math>S</math> on the target (which can be as small as the diffraction-limited spot size, but can also be larger), and the target is at a range of <math>R</math>, and if the two-photon absorption coefficient is <math>\alpha_2</math>, the power that hits the target will be <ref name=Prabhakaran_2012>https://www.sciencedirect.com/science/article/pii/B9780444533494002077 P.Prabhakaran, T.D.Kim, and K.S.Lee, in Polymer Science: A Comprehensive Reference, 2012</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
P = \frac{P_0}{1 + (4/\pi) \, (\alpha_2 \, P_0 \, R)/(S \, D)}
</math></div>
So if <math>(4/\pi) \, (\alpha_2 \, P_0 \, R)/(S \, D)</math> is much less than 1, two photon absorption will be negligible. If it is much larger than 1, two photon absorption will mean that most of your beam never gets where you want it to go.

The actual two-photon absorption cross sections of oxygen and nitrogen are difficult to find. However, based on typical two-photon cross sections of other atoms and molecules, the quantity <math>(4/\pi) \, \alpha_2 </math> for sea level air on earth can be expected to be somewhere in the range of <math>10^{-7} \mbox{cm}/\mbox{W}</math> to <math>10^{-10} \mbox{cm}/\mbox{W}</math>. If you are building a real-life ultraviolet laser death ray, you will need to get the correct value for the wavelength you are using. However, for the purposes of fiction just choosing <math>(4/\pi) \, \alpha_2 = 10^{-9} \mbox{cm}/\mbox{W}</math> won’t be too wrong. This <math>(4/\pi) \, \alpha_2 </math> will be proportional to the atmospheric density.

==Credit==
Author: Luke Campbell

==References==

[[Category:Lasers]][[Category:Physics]]

Lasers and the electromagnetic spectrum

2022-08-26T15:42:42Z

Sevoris:

The color of light you choose will have a big effect on what your laser beam can do. Let’s discuss various colors of light, both visible and invisible, to get an overview of what each is good for and what limitations each may have. After all, choosing the right color is the key to really making your laser <i>shine</i>.

==The Nature of Light==
You knew we had to go through this, didn’t you? Lasers are made of light, so we need some understanding of some of the basic physics of light in order to really grok lasers. Light is made when an accelerating charged particle shakes off some of its electric or magnetic field. These bits of free field naturally end up re-making themselves in such a way that they fly off and propagate through space as a wave. All the stuff that you see is actually wiggling bits of electric and magnetic fields continually re-making themselves almost like they were dragging themselves along by their bootstraps.

As you might guess from the name, these wiggling waves zip around at the speed of light. But we need to be a wee bit careful here, because light goes a bit slower through a transparent medium like water or glass or even air than it does through the vacuum of space. In air, light moves so close to the speed of light in vacuum that we can usually ignore the difference. But in transparent condensed materials like water or glass, the light might only be going <math>\frac{2}{3}</math> or <math>\frac{1}{2}</math> or even less of the speed in vacuum.

As a wave, light has a number of properties that can be used to describe it. One is the wavelength - how far it is from the crest of the wave to the crest of the next cycle. This is closely related to the frequency - if you are staying in one spot, how many times does the light wave go up and down in a given amount of time. If you multiply the frequency and wavelength together, you will get the speed of the wave. So with a bit of algebra and knowing that light moves at the speed of light, if you know the wavelength you can find the frequency, and vice versa. When light goes from one material to another, the frequency stays the same but the wavelength changes. Many people characterize light by its wavelength - almost always they are talking about its vacuum wavelength, not the wavelength it has in any particular medium.

The only other physical things that describe a pure wave of light are its direction of travel and the direction the light is vibrating. This latter direction is called the <i>polarization</i> of the light and is always perpendicular to the direction of travel. All other physical properties can be found by knowing the frequency, direction, and polarization.
<ref name=Jackson1975>J. D. Jackson, “Classical Electrodynamics”, John Wiley & Sons, New York (1975)</ref>
<ref name=Griffiths1989>David J. Griffiths, “Introduction to Electrodynamics”, Prentice Hall, Englewood Cliffs, New Jersey (1989)</ref>
<ref name=HRW2005>Halliday, Resnick, and Walker, “Fundamentals of Physics”, John Wiley & Sons (2005)</ref>
<ref name=Giancoli1988>Douglas C. Giancoli, “Physics for Scientists and Engineers, Second Edition”, Prentice Hall, Englewood Cliffs, New Jersey (1988)</ref>

In particular, the color of the light depends on its frequency. High frequency (and thus short wavelength) visible light looks violet, and as we go to lower frequencies (and thus longer wavelengths) the light goes across the rainbow of colors from violet to blue to green to yellow to orange until the lowest frequency light we can see is red. But it doesn’t stop there. There are a great many invisible colors of light that we can’t see, which we will describe below.

One other characteristic commonly used to describe very high frequency light is the energy-per-photon. A photon is the smallest possible amount of vibration possible, as dictated by quantum mechanics. The energy in this minimum amount of electromagnetic radiation is the frequency of the light times a constant of nature called Planck’s constant.
<ref name=Ohanian1990>Hans C. Ohanian, “Principles of Qunatum Mechanics”, Prentice Hall, Englewood Cliffs, New Jersey (1990)</ref>
Again, knowing any one of the frequency, vacuum wavelength, or energy-per-photon lets you find all the others. Because higher frequency light has more energy-per-photon, a pulse of a given energy will have fewer photons in it the higher the frequency of the light. But having a higher energy-per-photon does not mean you automatically get more energy in your beam!

==Radio==
Radio waves are essentially useless for lasers. Sure, they get through the air okay, but [[Diffraction|diffraction]] makes them almost impossible to focus enough to cause damage. So without further ado, we’ll move on.
<table border=1>
<tr><td>Color <td>Frequency <td>Wavelength <td>Energy
<tr><td>Radio <td>30 Hz - 30 MHz <td>10,000 km - 1 m <td>< 1.25 μeV
</table>

==Microwaves==
Microwaves, at least, focus better than radio waves. They still don’t [[Diffraction|focus]] well enough to make practical weapons - at least not if your intent is cooking or burning or blasting your enemy. They are useful at projecting destructive currents into electronic circuits. Such high power microwave devices operating in a counter-electronics role are usually considered a different class of weapon than lasers, so we will leave them for now.
<table border=1>
<tr><td>Color <td>Frequency <td>Wavelength <td>Energy
<tr><td>Microwave <td>30 MHz - 300 GHz <td>1 m - 1 mm <td>1.25 μeV - 1.25 meV
</table>

==Terahertz waves==
Terahertz waves are all the rage these days for remote scanning and a new window for material spectroscopy. Unfortunately, they are [[Attenuation#Absorption|absorbed]] by the air within a few tens of meters, so they are a poor choice for a weapon.
<table border=1>
<tr><td>Color <td>Frequency <td>Wavelength <td>Energy
<tr><td>Terahertz wave <td>300 GHz - 10 THz <td>1 mm - 30 μm <td>1.25 meV - 40 meV
</table>

==Far infrared==
Far infrared is a sort of orphan band of the electromagnetic spectrum, because it is hard for us to get sources and detectors in this range. As a result, we don’t have a lot of experience with what it can do. In principle, your high-tech sci-fi society could be able to make fiercely high power far infrared lasers. But they’re still not a good choice for a weapon because the [[Diffraction|diffraction]] limit makes them hard to focus to damaging intensities without huge focal apertures and they’d have bad issues with [[Dielectric Breakdown|cascade breakdown]] of the air.
<table border=1>
<tr><td>Color <td>Frequency <td>Wavelength <td>Energy
<tr><td>Far infrared <td>10 THz - 20 THz <td>30 μm - 15 μm <td>40 meV - 80 meV
</table>

==Long-wave infrared==
This is the electromagnetic band where most thermal radiation from room-temperature and body-temperature objects occur. When someone is seeing heat with infrared vision, this is the color they are seeing with. The atmosphere is quite [[Attenuation|transparent]] to long-wave infrared radiation. Unfortunately, these long wavelengths are still [[Diffraction|difficult to focus]] at any useful distance, and they tend to cause [[Dielectric Breakdown|cascade breakdown]] in the air when high powers are put into tight focal spots. The main reason they pop up in talking about lasers is that one of the first kinds of high power lasers, and still one of the cheapest and simplest to build, is the carbon dioxide laser which operates in the long-wave infrared.
<table border=1>
<tr><td>Color <td>Frequency <td>Wavelength <td>Energy
<tr><td>Long-wave infrared <td>20 THz - 38 THz <td>30 μm - 15 μm <td>80 meV - 160 meV
</table>

==Mid-wave infrared==
A lot of the mid-wave infrared spectrum is absorbed quickly by air. However, there is a “window” between 3.5 and 4 μm where the light can get through. This window was investigated by early laser weapon designers, using chemical deuterium fluoride lasers. Chemical laser weapons were nasty, toxin-spewing, noisy monstrosities of machines with abysmal beam quality and long logistics chains to supply their highly toxic, corrosive, flammable, and explosive chemicals. And even when they were the only game in town, the deuterium fluoride laser was replaced as soon as they could by chemical oxygen iodine lasers that at least operated in the near infrared and so could be [[Diffraction|focused three times as far]]. Today we have far better choices, so don’t expect mid-wave infrared lasers to get much love.
<table border=1>
<tr><td>Color <td>Frequency <td>Wavelength <td>Energy
<tr><td>Mid-wave infrared <td>38 THz - 100 THz <td>8 μm - 3 μm <td>160 meV - 410 meV
</table>

==Short-wave infrared==
Short-wave infrared is a good choice when you are looking for a color of light that [[Diffraction|focuses well]], [[Attenuation|can get through air]], can maintain a tight focus without [[Two-Photon Absorption|two-photon ionization]] messing it up, and that won’t pose a severe blinding hazard to anyone nearby. It is the shortest wavelength (and thus longest ranged) color of infrared that is eye-safe. You won’t get eye-safe colors again until you are up into the ultraviolet.

Some modern lasers can output high power short-wave infrared beams, primarily fiber lasers.
<table border=1>
<tr><td>Color <td>Frequency <td>Wavelength <td>Energy
<tr><td>Short-wave infrared <td>100 THz - 215 THz <td>3 μm - 1.4 μm <td>410 meV - 900 meV
</table>

==Near infrared==
This is the color that almost all modern combat lasers operate. The air is [[Attenuation|nicely transparent]] to light at this color, the beams [[Diffraction|focus well enough]], and you can get crazy high powers out of fiber lasers these days. Sure - look at the thing they are shooting and you might go blind, but there are bigger hazards in the military.
<table border=1>
<tr><td>Color <td>Frequency <td>Wavelength <td>Energy
<tr><td>Near infrared <td>215 THz - 430 THz <td>1.4 μm - 0.7 μm <td>900 meV - 1.8 eV
</table>

==Visible==
Yeah baby! Now we’re talkin’! Flashing beams lighting up the sky for stunning visual effects, strobing flashes where the beams hit. These are the beams you can really see! Except for the minor detail that if you ever do actually see one in person, there’s a good chance you won’t ever see anything again. But we’ll ignore that for the sake of a good special effects loaded sci-fi extravaganza.

[[Attenuation|The air is very transparent to visible light]]. It turns out that water is also at around its optimal transparency to visible light in the green, blue, and violet colors, so lasers of these colors might be the choice for underwater combat (or at least, underwater combat where you are close enough to see your enemy). If you want maximum range in air without [[Two-Photon Absorption|two photon absorption]] making your life miserable when your beam reaches close focus, visible light is the color choice for you.

Visible light is where you start having significant losses due to [[Attenuation#Scatter|Rayleigh scattering]] for light going all the way through an Earth-like atmosphere. Shorter wavelengths (the closer to violet) are scattered more than longer wavelengths (closer to red). But still, the shorter wavelengths focus better because of diffraction. When shooting straight down through Earth’s atmosphere, you still get the best laser intensity with violet light (even though your beam will have less energy, that energy will still be more concentrated). But as you shoot at more and more of an angle, the best choice goes toward longer and longer wavelengths. And for alien planets, all bets are off.
<table border=1>
<tr><td>Color <td>Frequency <td>Wavelength <td>Energy
<tr style=background-color:red><td>Red <td>430 THz - 480 THz <td>0.7 μm - 0.62 μm <td>1.8 eV - 2 eV
<tr style=background-color:orange><td>Orange <td>480 THz - 510 THz <td>0.62 μm - 0.59 μm <td>2 eV - 2.1 eV
<tr style=background-color:yellow><td>Yellow <td>510 THz - 530 THz <td>0.59 μm - 0.57 μm <td>2.1 eV - 2.2 eV
<tr style=background-color:green><td>Green <td>530 THz - 610 THz <td>0.57 μm - 0.49 μm <td>2.2 eV - 2.5 eV
<tr style=background-color:blue><td>Blue <td>610 THz - 670 THz <td>0.49 μm - 0.45 μm <td>2.5 eV - 2.7 eV
<tr style=background-color:violet><td>Violet <td>670 THz - 750 THz <td>0.45 μm - 0.4 μm <td>2.7 eV - 3.1 eV
</table>

==Near ultraviolet==
This region of the spectrum is made up of invisible colors with shorter wavelengths than we can see that [[Attenuation|can still go through air]]. But although these colors can get through sea level air, ozone in the upper atmosphere does a pretty good job of absorbing ultraviolet light with wavelengths shorter than 0.34 μm. So if you want to use your laser to shoot things on the ground from your spacecraft, choose a wavelength longer than 0.34 μm. Ultraviolet light is scattered more by air than visible light, which makes it more favorable to use visible light for shooting things on Earth from a spacecraft. But on an alien planet, the scales may tip in favor of ultraviolet. And if you are shooting down within the atmosphere, the optimum color depends on the range to the target. And there is also the issue that high powered ultraviolet pulses are likely to be absorbed due to [[Two-Photon Absorption|two-photon absorption]] before they get to their target.

Ultraviolet light is eye-safe at wavelengths shorter than 0.35 μm. Some ultraviolet-A colored light can get through window glass, but ultraviolet-B or C cannot. So if you are shooting your laser gun at someone on the other side of a window, choose ultraviolet-A or longer wavelengths.
<table border=1>
<tr><td>Color <td>Frequency <td>Wavelength <td>Energy
<tr><td>Ultraviolet-A <td>750 THz - 950 THz <td>0.4 μm - 0.315 μm <td>3.1 eV - 3.9 eV
<tr><td>Ultraviolet-B <td>950 THz - 1.1 PHz <td>0.315 μm - 0.28 μm <td>3.9 eV - 4.4 eV
<tr><td>Ultraviolet-C <td>1.1 PHz - 1.5 PHz <td>0.28 μm - 0.2 μm <td>4.4 eV - 6.2 eV
</table>

==Vacuum ultraviolet==
Wavelengths shorter than 0.2 μm can’t go through oxygen. Hence, these wavelengths are called “vacuum” frequencies because [[Attenuation|they can’t go through air]], only vacuum. Other atmospheres may let through slightly shorter wavelengths, but not by much. Anything shorter than 0.1 μm won’t be able to go through any matter (except perhaps a pure helium atmosphere - which can let stuff through down to 0.05 μm).

Still, as much as vacuum ultraviolet can’t be used in air, [[Diffraction|it focuses really well in vacuum]]. If you can make it in your laser, and if you can focus it, it’s a great choice for shooting things in space. In today’s world, our ultraviolet optics are not great and a lot of the light will get absorbed when we want it to be reflected. But a science fiction setting might have solved this, allowing their spaceships to mount devastating long range ultraviolet lasers.
<table border=1>
<tr><td>Color <td>Frequency <td>Wavelength <td>Energy
<tr><td>Vacuum ultraviolet <td>1.5 PHz - 30 PHz <td>0.2 μm - 10 nm <td>6.2 eV - 125 eV
</table>

==Soft x-ray==
The boundary between ultraviolet and x-ray is kind of fuzzy - there is no strict line with ultraviolet on one side and x-rays on the other. The lower energy-per-photon sorts of x-rays act a lot like vacuum ultraviolet. Sure, they [[Diffraction|focus better because of their shorter wavelengths]] but they still get [[Attenuation|almost immediately absorbed by air]]. The best way we can find to focus soft x-rays is by using a complicated grazing incidence mirror.
<table border=1>
<tr><td>Color <td>Frequency <td>Wavelength <td>Energy
<tr><td>Soft x-ray <td>30 PHz - 3 EHz <td>10 nm - 0.1 nm <td>125 eV - 12.5 keV
</table>

==Hard x-rays==
At higher energies-per-photon, x-rays start being able to go some distance through matter before they can be absorbed. There is no hard and fast rule on what is a soft x-ray and what is a hard x-ray. But these little photons can zip through tens of meters of air, or stand a reasonable chance of making it through a person’s body. This still makes them impractical for use as a weapon in an atmosphere, however. But they can really allow your spacecraft to reach out and touch someone if you can somehow figure out a way to make and focus these little guys. Grazing incidence telescopes can work, but become more and more difficult the more energetic the x-rays get.

If you shoot a hard x-ray beam through an enemy spacecraft, expect the people that were in the compartments the beam penetrated to die from radiation poisoning if the blast and heat don’t get them.
<table border=1>
<tr><td>Color <td>Frequency <td>Wavelength <td>Energy
<tr><td>Hard x-ray <td>3 EHz - 30 EHz <td>0.1 nm - 0.01 nm <td>12.5 keV - 125 keV
</table>

==Gamma rays==
The highest energy photons are called gamma rays. As with a lot of this high energy-per-photon light, there isn’t really a sharp distinction between high energy x-rays and low energy gamma rays. Gamma rays can get through several hundred meters of air. But don’t use a gamma ray laser if you are in air - the scattered radiation will come back to give you radiation sickness. Much like with hard x-ray, anyone even near where a gamma ray laser goes through matter will be dosed with dangerous levels of ionizing radiation. Also like hard x-rays, we have no idea how we would focus gamma rays.
<table border=1>
<tr><td>Color <td>Frequency <td>Wavelength <td>Energy
<tr><td>Gamma ray <td>> 30 EHz <td>< 0.01 nm <td>> 125 keV
</table>

==Credit==
Author: Luke Campbell

==References==

[[Category:Lasers]][[Category:Beams]][[Category:Physics]]

Category:Physics

2022-08-26T15:41:43Z

Sevoris: Created page with "Pages related to descriptions of pysics and their discussion."

Pages related to descriptions of pysics and their discussion.

Wormholes

2022-08-21T08:55:57Z

Sevoris: /* Semi-privileged wormhole networks */

Wormholes are hypothetical structures in space-time allowed by the general theory of relativity.
They provide short cuts through space-time that connect one region of space and time to another; potentially, these regions can be very distant from each other.
The symmetries in space-time of our universe mean that wormholes move like physical objects, and acquire the conserved (or in some cases approximately conserved) quantities of things that go into them and lose those quantities of things that come out – properties like energy, mass, momentum, angular momentum, and electric charge.

==What is a wormhole? The geometry of space-time and how to twist it into a pretzel==

We're going to have to start out with some pretty heady stuff. Like the very nature of existence. And what is reality, anyway?
Usually, space and time are so much an integral part of our existence that we don't even think of them at all. They simply form a backdrop on which all the interesting stuff plays out. But when we do think about them, we're so accustomed to living in a world where they exist and where distances are well defined and events can be uniquely identified based on <i>where</i> and <i>when</i> they happened (or are happening or will happen) that a lot of stuff seems so obvious that we don't stop to think about if it really has to be this way or if there are other possible ways of doing things.

===A toy model===

So lets consider a hypothetical scenario, where an ant can be in any of several states, represented by the squares in the pictures below. The ant can transition to any other state that has a numbered side with the same number as the state it is in. We will start on the left. If the ant chooses the "2" transition, we end up with the picture in the middle. If it then chooses the "10" transition, we end up with the state on the right.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Space_elements_1.png|frameless]]
<td width=400>[[File:Space_elements_2.png|frameless]]
<td width=400>[[File:Space_elements_3.png|frameless]]
</table>
But now notice that the behavior of this system is exactly the same as if we re-arrange the states into a grid, as shown below on the left. The disconnected states now become a space; an expanse of coordinates on which the ant can exist. Now, the ant's transition between states can be described as a trajectory as shown in the right picture below.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Space_elements_grid.png|frameless]]
<td width=255>[[File:Space_elements_grid_path.png|frameless]]
</table>

This gives us an inkling of how something like space-time might end up being assembled from individual events. But what if we don't have things match up into quite such a perfect grid? We
ll take our space elements and tile them into an infinite grid. Two sections of the grid are shown below. But now we remove two of those space-tiles, and we connect the edges adjacent to the removed tiles with a corresponding edge of the other removed tile (although, as shown in the figure, we don't have to keep the same orientation). Now the ant, entering side 1 (which we have color-coded red for convenience) comes out of side 1 of the other missing square. This is perhaps the most basic wormhole.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Space_elements_wormhole_grid_1.png|frameless]]
<td width=400>[[File:Space_elements_wormhole_grid_2.png|frameless]]
</table>

Of course, this simple assembly only stitches together a space. But the world we live in is one of space-time. As the ant transitions from one square to another, it is also transitioning further ahead in time. But when the ant passes through the wormhole, while it locally experiences no discontinuity that wormhole might connect to a different time as well as a different region of space. So the ant might emerge far in the past or future of when it stepped through.

===Manifolds and coordinate patches===

The toy model used above hopefully prepares you for some of the ideas to follow - one of how we can piece together a complete space and time by connecting together different pieces. But before we go further, lets define an important term: a <i>manifold</i> is a "shape" of a given number of dimensions where, if you zoom in close enough, you can always get to a scale where it looks like the shape is flat. We'll use this definition to help us describe how we can put space and time together to get a wormhole. But first, let's look at a few illustrative examples.

====One dimension====

The simplest one dimensional shape is a line. And a line is everywhere flat. So a line is a one dimensional manifold.

A one dimensional shape might seem to be a wiggly path if inscribed in two or more dimensions, but it has no <i>intrinsic</i> curvature. That is, if everything about your world is confined to that path, you can't tell that it is curved – all you can do is measure the distance you go along the path. This greatly limits the kinds of shapes that we can consider. So any curved path of infinite extent looks from the inside like a line, and so any infinite length path is also a manifold.

But there's another class of one dimensional shapes that are manifolds – loops. A loop is a path of finite length that connects back with itself. You can make one by drawing a closed curve in a higher dimensional space. Or you can take a straight line but then say that a given point <i>here</i> maps on to another point <i>there</i>, and all the points between <i>here</i> and <i>there</i> form a loop. When you are marching along the line starting at <i>there</i> and you reach <i>here</i> you are also at <i>there</i>, so if you go further you just end up back where you were. This last bit may seem like a pointless bit of pedantry that just confuses a simple problem, but it will be a very useful way of looking at things as we go on.

====Two dimensions====

The simplest two dimensional manifold is a plane. A plane is everywhere flat, and extends off to infinity. A small region around any point on a plane (naturally) locally looks like a plane, so it meets our definition of a manifold.

=====Curved manifolds=====

But in two dimensions, we can have something that we don't have in one dimension – intrinsic curvature. Consider a sphere. If you get close enough to the sphere, it will look like you are on a plane; we live on a world that is (approximately) spherical (if you neglect mountains and ocean basins and all that other stuff), but when you stand up and look around you it mostly looks flat. So a sphere is a manifold. But take a line on the surface of your sphere. Extend it out along the surface, drawing it as straight as possible. Eventually, it will go all the way around the sphere, dividing it in two, and meet itself. This path is called a great circle. The Earth's equator is (approximately) a great circle, as are the lines of longitude (but not lines of latitude, because they do not divide the Earth into two equal-sized halves). The straightest-possible path on a curved surface (like a great circle on a sphere) is called a <i>geodesic</i>. But to keep the jargon down, we'll just call them lines for this non-technical presentation, and hopefully this will keep things more intuitive and understandable to the layman.

In flat space, two lines that are initially parallel remain parallel, always remaining the same distance apart. But in a curved space, initially parallel lines do not remain the same distance apart. On a sphere, for example, any two great circles will intersect at two points. A sphere thus has intrinsic curvature.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Sphere_and_great_circles.png|frameless]]
<td>[[File:Hyperbolic_surface_and_lines.png|frameless]]
<td>[[File:Cylinder_and_lines.png|frameless]]
<tr>
<td width=400>Two great circles going around a sphere. Although initially parallel, they meet at two points.
<td width=400>Two initially parallel lines on a hyperbolic surface. As you get farther from the place where they are parallel, the lines diverge.
<td width=400>Lines that are initially parallel on a cylinder remain parallel everywhere.
</table>

On a sphere, parallel lines always end up converging. One example of a curved surface where parallel lines diverge is a hyperbolic surface. Like a sphere, a hyperbolic surface is a manifold. But it has the opposite sense of curvature.

Now consider a cylinder. Again, a cylinder is a manifold. A cylinder can be made by rolling up a plane. Like our construction of a loop, this can be done by simply taking a long rectangular section of a plane and saying that two of the parallel edges are actually the same (this is essentially what you do if you roll up a sheet of paper so that two of its parallel edges touch, and then pretend that they are merged together where they meet). This "rolling up" preserves distances, so any two parallel lines on the plane remain the same distance apart on the cylinder (think about that sheet of paper, and if you drew two parallel lines on it - they will stay the same distance apart when the paper is rolled up). A cylinder has <i>extrinsic</i> curvature – you can tell that it is curved when looking at it from three dimensions. But it has no <i>intrinsic</i> curvature – if you are confined to the surface of the cylinder it behaves locally as if it is flat.

A manifold that is flat obeys all the rules of Euclidean geometry (although you'll need to go to the three dimensional case for dealing with volumes), and the standard Cartesian coordinate system (representing position by two orthogonal coordinates, commonly called <math>x</math> and <math>y</math>) can be used to specify any point. A manifold that is not flat violates Euclid's fifth postulate; the geometry of the manifold will be <i>non-Euclidean</i> (somewhere, H. P. Lovecraft is quietly having a mental breakdown). It cannot be represented by Cartesian coordinates, but different sets of coordinates – or perhaps patches of multiple coordinate systems that apply in different places – can be defined instead to locate points.

=====Topology: simply and multiply connected manifolds=====
If you take a plane or a sphere, any closed loop on the manifold can be shrunk in a continuous manner until it vanishes. This is called <i>simply connected</i>. But not all manifolds are simply connected. Those which are not simply connected are called <i>multiply connected</i>. The most basic multiply connected manifold is a torus, like a bagel or doughnut or bicycle inner tube. While there are closed loops on a torus that can be shrunk to a point and made to vanish, those that go all the way around the torus tube cannot – they'll get caught up circling the tube. Likewise, loops that go all the way along the tube will get caught up on the central hole if they are shrunk and again cannot be made to go to a point.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:torus_and_loops.png|frameless]]
<td width=400>[[File:Topological_torus_and_loops.png|frameless]]
<tr>
<td colspan=2>Three classes of loops on a torus. On the left is a standard geometrical torus; on the right is a topological torus that is flat but has opposite edges identified so that they are connected. The blue loop doesn't go around the torus, it can be continuously deformed to shrink to a point and vanish. The red line goes around the torus the long way. If you shrink it, it will get caught up going around the hole in the middle and get stuck there; it cannot be continuously deformed to make it any smaller. The green line loops around the torus from the outside to the inside and back. Again, if you shrink it the line gets caught up circling the torus tube and can't be made any smaller.
</table>

And this brings us to another interesting kind of manifold. You can get a manifold that is flat but has the topology of a torus. You can do this by taking a rectangle and saying that the opposite sides are actually the same. This gives you the effect of a "wrap-around screen" like on that old arcade game Asteroids – if you go off the edge on one side, you just come back on the opposite side at the same height. in physics, this is known as "periodic boundary conditions". Closed loops that go all the way around one way or the other cannot be shrunk to a point. But this kind of surface <i>cannot</i> be embedded in three-dimensional space! We might have grown used to surfaces that we can envision existing in the space we live in, but there are manifolds, even fairly simple two-dimensional manifolds, that cannot be embedded in higher-dimensional spaces like that. Remember this when we get to actual wormholes.

You can have multiply connected manifolds that are not toruses, even though they might have the same topology; any manifold with one or more "loops" or "handles" is multiply connected.

=====Non-orientable manifolds=====

And there is one other interesting kind of manifold that is relevant to wormholes – manifolds that are <i>non-orientable</i>. An orientable manifold is one where you can define a consistent sense of clockwise and counter-clockwise everywhere on the manifold. If you start with a clock spinning clockwise, or a right-handed glove, or a copy of Lewis Carrol's "Through the Looking-Glass, and What Alice Found There", you can move it anywhere on the surface and if it meets back up with itself it will still be spinning clockwise, or right handed, or normally readable. But in a non-orientable surface, there are ways you can move the object so that when you get it back to where it originally was the clock is spinning counter-clockwise, or the glove is left-handed, or the book is written in mirror writing. The simplest example of such a surface is a Mobius strip. As shown below, if you move an oriented circle around the strip, it comes back with the opposite orientation.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Topological_Mobius_strip.png|frameless]]
<td width=400>[[File:Mobius_twist.png|frameless]]
<td width=400>[[File:Mobius_strip.png|frameless]]
<tr>
<td colspan=3>A Mobius strip … a strip of surface with a twist so the edges where it connects meet up "the wrong way around". On the right is a 2D strip where you connect along the short edge matching the 1's together and the 2's together. The center shows the same strip with a twist in it so you can line the edges up the way you are supposed to, demonstrating how the twist reverses the orientation of the spinning circles. An embedded image in 3D space with the edges connected together is shown on the left. You cannot define a consistent orientation to this surface – starting at the red counter-clockwise circle, if you move the circle continuously along the surface when you get back where you started you will find it is going clockwise!
</table>

====Three dimensions====

The most basic three-dimensional manifold is a flat space. This is, well, just a normal space. You can go off in any combination of three independent directions (left-right, up-down, front-back), and because it is flat, initially parallel lines stay parallel. It all behaves much as you would expect three dimensional spaces to behave.

But you can get variations on this. They are harder to visualize than two-dimensional cases, because we don't have the brainspace for processing curved spaces inside of a four-dimensional hyperspace. But you can do things like identifying two planes to be the same, such that if you go far enough in one direction you can end back up where you started again (akin to a cylinder), or make them curved such that distances are distorted and lines don't remain parallel and maybe you can put larger things inside of smaller surfaces. There are 3D equivalents of spherical and hyperbolic surfaces (called, naturally enough, spherical and hyperbolic spaces). There are even non-orientable spaces.

One way of trying to envision a curved space, and to gain some intuition about them, is to take a two-dimensional slice of that curved space and embed that in a flat three-dimensional space. This is called an embedding diagram. Remember that this is a reduced representation that throws away a lot of the information (unless there are special symmetries you can exploit) and is not actually the thing being described.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Space_with_embedded_plane.png|frameless]]
<td width=400>[[File:Embedded_plane.png|frameless]]
<td width=400>[[File:Embedding_diagram_example.png|frameless]]
<td width=400>[[File:Embedding_diagram_example_very_curved.png|frameless]]
<tr>
<td>A space with an embedded plane. We show a sphere in this space, with the plane bisecting the sphere.
<td>Just the embedded plane. Note how in a spherically symmetric manifold (like this space), a circle on the embedding diagram represents a complete 2-dimensional spherical surface.
<td>An embedding diagram of a "dimpled" space. Just as before, the closed circles represent closed spherical surfaces. It is just that in this case, the distance between two concentric spheres of radius <math>R_1</math> and <math>R_2</math> is longer than the usual <math>R_1 - R_2</math> because of the spatial curvature, as shown by the increasing distance along the dimple's surface.
<td>An embedding diagram of an extremely curved section of space. In this example, there are several places where we have a bigger sphere inside of a smaller one! Curved space is weird like that.
</table>

Remember that not every manifold can be embedded, so not every curved space can be represented by an embedding diagram. But you can take patches of the curved space and show them by embedding diagrams, and then describe how these patches stitch together to create a contiguous surface.

====Space-time: three space dimensions and one time dimension====

This is the world we live in. So this is the class of manifolds that we really need to use when making a wormhole. Sometimes it's abbreviated as 3+1D (as opposed to 1D, 2D, 3D, etc.).

The basic, infinite, flat space-time manifold is often called Minkowski space. I know, it's a big word using scary foreign-sounding names. But all it means is that its a 3+1D manifold that goes on forever and is flat. In flat space-time, everything obeys the rules of special relativity. Which means it really doesn't work when masses become significant enough to make gravity.

Our best model of how gravity works is general relativity. In this theory, the curvature of space-time is created by mass. At least, that's what they always say in the over-simplified pop-sci summaries of general relativity. Actually, space-time curvature is created by all the components of the stress-energy tensor. It's just that in almost all cases, mass is the most significant part of this. But momentum, stresses, pressures, and shear forces all have contributions as well. In general relativity, physical objects move along the straightest possible lines in this curved space-time. The way the curvature causes the lines to bend and converge is what leads to the appearance of gravity as a force if the curvature is sufficiently weak.<ref name="Misner_Thorne_and_Wheeler">Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler, "Gravitation", W. H. Freemann and Company, New York (1973)</ref>

The most basic curved space-time manifold is the Schwarzschild geometry, which is what you get when you have an uncharged, non-spinning spherical mass doing its distortion thing on space-time. The Schwarzschild geometry is <i>static</i> meaning it does not change in time, and is <i>spherically symmetric</i>, meaning that it has a definite center and it doesn't matter which direction you are relative to that center – it is independent of angle. If the mass becomes dense enough, space-time gets so distorted that an event horizon forms. Nothing that goes into the event horizon can ever come back out, and the inevitability of the geometry crushes the mass down to an infinitely dense point at the center called a singularity. Nothing on the inside of the horizon can ever escape. It forms a <i>black hole</i>, that forever traps any light or matter that enters it. For masses that don't exceed this critical density, the gravitational distortion outside of the object is the same as for a black hole of equal mass, but there is no event horizon. The object can still be seen and interacted with.<ref name="Misner_Thorne_and_Wheeler"></ref>

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Schwarzschild_star.png|frameless]]
<td width=400>[[File:Schwarzschild_black_hole.png|frameless]]
<tr>
<td>An embedding diagram of the Schwarzschild geometry of a dense, massive star. The physical extent of the star is shown in blue.
<td>An embedding diagram of the Schwarzschild geometry of a black hole of the same mass as the star to the left. The event horizon is shown in red.
</table>

There is one distinction between space-time manifolds that will become important later. Some manifolds have an isolated region of high curvature (corresponding to a concentration of mass, or at least of stress-energy) but as you go farther away from the strongly curved region the space-time becomes increasingly flat. If you can always go out far enough away from a curved region that space-time becomes flatter that whatever flatness criterion you choose, no matter what direction you choose to leave the curved region, the manifold is called <i>asymptotically flat</i>. For our purposes, if you can go far enough away that the Newtonian approximation to general relativity is accurate, and gravity can be described as a force rather than requiring a geometric description, then the geometry can be considered asymptotically flat. You can also allow gravitational waves in your asymptotically flat space-time – their very small space-time curvature is not enough to cause problems and they can still be described by a linearized version of gravity. Both the Schwarzschild and the Minkowski geometries are asymptotically flat. Spherical or hyperbolic geometries are not asymptotically flat. Planets, stars, galaxies, neutron stars, black holes, and galactic clusters are all approximately asymptotically flat; but the universe as a whole is <i>not</i>. As long as we confine our attention to merely galactic clusters, we can make the approximation of asymptotic flatness.

===Making a wormhole===

Now that we have covered some of the introductory material, we can get down to the main course. To make a wormhole, take two regions of space-time, usually far apart and usually asymptotically flat (although neither of these is strictly required) and connect them together. That's it. You're done. Sounds easy, doesn't it?

Note that if you are connecting two regions in the same universe, your new geometry will be multiply connected. A loop that goes through the wormhole and then connects back up outside the wormhole cannot be shrunk to a point. There will now be two equivalent ways to get between any two points – the path that goes through your usual space-time, and the path that goes through the wormhole. If you use a wormhole to connect two separate universes, the combination of the two universes and the wormhole can still be simply connected.

In general relativity, you can distort the geometry of space-time with stress-energy, but there is no way to change its topology; that is, no way to turn a simply connected universe into one that is multiply connected, or to add or remove more "loops" or "handles" to an already multiply connected universe. Such changes, if they are possible at all, would only be possible in the realm of quantum gravity and we do not know enough about quantum gravity to know if it would be possible or not. So, it might not be possible to make intra-universal wormholes at all. Or maybe quantum gravity does allow topology changes in space-time and there are ways to connect different parts of our universe together with wormholes. On the plus side, if you can't change the topology of space-time, you can always bud out a new universe from our own – perhaps by triggering the inflationary field that started our universe. It will be connected to ours by a wormhole. And because the topology can't change, that wormhole would never be able to break. We would have an eternal bridge to the new universe. So a system of physics that does not allow topology changes still allows wormholes to other universes.

==Common kinds of wormholes==

===Spherically symmetric wormholes===

General relativity involves a lot of non-trivial math. You need to deal with four-dimensional curvature tensors with up to 20 independent components, and figure out how they relate to a given distribution of matter, fields, energy, stresses, and material flows. If you can reduce the number of dimensions you need to keep track of, it can simplify things a lot. One way of doing this is assuming your space-time geometry does not change with time (it is static), and looks the same in all directions when viewed from a central point (it is spherically symmetric). This makes things a lot easier to analyze.

Astute readers will have noticed that these conditions are satisfied by the Schwarzschild geometry. In fact, if you play around with the Schwarzschild geometry enough, you will find that you can even get a wormhole-like solution. At the event horizon, you can connect your black hole solution to a time-reversed structure called a white hole. This is called an Einstein-Rosen bridge. Unfortunately, it has some issues. For one, it can't form naturally from infalling matter creating a black hole. For another, going through an Einstein-Rosen bridge involves passing through an event horizon, so the trip is necessarily one way. And finally, you can show that even the smallest amount of matter or radiation falling into the Einstein-Rosen bridge leads to it collapsing before the matter or radiation can pass through, trapping it inside the black hole. So this is not looking particularly useful.

So what are we going to do? Well, the Einstein-Rosen bridge, like a lot of early work in general relativity, was made by assuming a distribution of stress-energy (in this case, matter) that was assumed to be physically plausible, and finding out what the resulting geometry would be. But we want a wormhole, dangit! So what happens if we just demand a static tunnel connecting two asymptotically flat regions of space-time? If you do this, you get a Morris-Thorne wormhole<ref name="MorrisThorne">Michael S. Morris and Kip S. Thorne, "Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity", American Journal of Physics <b>56</b>(5), 395-412, May 1988</ref>. You still have the relation between stress-energy and geometry, but now it is your assumed geometry dictating the stress-energy rather than the other way around. One difficulty with this method is that the distribution of stress-energy you end up with might not end up seeming physically plausible. In particular, it can be shown that any wormhole must have regions in it with negative energy density<ref name="Visser_Lorentzian_wormholes">Matt Visser, "Lorentzian Wormholes: From Einstein to Hawking", Springer/AIP Press (1996)</ref>, and these spherically symmetric wormholes are no different. We will discuss the issues around and solutions to the negative energy density problem later.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Morris_Thorne_wormhole.png|frameless]]
<tr>
<td>An embedding diagram of a Morris-Thorne wormhole. For this wormhole, we just took two copies of the Schwarzschild geometry from above and connected the middle "dimples" with a cylindrical patch of space-time.
</table>

The places where you enter a wormhole are called it's <i>mouths</i>, and most wormholes have two. The space-time tunnel between the mouths is called the wormhole <i>throat</i>.

====Some common problems with spherically symmetric wormholes====

if you don't have your distribution of matter and energy going off to infinity, then eventually if you go far enough away your geometry will reduce to the Schwarzschild geometry. A good many published examples of such wormholes are actually Schwarzschild-like up until pretty close to where the event horizon would be. The first problem you might encounter, then, is the formation of a horizon. A single horizon makes any trip one-way. Going into it on the side with the horizon would be indistinguishable from entering a black hole - except that you pop out somewhere else instead of getting removed from existence at the singularity. And because of the horizon, you couldn't ever go back. If both sides have a horizon, you simply can't get out of it. Why do you even have this thing, anyway?

The strong curvature as you get close to the throat can also produce very strong tidal forces. Due to the gravitational forces, if you go in feet-first your feet could be pulled much harder than your head as you go in and you could end up getting pulled apart – spaghettified is the technical term. In addition to these pure spatial tides, it turns out that you also have velocity-dependent tides. These tides are transverse to the radial direction - if you dive in really fast, you will be squeezed even harder from side to size.

Early attempts to limit the tidal forces while still keeping a throat large enough to fit a person through resulted in very large masses. Like planetary scale masses or even solar scale masses. This would mean you would need to spend a lot of propellant de-orbiting into the wormhole and then rocketing your way back out. Some of these early wormholes also had issues like the extreme gravity blue-shifting the cosmic microwave background to blow-torch intensity x-rays. Sometimes, attempts to keep the tides down also lead to really large throat sizes. Like the distance from the Sun to the Earth sizes. Or even light year sizes. These gigantic wormholes would be rather inconvenient.

None of these are necessarily intrinsic limitations to spherical wormholes. But it does indicate that a perfectly reasonable assumption for the wormhole tech in your science fiction universe could involve wormholes with Jupiter-level masses, extreme tides, and other fun details.

===Thin shell wormholes===

Take a shape located somewhere in flat (Minkowski) space-time. The important part of this shape is that it must be a closed surface that completely encloses a given volume. Now go someplace else in space-time and take an identical shape. Remove all of the space-time from inside both shapes, and then say that the corresponding parts of the two surfaces of the shapes are actually the same points. Now you have a thin shell wormhole. The throat region is infinitesimally short – the section that corresponds to the wormhole is just that surface of your shape, hence the name "thin shell".

The first thin shell wormholes proposed were simple polyhedra <ref name="Polyhedral_wormholes">Matt Visser, "Traversable wormholes: Some simple examples", Physical Review D vol. 29 no. 10 pages 3182-3184 (1989)</ref>. These are attractive choices for a potential wormhole-user because the flat faces of a polyhedral thin-shell wormhole have no tidal effects whatsoever. Further, the flat faces do not have any stress-energy either. It is simply flat space-time, no different from any other area of flat space-time. Going through the face of a polyhedral wormhole is no different from going through the flat space-time of the doorway separating your hallway from your bedroom. The negative energy stuff is concentrated entirely in the edges and corners of the polyhedron. And hoo boy, do you need a lot of it. For a cubic wormhole, each edge requires approximately <math>-1.52 \times 10^{43}</math> J/m of length. This is <math>-1.69 \times 10^{26}</math> kg/m, or close to a tenth of a Jupiter mass per meter of edge length. It has been proposed that perhaps cosmic strings – defects of the primordial universe where things couldn't line up the right way and left a linear flaw in reality – with negative tension might be able to meet this requirement and hold open a polyhedral wormhole.

But there's a strange thing when you consider the mass that these wormholes require. They are embedded in flat space-time. Which means that outside of that thin shell they have, they induce no curvature on the space-time around them. Remember that mass causes space-time to curve, dimpling up as in the Schwarzschild geometry shown earlier. Because these wormholes induce no far-flung curvature to imprint on the surrounding space-time, they have zero mass (technically ADM mass). And because mass is equivalent to energy, it should technically be possible to make one of these for no energy input at all from those of us in flat space-time. Forces are the rate of change of energy with distance, pressures are the rate of change of energy with volume. But because the energy is always zero no matter what size they are (as measured from out in flat space-time), there would not be any force or pressure needed to hold them open. The extreme conditions of their interior contrasts with the relatively benign conditions once away from the thin shell (or, indeed, within any flat area within the shell).

A curious feature of polyhedral wormholes is that each face of the polyhedron might connect to a <i>different</i> polyhedron. You might, for example, have a collection of seven cubical wormhole mouths, and each of the six faces of the wormhole you are facing connect to a different one of the six remaining mouths.

For more general thin-shell wormholes, you just need to smear that negative mass out a bit. As long as you keep a flat area to go through, travelers will not experience any strange tides or encounter space-warping exotic energies. You could, for example, have your standard circular portal beloved of fiction. Or you could have any other arbitrary shape you wanted.

Thin shell wormholes are what you get when you take a mathematical limit; they would be an approximation of a more physical distribution of stress and energy. Any realizable version would have a finite, although possible small, thickness to its shell and hence a finite, although possible small, throat length. As we will discuss later, they would likely also have a positive, although possible small, mass. But as an approximation, and as a valid solution to Einstein's field equations in general relativity, they allow us to explore many aspects of wormhole physics using a relatively simple model that is easy to calculate, and can lead to a lot of interesting inspiration for fictional ideas of wormholes.

====Tidal distortion during transit====

Consider a thin shell wormhole whose shell-shape is everywhere convex - no flat areas this time. For convenience, let's make it a sphere. As a traveler passes through the shell, the sections of his body that pass the shell must suddenly "pop" from curving one way to curving the other way, as illustrated below:

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Velocity_dependent_tides_pre_transit.png|frameless]]
<td width=400>[[File:Velocity_dependent_tides_in_transit.png|frameless]]
<tr>
<td>A traveler (left rectangular shape) about to go through a spherical thin-shell wormhole (here represented as circles, with the interior volume blacked out to indicate that there is no space-time there and nothing can exist in that region).
<td>The traveler in the process of passing through the wormhole. Note that on the wormhole shell, the portion of the traveler to the left has a leftward-curvature, while the portion on the right has a rightward curvature. As the traveler passes through, each section that goes through the shell immediately jumps from left-curving to right-curving. This will induce stresses in the traveler as his interior is deformed.
</table>

it is clear that as the curvature of the wormhole becomes large compared to the size of the traveler, the distortion will decrease. In the limit of a flat section of the surface, no distortion is experienced at all.

A consequence of this is that something going through a curved section of a thin shell wormhole will experience stresses (distortions) and strains (forces). The elastic energy stored by a strained object is proportional to the product of the stress and the strain, added up over all of its volume. Assuming that the object going in to the wormhole is initially in its relaxed state, it will require energy to push it into the wormhole shell. If it enters with insufficient initial kinetic energy to get it through or external forces pushing on it, it will just bounce off.

===Cylindrical and toroidal wormholes===

In principle, you can have a wormhole with cylindrical symmetry
<ref>Kirill A. Bronnikov and José P. S. Lemos, "Cylindrical wormholes", arXiv:0902.2360 [gr-qc] 24 Feb 2009</ref>
. This forms a valid solution to Einsteins field equations in general relativity. However, a true cylindrical wormhole would be infinite in length. So they probably don't exist naturally, and you won't be able to build one. Unlike all the other wormholes discussed here, they are not asymptotically flat. Unlike wormholes of finite extent, not all cylindrical wormholes require negative energy regions. However, it has been shown that negative energy stuff is needed if the geometry radially far enough away from the wormhole approaches the flat space-time of our usual universe, so a universe with a cylindrical wormhole without negative energy would not look like our own.

So the thought is that maybe you can get close enough to a cylindrical wormhole by making a really long wormhole that bends ever so slightly and comes back and meets itself, forming an extremely stringy torus. The general case for this has not yet been worked out, but a thin shell approximation of a toroidal wormhole has been developed
<ref>Vladimir Dzhunushaliev, Vladimir Folomeev, Burkhard Kleihaus, and Jutta Kunz, "Thin-shell toroidal wormhole", arXiv:1901.07545v2 [gr-qc] 28 Jan 2019</ref>
.
These thin shell toriodal wormholes have been shown to be stable to perturbations, one nice benefit that has not been established for a lot of other kinds of wormholes. They do require some negative energy regions on their surface, but there will also be some regions that do not contain negative energy.

===Non-orientable wormholes===

In the discussion on non-orientable manifolds, we showed how in these manifolds you can't properly determine the clockwise - counter-clockwise orientation of objects or motions, or the orientation of objects with respect to their mirror reflections. If you move an object along certain paths, when it comes back it might come back the wrong way around. So it is probably no surprise that a non-orientable wormhole turns the universe into a non-orientable manifold. Something going through a non-orientable wormhole comes out looking like it has been mirror reflected.

Normally this seems like it would be rather boring and useless, but for an obscure constraint from quantum physics called the CPT theorem. This theorem, which holds for all physical phenomena, means that the product of the discrete symmetries of charge conjugation (C), parity transformation (P), and time reversal (T) always returns you to your original state. Huh? What does that mean? Well, parity transformation is equivalent to a mirror reflection; so passing through a non-orientable wormhole means parity is inverted. This means that exactly one of the other two symmetries must also be inverted for the CPT theorem to hold. The thing coming out on the other end is still manifestly going forward in time, so it must be charge conjugated, whatever that is.

Charge conjugation means you turn all particles into their antiparticles. So anything passing through a non-orientable wormhole emerges on the other end made entirely out of antimatter.

While this particular turn of events might discourage people from passing through, it does offer obvious application for energy generation, space propulsion, and weaponry.

==Exotic energy conditions==

It is fairly easy to see than any wormhole requires negative energy densities (or more generally, regions where the tension is higher than the energy in natural units) to exist. Take a look at any of the wormhole diagrams above – the spherically symmetric ones will probably make it the most clear. If you send a parallel ray of stuff – maybe particles or light or golf balls – through the wormhole, and trace out the straightest possible path of all of those parallel lines, you will see that after they go through the wormhole all those initially parallel trajectories will now be diverging. Positive energy (such as mass) causes attractive forces, drawing trajectories toward it and making the paths converging. To get diverging paths you need the opposite – repulsive gravity – which you can get from negative energy densities or very high tensions. Having the tension be higher than the energy density turns out to be equivalent to a negative energy density – in some frames of reference observers will measure a negative energy density there.

A lot of physicists used to be reflexively dismissive of negative energy densities. After all, they can't really exist; all energy in the real world is positive.

Except that it is not. There are certain odd cases that allow for negative energy densities. One of these is the Casimir vacuum. A conducting boundary cannot support an electric field parallel to that boundary (if there was such a field, charge would flow under the force of the field until the parallel component of the field is screened out). So if you have two parallel conductive plates, the only electromagnetic waves that you can get between them require their fields to vanish at the plates. This restricts the number of ways the electromagnetic field can vibrate, called modes. In quantum mechanics, merely the potential for a vibration is associated with a certain energy (called <i>zero point energy</i>) even if no quanta of vibration are actually present. By making these modes impossible, they cannot contribute their zero point energy to the region between the plates. This lowers the total energy of the space between the plates compared to the region outside the plates (or with the plates absent). Because the energy of empty space is zero, this means that the region between these Casimir plates has negative energy. The Casimir effect has been measured in the laboratory. It is fairly weak, but it is present. In fact, not long after traversable wormholes were first described, one proposal suggested holding open the wormhole using the Casimir effect<ref name=MirrosThorneYurtsever">Michael S. Morris, Kip S. Thorne, and Ulvi Turtsever, "Wormholes, Time Machines, and the Weak Energy Condition", Physical Review Letters, Volume 61, Number 13, 26 September 1988, pages 1446-1449</ref>

Another situation that gives rise to a negative energy density is a <i>squeezed vacuum</i>. A squeezed state is a way of manipulating a quantum oscillator (like a pulse of light) to get around the uncertainty relations of quantum mechanics to increase the precision in one variable by increasing the uncertainty in its conjugate variable. Applying this operator reduces the energy of the oscillator, and if this is applied to the vacuum state, which already has zero energy, you can end up with regions of negative energy.

It is worth noting that not just any old negative energy region of exotic space-time will work. In particular, dark energy is a condition of the universe we live in where even bare vacuum itself has a small amount of negative energy. The current best model we have for this is as a cosmological constant - where every part of the universe has an equal density of negative energy and a related amount of pressure. But the wormhole allowing contributions to space-time curvature of that negative energy are exactly cancelled by the wormhole-denying contributions of the pressure. So according to our best current models, dark energy will not support a wormhole.

Also, the Casimir vacuum described above is also problematic. The exotic vacuum between the plates does have the necessary properties (if not magnitudes) to support a wormhole, but this is generally far overwhelmed by the immediately adjacent positive mass matter making up the conductive plates. The technical term for the criterion that determines whether you can have a wormhole or not is the Averaged Null Energy Condition (ANEC). If you project a ray through space-time that follows the path that light (in a vacuum) could take, and do some math along that ray and add up all the contributions from all the places along that ray, then if your result is zero or positive then the ANEC is said to be satisfied. If this is the case, you can't get a wormhole. If the ANEC is less than zero, it is said to be violated and in principle you could use that weird exotic energy stuff to prop open a wormhole's throat. There are other so-called energy conditions, and if you have regions of space-time that also violate those energy conditions they can be used to hold open wormholes as well - but all of these also violate the ANEC, so the ANEC is all you need to consider.

===So just how do I get the ANEC-violating stuff I need?===

The short answer is, we don't know.

As we already said, a cosmological constant form of dark energy won't do it (although different ideas about dark energy, such as phantom energy, could conceivably work). The inflationary vacuum thought to have existed very briefly during the first instances of our universe's existence is also usually thought to be a cosmological constant (although one that stopped existing after a very brief period of time, so I guess it is less than constant), so the inflationary vacuum probably won't work either.

If you can get the Casimir vacuum without the massive conducting plates that bound it, you might be able to do something with that. For example, the twisted geometry of space-time itself in a wormhole throat imposes boundary conditions that restrict the modes of vibration inside of it similar to the Casimir effect. This is called the <i>topological Casimir effect</i>, potentially it could be used to help support a wormhole. Also, although the ANEC is satisfied if the probing space-time ray goes through the Casimir plates, a ray that goes between the plates but never touches them will violate the ANEC. Can you use this to hold open a wormhole? Maybe a sufficiently clever person will eventually figure out how to do so.

Even more serendipitous, it turns out that gravity itself naturally squeezes the vacuum
<ref name="gravitationally_squeezed_vacuum">David Hochberg and Thomas W. Kephart, "Lorentzian wormholes from the gravitationally squeezed vacuum", Physics Letters B 268, 377-383 (1991)</ref>
. If you have the curved space-time of gravity, all of the modes of oscillation in it become squeezed states. So it might be possible that the curved space-time of a wormhole can squeeze the vacuum enough to support itself. Or a combination of gravitational vacuum squeezing and the topological Casimir vacuum from the wormhole's geometry constraints might be all that is needed. At least one proposal has suggested a class of wormholes that can be self supporting in such a way
<ref>S. V. Krasnikov, "Toward a Traversable Wormhole", arXiv:gr-qc/0003092v1 22 Mar 2000</ref>
<ref>S. Krasnikov, "Traversible Wormhole", Physical Review D, volume 62, article 084028 (2000)</ref>,
although the throat of these wormholes is very "wrinkly" or "crumpled up" and it is unclear what the effects of that would be on anyone passing through.

====Quantum energy inequalities====

When you apply quantum field theory to what happens when you have a region where the quantum fields produce local negative energy densities, you end up with a limit of how much negative energy stuff you can have before it has to be balanced out by an even greater amount of nearby positive energy stuff
<ref>[https://web.archive.org/web/20060206055950/http://maths.york.ac.uk/www/PhysicsQIneq.htm|Quantum Energy Inequalities]</ref>
<ref>[http://www.phys.lsu.edu/mog/mog20/node16.html|Quantum field theory on curved spacetime at the Erwin Schrödinger Institute]</ref>
<ref name="Fewster_Roman_2005">Christopher J. Fewster and Thomas A. Roman, "On wormholes with arbitrarily small quantities of exotic matter", Physical Review D <b>72</b>, 044023 (2005)</ref>
.

====Matter with negative mass====

It might be tempting to try to solve the negative energy density requirement for wormholes by introducing some sort of matter that naturally has negative mass.
However, negative mass would introduce all kinds of problems to the smooth operation of how the world works. For one thing, if negative mass (or equivalently energy) can exist, why doesn't it just spontaneously pop out of empty space accompanied by an equal magnitude of positive mass (or energy)? For another, negative masses lead to all kinds of runaway instabilities. For example, imagine what happens if you push on an object with a negative mass. You exert a force on it with your hand. But by Newton's second law of motion, an object's acceleration is proportional to its mass. So the negative mass thing will accelerate backwards, against the direction of the force, and into your hand. Which means you will be pushing even harder on it, which makes it push back even harder than before. You have a runaway process that blows up until your hand breaks, and any fragments of flesh and bone and sinew that were in the way of the negative mass thing have been flung off.

Or consider a negative mass thing with an electric charge. If you put something with an electric charge in an electric field the field will exert a force on the charge. For a normal positive mass charged thing, it will move in the direction of the force. The work done is the force in the direction of the displacement in position, so because the force and motion of the object are in the same direction, the field does work on the charged object and energy flows from the field into the kinetic energy of the charged object. But if the charged object has negative mass it will begin to move opposite to the direction of the force. This will do work on the field instead of the object, increasing the energy of the field. The energy comes from the kinetic energy of the negative mass charged object – but because kinetic energy is proportional to mass and increases with increasing speed, as the negative mass object loses kinetic energy it gains speed, going ever faster. Both the field strength and the speed of the object increase without bound. If the electric field originates from the electric part of an electromagnetic wave, it will vary sinusoidally with time and space. As the negative mass object is forced into a sinusoidal trajectory opposing the electric force on it, it will amplify that wave making it ever more intense. A single, isolated charge exposed to an electromagnetic wave will eventually have its inertia keep it moving from the previously imposed force even after the force has switched direction, so if the negative mass is free with no other forces acting on it there will be no net amplification. But if there are any other damping forces, or internal arrangements of bound charge that can resonate with the wave, then instead of being absorbed (as would happen with a positive mass), the wave will be amplified indefinitely and the ever-increasing energy of the wave will come from the ever increasing temperature of the negative mass thing (because the mass is negative, the thermal motion of temperature gives it a negative energy whose magnitude only increases as the temperature rises).

And, of course, the quantum energy inequalities indicate that you can't just have isolated lumps of negative mass stuff floating around.

So just having lumps of stuff that have a negative mass is probably not realistic, and it were possible it would lead to all kinds of universe-shattering cataclysms. Hopefully, we can get wormholes to stay open just using the exotic vacuum states from the Casimir effect and squeezed vacuums.

===Reducing or eliminating ANEC violating stuff===

As soon as people figured out that the wormholes they want needed stuff they couldn't get, they set about trying to find ways to use as little of the stuff they couldn't get as possible. A number of attempts have been made to hammer out a geometry for spherical wormholes that are stable, meet the quantum energy inequalities, and allow a person to pass through them in a reasonable length of time without being shredded by tides
<ref name="Fewster_Roman_2005"></ref>
<ref name="Visser_Kar_Dadhich">Matt Visser, Sayan Kar, and Naresh Dadhich, "Traversable Wormholes with Arbitrarily Small Energy Condition Violations", Physical Review Letters Vol. 90 No. 20 article 201102 (2003)</ref>
<ref name="Kuhfittig">Peter K. F. Kuhfittig, "More on wormholes supported by small amounts of exotic matter", Physical Review D <b>73</b>, 084014 (2006); Peter K. F. Kuhfittig, "Wormholes supported by small amounts of exotic matter: some corrections", arXiv:gr-qc/0508060v1 15 Aug 2005</ref>
.
Often, these resulting structures end up having extreme dimensions, such as throat sizes on the order of light years
<ref name="Fewster_Roman_2005"></ref>
<ref>Remo Garattini, "Generalized Absurdly Benign Traversable Wormholes powered by Casimir Energy", arXic:2008.05901v1 [gr-qc] 12 Aug 2020</ref>

Thin shell wormholes have also been the subject of investigation. One study <ref name="Mazharimousavi_Halilsoy">S. Habib Mazharimousavi, M. Halilsoy, "3 + 1-dimensional thin shell wormhole with deformed throat can be supported by normal matter", Eur. Phys. J. C (2015) 75:271 DOI 10.1140/epjc/s10052-015-3506-6</ref> found a number of thin shell wormholes with sharp-edged shapes could be supported entirely without any exotic energy. However, this only applies if the edges are infinitely sharp. A finite radius of curvature at the edges would require negative energy at those edges.

If the wormhole is taken have the properties of a class of subatomic particles called a fermion, then if the ratio of the electric charge to the mass is sufficiently large you can get a wormhole that can remain open without any negative energy regions at all
<ref name="Blazquez-Salcedo_Knoll_Radu">Jose Luis Blázquez-Salcedo, Christian Knoll, and Eugen Radu, “Transversable wormholes in Einstein-Dirac-Maxwell theory”, Physical Review Letters <b>126</b>, 101102 (2021); ArXiv: 2010.07317v2 [gr-qc].</ref>
.

==Fields and conserved properties==

Our best theories of how the world works – quantum mechanics and general relativity – predict that certain things are [[Conservation_Laws:_Limits_to_Cheating|conserved]]. For asymptotically flat space-times, these quantities are energy, linear momentum (or just momentum), angular momentum, and electric charge. Furthermore, in the non-relativistic limit mass is conserved independently of non-mass energy.

And not only are these things conserved, but they are always conserved <i>locally</i>. This means you can't just get rid of, say, some energy in one place and expect things to balance out by having that energy appear somewhere else.

What does it mean for something to be conserved? Mathematically, it means that the thing obeys the continuity equation. There are various ways of writing the continuity equation, all equivalent, but basically it boils down to the following: if you have a region of space surrounded by a closed surface, then the amount of conserved stuff inside that surface can only increase if some of the stuff enters through the surface and it can only decrease if some of the stuff leaves through the surface. Makes sense, right? If you have three loaves of bread in a room, then if the deliveryman brings in another two loaves you will have five loaves. But unlike loaves of bread, which can be baked or eaten or grow moldy or burn up or be broken into crumbs, the only way to get energy or electric charge or the other conserved things into or out of the room is to have them come into or go out of the room.

Each of these conserved quantities has an associated field. Electric charges create an electric field. Energy and momentum and angular momentum each create a distribution of the curvature tensor of space-time. In the non-relativistic limit, this is the mass part of the energy creating the gravitational field.

There is a relationship between the amount of a conserved quantity in our hypothetical closed surface and the net amount of field that penetrates that surface. This is called [https://en.wikipedia.org/wiki/Gauss%27s_law <i>Gauss's law</i>]. Basically, it says that if you add up all of the amount of field that goes out of the surface, and subtract off all of the field that goes back into the surface, the resulting amount will be directly proportional to the amount of the conserved stuff in there. Do you start off with a kilogram of mass inside your room? Then you have a certain amount of gravitational field from that mass that leaks out of the room. Add another kilo for two kilos total, and the amount of gravitational field leaving is also doubled. The distribution of where that field comes out may change, but as long as you don't take that mass out of the room or add more mass in the room, the total amount of field leaking out won't change. And any gravitational field from masses outside of the room that leaks into the room must also leak back out again <i>somewhere</i>. Those masses outside the room will not affect the net amount of field leaking through the surface. What this means is that you can determine the amount of conserved stuff inside a surface if you know the amount of field going through that surface.

This can be visualized in the field line approximation. The field around a positive charge can be approximated as a series of directed lines that radiate away from the charge. The field around a negative charge in this approximation is a series of lines that converge on the charge (for non-relativistic gravitation, you only have positive charges because mass is always positive. We'll just ignore the fact that wormholes require negative energy density stuff in order to exist which will lead to negative mass – we're just trying to get across the basic idea here). Field lines can only ever start on a positive charge and only ever end on a negative charge. The number of lines radiating away or converging on the charge is proportional to the size of that charge (so if you double the charge, you double the number of lines that connect to it). Field lines propagating through space bend away from a positive charge and bend toward a negative charge.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=322>[[File:charges_positive.png|frameless]]
<td width=322>[[File:charges_negative.png|frameless]]
<td width=322>[[File:charges_dipole.png|frameless]]
<tr>
<td>The field lines radiating away from a positive electric charge.
<td>The field lines converging on a negative electric charge.
<td>The field lines emerging from a positive charge and curving around to go into a negative charge of the same magnitude. This configuration, with a positive and negative charge of the same magnitude, is called a <i>dipole</i>.
</table>

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=322>[[File:charges_positive_Gauss_law.png|frameless]]
<td width=322>[[File:charges_negative_Gauss_law.png|frameless]]
<td width=322>[[File:charges_dipole_Gauss_law.png|frameless]]
<td width=322>[[File:charges_exterior_dipole_Gauss_law.png|frameless]]
<tr>
<td>A closed surface encloses a positive charge. A net amount of field lines exit the surface.
<td>A closed surface encloses a negative charge. A net amount of field lines enter the surface.
<td>A closed surface encloses a dipole. The total amount of charge inside the surface is zero because the negative charge cancels the positive charge. No net amount of field goes through the surface - just as many field lines enter as leave.
<td>A closed surface that has no charge inside of it, but various charges outside of it (in this case a dipole). Again, no net amount of field goes through the surface - every field line that enters comes back out somewhere else.
</table>

Now consider the case of an isolated positive charge, like one one shown above, that is far away from anything else. We will bring up one mouth of a wormhole close to the charge. Because the field lines going out of the charge cannot break, they cannot go through the wormhole. Instead, they will curve around the wormhole mouth. As the charge enters the wormhole mouth and travels into its throat, it drags its field lines along with it. With all of its field lines entering the mouth, that mouth looks like it has a net positive charge. As the charge leaves the other mouth, it continues to drag ints field line out of the wormhole - the field lines that start on the charge now curve around to enter the wormhole mouth. The mouth that the charge exited from now looks like it has a negative charge.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Wormhole_with_charges_initial.png|frameless]]
<td width=400>[[File:Wormhole_with_charges_inside.png|frameless]]
<td width=400>[[File:Wormhole_with_charges_passed_through.png|frameless]]
<td width=400>[[File:Wormhole_with_charges_long_gone.png|frameless]]
<tr>
<td>The charge approaches the wormhole.
<td>The charge is inside the wormhole. The mouth of the wormhole that the charge entered looks like it has a positive charge.
<td>The charge has passed through the wormhole. The mouth that the charge entered still looks like it has a positive charge, but now the mouth that the charge came out of looks like it has a negative charge.
<td>The positive charge has now moved very far away. Both mouths of the wormhole still appear to be charged.
</table>

This can also be understood by the continuity equation. Picture an imaginary closed surface around the mouth of the wormhole. This is like the case of the room, before, except that <i>now there is a way out of the enclosed volume that does not pass through the surface</i> – by going through the wormhole! Any conserved quantity that goes into the surface adds its quantity to the stuff in the surface. If it goes through the wormhole, it does not come out through the surface. The conserved quantity inside that surface stays the same! To an outside observer, it looks like the wormhole mouth has that quantity.

Either way you look at it, the results are the same. A wormhole mouth acquires any conserved quantity of the things that enter it, and loses any conserved quantity of the stuff that exits from it.
<ul>
<li> A wormhole mouth acquires the <i>energy</i> of anything that goes into it. Its energy decreases by the energy of anything that goes out of it.
<li> In the non-relativistic limit, this means that a wormhole mouth acquires the <i>mass</i> of anything that goes into it. Its mass decreases by the mass of anything that goes out of it.
<li> A wormhole mouth acquires the <i>electric charge</i> of anything that goes into it. It gains negative the electric charge of anything that goes out of it.
<li> A wormhole mouth acquires the <i>momentum</i> vector of anything that goes into it. It gains the negative of the momentum vector of anything that goes out of it.
<li> A wormhole mouth acquires the <i>angular momentum</i> vector of anything that goes into it. It gains the negative of the angular momentum vector of anything that goes out of it.
</ul>

This perhaps should not be a surprising result – other general relativistic space-time distortions also exhibit this behavior. A black hole, for example, gains the mass, charge, momentum, and angular momentum of anything that goes into it.

You can use these same arguments to find that the same results apply to any asymptotically flat manifolds of space-time. They do not necessarily hold for space-time geometries that are not asymptotically flat. If the gravitational distortion of a wormhole extends infinitely far away, you will not be able to even define its total energy or momentum or angular momentum, so it will be impossible to say if they are conserved. But do note that all of the wormhole examples that we have discussed so far (except for the cylindrical wormhole of infinite extent) are asymptotically flat, and will obey the local conservation of all of these quantities.

===Wormhole dynamics===

A wormhole mouth has the gravitational field and conserved properties of mass, momentum, and angular momentum. By the equivalence principle of general relativity, this must mean that it behaves as if it had those properties. So a wormhole mouth will follow ballistic trajectories in vacuum through space-time as if it were an object in free fall. If it is around a concentrated mass, like a planet or sun, it will have a Keplerian orbit around that object (or, if the wormhole has more mass than the planet or sun, perhaps it would be more reasonable to say that the planet or sun would orbit it). The mass of the wormhole mouth will resist acceleration like any other mass, according to Newton's laws of motion. Force is the time rate of change of momentum, so as the wormhole mouth's momentum changes it will experience forces, and these will change its trajectory in the same way as if that momentum (or forces) acted on a material object with the same mass. And if the wormhole mouth has an electric charge, it will experience forces from electric and magnetic fields; again, these can change its trajectory.

In other words, a wormhole mouth acts like a physical object of the same mass, charge, energy, momentum, and angular momentum.

The motion of the wormhole mouths through our normal space does not affect their relative separation through the wormhole's throat. That can remain the same length regardless of the behavior of the mouths in our normal universe.

As we covered, if an object enters a wormhole mouth that mouth gains the object's momentum. This will give the wormhole a "kick" changing its motion in the direction that the entering object was originally moving. The dynamics work out the same as if the wormhole were a sticky blob and the object collided and stuck to it. In both cases, you get a final object with the combined masses of both initial things (the mouth and the entering object) that has been knocked off course a bit by the momentum of the entering object hitting it.

If that object comes out of the other mouth, then as that mouth gains the opposite of the objects momentum it gets a recoil kick in the opposite direction. Again, this results in dynamics that are similar to things we already know about – like a gun with a bullet loaded firing the bullet. As the bullet leaves, the gun + bullet system loses the mass of the bullet, and as the bullet shoots away the gun recoils in the opposite direction.

So now consider what happens if you direct a constant stream of material through a wormhole. As the jet of material exits its mouth, that mouth loses mass at the same rate as the mass flow rate of the jet. It also continually gains momentum by the recoil kick of the gas. These dynamics are identical to that of a rocket - as the propellant is ejected, the rocket loses the propellant's mass and the recoil momentum from the escaping gas jet pushes the rocket in the opposite direction. So by shooting stuff through a wormhole you can turn it into a rocket, with dynamics identical to a rocket, obeying the [https://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation Tsiolkovsky rocket equation], and everything else that happens with rockets.

===Does gravity go through a wormhole?===

The above diagrams of the electric field of a charge as it approaches a wormhole mouth also hold for the gravitational field of a mass that is near a wormhole mouth. In the same way that the electric field cannot enter the wormhole's interior or leak through to the other side unless the charge enters the wormhole or goes through to the other side, so to will the gravitational field of a nearby mass curve around the wormhole. The gravity of a nearby planet will fall to zero inside the wormhole throat and will not affect those near the opposite mouth.

===Can waves go through a wormhole?===

So if gravity won't go through a wormhole, and static electric and magnetic fields can't go through a wormhole, then what about light? That's made up of electricity and magnetism, right?

So, turns out that light can go through a wormhole. So can other electromagnetic waves like radio and x-rays. So can gravitational waves. The rules for drawing field lines only really work for static charges; for radiating waves you also need to include the parts about how the changing field generates more field. In the end, waves that are smaller in wavelength than the size of the wormhole can go through just fine. You'll get some [[Diffraction|diffraction]] around the mouths, but for wavelengths much smaller than the mouth size even that might not be noticable.

===Can negative mass wormholes exist?===

So a wormhole mouth loses the mass of anything that comes out of it. What happens if you send something through a wormhole with more mass than the mouth which it exits from? This rule says that the mass must become negative, but as we saw earlier this leads to all kinds of problems (especially because the wormhole can also be charged) and is likely to be forbidden by the laws of nature.

If a wormhole's mouth can reach negative mass, you can also get other interesting instabilities. The positive mass mouth will attract matter into it, which will pass through, increasing the mass of the positive mass mouth and making the mass of the negative mass mouth even more negative. Meanwhile the gravity of the negative mass mouth will repel matter, preventing any return flow. Both the negative and positive masses will gradually increase over time.

So what's an aspiring science fiction writer to do? If you have a particular desire for negative mass things in the fiction you create, go ahead and allow it. Just be aware of the consequences. Otherwise, it is probably easier to say that negative mass wormhole mouths can't exist, and justify that with the quantum energy inequalities. Which, of course, brings up the question of what happens when you try to push an object through a wormhole that would make the mass of one of its mouths go negative? Two obvious possibilities are that either you get a back-reaction force that makes it so you can't ever make something go through the mouth to make its mass negative (perhaps by the formation of a horizon as the mass approaches zero), or the wormhole collapses before the mass can ever reach zero. Perhaps inventive science fiction authors and fans can come up with other plausible options as well.

==Wormholes and time travel==

Wormholes connect across both space and time. So it is natural to consider the possibility of using wormholes for time travel.

Now maybe you don't want to have to deal with time travel in your fiction. It introduces all kinds of opportunities for paradoxes, and you have to keep track of plot lines in a self-consistent way that go back and forth through time, influencing themselves in complicated ways. Okay, no problem. Lets just have our wormholes always connect to the same time. Easy right? In fact, it is natural to justify this. Physics is local, so if you can make wormholes you probably have to create both ends right next to each other at the same time. You can then move the wormhole ends away from each other – maybe putting one end on an interstellar spacecraft or something – but they stay at the same time. Right?

Right?

Except ...

In relativity, motion not only affects how fast you go through space, but also how fast you go through time. When things move, they experience time dilation, so if you move one wormhole rapidly far away and then bring it back, the time dilation it experienced will naturally allow it to form a time machine so you can send things and people back to interfere with their own pasts. Even worse, time can also be slowed down by gravity, so if your wormholes are in areas affected differently by your galaxy's own gravity, or around stars of different mass, or just at different elevations on the same planet, the rate they experience time will be different. Temporal complications ahoy!

Let's look at how this works in more detail. We'll take our standard science fiction empire, called The Empire. Like all empires, it has a central area of authority, the metropole (which we will place on a planet called Metropole), which extracts wealth from various colonies and subjugated client states for its own enrichment in the form of taxes, tribute, and forced favorable trading opportunities. Let's give our Empire two colony worlds which we'll call Colony A and Colony B (these Empire folks sure aren't very creative in their naming, are they?). Just for convenience, we'll put Metropole, Colony A, and Colony B all 100 light years away from each other, in an equilateral triangle.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:WormholesTimeTravel graphic1 wb.svg|600px|frameless]]
<td>[[File:WormholesTimeTravel graphic2 wb.svg|600px|frameless]]
<tr>
<td width=400>Metropole and its two colonies A and B
<td width=400>Metropole sending a wormhole to Colony A.
</table>

So Metropole launches a wormhole mouth out to Colony A. They want to get to Colony A quickly, so they'll make the wormhole go really fast. Let's say they fling it out at 99.9999% the speed of light. At this speed, it takes 100.0001 years for the wormhole mouth to reach its destination in the reference frame of Metropole and Colony A. But due to relativistic time dilation, in the reference frame of the projected wormhole mouth it only takes 0.1414 years to go from Metropole to Colony A. Because the techs at Metropole can look through the wormhole, they will see that 0.1414 years after launch it arrives at its destination. At that point, brave explorers from Metropole can go through and set foot on Colony A – they only need wait about a month and a half to feel alien soil under their feet, rather than a century. In Metropole's reference frame, going through the wormhole takes you 100 light years away, and 99.8596 years into the future. Going the other way, from Colony A to Metropole, takes you 100 light years away and 99.8596 years into the past.

[[File:WormholesTimeTravel graphic3 wb.svg|800px|frameless]]

One might think that these time warps would let you engage in all kinds of time travel. It is easy to see that the Metropole—Colony A situation described here doesn't allow these kinds of shenanigans. For practical purposes, you only have a time machine when you can go back to the place you left at a time before you left. And you can't do that here. Go from Colony A to Metropole and you go back in time 99.8596 years. Go back to Colony A through the wormhole, and you go forward in time the same amount, plus any time you spent on Metropole, so you get back after you left. If you go back through flat space-time, it will always take at least 100 years since you can't go faster than the speed of light so you also get back after you left. No paradoxes for you!

However, it is easy to imagine situations where a wormhole, or a configuration of wormholes, does make a time machine. For example, what happens if we immediately turn around and send the wormhole mouth from Colony A back to Metropole? In the reference frame of Metropole, the wormhole comes back with its "Return to sender" sticker 200.0002 years after it was launched. But going through from the left-behind mouth to the round trip mouth will take you 199.7192 years into the future. And if you step into the round trip mouth, you will go back in time by 199.7192 years into your own past. You now have a real honest to goodness time machine, complete with paradoxes.

[[File:WormholesTimeTravel graphic6´4 wb.svg|800px|frameless]]

Or maybe the wormhole mouth stays at Colony A. And Metropole sends another one to Colony B, that also goes 99.8596 years into the future. Now Colony A sends a wormhole to Colony B. This wormhole also goes 99.8564 years into the future as a consequence of its trip. This means if a traveler at Colony B went through the Colony A wormhole he would go back in time 99.8564 years. Then going from Colony A to Metropole he would go back in time another 99.8564 years. Then he could go from Metropole to Colony B and go forward in time 99.8564 years. The net result is that he ended up back where he started nearly a century before he left.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:WormholesTimeTravel graphic5 wb.svg|600px|frameless]]
<td>[[File:WormholesTimeTravel graphic6 wb.svg|600px|frameless]]
</table>

And it is not just motion that causes time dilation. Being different depths into a gravitational potential does this as well. If Colony A is deeper into the gravity well of the galaxy, or orbits a heavier star, it will experience more gravitational time dilation than Metropole. So eventually it will build up more than 100 years of time slip between the two end of the wormhole and you can have a time machine.

So is time travel inevitable if you have wormholes? What a nuisance!

Well, maybe not. Here's why. Think about what happens when the Colony A – Colony B wormhole has gone just far enough that a light signal going through the wormholes can get back to where it left just as it is leaving. Now, since the propagating signal and the newly transmitted signal are both leaving at the same time, you have double the intensity. So this doubled intensity signal goes around and meets itself again, quadrupling its intensity. And so on. At this point, just as the configuration is on the verge of becoming a time machine, it becomes a perfect resonator for light signals, which then build up to arbitrarily high intensities until something breaks and you don't have an incipient time machine any more.

Now, this won't always work if you just consider light signals. Light can get defocused or sent off in odd directions or something so that you don't have a good path back until after you end up with a time machine. But if you look at what happens in semiclassical quantum gravity, will get a buildup in the amplitude of quantum fluctuations as soon as you start to get a time machine, called <i>vacuum polarization</i>. These fluctuations are expected to build up to sufficient amplitude as to destroy the time machine to be, at least for the few simple cases studied. Or perhaps it just causes the wormholes to bounce away from each other so as to prevent time machine formation. This has not yet been shown to occur in general. But to does seem to happen for all possible configurations of one or two wormholes about to become time machines
<ref name="Visser_Lorentzian_wormholes"></ref>
.

But let us assume for the moment that wormholes cannot form time machines in any configuration due to this vacuum polarization mechanism. Now there are limits on just how you can place and move your wormholes. You don't want to accidentally almost form a time machine and break your transportation network. There are several ways to do this:
<ul>
<li> Move your wormholes slowly, so as not to build up much time dilation (or <i>time lag</i>, if you will).
<li> Don't form closed loops in your wormhole transportation network. The less distance you need to travel to go all the way around a loop to get back where you started, the less time lag is needed to make a time machine.
<li> "Discharge" the time lag by periodically rapidly moving the mouth of a wormhole that is "behind" in time in such a way that it comes back to where it was. Perhaps you can put a charge on it by sending a [[Particle_Beam_Weapons|particle beam]] through it, and then accelerate it up to relativistic speeds in a synchrotron for a while.
<li> Send new wormholes that you are going to place through the existing wormhole network. This way they acquire the same time lag across their mouths as all of the wormholes they traverse. Until they build up some additional time lag of their own, you provide a perfect return trip without making a time machine.
</ul>

In conclusion, if you want a story or setting that involves time travel, you can plausibly still use wormholes somehow to allow that (although it might take loops of several of them). If you want a story that avoids the complications of time travel, you can still plausibly have wormholes in that setting (although you now need to beware of the limitations on your wormhole transportation network imposed by the no time machine rule).

==Practical implications==

===Getting places===

Usually in science fiction, people want wormholes because they want to be able to get to cool places far away without taking forever. Very often, this involves the technology (or the protagonist, with some kind of psychic power) "opening" a wormhole from the current location to the desired location when there wasn't one before.

Sadly, the ability to just create a wormhole to anywhere you want to seems unlikely. Physics tends to be local, which means that if you can make a wormhole both ends will emerge right next to each other. You will need to move the ends to where you want them to be using other methods (mailing them by post, putting them on a spacecraft, etc.). As was mentioned earlier, if you shoot a stream of stuff through a wormhole, the end that the stuff comes out of becomes a rocket. So one way to move an end of a wormhole is to shoot stuff through the wormhole. Because you can leave all the big heavy equipment at home (generators, rocket motors, cooling systems, and everything else), if you can make your wormhole light enough this could be a low cost and efficient method of getting it places. Remember that you can exploit time dilation to make it seem like you can get someplace in much less time than you would expect light to take going to that place (at the expense of taking you some time forward in the future as well when you step through). If you can get your wormhole mass down to a few grams, or even several kilograms, using powerful lasers to shoot through the wormhole to make a photon rocket could be a highly efficient way to go places at relativistic speeds. Perhaps it could even fuel itself by collecting mass from the interstellar medium (ISM) as it flies along, and it might be able to brake against the ISM or solar wind of the destination star system to slow down, reducing mass loss even further.

===Keeping time===

We have already shown that wormholes connect across both space and time, so that a trip between star systems could take you hundreds of years into the future, and the return trip takes you hundreds of years back in time. And this is even before we throw in how time slips between planets when considering relativistic time dilation due to different speeds and gravitational potentials.

Fortunately, all the weirdness of different time rates and going backward and forward in time can be ignored by the average person. This is because you never need to go from one world to another, or back, across the vast gulfs of interstellar space. You just take the wormhole between them. All you ever need to worry about is the coordinate frame that goes across the wormhole. When considering this reference frame, you're not hopping all over the place in time. If it takes ten minutes to cross the wormhole between the two planets, when you get to your destination world the clocks will read ten minutes later than they did when you left your departure world. By coordinating their time-keeping across the wormhole network, all of the worlds of the network can agree on a common time to coordinate their activities. This is all travelers ever need to worry about, and they can then ignore all the relativistic weirdness. Your network engineers will still need to keep track of relative time drift and how close a given configuration is getting to a time loop. But unless your protagonist is a network engineer, they can just ignore all that stuff. And, as an author, so can you! Assume your engineers are competent, you have good regulatory bodies and standards institutions, and don't worry about any of this "time travel" that doesn't actually let you cause paradoxes.

===Keeping wormholes away from planets===

By now, people expect science fiction to have spaceships. Wormholes offer a convenient excuse to have spaceships, because it provides a way for those spaceships to get to other worlds that are not in our solar system. But if you have wormholes, sometimes you need to ask why you still have spaceships? Why not just put one end on Earth and the other on an alien planet and step through to get where you are going?

Sometimes there are good reasons for this. If your wormhole has as much mass as Jupiter, you are not just going to keep it in Topeka, Kansas. That would kind of end the Earth. So you would put it in orbit around the Sun, and send spaceships from Earth out to the wormhole and through it to wherever it goes.

But all too often we see authors using special pleading to pigeonhole in their spaceships without thinking about good physics. Things like "it can't work in atmosphere" (use an airlock), "it can't work within a certain number of planetary radii" (why? There is no physical plausibility to such a restriction), "It can't work in a gravitational field of more than X" (violates the equivalence principle), and so on. But if you insist on having spaceships, and you also insist on wormholes that mass less than a planet, there are a few plausible ways to justify your preconceived ideas that at least plausibly align with physics.

For example, the closest you can plausibly get to "X number of planetary radii" or "gravity field less than X" is a criterion that relies on local measurements of space-time curvature only. And it turns out that this quantity is the tidal forces (curvature is a direct measurement of tides). Tidal forces are going to be proportional to the mass of the nearby thing causing the tides, and inversely proportional to the cube of the distance. If <math>R</math> is a measure of your space-time curvature, <math>M</math> is your object's mass, and <math>r</math> is the distance from the object to the wormhole, then

<div align=center><math>R \propto \frac{M}{r^3}</math>.</div>

So if you want your wormholes to be at least ten planetary radii from Earth (<math>r = 64 \times 10^6</math> m, <math>M = 6 \times 10^{24}</math> kg), then the curvature limit is <math>R = 23</math> kg/m³ in some convenient system of units where we don't have to include constants of proportionality. But now suppose you want to send a <math>26 \times 10^6</math> kg ore freighter through that wormhole. You'll only be able to get your freighter within about 100 meters of the wormhole before the freighter's tides collapse the wormhole. This <i>might</i> work if the freighter were really long and skinny, much longer than 100 meters, to reduce its transverse tides; or if the wormhole was much larger than 100 meters in diameter. But in general, note that the curvature and tides have units of density, so if the average density in any volume around the wormhole exceeds the critical density – oops, there it goes.

Also, having a limit on the geometric curvature of space-time similar to what you get around mere planets doesn't end up being very plausible – the curvatures in the wormhole itself exceed that by so very many orders of magnitude that having so stringent of a curvature tolerance seems very implausible, and just dancing on the edge of disaster. For example, if your wormhole is about 100 meters across, its curvature in these same units will be approximately 500 billion trillion kg/m³. Keeping the tides to within 23 kg/m³ is the equivalent of an engineering tolerance of one part in 25 billion trillion.

So if you don't really have a good way to limit where you can place a wormhole based on the nearby "gravity", are there other ways you can exclude them from the vicinity of planets?

Safety might be one good reason. We have already discussed various ways a wormhole might collapse (forming a time machine, having its mass go negative). What happens when a wormhole collapses? If there are any parts of the wormhole or things in the wormhole that are charged or made of charged particles (like ordinary matter) you could justify a considerable fraction of the wormhole's mass-energy being converted to electromagnetic radiation. In addition, if the wormhole collapses into a black hole it will evaporate away its mass-energy primarily in the form of electromagnetic waves as it undergoes Hawking radiation. Large black holes will last a long time, but if the black hole left behind has a mass of 500 tons or less it will last less than 10 seconds. And 500 tons worth of energy delivered in 10 seconds would do quite a number on a planet. These sorts of considerations could lead to a regulatory environment that require wormholes to be located several light seconds away from inhabited places (like planets).

===Wormholes on planets===

But sometimes you might want to have the wormhole on a planet. Perhaps you are envisioning one of those circular or oval fantasy portals, looking like a window to an exotic foreign land, and when you step through you are actually there. We already know that thin-shell wormholes can be very low mass, and can have a circular shape. So that should work, right?

There are at least two issues to consider here. One is that even if both wormhole ends are on the same planet, natural variations in atmospheric pressure due to weather will lead to very significant differences in air pressure from one side of the wormhole to another (unless the wormholes are so close that they are within a small distance of the extent of the same weather system – say, within a few km of each other). This pressure difference will drive severe winds through the wormhole, usually comparable to the winds of a hurricane (although there will be periods of relative calm when the pressure at both wormholes just happens to be at the same pressure). This effect will only be exacerbated if the wormhole mouths are at different altitudes, due to the decrease in air pressure with altitude. And if the wormhole connects between different worlds, the effect will be much worse. In addition to making travel inconvenient and possibly damaging the equipment, the winds will produce large uncontrollable mass flows across the wormhole (air on earth has a density of about 1 kg/m³, so if you have 50 m/s winds blowing through a 2 meter diameter portal you have a mass flow of about 160 kg/s through your wormhole). This will make it difficult for mass balance, and keeping both ends having high enough masses that neither gets so close to zero that the wormhole breaks or forms a horizon or goes negative mass and causes a runaway planetary catastrophe or whatever wormholes do when one end gets driven to near negative mass by the stuff leaving it.

As a consequence, planetary wormhole mouths will probably be kept in airlocks.

The other issue is that different latitudes on a planet will be rotating at different speeds; at the poles they won't be moving at all while at the equator they will be going a full planetary circumference every day. This means wormhole mouths at different latitudes will be building up different time dilations with respect to each other. In addition, wormholes at different altitudes will experience different gravitational time dilation rates. If you let this go on for long enough, the differences in time across the wormhole mouths will form a time machine. Whether this collapses the wormholes, makes them "bounce" apart from each other (likely ruining a lot of property and infrastructure in the process), or just forms a time loop that leads to various inconvenient paradoxes depends on the assumptions you made for the physics of your world. But it will probably be something you will want to avoid. Re-balancing the time differences between wormhole ends will probably need to be done every few decades.

===Wormholes on spaceships===

The necessity for radiators on spacecraft is really annoying. Dealing with all that waste heat is not fun, and these issues are not awesome kewl stuff like gigawatts and megatons and hundreds of g's of thrust and other things that let you show everyone how much your spaceships totally rule and everyone else's spacecraft totally drool. Instead you need these big fragile stupid-looking things hanging off the side of your ship, ruining your aesthetics and efficiency and being vulnerable to your enemies blowing them up, and lighting you up so bright that you're easily visible from the other side of the solar system. Hey! I know! Let's put a tiny wormhole on our spaceships so we can just send all the heat someplace else. Great idea! That'll work, right?

Actually, yes. If you have small enough wormholes, it will work great. But there are consequences.

So you're running your coolant loops from your reactor through the wormhole to some cooling towers on your home planet. Much better than radiators. But – why do you still have that reactor on your spaceship? It's big and heavy and needs lots of big and heavy shielding and poses nasty radiation hazards. Surely you could leave it behind on your home planet as well, next to those cooling towers, and just run a power line through the wormhole? You'll be able to run a bigger, more powerful reactor that way anyway, and that extra weight won't be bogging your ship down.

And why does your spaceship have its lasers on board? You can have a much bigger laser on the planet that you don't need to lug around. Just shine the laser beam through the wormhole and re-direct it with mirrors.

And all of your missiles? Might as well just feed them through the wormhole when you need to fire them.

And then do you really need all that life support equipment? Just run an HVAC and plumbing that sends fresh air and water to the spacecraft and takes used air and water off. And remember to send the occasional snack, too.

Oh, but wait. Why is all your crew even on the spacecraft again? They have a wormhole to the spaceship. You can just have all your crew run the spacecraft remotely from a mission control station. Most of your equipment is on the planet anyway, so most of the engineering staff is already here. Now you don't actually need to put people in harm's way. If someone does need to fix something just put them in a space suit and send them through, let them do their job, then bring them back. Most of that can be done by a tele-operated robot, in any event. And being able to go back to their families when their shift is over is great for morale.

Hmmm, your sensors could just look through the wormhole instead of having all the vulnerable and heavy and expensive equipment actually on the spacecraft. And as long as you are doing that, you can just leave your laser beam pointer at home too, and aim the laser by pointing it through the wormhole.

So now what's actually left in your actual spacecraft? Oh, the rocket itself. But remember how we already talked about how we could just keep the rocket engine and propellant on the planet and shoot the rocket jet through the wormhole, and it will move just like a rocket? Yeah, so no rocket either.

Congratulations, you've successfully gotten rid of your spaceship. The wormhole itself is your spaceship. It's all you need, it's safer, it's more capable, it's stealthier, and just better all around than some contraption of steel and Mylar and fissioning uranium fuel rods and fragile people going through space.

===Adding mass to wormholes===

So you just made a pair of wormholes. Chances are, you pulled them apart as near Planck mass objects, with about 20 micrograms of mass. Well, that's not very useful! What if you want to send something through that's bigger than 20 micrograms? You're going to need to add mass to your wormhole mouths.

A similar issue comes up if you project a wormhole at some far away planet or star system using the wormhole-rocket trick already discussed, maybe by using powerful lasers to make a photon rocket. To be practical, your projected wormhole mouth will need to be very low mass. But once you get it to where you want to go, you'll need to give the mouth enough mass to let your explorers and colonists and equipment through.

A simple way to do this is to simply suck stuff into the wormhole. This adds mass to the wormhole mouth the stuff comes through. Of course, if you take it out from the other side, that other mouth will lose that much mass, and if it didn't have than much mass to begin with you'll likely have problems. So just leave that stuff in there! Shove it off to the side. Let it clutter up the unused areas out of the main traffic route in the way that basements and attics and the unused spaces of garages accumulate all that old junk that you never use any more but don't want to throw away. Build a wall inside of your wormhole, so travelers don't have to see all the asteroidal rock or super salty water or very surprised natives or whatever it was you scooped up on your alien world. Shove it into the undesirable high curvature places with nasty tides. This wall can also help keep the legitimate passengers away from the inner working of the wormhole, the strange energies that support it, the cables and pipes and crawlspaces that only your maintenance workers should access anyway, and the extreme space-time curvatures that would rip the passengers to shreds if they ever were directly exposed to them.

===Tiny wormholes===

Wormholes that are too small to fit people through could still be useful for sending information. You could use them to rapidly communicate over long distances. If the wormholes were light enough and cheap enough, they might even replace cell phones and laggy land lines with super high bit rate connections.

You could also send energy through tiny wormholes, in the form of light or electric current. Storms that knock down power lines no longer need to cut off your home's electricity. You could also use them to power machinery that need light weight and high power output – electric transportation like airplanes, shipping, or trucking; directed energy weapons; or high specific impulse rockets.

===Causality denial, causality attacks===

We have shown that moving wormholes around can lead to time travel, and that there are plausible physical mechanisms that can prevent time-travel allowing wormhole configurations from happening. People being people, you just know that if this is possible, that it will not be long before these effects are used in politics and warfare.

In a network of wormholes that is about to create a time machine, one likely resolution is that things break until a time machine is no longer possible. In any chain stressed to its point of failure, it is the weakest link that breaks. So in a path through wormholes and normal space-time that is about to form a time machine, it is the weakest wormhole that will collapse (or otherwise have something bad happen to it). So if your rival polity has a wormhole that allows then easy trade with a rich world, and you want to edge in on that trade and simultaneously deny that trade to your rival, you can just send a bigger, stronger wormhole in such a way that forces a time loop. Now your rival's wimpy wormhole breaks, and you get access to the trade opportunities with the world! This is the basics of a causality attack – use the causality-enforcing properties of wormholes to attack your enemy's wormhole infrastructure.

But what if your rival polity anticipated this? What if they have strong wormholes, with an arrangement in space-time that blocks off easy access of your wormholes to a profitable region of space? This is a causality denial action, or a causality fortress. By controlling the time displacement of the wormholes in their network, a polity can control what additional connections can be made. This can be used to enforce borders, prevent infiltration, or restrict civilian wormholes to legitimate uses.

===Moving one wormhole through another wormhole===

If you have a wormhole mouth, you can move it through another wormhole as long as the mouth (and any supporting structures) is small enough to fit through the throat of the wormhole it is going through. The geometry all works out. In fact, this has a rather nice side-effect. If you are looking through the wormhole being moved from back home, you only see it take as long to go through the other wormhole as a wormhole transit usually takes even if it got projected far into the future or past. So the moved wormhole acquires the same time differences as the wormhole it just went through (added on to any time differences it already had, it doesn't get reset or anything). This can let you easily build wormhole networks that connect to themselves in loops. It also makes the network more susceptible to breaking, because there is much less leeway for error with a closed loop as far as making a time machine and breaking the loop. So it can be done, but there are consequences.

===Dropping a wormhole into a black hole===

In principle, if you have a wormhole connection to the inside of a black hole, you can get information out from inside the event horizon. The theorem that prevents things from escaping a black hole was made under the assumption that the energy can't go negative. And as we've already seen, wormholes require regions of negative energy to work. You still wouldn't be able to see further toward the singularity at the center than the wormhole's location. And the wormhole would be inevitably dragged into the singularity in a finite amount of time. But, in principle, during that time you could see what is happening in the black hole.

But there's a big catch with this. You can't actually get the wormhole into the black hole. Not if you assume that time machines can't form, anyway. From the point of view of the wormhole mouth falling into the black hole, it will pass through the event horizon in a finite amount of time. From the point of view of someone watching from outside, the infalling mouth gets more and more time dilated to the point that it never crosses the horizon. This time dilation sets up the conditions for a time machine, and any time machine preventing physics happens on the wormhole before it can ever get in.

== Possible Networks ==
=== Acyclic wormhole networks ===

Recall that wormholes are space-time connections. They bridge space as well as time. Assuming we want to build this network in a stable manner, we want to avoid Closed Timelike Curves from appearing. Let us start with some simple anaylsis of consequences on this. Assume projected wormholes have one mouth sent at relativistic speeds to a distant target and the other mouth is kept at rest at home. From the perspective of the dispatching party, this will lead to the new connection opening very quickly. Done again and again, large volumes of space can be accessed in a short time, a powerful advantage.<br />

[[File:Wormholes PartialOrder.svg|400px|thumb|The partial order relationship of the wormhole connection]]

Every projected connection thus extends onto the future. Then we must avoid forming loops, or more generally "backtracking" back towards a dispatched node. This nets us a graph with no closed loops - an acyclical graph, also called a tree. Since connections may not return to previously established nodes if they are time-shifting from the past into the future, we also get a partial order: nodes <math display="inline">u</math> that dispatch wormholes have a higher order over the nodes <math display="inline">v</math> they establish: <math display="inline">u < v</math>. This lets us establish a direction on the whole graph. For a particular node <math display="inline">n</math>, its '''parent node''' established it, <math display="inline">parent(n) < n</math>, and it in turn establishes '''children nodes''', <math display="inline">n < child(n)</math>. The node that only has children nodes is the tree's '''root node'''. Nodes that have no children are '''leafs''' of the tree. The trees '''height''' is the longest outward path one can take from the root to a leaf.<br />

In terms of traversal, such nodes have some interesting and useful properties. Since wormhole traversals may be assumed to be effectively instantaneously bridging large distances, travel time is mostly defined by the travel distance between wormhole mouths. And as such, we want to minimize the amount of transits we have to make to get from a node <math display="inline">A</math> to a node <math display="inline">B</math> the fastest. It turns out, in order to achieve this, we want to have as large a choice of connections at each node. In the ideal case, any node except root can be reached in two transits: one transit to root, and from there to the other node. In the less ideal case we have to make one further stop-over, choosing from a number of connections at a second node to reach our destinations. The less ideal case from that sees us make two transits through such "switch yards". Assuming each node except the leafs in such a wormhole network has exactly <math display="inline">k</math> connections and there are <math display="inline">n</math> nodes in the whole network, we can reach any destination from root in <math display="inline">\log_k (n)</math> choices of wormhole transit. This is a powerful way to get to connections quickly through centralization. (In computer science, such k-nary trees are of great interest for quickly search-able data structures for this reason.) Configurations that are less "balanced" net appropriately lesser advantages and longer transit times.<br />

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Wormholes AcyclicGraphExample.svg |600px|frameless]]
<td>[[File:Wormholes Trees.svg|600px|frameless]]
<tr>
<td width=400>An example of an acyclic wormhole network in space-time.
<td width=400>The connected tree graph of the network exampke, and a k-nary tree example.
</table>

Also, traffic to most other nodes ''has'' to go through a root-wards node. This gives these nodes significant presence in the larger network. Routing through them is unavoidable if you want to reach most of the rest of the network. Root nodes can profit from taxation, services to travelers, control of routing infrastructure for information, and so on. But there is also a catch. Assuming leaf nodes each send the same amount of traffic inward. The amount of traffic load increases polynomial as you go towards root. Your wormhole infrastructure (and what transports cargo between wormholes, in space or on planets) has to cope with this scaling. Congestion might become an issue.<br />

=== Mesh wormhole networks ===

The acyclic network may have some bothersome problems. The central routing focuses power and can keep transit times down quite significantly depending on the specific network infrastructure, but it’s also congestion-prone, and some close locations may be widely separated on the network. Or maybe you want to avoid centralized routing dependencies for political reasons. Root might like being root; everyone else being subject to root’s whims, not so much. So, can we build networks where there are two routes between the same two nodes? We can, but it comes with some necessary adjustment.<br />

Recall why the acyclic network is acyclic: to avoid forming CTCs due to the time connection between future and past from the projected wormholes. But if we let both wormhole mouths experience the same time dilation, the problem that ultimately leads to CTCs - one connection routing displacing further into the future than the other connection, and thus allowing one to arrive at a node before ones departure - is avoided entirely, since there is no time displacement to be accumulated. Thus we get meshes, where there may be many connections and cycles between nodes. An observer at one node watching for his transition from another node 10 light years away would have to wait for ten years to watch his light arrive. Closed timelike loops are not possible so long as this network is kept balanced.<br />

[[File:Wormholes MeshNetwork.svg|thumb|center|700px|An example of such a mesh network]]

The disadvantage - this network must take great care to ensure that the time dilation that inenvitably occurs due to different star velocities and the galactic gravitational field don’t accidentally create a time shift and CTC in the network. And also we cannot exploit time dilation to open connections very quickly, from the home node’s perspective. One has to wait out the full years it takes the wormhole’s far mouth to get to a remote destination, before the connection can be used.<br />

=== Semi-privileged wormhole networks ===

The interesting thing is, it is very much possible to build networks that make use both of relativistic projection to seemingly accelerate the network construction time, ''and'' contain cycles, with alternate routes to get to another node. The acyclic construction is a strategy to avoid CTC formation, but we should remember the ultimate constraint reason: we avoid cycles so that there are no connections which, through a certain time shift on them, a CTC could be formed. Acyclic construction is one strategy. Only building cyclic connections with careful time shift balancing is another!<br />

The foundation for such networks is an acyclic "base graph" that may be isolated from the larger network graph. We also call this the '''red graph'''. The base graph maintains all the partially ordered properties of the simple acyclic graph. We can call this the "privileged" graph, since its nodes and their connections define the principle positions in space-time of the network, and we must avoid generating CTCs on this definition. How do we avoid forming CTCs but still create alternative paths and thus loops?<br />

Let’s assume we have two nodes <math display="inline">A</math> and <math display="inline">B</math> ten light years apart. They both had connections established from a parent node <math display="inline">P</math>. Those connections are time-shifted over 99 years and part of the privileged base graph.

We couldn’t build a time-shifted connection between <math display="inline">A</math> and <math display="inline">B</math> without causing a CTC to be formed. But we are entirely fine sending signals '''Fast-as-Light''' from <math display="inline">A</math> to <math display="inline">B</math> and back. This is not a violation. Now consider: how does a wormhole with no time shift across it behave? In the mesh network case, such connections don’t cause CTCs to appear. And it doesn’t cause one to appear here either. How can we build such connections? We can of course use projection with relativistic speeds at both ends, like how we would build Mesh networks. But there is an even faster solution. If we can move wormholes through wormholes, we can shift the new mouths through the existing network to create this connection!<br />

The main catch is that we have to tightly watch out for temporal drift on such new connections. Through adding such edges that overall form a "blue graph" on top of the red graph, we are creating loops on the red graph underpinning the network. If one end of a wormhole mouth drifts more in time than the other (which will inevitably happen because stars won’t have the same mass, experience the same galactic gravity, or have the same velocity relative to each other) the conditions for a CTC will start to appear. The only way to pre-empt this is with continuous maintenance, taking connections offline and adjusting their temporal position using synchrotons, and having "safety space-time" spacing between wormholes. In this way, there is some "room" where signals can arrive faster and faster, but not before they were sent, allowing the connection to be used most of the time with only short intervals of adjustment.<br />

This works even when both ends connected don’t have the same time distance from the root node - the important thing is that the new shortcut accomodates this difference with a time shift of its own. We can calculate this difference easily. For two non-root nodes <math display="inline">A,B</math>, for each node we sum the temporal difference over the path from root to the nodes along the red graph, then subtract the difference from each other. The resulting remaining temporal difference is that the new connection on the blue graphh must have to keep the network stable. This means we can even do things like connect systems far out directly to core systems, or bridge from one edge of the graph to the other - arbitrary routings are possible. These routings can allow faster signal transmission and mass transport than through root, mitigating bandwith issues. The main problem is to watch out for network stability. As nodes accumulate, CTC avoidance becomes more and more delicate, and safety spacing must be maintained diligently. This can add to the effective travel time through such alternate connections, and thus maintains some of the "privilege" of the acyclic base graph.<br />

[[File:Wormholes BuildingCyclics.svg|800px|center]]

Arguably this network configuration is the most powerful, marrying many advantages and removing some disadvantages. It is however also bound to the need for safety spacing. Depending on your infrastructure assumptions, this can pose issues. If you want to implement serious stretches of safety space, wormhole links not part of the acyclic base graph would likely have to be located at interplanetary distances away from the main links, with light hours to light days of safety spacing. If your main network runs on planetary trains, this is rather a bother. Adjusting the internal throat length of wormholes can help in this regard, but of course now poses the issues of how to safely navigate such a long wormhole. Maybe you could extend something like space elevator tethers through such stretchy throats. With no gravity inside the wormhole itself, such cables could probably stretch quite significant distances. If your culture is handling wormholes mostly in space, you still have to account for added travel times through the safety spacing. Communications links would probably be easier connected between arbitrary nodes than macroscopic, craft-traversable wormholes.

[[File:Wormholes ComplexExample.svg|800px|thumb|center|An example of a complex semi-privileged graph wormhole network]]

==Credit==
Authors: Luke Campbell and Sevoris

==References==

File:Wormholes ComplexExample.svg

2022-08-21T08:54:09Z

Sevoris: A complex example of a semi-privileged wormhole network, involving multiple cyclics. Created for Galactic Library by Sevoris Doe.

== Summary ==
A complex example of a semi-privileged wormhole network, involving multiple cyclics.

Created for Galactic Library by Sevoris Doe.

File:Wormholes BuildingCyclics.svg

2022-08-21T08:50:49Z

Sevoris: Building a cyclic connection without causing a CTC. Created by Sevoris Doe for Galactic Library.

== Summary ==
Building a cyclic connection without causing a CTC.

Created by Sevoris Doe for Galactic Library.

Wormholes

2022-08-21T08:50:04Z

Sevoris: /* Mesh wormhole networks */

Wormholes are hypothetical structures in space-time allowed by the general theory of relativity.
They provide short cuts through space-time that connect one region of space and time to another; potentially, these regions can be very distant from each other.
The symmetries in space-time of our universe mean that wormholes move like physical objects, and acquire the conserved (or in some cases approximately conserved) quantities of things that go into them and lose those quantities of things that come out – properties like energy, mass, momentum, angular momentum, and electric charge.

==What is a wormhole? The geometry of space-time and how to twist it into a pretzel==

We're going to have to start out with some pretty heady stuff. Like the very nature of existence. And what is reality, anyway?
Usually, space and time are so much an integral part of our existence that we don't even think of them at all. They simply form a backdrop on which all the interesting stuff plays out. But when we do think about them, we're so accustomed to living in a world where they exist and where distances are well defined and events can be uniquely identified based on <i>where</i> and <i>when</i> they happened (or are happening or will happen) that a lot of stuff seems so obvious that we don't stop to think about if it really has to be this way or if there are other possible ways of doing things.

===A toy model===

So lets consider a hypothetical scenario, where an ant can be in any of several states, represented by the squares in the pictures below. The ant can transition to any other state that has a numbered side with the same number as the state it is in. We will start on the left. If the ant chooses the "2" transition, we end up with the picture in the middle. If it then chooses the "10" transition, we end up with the state on the right.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Space_elements_1.png|frameless]]
<td width=400>[[File:Space_elements_2.png|frameless]]
<td width=400>[[File:Space_elements_3.png|frameless]]
</table>
But now notice that the behavior of this system is exactly the same as if we re-arrange the states into a grid, as shown below on the left. The disconnected states now become a space; an expanse of coordinates on which the ant can exist. Now, the ant's transition between states can be described as a trajectory as shown in the right picture below.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Space_elements_grid.png|frameless]]
<td width=255>[[File:Space_elements_grid_path.png|frameless]]
</table>

This gives us an inkling of how something like space-time might end up being assembled from individual events. But what if we don't have things match up into quite such a perfect grid? We
ll take our space elements and tile them into an infinite grid. Two sections of the grid are shown below. But now we remove two of those space-tiles, and we connect the edges adjacent to the removed tiles with a corresponding edge of the other removed tile (although, as shown in the figure, we don't have to keep the same orientation). Now the ant, entering side 1 (which we have color-coded red for convenience) comes out of side 1 of the other missing square. This is perhaps the most basic wormhole.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Space_elements_wormhole_grid_1.png|frameless]]
<td width=400>[[File:Space_elements_wormhole_grid_2.png|frameless]]
</table>

Of course, this simple assembly only stitches together a space. But the world we live in is one of space-time. As the ant transitions from one square to another, it is also transitioning further ahead in time. But when the ant passes through the wormhole, while it locally experiences no discontinuity that wormhole might connect to a different time as well as a different region of space. So the ant might emerge far in the past or future of when it stepped through.

===Manifolds and coordinate patches===

The toy model used above hopefully prepares you for some of the ideas to follow - one of how we can piece together a complete space and time by connecting together different pieces. But before we go further, lets define an important term: a <i>manifold</i> is a "shape" of a given number of dimensions where, if you zoom in close enough, you can always get to a scale where it looks like the shape is flat. We'll use this definition to help us describe how we can put space and time together to get a wormhole. But first, let's look at a few illustrative examples.

====One dimension====

The simplest one dimensional shape is a line. And a line is everywhere flat. So a line is a one dimensional manifold.

A one dimensional shape might seem to be a wiggly path if inscribed in two or more dimensions, but it has no <i>intrinsic</i> curvature. That is, if everything about your world is confined to that path, you can't tell that it is curved – all you can do is measure the distance you go along the path. This greatly limits the kinds of shapes that we can consider. So any curved path of infinite extent looks from the inside like a line, and so any infinite length path is also a manifold.

But there's another class of one dimensional shapes that are manifolds – loops. A loop is a path of finite length that connects back with itself. You can make one by drawing a closed curve in a higher dimensional space. Or you can take a straight line but then say that a given point <i>here</i> maps on to another point <i>there</i>, and all the points between <i>here</i> and <i>there</i> form a loop. When you are marching along the line starting at <i>there</i> and you reach <i>here</i> you are also at <i>there</i>, so if you go further you just end up back where you were. This last bit may seem like a pointless bit of pedantry that just confuses a simple problem, but it will be a very useful way of looking at things as we go on.

====Two dimensions====

The simplest two dimensional manifold is a plane. A plane is everywhere flat, and extends off to infinity. A small region around any point on a plane (naturally) locally looks like a plane, so it meets our definition of a manifold.

=====Curved manifolds=====

But in two dimensions, we can have something that we don't have in one dimension – intrinsic curvature. Consider a sphere. If you get close enough to the sphere, it will look like you are on a plane; we live on a world that is (approximately) spherical (if you neglect mountains and ocean basins and all that other stuff), but when you stand up and look around you it mostly looks flat. So a sphere is a manifold. But take a line on the surface of your sphere. Extend it out along the surface, drawing it as straight as possible. Eventually, it will go all the way around the sphere, dividing it in two, and meet itself. This path is called a great circle. The Earth's equator is (approximately) a great circle, as are the lines of longitude (but not lines of latitude, because they do not divide the Earth into two equal-sized halves). The straightest-possible path on a curved surface (like a great circle on a sphere) is called a <i>geodesic</i>. But to keep the jargon down, we'll just call them lines for this non-technical presentation, and hopefully this will keep things more intuitive and understandable to the layman.

In flat space, two lines that are initially parallel remain parallel, always remaining the same distance apart. But in a curved space, initially parallel lines do not remain the same distance apart. On a sphere, for example, any two great circles will intersect at two points. A sphere thus has intrinsic curvature.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Sphere_and_great_circles.png|frameless]]
<td>[[File:Hyperbolic_surface_and_lines.png|frameless]]
<td>[[File:Cylinder_and_lines.png|frameless]]
<tr>
<td width=400>Two great circles going around a sphere. Although initially parallel, they meet at two points.
<td width=400>Two initially parallel lines on a hyperbolic surface. As you get farther from the place where they are parallel, the lines diverge.
<td width=400>Lines that are initially parallel on a cylinder remain parallel everywhere.
</table>

On a sphere, parallel lines always end up converging. One example of a curved surface where parallel lines diverge is a hyperbolic surface. Like a sphere, a hyperbolic surface is a manifold. But it has the opposite sense of curvature.

Now consider a cylinder. Again, a cylinder is a manifold. A cylinder can be made by rolling up a plane. Like our construction of a loop, this can be done by simply taking a long rectangular section of a plane and saying that two of the parallel edges are actually the same (this is essentially what you do if you roll up a sheet of paper so that two of its parallel edges touch, and then pretend that they are merged together where they meet). This "rolling up" preserves distances, so any two parallel lines on the plane remain the same distance apart on the cylinder (think about that sheet of paper, and if you drew two parallel lines on it - they will stay the same distance apart when the paper is rolled up). A cylinder has <i>extrinsic</i> curvature – you can tell that it is curved when looking at it from three dimensions. But it has no <i>intrinsic</i> curvature – if you are confined to the surface of the cylinder it behaves locally as if it is flat.

A manifold that is flat obeys all the rules of Euclidean geometry (although you'll need to go to the three dimensional case for dealing with volumes), and the standard Cartesian coordinate system (representing position by two orthogonal coordinates, commonly called <math>x</math> and <math>y</math>) can be used to specify any point. A manifold that is not flat violates Euclid's fifth postulate; the geometry of the manifold will be <i>non-Euclidean</i> (somewhere, H. P. Lovecraft is quietly having a mental breakdown). It cannot be represented by Cartesian coordinates, but different sets of coordinates – or perhaps patches of multiple coordinate systems that apply in different places – can be defined instead to locate points.

=====Topology: simply and multiply connected manifolds=====
If you take a plane or a sphere, any closed loop on the manifold can be shrunk in a continuous manner until it vanishes. This is called <i>simply connected</i>. But not all manifolds are simply connected. Those which are not simply connected are called <i>multiply connected</i>. The most basic multiply connected manifold is a torus, like a bagel or doughnut or bicycle inner tube. While there are closed loops on a torus that can be shrunk to a point and made to vanish, those that go all the way around the torus tube cannot – they'll get caught up circling the tube. Likewise, loops that go all the way along the tube will get caught up on the central hole if they are shrunk and again cannot be made to go to a point.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:torus_and_loops.png|frameless]]
<td width=400>[[File:Topological_torus_and_loops.png|frameless]]
<tr>
<td colspan=2>Three classes of loops on a torus. On the left is a standard geometrical torus; on the right is a topological torus that is flat but has opposite edges identified so that they are connected. The blue loop doesn't go around the torus, it can be continuously deformed to shrink to a point and vanish. The red line goes around the torus the long way. If you shrink it, it will get caught up going around the hole in the middle and get stuck there; it cannot be continuously deformed to make it any smaller. The green line loops around the torus from the outside to the inside and back. Again, if you shrink it the line gets caught up circling the torus tube and can't be made any smaller.
</table>

And this brings us to another interesting kind of manifold. You can get a manifold that is flat but has the topology of a torus. You can do this by taking a rectangle and saying that the opposite sides are actually the same. This gives you the effect of a "wrap-around screen" like on that old arcade game Asteroids – if you go off the edge on one side, you just come back on the opposite side at the same height. in physics, this is known as "periodic boundary conditions". Closed loops that go all the way around one way or the other cannot be shrunk to a point. But this kind of surface <i>cannot</i> be embedded in three-dimensional space! We might have grown used to surfaces that we can envision existing in the space we live in, but there are manifolds, even fairly simple two-dimensional manifolds, that cannot be embedded in higher-dimensional spaces like that. Remember this when we get to actual wormholes.

You can have multiply connected manifolds that are not toruses, even though they might have the same topology; any manifold with one or more "loops" or "handles" is multiply connected.

=====Non-orientable manifolds=====

And there is one other interesting kind of manifold that is relevant to wormholes – manifolds that are <i>non-orientable</i>. An orientable manifold is one where you can define a consistent sense of clockwise and counter-clockwise everywhere on the manifold. If you start with a clock spinning clockwise, or a right-handed glove, or a copy of Lewis Carrol's "Through the Looking-Glass, and What Alice Found There", you can move it anywhere on the surface and if it meets back up with itself it will still be spinning clockwise, or right handed, or normally readable. But in a non-orientable surface, there are ways you can move the object so that when you get it back to where it originally was the clock is spinning counter-clockwise, or the glove is left-handed, or the book is written in mirror writing. The simplest example of such a surface is a Mobius strip. As shown below, if you move an oriented circle around the strip, it comes back with the opposite orientation.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Topological_Mobius_strip.png|frameless]]
<td width=400>[[File:Mobius_twist.png|frameless]]
<td width=400>[[File:Mobius_strip.png|frameless]]
<tr>
<td colspan=3>A Mobius strip … a strip of surface with a twist so the edges where it connects meet up "the wrong way around". On the right is a 2D strip where you connect along the short edge matching the 1's together and the 2's together. The center shows the same strip with a twist in it so you can line the edges up the way you are supposed to, demonstrating how the twist reverses the orientation of the spinning circles. An embedded image in 3D space with the edges connected together is shown on the left. You cannot define a consistent orientation to this surface – starting at the red counter-clockwise circle, if you move the circle continuously along the surface when you get back where you started you will find it is going clockwise!
</table>

====Three dimensions====

The most basic three-dimensional manifold is a flat space. This is, well, just a normal space. You can go off in any combination of three independent directions (left-right, up-down, front-back), and because it is flat, initially parallel lines stay parallel. It all behaves much as you would expect three dimensional spaces to behave.

But you can get variations on this. They are harder to visualize than two-dimensional cases, because we don't have the brainspace for processing curved spaces inside of a four-dimensional hyperspace. But you can do things like identifying two planes to be the same, such that if you go far enough in one direction you can end back up where you started again (akin to a cylinder), or make them curved such that distances are distorted and lines don't remain parallel and maybe you can put larger things inside of smaller surfaces. There are 3D equivalents of spherical and hyperbolic surfaces (called, naturally enough, spherical and hyperbolic spaces). There are even non-orientable spaces.

One way of trying to envision a curved space, and to gain some intuition about them, is to take a two-dimensional slice of that curved space and embed that in a flat three-dimensional space. This is called an embedding diagram. Remember that this is a reduced representation that throws away a lot of the information (unless there are special symmetries you can exploit) and is not actually the thing being described.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Space_with_embedded_plane.png|frameless]]
<td width=400>[[File:Embedded_plane.png|frameless]]
<td width=400>[[File:Embedding_diagram_example.png|frameless]]
<td width=400>[[File:Embedding_diagram_example_very_curved.png|frameless]]
<tr>
<td>A space with an embedded plane. We show a sphere in this space, with the plane bisecting the sphere.
<td>Just the embedded plane. Note how in a spherically symmetric manifold (like this space), a circle on the embedding diagram represents a complete 2-dimensional spherical surface.
<td>An embedding diagram of a "dimpled" space. Just as before, the closed circles represent closed spherical surfaces. It is just that in this case, the distance between two concentric spheres of radius <math>R_1</math> and <math>R_2</math> is longer than the usual <math>R_1 - R_2</math> because of the spatial curvature, as shown by the increasing distance along the dimple's surface.
<td>An embedding diagram of an extremely curved section of space. In this example, there are several places where we have a bigger sphere inside of a smaller one! Curved space is weird like that.
</table>

Remember that not every manifold can be embedded, so not every curved space can be represented by an embedding diagram. But you can take patches of the curved space and show them by embedding diagrams, and then describe how these patches stitch together to create a contiguous surface.

====Space-time: three space dimensions and one time dimension====

This is the world we live in. So this is the class of manifolds that we really need to use when making a wormhole. Sometimes it's abbreviated as 3+1D (as opposed to 1D, 2D, 3D, etc.).

The basic, infinite, flat space-time manifold is often called Minkowski space. I know, it's a big word using scary foreign-sounding names. But all it means is that its a 3+1D manifold that goes on forever and is flat. In flat space-time, everything obeys the rules of special relativity. Which means it really doesn't work when masses become significant enough to make gravity.

Our best model of how gravity works is general relativity. In this theory, the curvature of space-time is created by mass. At least, that's what they always say in the over-simplified pop-sci summaries of general relativity. Actually, space-time curvature is created by all the components of the stress-energy tensor. It's just that in almost all cases, mass is the most significant part of this. But momentum, stresses, pressures, and shear forces all have contributions as well. In general relativity, physical objects move along the straightest possible lines in this curved space-time. The way the curvature causes the lines to bend and converge is what leads to the appearance of gravity as a force if the curvature is sufficiently weak.<ref name="Misner_Thorne_and_Wheeler">Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler, "Gravitation", W. H. Freemann and Company, New York (1973)</ref>

The most basic curved space-time manifold is the Schwarzschild geometry, which is what you get when you have an uncharged, non-spinning spherical mass doing its distortion thing on space-time. The Schwarzschild geometry is <i>static</i> meaning it does not change in time, and is <i>spherically symmetric</i>, meaning that it has a definite center and it doesn't matter which direction you are relative to that center – it is independent of angle. If the mass becomes dense enough, space-time gets so distorted that an event horizon forms. Nothing that goes into the event horizon can ever come back out, and the inevitability of the geometry crushes the mass down to an infinitely dense point at the center called a singularity. Nothing on the inside of the horizon can ever escape. It forms a <i>black hole</i>, that forever traps any light or matter that enters it. For masses that don't exceed this critical density, the gravitational distortion outside of the object is the same as for a black hole of equal mass, but there is no event horizon. The object can still be seen and interacted with.<ref name="Misner_Thorne_and_Wheeler"></ref>

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Schwarzschild_star.png|frameless]]
<td width=400>[[File:Schwarzschild_black_hole.png|frameless]]
<tr>
<td>An embedding diagram of the Schwarzschild geometry of a dense, massive star. The physical extent of the star is shown in blue.
<td>An embedding diagram of the Schwarzschild geometry of a black hole of the same mass as the star to the left. The event horizon is shown in red.
</table>

There is one distinction between space-time manifolds that will become important later. Some manifolds have an isolated region of high curvature (corresponding to a concentration of mass, or at least of stress-energy) but as you go farther away from the strongly curved region the space-time becomes increasingly flat. If you can always go out far enough away from a curved region that space-time becomes flatter that whatever flatness criterion you choose, no matter what direction you choose to leave the curved region, the manifold is called <i>asymptotically flat</i>. For our purposes, if you can go far enough away that the Newtonian approximation to general relativity is accurate, and gravity can be described as a force rather than requiring a geometric description, then the geometry can be considered asymptotically flat. You can also allow gravitational waves in your asymptotically flat space-time – their very small space-time curvature is not enough to cause problems and they can still be described by a linearized version of gravity. Both the Schwarzschild and the Minkowski geometries are asymptotically flat. Spherical or hyperbolic geometries are not asymptotically flat. Planets, stars, galaxies, neutron stars, black holes, and galactic clusters are all approximately asymptotically flat; but the universe as a whole is <i>not</i>. As long as we confine our attention to merely galactic clusters, we can make the approximation of asymptotic flatness.

===Making a wormhole===

Now that we have covered some of the introductory material, we can get down to the main course. To make a wormhole, take two regions of space-time, usually far apart and usually asymptotically flat (although neither of these is strictly required) and connect them together. That's it. You're done. Sounds easy, doesn't it?

Note that if you are connecting two regions in the same universe, your new geometry will be multiply connected. A loop that goes through the wormhole and then connects back up outside the wormhole cannot be shrunk to a point. There will now be two equivalent ways to get between any two points – the path that goes through your usual space-time, and the path that goes through the wormhole. If you use a wormhole to connect two separate universes, the combination of the two universes and the wormhole can still be simply connected.

In general relativity, you can distort the geometry of space-time with stress-energy, but there is no way to change its topology; that is, no way to turn a simply connected universe into one that is multiply connected, or to add or remove more "loops" or "handles" to an already multiply connected universe. Such changes, if they are possible at all, would only be possible in the realm of quantum gravity and we do not know enough about quantum gravity to know if it would be possible or not. So, it might not be possible to make intra-universal wormholes at all. Or maybe quantum gravity does allow topology changes in space-time and there are ways to connect different parts of our universe together with wormholes. On the plus side, if you can't change the topology of space-time, you can always bud out a new universe from our own – perhaps by triggering the inflationary field that started our universe. It will be connected to ours by a wormhole. And because the topology can't change, that wormhole would never be able to break. We would have an eternal bridge to the new universe. So a system of physics that does not allow topology changes still allows wormholes to other universes.

==Common kinds of wormholes==

===Spherically symmetric wormholes===

General relativity involves a lot of non-trivial math. You need to deal with four-dimensional curvature tensors with up to 20 independent components, and figure out how they relate to a given distribution of matter, fields, energy, stresses, and material flows. If you can reduce the number of dimensions you need to keep track of, it can simplify things a lot. One way of doing this is assuming your space-time geometry does not change with time (it is static), and looks the same in all directions when viewed from a central point (it is spherically symmetric). This makes things a lot easier to analyze.

Astute readers will have noticed that these conditions are satisfied by the Schwarzschild geometry. In fact, if you play around with the Schwarzschild geometry enough, you will find that you can even get a wormhole-like solution. At the event horizon, you can connect your black hole solution to a time-reversed structure called a white hole. This is called an Einstein-Rosen bridge. Unfortunately, it has some issues. For one, it can't form naturally from infalling matter creating a black hole. For another, going through an Einstein-Rosen bridge involves passing through an event horizon, so the trip is necessarily one way. And finally, you can show that even the smallest amount of matter or radiation falling into the Einstein-Rosen bridge leads to it collapsing before the matter or radiation can pass through, trapping it inside the black hole. So this is not looking particularly useful.

So what are we going to do? Well, the Einstein-Rosen bridge, like a lot of early work in general relativity, was made by assuming a distribution of stress-energy (in this case, matter) that was assumed to be physically plausible, and finding out what the resulting geometry would be. But we want a wormhole, dangit! So what happens if we just demand a static tunnel connecting two asymptotically flat regions of space-time? If you do this, you get a Morris-Thorne wormhole<ref name="MorrisThorne">Michael S. Morris and Kip S. Thorne, "Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity", American Journal of Physics <b>56</b>(5), 395-412, May 1988</ref>. You still have the relation between stress-energy and geometry, but now it is your assumed geometry dictating the stress-energy rather than the other way around. One difficulty with this method is that the distribution of stress-energy you end up with might not end up seeming physically plausible. In particular, it can be shown that any wormhole must have regions in it with negative energy density<ref name="Visser_Lorentzian_wormholes">Matt Visser, "Lorentzian Wormholes: From Einstein to Hawking", Springer/AIP Press (1996)</ref>, and these spherically symmetric wormholes are no different. We will discuss the issues around and solutions to the negative energy density problem later.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Morris_Thorne_wormhole.png|frameless]]
<tr>
<td>An embedding diagram of a Morris-Thorne wormhole. For this wormhole, we just took two copies of the Schwarzschild geometry from above and connected the middle "dimples" with a cylindrical patch of space-time.
</table>

The places where you enter a wormhole are called it's <i>mouths</i>, and most wormholes have two. The space-time tunnel between the mouths is called the wormhole <i>throat</i>.

====Some common problems with spherically symmetric wormholes====

if you don't have your distribution of matter and energy going off to infinity, then eventually if you go far enough away your geometry will reduce to the Schwarzschild geometry. A good many published examples of such wormholes are actually Schwarzschild-like up until pretty close to where the event horizon would be. The first problem you might encounter, then, is the formation of a horizon. A single horizon makes any trip one-way. Going into it on the side with the horizon would be indistinguishable from entering a black hole - except that you pop out somewhere else instead of getting removed from existence at the singularity. And because of the horizon, you couldn't ever go back. If both sides have a horizon, you simply can't get out of it. Why do you even have this thing, anyway?

The strong curvature as you get close to the throat can also produce very strong tidal forces. Due to the gravitational forces, if you go in feet-first your feet could be pulled much harder than your head as you go in and you could end up getting pulled apart – spaghettified is the technical term. In addition to these pure spatial tides, it turns out that you also have velocity-dependent tides. These tides are transverse to the radial direction - if you dive in really fast, you will be squeezed even harder from side to size.

Early attempts to limit the tidal forces while still keeping a throat large enough to fit a person through resulted in very large masses. Like planetary scale masses or even solar scale masses. This would mean you would need to spend a lot of propellant de-orbiting into the wormhole and then rocketing your way back out. Some of these early wormholes also had issues like the extreme gravity blue-shifting the cosmic microwave background to blow-torch intensity x-rays. Sometimes, attempts to keep the tides down also lead to really large throat sizes. Like the distance from the Sun to the Earth sizes. Or even light year sizes. These gigantic wormholes would be rather inconvenient.

None of these are necessarily intrinsic limitations to spherical wormholes. But it does indicate that a perfectly reasonable assumption for the wormhole tech in your science fiction universe could involve wormholes with Jupiter-level masses, extreme tides, and other fun details.

===Thin shell wormholes===

Take a shape located somewhere in flat (Minkowski) space-time. The important part of this shape is that it must be a closed surface that completely encloses a given volume. Now go someplace else in space-time and take an identical shape. Remove all of the space-time from inside both shapes, and then say that the corresponding parts of the two surfaces of the shapes are actually the same points. Now you have a thin shell wormhole. The throat region is infinitesimally short – the section that corresponds to the wormhole is just that surface of your shape, hence the name "thin shell".

The first thin shell wormholes proposed were simple polyhedra <ref name="Polyhedral_wormholes">Matt Visser, "Traversable wormholes: Some simple examples", Physical Review D vol. 29 no. 10 pages 3182-3184 (1989)</ref>. These are attractive choices for a potential wormhole-user because the flat faces of a polyhedral thin-shell wormhole have no tidal effects whatsoever. Further, the flat faces do not have any stress-energy either. It is simply flat space-time, no different from any other area of flat space-time. Going through the face of a polyhedral wormhole is no different from going through the flat space-time of the doorway separating your hallway from your bedroom. The negative energy stuff is concentrated entirely in the edges and corners of the polyhedron. And hoo boy, do you need a lot of it. For a cubic wormhole, each edge requires approximately <math>-1.52 \times 10^{43}</math> J/m of length. This is <math>-1.69 \times 10^{26}</math> kg/m, or close to a tenth of a Jupiter mass per meter of edge length. It has been proposed that perhaps cosmic strings – defects of the primordial universe where things couldn't line up the right way and left a linear flaw in reality – with negative tension might be able to meet this requirement and hold open a polyhedral wormhole.

But there's a strange thing when you consider the mass that these wormholes require. They are embedded in flat space-time. Which means that outside of that thin shell they have, they induce no curvature on the space-time around them. Remember that mass causes space-time to curve, dimpling up as in the Schwarzschild geometry shown earlier. Because these wormholes induce no far-flung curvature to imprint on the surrounding space-time, they have zero mass (technically ADM mass). And because mass is equivalent to energy, it should technically be possible to make one of these for no energy input at all from those of us in flat space-time. Forces are the rate of change of energy with distance, pressures are the rate of change of energy with volume. But because the energy is always zero no matter what size they are (as measured from out in flat space-time), there would not be any force or pressure needed to hold them open. The extreme conditions of their interior contrasts with the relatively benign conditions once away from the thin shell (or, indeed, within any flat area within the shell).

A curious feature of polyhedral wormholes is that each face of the polyhedron might connect to a <i>different</i> polyhedron. You might, for example, have a collection of seven cubical wormhole mouths, and each of the six faces of the wormhole you are facing connect to a different one of the six remaining mouths.

For more general thin-shell wormholes, you just need to smear that negative mass out a bit. As long as you keep a flat area to go through, travelers will not experience any strange tides or encounter space-warping exotic energies. You could, for example, have your standard circular portal beloved of fiction. Or you could have any other arbitrary shape you wanted.

Thin shell wormholes are what you get when you take a mathematical limit; they would be an approximation of a more physical distribution of stress and energy. Any realizable version would have a finite, although possible small, thickness to its shell and hence a finite, although possible small, throat length. As we will discuss later, they would likely also have a positive, although possible small, mass. But as an approximation, and as a valid solution to Einstein's field equations in general relativity, they allow us to explore many aspects of wormhole physics using a relatively simple model that is easy to calculate, and can lead to a lot of interesting inspiration for fictional ideas of wormholes.

====Tidal distortion during transit====

Consider a thin shell wormhole whose shell-shape is everywhere convex - no flat areas this time. For convenience, let's make it a sphere. As a traveler passes through the shell, the sections of his body that pass the shell must suddenly "pop" from curving one way to curving the other way, as illustrated below:

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Velocity_dependent_tides_pre_transit.png|frameless]]
<td width=400>[[File:Velocity_dependent_tides_in_transit.png|frameless]]
<tr>
<td>A traveler (left rectangular shape) about to go through a spherical thin-shell wormhole (here represented as circles, with the interior volume blacked out to indicate that there is no space-time there and nothing can exist in that region).
<td>The traveler in the process of passing through the wormhole. Note that on the wormhole shell, the portion of the traveler to the left has a leftward-curvature, while the portion on the right has a rightward curvature. As the traveler passes through, each section that goes through the shell immediately jumps from left-curving to right-curving. This will induce stresses in the traveler as his interior is deformed.
</table>

it is clear that as the curvature of the wormhole becomes large compared to the size of the traveler, the distortion will decrease. In the limit of a flat section of the surface, no distortion is experienced at all.

A consequence of this is that something going through a curved section of a thin shell wormhole will experience stresses (distortions) and strains (forces). The elastic energy stored by a strained object is proportional to the product of the stress and the strain, added up over all of its volume. Assuming that the object going in to the wormhole is initially in its relaxed state, it will require energy to push it into the wormhole shell. If it enters with insufficient initial kinetic energy to get it through or external forces pushing on it, it will just bounce off.

===Cylindrical and toroidal wormholes===

In principle, you can have a wormhole with cylindrical symmetry
<ref>Kirill A. Bronnikov and José P. S. Lemos, "Cylindrical wormholes", arXiv:0902.2360 [gr-qc] 24 Feb 2009</ref>
. This forms a valid solution to Einsteins field equations in general relativity. However, a true cylindrical wormhole would be infinite in length. So they probably don't exist naturally, and you won't be able to build one. Unlike all the other wormholes discussed here, they are not asymptotically flat. Unlike wormholes of finite extent, not all cylindrical wormholes require negative energy regions. However, it has been shown that negative energy stuff is needed if the geometry radially far enough away from the wormhole approaches the flat space-time of our usual universe, so a universe with a cylindrical wormhole without negative energy would not look like our own.

So the thought is that maybe you can get close enough to a cylindrical wormhole by making a really long wormhole that bends ever so slightly and comes back and meets itself, forming an extremely stringy torus. The general case for this has not yet been worked out, but a thin shell approximation of a toroidal wormhole has been developed
<ref>Vladimir Dzhunushaliev, Vladimir Folomeev, Burkhard Kleihaus, and Jutta Kunz, "Thin-shell toroidal wormhole", arXiv:1901.07545v2 [gr-qc] 28 Jan 2019</ref>
.
These thin shell toriodal wormholes have been shown to be stable to perturbations, one nice benefit that has not been established for a lot of other kinds of wormholes. They do require some negative energy regions on their surface, but there will also be some regions that do not contain negative energy.

===Non-orientable wormholes===

In the discussion on non-orientable manifolds, we showed how in these manifolds you can't properly determine the clockwise - counter-clockwise orientation of objects or motions, or the orientation of objects with respect to their mirror reflections. If you move an object along certain paths, when it comes back it might come back the wrong way around. So it is probably no surprise that a non-orientable wormhole turns the universe into a non-orientable manifold. Something going through a non-orientable wormhole comes out looking like it has been mirror reflected.

Normally this seems like it would be rather boring and useless, but for an obscure constraint from quantum physics called the CPT theorem. This theorem, which holds for all physical phenomena, means that the product of the discrete symmetries of charge conjugation (C), parity transformation (P), and time reversal (T) always returns you to your original state. Huh? What does that mean? Well, parity transformation is equivalent to a mirror reflection; so passing through a non-orientable wormhole means parity is inverted. This means that exactly one of the other two symmetries must also be inverted for the CPT theorem to hold. The thing coming out on the other end is still manifestly going forward in time, so it must be charge conjugated, whatever that is.

Charge conjugation means you turn all particles into their antiparticles. So anything passing through a non-orientable wormhole emerges on the other end made entirely out of antimatter.

While this particular turn of events might discourage people from passing through, it does offer obvious application for energy generation, space propulsion, and weaponry.

==Exotic energy conditions==

It is fairly easy to see than any wormhole requires negative energy densities (or more generally, regions where the tension is higher than the energy in natural units) to exist. Take a look at any of the wormhole diagrams above – the spherically symmetric ones will probably make it the most clear. If you send a parallel ray of stuff – maybe particles or light or golf balls – through the wormhole, and trace out the straightest possible path of all of those parallel lines, you will see that after they go through the wormhole all those initially parallel trajectories will now be diverging. Positive energy (such as mass) causes attractive forces, drawing trajectories toward it and making the paths converging. To get diverging paths you need the opposite – repulsive gravity – which you can get from negative energy densities or very high tensions. Having the tension be higher than the energy density turns out to be equivalent to a negative energy density – in some frames of reference observers will measure a negative energy density there.

A lot of physicists used to be reflexively dismissive of negative energy densities. After all, they can't really exist; all energy in the real world is positive.

Except that it is not. There are certain odd cases that allow for negative energy densities. One of these is the Casimir vacuum. A conducting boundary cannot support an electric field parallel to that boundary (if there was such a field, charge would flow under the force of the field until the parallel component of the field is screened out). So if you have two parallel conductive plates, the only electromagnetic waves that you can get between them require their fields to vanish at the plates. This restricts the number of ways the electromagnetic field can vibrate, called modes. In quantum mechanics, merely the potential for a vibration is associated with a certain energy (called <i>zero point energy</i>) even if no quanta of vibration are actually present. By making these modes impossible, they cannot contribute their zero point energy to the region between the plates. This lowers the total energy of the space between the plates compared to the region outside the plates (or with the plates absent). Because the energy of empty space is zero, this means that the region between these Casimir plates has negative energy. The Casimir effect has been measured in the laboratory. It is fairly weak, but it is present. In fact, not long after traversable wormholes were first described, one proposal suggested holding open the wormhole using the Casimir effect<ref name=MirrosThorneYurtsever">Michael S. Morris, Kip S. Thorne, and Ulvi Turtsever, "Wormholes, Time Machines, and the Weak Energy Condition", Physical Review Letters, Volume 61, Number 13, 26 September 1988, pages 1446-1449</ref>

Another situation that gives rise to a negative energy density is a <i>squeezed vacuum</i>. A squeezed state is a way of manipulating a quantum oscillator (like a pulse of light) to get around the uncertainty relations of quantum mechanics to increase the precision in one variable by increasing the uncertainty in its conjugate variable. Applying this operator reduces the energy of the oscillator, and if this is applied to the vacuum state, which already has zero energy, you can end up with regions of negative energy.

It is worth noting that not just any old negative energy region of exotic space-time will work. In particular, dark energy is a condition of the universe we live in where even bare vacuum itself has a small amount of negative energy. The current best model we have for this is as a cosmological constant - where every part of the universe has an equal density of negative energy and a related amount of pressure. But the wormhole allowing contributions to space-time curvature of that negative energy are exactly cancelled by the wormhole-denying contributions of the pressure. So according to our best current models, dark energy will not support a wormhole.

Also, the Casimir vacuum described above is also problematic. The exotic vacuum between the plates does have the necessary properties (if not magnitudes) to support a wormhole, but this is generally far overwhelmed by the immediately adjacent positive mass matter making up the conductive plates. The technical term for the criterion that determines whether you can have a wormhole or not is the Averaged Null Energy Condition (ANEC). If you project a ray through space-time that follows the path that light (in a vacuum) could take, and do some math along that ray and add up all the contributions from all the places along that ray, then if your result is zero or positive then the ANEC is said to be satisfied. If this is the case, you can't get a wormhole. If the ANEC is less than zero, it is said to be violated and in principle you could use that weird exotic energy stuff to prop open a wormhole's throat. There are other so-called energy conditions, and if you have regions of space-time that also violate those energy conditions they can be used to hold open wormholes as well - but all of these also violate the ANEC, so the ANEC is all you need to consider.

===So just how do I get the ANEC-violating stuff I need?===

The short answer is, we don't know.

As we already said, a cosmological constant form of dark energy won't do it (although different ideas about dark energy, such as phantom energy, could conceivably work). The inflationary vacuum thought to have existed very briefly during the first instances of our universe's existence is also usually thought to be a cosmological constant (although one that stopped existing after a very brief period of time, so I guess it is less than constant), so the inflationary vacuum probably won't work either.

If you can get the Casimir vacuum without the massive conducting plates that bound it, you might be able to do something with that. For example, the twisted geometry of space-time itself in a wormhole throat imposes boundary conditions that restrict the modes of vibration inside of it similar to the Casimir effect. This is called the <i>topological Casimir effect</i>, potentially it could be used to help support a wormhole. Also, although the ANEC is satisfied if the probing space-time ray goes through the Casimir plates, a ray that goes between the plates but never touches them will violate the ANEC. Can you use this to hold open a wormhole? Maybe a sufficiently clever person will eventually figure out how to do so.

Even more serendipitous, it turns out that gravity itself naturally squeezes the vacuum
<ref name="gravitationally_squeezed_vacuum">David Hochberg and Thomas W. Kephart, "Lorentzian wormholes from the gravitationally squeezed vacuum", Physics Letters B 268, 377-383 (1991)</ref>
. If you have the curved space-time of gravity, all of the modes of oscillation in it become squeezed states. So it might be possible that the curved space-time of a wormhole can squeeze the vacuum enough to support itself. Or a combination of gravitational vacuum squeezing and the topological Casimir vacuum from the wormhole's geometry constraints might be all that is needed. At least one proposal has suggested a class of wormholes that can be self supporting in such a way
<ref>S. V. Krasnikov, "Toward a Traversable Wormhole", arXiv:gr-qc/0003092v1 22 Mar 2000</ref>
<ref>S. Krasnikov, "Traversible Wormhole", Physical Review D, volume 62, article 084028 (2000)</ref>,
although the throat of these wormholes is very "wrinkly" or "crumpled up" and it is unclear what the effects of that would be on anyone passing through.

====Quantum energy inequalities====

When you apply quantum field theory to what happens when you have a region where the quantum fields produce local negative energy densities, you end up with a limit of how much negative energy stuff you can have before it has to be balanced out by an even greater amount of nearby positive energy stuff
<ref>[https://web.archive.org/web/20060206055950/http://maths.york.ac.uk/www/PhysicsQIneq.htm|Quantum Energy Inequalities]</ref>
<ref>[http://www.phys.lsu.edu/mog/mog20/node16.html|Quantum field theory on curved spacetime at the Erwin Schrödinger Institute]</ref>
<ref name="Fewster_Roman_2005">Christopher J. Fewster and Thomas A. Roman, "On wormholes with arbitrarily small quantities of exotic matter", Physical Review D <b>72</b>, 044023 (2005)</ref>
.

====Matter with negative mass====

It might be tempting to try to solve the negative energy density requirement for wormholes by introducing some sort of matter that naturally has negative mass.
However, negative mass would introduce all kinds of problems to the smooth operation of how the world works. For one thing, if negative mass (or equivalently energy) can exist, why doesn't it just spontaneously pop out of empty space accompanied by an equal magnitude of positive mass (or energy)? For another, negative masses lead to all kinds of runaway instabilities. For example, imagine what happens if you push on an object with a negative mass. You exert a force on it with your hand. But by Newton's second law of motion, an object's acceleration is proportional to its mass. So the negative mass thing will accelerate backwards, against the direction of the force, and into your hand. Which means you will be pushing even harder on it, which makes it push back even harder than before. You have a runaway process that blows up until your hand breaks, and any fragments of flesh and bone and sinew that were in the way of the negative mass thing have been flung off.

Or consider a negative mass thing with an electric charge. If you put something with an electric charge in an electric field the field will exert a force on the charge. For a normal positive mass charged thing, it will move in the direction of the force. The work done is the force in the direction of the displacement in position, so because the force and motion of the object are in the same direction, the field does work on the charged object and energy flows from the field into the kinetic energy of the charged object. But if the charged object has negative mass it will begin to move opposite to the direction of the force. This will do work on the field instead of the object, increasing the energy of the field. The energy comes from the kinetic energy of the negative mass charged object – but because kinetic energy is proportional to mass and increases with increasing speed, as the negative mass object loses kinetic energy it gains speed, going ever faster. Both the field strength and the speed of the object increase without bound. If the electric field originates from the electric part of an electromagnetic wave, it will vary sinusoidally with time and space. As the negative mass object is forced into a sinusoidal trajectory opposing the electric force on it, it will amplify that wave making it ever more intense. A single, isolated charge exposed to an electromagnetic wave will eventually have its inertia keep it moving from the previously imposed force even after the force has switched direction, so if the negative mass is free with no other forces acting on it there will be no net amplification. But if there are any other damping forces, or internal arrangements of bound charge that can resonate with the wave, then instead of being absorbed (as would happen with a positive mass), the wave will be amplified indefinitely and the ever-increasing energy of the wave will come from the ever increasing temperature of the negative mass thing (because the mass is negative, the thermal motion of temperature gives it a negative energy whose magnitude only increases as the temperature rises).

And, of course, the quantum energy inequalities indicate that you can't just have isolated lumps of negative mass stuff floating around.

So just having lumps of stuff that have a negative mass is probably not realistic, and it were possible it would lead to all kinds of universe-shattering cataclysms. Hopefully, we can get wormholes to stay open just using the exotic vacuum states from the Casimir effect and squeezed vacuums.

===Reducing or eliminating ANEC violating stuff===

As soon as people figured out that the wormholes they want needed stuff they couldn't get, they set about trying to find ways to use as little of the stuff they couldn't get as possible. A number of attempts have been made to hammer out a geometry for spherical wormholes that are stable, meet the quantum energy inequalities, and allow a person to pass through them in a reasonable length of time without being shredded by tides
<ref name="Fewster_Roman_2005"></ref>
<ref name="Visser_Kar_Dadhich">Matt Visser, Sayan Kar, and Naresh Dadhich, "Traversable Wormholes with Arbitrarily Small Energy Condition Violations", Physical Review Letters Vol. 90 No. 20 article 201102 (2003)</ref>
<ref name="Kuhfittig">Peter K. F. Kuhfittig, "More on wormholes supported by small amounts of exotic matter", Physical Review D <b>73</b>, 084014 (2006); Peter K. F. Kuhfittig, "Wormholes supported by small amounts of exotic matter: some corrections", arXiv:gr-qc/0508060v1 15 Aug 2005</ref>
.
Often, these resulting structures end up having extreme dimensions, such as throat sizes on the order of light years
<ref name="Fewster_Roman_2005"></ref>
<ref>Remo Garattini, "Generalized Absurdly Benign Traversable Wormholes powered by Casimir Energy", arXic:2008.05901v1 [gr-qc] 12 Aug 2020</ref>

Thin shell wormholes have also been the subject of investigation. One study <ref name="Mazharimousavi_Halilsoy">S. Habib Mazharimousavi, M. Halilsoy, "3 + 1-dimensional thin shell wormhole with deformed throat can be supported by normal matter", Eur. Phys. J. C (2015) 75:271 DOI 10.1140/epjc/s10052-015-3506-6</ref> found a number of thin shell wormholes with sharp-edged shapes could be supported entirely without any exotic energy. However, this only applies if the edges are infinitely sharp. A finite radius of curvature at the edges would require negative energy at those edges.

If the wormhole is taken have the properties of a class of subatomic particles called a fermion, then if the ratio of the electric charge to the mass is sufficiently large you can get a wormhole that can remain open without any negative energy regions at all
<ref name="Blazquez-Salcedo_Knoll_Radu">Jose Luis Blázquez-Salcedo, Christian Knoll, and Eugen Radu, “Transversable wormholes in Einstein-Dirac-Maxwell theory”, Physical Review Letters <b>126</b>, 101102 (2021); ArXiv: 2010.07317v2 [gr-qc].</ref>
.

==Fields and conserved properties==

Our best theories of how the world works – quantum mechanics and general relativity – predict that certain things are [[Conservation_Laws:_Limits_to_Cheating|conserved]]. For asymptotically flat space-times, these quantities are energy, linear momentum (or just momentum), angular momentum, and electric charge. Furthermore, in the non-relativistic limit mass is conserved independently of non-mass energy.

And not only are these things conserved, but they are always conserved <i>locally</i>. This means you can't just get rid of, say, some energy in one place and expect things to balance out by having that energy appear somewhere else.

What does it mean for something to be conserved? Mathematically, it means that the thing obeys the continuity equation. There are various ways of writing the continuity equation, all equivalent, but basically it boils down to the following: if you have a region of space surrounded by a closed surface, then the amount of conserved stuff inside that surface can only increase if some of the stuff enters through the surface and it can only decrease if some of the stuff leaves through the surface. Makes sense, right? If you have three loaves of bread in a room, then if the deliveryman brings in another two loaves you will have five loaves. But unlike loaves of bread, which can be baked or eaten or grow moldy or burn up or be broken into crumbs, the only way to get energy or electric charge or the other conserved things into or out of the room is to have them come into or go out of the room.

Each of these conserved quantities has an associated field. Electric charges create an electric field. Energy and momentum and angular momentum each create a distribution of the curvature tensor of space-time. In the non-relativistic limit, this is the mass part of the energy creating the gravitational field.

There is a relationship between the amount of a conserved quantity in our hypothetical closed surface and the net amount of field that penetrates that surface. This is called [https://en.wikipedia.org/wiki/Gauss%27s_law <i>Gauss's law</i>]. Basically, it says that if you add up all of the amount of field that goes out of the surface, and subtract off all of the field that goes back into the surface, the resulting amount will be directly proportional to the amount of the conserved stuff in there. Do you start off with a kilogram of mass inside your room? Then you have a certain amount of gravitational field from that mass that leaks out of the room. Add another kilo for two kilos total, and the amount of gravitational field leaving is also doubled. The distribution of where that field comes out may change, but as long as you don't take that mass out of the room or add more mass in the room, the total amount of field leaking out won't change. And any gravitational field from masses outside of the room that leaks into the room must also leak back out again <i>somewhere</i>. Those masses outside the room will not affect the net amount of field leaking through the surface. What this means is that you can determine the amount of conserved stuff inside a surface if you know the amount of field going through that surface.

This can be visualized in the field line approximation. The field around a positive charge can be approximated as a series of directed lines that radiate away from the charge. The field around a negative charge in this approximation is a series of lines that converge on the charge (for non-relativistic gravitation, you only have positive charges because mass is always positive. We'll just ignore the fact that wormholes require negative energy density stuff in order to exist which will lead to negative mass – we're just trying to get across the basic idea here). Field lines can only ever start on a positive charge and only ever end on a negative charge. The number of lines radiating away or converging on the charge is proportional to the size of that charge (so if you double the charge, you double the number of lines that connect to it). Field lines propagating through space bend away from a positive charge and bend toward a negative charge.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=322>[[File:charges_positive.png|frameless]]
<td width=322>[[File:charges_negative.png|frameless]]
<td width=322>[[File:charges_dipole.png|frameless]]
<tr>
<td>The field lines radiating away from a positive electric charge.
<td>The field lines converging on a negative electric charge.
<td>The field lines emerging from a positive charge and curving around to go into a negative charge of the same magnitude. This configuration, with a positive and negative charge of the same magnitude, is called a <i>dipole</i>.
</table>

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=322>[[File:charges_positive_Gauss_law.png|frameless]]
<td width=322>[[File:charges_negative_Gauss_law.png|frameless]]
<td width=322>[[File:charges_dipole_Gauss_law.png|frameless]]
<td width=322>[[File:charges_exterior_dipole_Gauss_law.png|frameless]]
<tr>
<td>A closed surface encloses a positive charge. A net amount of field lines exit the surface.
<td>A closed surface encloses a negative charge. A net amount of field lines enter the surface.
<td>A closed surface encloses a dipole. The total amount of charge inside the surface is zero because the negative charge cancels the positive charge. No net amount of field goes through the surface - just as many field lines enter as leave.
<td>A closed surface that has no charge inside of it, but various charges outside of it (in this case a dipole). Again, no net amount of field goes through the surface - every field line that enters comes back out somewhere else.
</table>

Now consider the case of an isolated positive charge, like one one shown above, that is far away from anything else. We will bring up one mouth of a wormhole close to the charge. Because the field lines going out of the charge cannot break, they cannot go through the wormhole. Instead, they will curve around the wormhole mouth. As the charge enters the wormhole mouth and travels into its throat, it drags its field lines along with it. With all of its field lines entering the mouth, that mouth looks like it has a net positive charge. As the charge leaves the other mouth, it continues to drag ints field line out of the wormhole - the field lines that start on the charge now curve around to enter the wormhole mouth. The mouth that the charge exited from now looks like it has a negative charge.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Wormhole_with_charges_initial.png|frameless]]
<td width=400>[[File:Wormhole_with_charges_inside.png|frameless]]
<td width=400>[[File:Wormhole_with_charges_passed_through.png|frameless]]
<td width=400>[[File:Wormhole_with_charges_long_gone.png|frameless]]
<tr>
<td>The charge approaches the wormhole.
<td>The charge is inside the wormhole. The mouth of the wormhole that the charge entered looks like it has a positive charge.
<td>The charge has passed through the wormhole. The mouth that the charge entered still looks like it has a positive charge, but now the mouth that the charge came out of looks like it has a negative charge.
<td>The positive charge has now moved very far away. Both mouths of the wormhole still appear to be charged.
</table>

This can also be understood by the continuity equation. Picture an imaginary closed surface around the mouth of the wormhole. This is like the case of the room, before, except that <i>now there is a way out of the enclosed volume that does not pass through the surface</i> – by going through the wormhole! Any conserved quantity that goes into the surface adds its quantity to the stuff in the surface. If it goes through the wormhole, it does not come out through the surface. The conserved quantity inside that surface stays the same! To an outside observer, it looks like the wormhole mouth has that quantity.

Either way you look at it, the results are the same. A wormhole mouth acquires any conserved quantity of the things that enter it, and loses any conserved quantity of the stuff that exits from it.
<ul>
<li> A wormhole mouth acquires the <i>energy</i> of anything that goes into it. Its energy decreases by the energy of anything that goes out of it.
<li> In the non-relativistic limit, this means that a wormhole mouth acquires the <i>mass</i> of anything that goes into it. Its mass decreases by the mass of anything that goes out of it.
<li> A wormhole mouth acquires the <i>electric charge</i> of anything that goes into it. It gains negative the electric charge of anything that goes out of it.
<li> A wormhole mouth acquires the <i>momentum</i> vector of anything that goes into it. It gains the negative of the momentum vector of anything that goes out of it.
<li> A wormhole mouth acquires the <i>angular momentum</i> vector of anything that goes into it. It gains the negative of the angular momentum vector of anything that goes out of it.
</ul>

This perhaps should not be a surprising result – other general relativistic space-time distortions also exhibit this behavior. A black hole, for example, gains the mass, charge, momentum, and angular momentum of anything that goes into it.

You can use these same arguments to find that the same results apply to any asymptotically flat manifolds of space-time. They do not necessarily hold for space-time geometries that are not asymptotically flat. If the gravitational distortion of a wormhole extends infinitely far away, you will not be able to even define its total energy or momentum or angular momentum, so it will be impossible to say if they are conserved. But do note that all of the wormhole examples that we have discussed so far (except for the cylindrical wormhole of infinite extent) are asymptotically flat, and will obey the local conservation of all of these quantities.

===Wormhole dynamics===

A wormhole mouth has the gravitational field and conserved properties of mass, momentum, and angular momentum. By the equivalence principle of general relativity, this must mean that it behaves as if it had those properties. So a wormhole mouth will follow ballistic trajectories in vacuum through space-time as if it were an object in free fall. If it is around a concentrated mass, like a planet or sun, it will have a Keplerian orbit around that object (or, if the wormhole has more mass than the planet or sun, perhaps it would be more reasonable to say that the planet or sun would orbit it). The mass of the wormhole mouth will resist acceleration like any other mass, according to Newton's laws of motion. Force is the time rate of change of momentum, so as the wormhole mouth's momentum changes it will experience forces, and these will change its trajectory in the same way as if that momentum (or forces) acted on a material object with the same mass. And if the wormhole mouth has an electric charge, it will experience forces from electric and magnetic fields; again, these can change its trajectory.

In other words, a wormhole mouth acts like a physical object of the same mass, charge, energy, momentum, and angular momentum.

The motion of the wormhole mouths through our normal space does not affect their relative separation through the wormhole's throat. That can remain the same length regardless of the behavior of the mouths in our normal universe.

As we covered, if an object enters a wormhole mouth that mouth gains the object's momentum. This will give the wormhole a "kick" changing its motion in the direction that the entering object was originally moving. The dynamics work out the same as if the wormhole were a sticky blob and the object collided and stuck to it. In both cases, you get a final object with the combined masses of both initial things (the mouth and the entering object) that has been knocked off course a bit by the momentum of the entering object hitting it.

If that object comes out of the other mouth, then as that mouth gains the opposite of the objects momentum it gets a recoil kick in the opposite direction. Again, this results in dynamics that are similar to things we already know about – like a gun with a bullet loaded firing the bullet. As the bullet leaves, the gun + bullet system loses the mass of the bullet, and as the bullet shoots away the gun recoils in the opposite direction.

So now consider what happens if you direct a constant stream of material through a wormhole. As the jet of material exits its mouth, that mouth loses mass at the same rate as the mass flow rate of the jet. It also continually gains momentum by the recoil kick of the gas. These dynamics are identical to that of a rocket - as the propellant is ejected, the rocket loses the propellant's mass and the recoil momentum from the escaping gas jet pushes the rocket in the opposite direction. So by shooting stuff through a wormhole you can turn it into a rocket, with dynamics identical to a rocket, obeying the [https://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation Tsiolkovsky rocket equation], and everything else that happens with rockets.

===Does gravity go through a wormhole?===

The above diagrams of the electric field of a charge as it approaches a wormhole mouth also hold for the gravitational field of a mass that is near a wormhole mouth. In the same way that the electric field cannot enter the wormhole's interior or leak through to the other side unless the charge enters the wormhole or goes through to the other side, so to will the gravitational field of a nearby mass curve around the wormhole. The gravity of a nearby planet will fall to zero inside the wormhole throat and will not affect those near the opposite mouth.

===Can waves go through a wormhole?===

So if gravity won't go through a wormhole, and static electric and magnetic fields can't go through a wormhole, then what about light? That's made up of electricity and magnetism, right?

So, turns out that light can go through a wormhole. So can other electromagnetic waves like radio and x-rays. So can gravitational waves. The rules for drawing field lines only really work for static charges; for radiating waves you also need to include the parts about how the changing field generates more field. In the end, waves that are smaller in wavelength than the size of the wormhole can go through just fine. You'll get some [[Diffraction|diffraction]] around the mouths, but for wavelengths much smaller than the mouth size even that might not be noticable.

===Can negative mass wormholes exist?===

So a wormhole mouth loses the mass of anything that comes out of it. What happens if you send something through a wormhole with more mass than the mouth which it exits from? This rule says that the mass must become negative, but as we saw earlier this leads to all kinds of problems (especially because the wormhole can also be charged) and is likely to be forbidden by the laws of nature.

If a wormhole's mouth can reach negative mass, you can also get other interesting instabilities. The positive mass mouth will attract matter into it, which will pass through, increasing the mass of the positive mass mouth and making the mass of the negative mass mouth even more negative. Meanwhile the gravity of the negative mass mouth will repel matter, preventing any return flow. Both the negative and positive masses will gradually increase over time.

So what's an aspiring science fiction writer to do? If you have a particular desire for negative mass things in the fiction you create, go ahead and allow it. Just be aware of the consequences. Otherwise, it is probably easier to say that negative mass wormhole mouths can't exist, and justify that with the quantum energy inequalities. Which, of course, brings up the question of what happens when you try to push an object through a wormhole that would make the mass of one of its mouths go negative? Two obvious possibilities are that either you get a back-reaction force that makes it so you can't ever make something go through the mouth to make its mass negative (perhaps by the formation of a horizon as the mass approaches zero), or the wormhole collapses before the mass can ever reach zero. Perhaps inventive science fiction authors and fans can come up with other plausible options as well.

==Wormholes and time travel==

Wormholes connect across both space and time. So it is natural to consider the possibility of using wormholes for time travel.

Now maybe you don't want to have to deal with time travel in your fiction. It introduces all kinds of opportunities for paradoxes, and you have to keep track of plot lines in a self-consistent way that go back and forth through time, influencing themselves in complicated ways. Okay, no problem. Lets just have our wormholes always connect to the same time. Easy right? In fact, it is natural to justify this. Physics is local, so if you can make wormholes you probably have to create both ends right next to each other at the same time. You can then move the wormhole ends away from each other – maybe putting one end on an interstellar spacecraft or something – but they stay at the same time. Right?

Right?

Except ...

In relativity, motion not only affects how fast you go through space, but also how fast you go through time. When things move, they experience time dilation, so if you move one wormhole rapidly far away and then bring it back, the time dilation it experienced will naturally allow it to form a time machine so you can send things and people back to interfere with their own pasts. Even worse, time can also be slowed down by gravity, so if your wormholes are in areas affected differently by your galaxy's own gravity, or around stars of different mass, or just at different elevations on the same planet, the rate they experience time will be different. Temporal complications ahoy!

Let's look at how this works in more detail. We'll take our standard science fiction empire, called The Empire. Like all empires, it has a central area of authority, the metropole (which we will place on a planet called Metropole), which extracts wealth from various colonies and subjugated client states for its own enrichment in the form of taxes, tribute, and forced favorable trading opportunities. Let's give our Empire two colony worlds which we'll call Colony A and Colony B (these Empire folks sure aren't very creative in their naming, are they?). Just for convenience, we'll put Metropole, Colony A, and Colony B all 100 light years away from each other, in an equilateral triangle.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:WormholesTimeTravel graphic1 wb.svg|600px|frameless]]
<td>[[File:WormholesTimeTravel graphic2 wb.svg|600px|frameless]]
<tr>
<td width=400>Metropole and its two colonies A and B
<td width=400>Metropole sending a wormhole to Colony A.
</table>

So Metropole launches a wormhole mouth out to Colony A. They want to get to Colony A quickly, so they'll make the wormhole go really fast. Let's say they fling it out at 99.9999% the speed of light. At this speed, it takes 100.0001 years for the wormhole mouth to reach its destination in the reference frame of Metropole and Colony A. But due to relativistic time dilation, in the reference frame of the projected wormhole mouth it only takes 0.1414 years to go from Metropole to Colony A. Because the techs at Metropole can look through the wormhole, they will see that 0.1414 years after launch it arrives at its destination. At that point, brave explorers from Metropole can go through and set foot on Colony A – they only need wait about a month and a half to feel alien soil under their feet, rather than a century. In Metropole's reference frame, going through the wormhole takes you 100 light years away, and 99.8596 years into the future. Going the other way, from Colony A to Metropole, takes you 100 light years away and 99.8596 years into the past.

[[File:WormholesTimeTravel graphic3 wb.svg|800px|frameless]]

One might think that these time warps would let you engage in all kinds of time travel. It is easy to see that the Metropole—Colony A situation described here doesn't allow these kinds of shenanigans. For practical purposes, you only have a time machine when you can go back to the place you left at a time before you left. And you can't do that here. Go from Colony A to Metropole and you go back in time 99.8596 years. Go back to Colony A through the wormhole, and you go forward in time the same amount, plus any time you spent on Metropole, so you get back after you left. If you go back through flat space-time, it will always take at least 100 years since you can't go faster than the speed of light so you also get back after you left. No paradoxes for you!

However, it is easy to imagine situations where a wormhole, or a configuration of wormholes, does make a time machine. For example, what happens if we immediately turn around and send the wormhole mouth from Colony A back to Metropole? In the reference frame of Metropole, the wormhole comes back with its "Return to sender" sticker 200.0002 years after it was launched. But going through from the left-behind mouth to the round trip mouth will take you 199.7192 years into the future. And if you step into the round trip mouth, you will go back in time by 199.7192 years into your own past. You now have a real honest to goodness time machine, complete with paradoxes.

[[File:WormholesTimeTravel graphic6´4 wb.svg|800px|frameless]]

Or maybe the wormhole mouth stays at Colony A. And Metropole sends another one to Colony B, that also goes 99.8596 years into the future. Now Colony A sends a wormhole to Colony B. This wormhole also goes 99.8564 years into the future as a consequence of its trip. This means if a traveler at Colony B went through the Colony A wormhole he would go back in time 99.8564 years. Then going from Colony A to Metropole he would go back in time another 99.8564 years. Then he could go from Metropole to Colony B and go forward in time 99.8564 years. The net result is that he ended up back where he started nearly a century before he left.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:WormholesTimeTravel graphic5 wb.svg|600px|frameless]]
<td>[[File:WormholesTimeTravel graphic6 wb.svg|600px|frameless]]
</table>

And it is not just motion that causes time dilation. Being different depths into a gravitational potential does this as well. If Colony A is deeper into the gravity well of the galaxy, or orbits a heavier star, it will experience more gravitational time dilation than Metropole. So eventually it will build up more than 100 years of time slip between the two end of the wormhole and you can have a time machine.

So is time travel inevitable if you have wormholes? What a nuisance!

Well, maybe not. Here's why. Think about what happens when the Colony A – Colony B wormhole has gone just far enough that a light signal going through the wormholes can get back to where it left just as it is leaving. Now, since the propagating signal and the newly transmitted signal are both leaving at the same time, you have double the intensity. So this doubled intensity signal goes around and meets itself again, quadrupling its intensity. And so on. At this point, just as the configuration is on the verge of becoming a time machine, it becomes a perfect resonator for light signals, which then build up to arbitrarily high intensities until something breaks and you don't have an incipient time machine any more.

Now, this won't always work if you just consider light signals. Light can get defocused or sent off in odd directions or something so that you don't have a good path back until after you end up with a time machine. But if you look at what happens in semiclassical quantum gravity, will get a buildup in the amplitude of quantum fluctuations as soon as you start to get a time machine, called <i>vacuum polarization</i>. These fluctuations are expected to build up to sufficient amplitude as to destroy the time machine to be, at least for the few simple cases studied. Or perhaps it just causes the wormholes to bounce away from each other so as to prevent time machine formation. This has not yet been shown to occur in general. But to does seem to happen for all possible configurations of one or two wormholes about to become time machines
<ref name="Visser_Lorentzian_wormholes"></ref>
.

But let us assume for the moment that wormholes cannot form time machines in any configuration due to this vacuum polarization mechanism. Now there are limits on just how you can place and move your wormholes. You don't want to accidentally almost form a time machine and break your transportation network. There are several ways to do this:
<ul>
<li> Move your wormholes slowly, so as not to build up much time dilation (or <i>time lag</i>, if you will).
<li> Don't form closed loops in your wormhole transportation network. The less distance you need to travel to go all the way around a loop to get back where you started, the less time lag is needed to make a time machine.
<li> "Discharge" the time lag by periodically rapidly moving the mouth of a wormhole that is "behind" in time in such a way that it comes back to where it was. Perhaps you can put a charge on it by sending a [[Particle_Beam_Weapons|particle beam]] through it, and then accelerate it up to relativistic speeds in a synchrotron for a while.
<li> Send new wormholes that you are going to place through the existing wormhole network. This way they acquire the same time lag across their mouths as all of the wormholes they traverse. Until they build up some additional time lag of their own, you provide a perfect return trip without making a time machine.
</ul>

In conclusion, if you want a story or setting that involves time travel, you can plausibly still use wormholes somehow to allow that (although it might take loops of several of them). If you want a story that avoids the complications of time travel, you can still plausibly have wormholes in that setting (although you now need to beware of the limitations on your wormhole transportation network imposed by the no time machine rule).

==Practical implications==

===Getting places===

Usually in science fiction, people want wormholes because they want to be able to get to cool places far away without taking forever. Very often, this involves the technology (or the protagonist, with some kind of psychic power) "opening" a wormhole from the current location to the desired location when there wasn't one before.

Sadly, the ability to just create a wormhole to anywhere you want to seems unlikely. Physics tends to be local, which means that if you can make a wormhole both ends will emerge right next to each other. You will need to move the ends to where you want them to be using other methods (mailing them by post, putting them on a spacecraft, etc.). As was mentioned earlier, if you shoot a stream of stuff through a wormhole, the end that the stuff comes out of becomes a rocket. So one way to move an end of a wormhole is to shoot stuff through the wormhole. Because you can leave all the big heavy equipment at home (generators, rocket motors, cooling systems, and everything else), if you can make your wormhole light enough this could be a low cost and efficient method of getting it places. Remember that you can exploit time dilation to make it seem like you can get someplace in much less time than you would expect light to take going to that place (at the expense of taking you some time forward in the future as well when you step through). If you can get your wormhole mass down to a few grams, or even several kilograms, using powerful lasers to shoot through the wormhole to make a photon rocket could be a highly efficient way to go places at relativistic speeds. Perhaps it could even fuel itself by collecting mass from the interstellar medium (ISM) as it flies along, and it might be able to brake against the ISM or solar wind of the destination star system to slow down, reducing mass loss even further.

===Keeping time===

We have already shown that wormholes connect across both space and time, so that a trip between star systems could take you hundreds of years into the future, and the return trip takes you hundreds of years back in time. And this is even before we throw in how time slips between planets when considering relativistic time dilation due to different speeds and gravitational potentials.

Fortunately, all the weirdness of different time rates and going backward and forward in time can be ignored by the average person. This is because you never need to go from one world to another, or back, across the vast gulfs of interstellar space. You just take the wormhole between them. All you ever need to worry about is the coordinate frame that goes across the wormhole. When considering this reference frame, you're not hopping all over the place in time. If it takes ten minutes to cross the wormhole between the two planets, when you get to your destination world the clocks will read ten minutes later than they did when you left your departure world. By coordinating their time-keeping across the wormhole network, all of the worlds of the network can agree on a common time to coordinate their activities. This is all travelers ever need to worry about, and they can then ignore all the relativistic weirdness. Your network engineers will still need to keep track of relative time drift and how close a given configuration is getting to a time loop. But unless your protagonist is a network engineer, they can just ignore all that stuff. And, as an author, so can you! Assume your engineers are competent, you have good regulatory bodies and standards institutions, and don't worry about any of this "time travel" that doesn't actually let you cause paradoxes.

===Keeping wormholes away from planets===

By now, people expect science fiction to have spaceships. Wormholes offer a convenient excuse to have spaceships, because it provides a way for those spaceships to get to other worlds that are not in our solar system. But if you have wormholes, sometimes you need to ask why you still have spaceships? Why not just put one end on Earth and the other on an alien planet and step through to get where you are going?

Sometimes there are good reasons for this. If your wormhole has as much mass as Jupiter, you are not just going to keep it in Topeka, Kansas. That would kind of end the Earth. So you would put it in orbit around the Sun, and send spaceships from Earth out to the wormhole and through it to wherever it goes.

But all too often we see authors using special pleading to pigeonhole in their spaceships without thinking about good physics. Things like "it can't work in atmosphere" (use an airlock), "it can't work within a certain number of planetary radii" (why? There is no physical plausibility to such a restriction), "It can't work in a gravitational field of more than X" (violates the equivalence principle), and so on. But if you insist on having spaceships, and you also insist on wormholes that mass less than a planet, there are a few plausible ways to justify your preconceived ideas that at least plausibly align with physics.

For example, the closest you can plausibly get to "X number of planetary radii" or "gravity field less than X" is a criterion that relies on local measurements of space-time curvature only. And it turns out that this quantity is the tidal forces (curvature is a direct measurement of tides). Tidal forces are going to be proportional to the mass of the nearby thing causing the tides, and inversely proportional to the cube of the distance. If <math>R</math> is a measure of your space-time curvature, <math>M</math> is your object's mass, and <math>r</math> is the distance from the object to the wormhole, then

<div align=center><math>R \propto \frac{M}{r^3}</math>.</div>

So if you want your wormholes to be at least ten planetary radii from Earth (<math>r = 64 \times 10^6</math> m, <math>M = 6 \times 10^{24}</math> kg), then the curvature limit is <math>R = 23</math> kg/m³ in some convenient system of units where we don't have to include constants of proportionality. But now suppose you want to send a <math>26 \times 10^6</math> kg ore freighter through that wormhole. You'll only be able to get your freighter within about 100 meters of the wormhole before the freighter's tides collapse the wormhole. This <i>might</i> work if the freighter were really long and skinny, much longer than 100 meters, to reduce its transverse tides; or if the wormhole was much larger than 100 meters in diameter. But in general, note that the curvature and tides have units of density, so if the average density in any volume around the wormhole exceeds the critical density – oops, there it goes.

Also, having a limit on the geometric curvature of space-time similar to what you get around mere planets doesn't end up being very plausible – the curvatures in the wormhole itself exceed that by so very many orders of magnitude that having so stringent of a curvature tolerance seems very implausible, and just dancing on the edge of disaster. For example, if your wormhole is about 100 meters across, its curvature in these same units will be approximately 500 billion trillion kg/m³. Keeping the tides to within 23 kg/m³ is the equivalent of an engineering tolerance of one part in 25 billion trillion.

So if you don't really have a good way to limit where you can place a wormhole based on the nearby "gravity", are there other ways you can exclude them from the vicinity of planets?

Safety might be one good reason. We have already discussed various ways a wormhole might collapse (forming a time machine, having its mass go negative). What happens when a wormhole collapses? If there are any parts of the wormhole or things in the wormhole that are charged or made of charged particles (like ordinary matter) you could justify a considerable fraction of the wormhole's mass-energy being converted to electromagnetic radiation. In addition, if the wormhole collapses into a black hole it will evaporate away its mass-energy primarily in the form of electromagnetic waves as it undergoes Hawking radiation. Large black holes will last a long time, but if the black hole left behind has a mass of 500 tons or less it will last less than 10 seconds. And 500 tons worth of energy delivered in 10 seconds would do quite a number on a planet. These sorts of considerations could lead to a regulatory environment that require wormholes to be located several light seconds away from inhabited places (like planets).

===Wormholes on planets===

But sometimes you might want to have the wormhole on a planet. Perhaps you are envisioning one of those circular or oval fantasy portals, looking like a window to an exotic foreign land, and when you step through you are actually there. We already know that thin-shell wormholes can be very low mass, and can have a circular shape. So that should work, right?

There are at least two issues to consider here. One is that even if both wormhole ends are on the same planet, natural variations in atmospheric pressure due to weather will lead to very significant differences in air pressure from one side of the wormhole to another (unless the wormholes are so close that they are within a small distance of the extent of the same weather system – say, within a few km of each other). This pressure difference will drive severe winds through the wormhole, usually comparable to the winds of a hurricane (although there will be periods of relative calm when the pressure at both wormholes just happens to be at the same pressure). This effect will only be exacerbated if the wormhole mouths are at different altitudes, due to the decrease in air pressure with altitude. And if the wormhole connects between different worlds, the effect will be much worse. In addition to making travel inconvenient and possibly damaging the equipment, the winds will produce large uncontrollable mass flows across the wormhole (air on earth has a density of about 1 kg/m³, so if you have 50 m/s winds blowing through a 2 meter diameter portal you have a mass flow of about 160 kg/s through your wormhole). This will make it difficult for mass balance, and keeping both ends having high enough masses that neither gets so close to zero that the wormhole breaks or forms a horizon or goes negative mass and causes a runaway planetary catastrophe or whatever wormholes do when one end gets driven to near negative mass by the stuff leaving it.

As a consequence, planetary wormhole mouths will probably be kept in airlocks.

The other issue is that different latitudes on a planet will be rotating at different speeds; at the poles they won't be moving at all while at the equator they will be going a full planetary circumference every day. This means wormhole mouths at different latitudes will be building up different time dilations with respect to each other. In addition, wormholes at different altitudes will experience different gravitational time dilation rates. If you let this go on for long enough, the differences in time across the wormhole mouths will form a time machine. Whether this collapses the wormholes, makes them "bounce" apart from each other (likely ruining a lot of property and infrastructure in the process), or just forms a time loop that leads to various inconvenient paradoxes depends on the assumptions you made for the physics of your world. But it will probably be something you will want to avoid. Re-balancing the time differences between wormhole ends will probably need to be done every few decades.

===Wormholes on spaceships===

The necessity for radiators on spacecraft is really annoying. Dealing with all that waste heat is not fun, and these issues are not awesome kewl stuff like gigawatts and megatons and hundreds of g's of thrust and other things that let you show everyone how much your spaceships totally rule and everyone else's spacecraft totally drool. Instead you need these big fragile stupid-looking things hanging off the side of your ship, ruining your aesthetics and efficiency and being vulnerable to your enemies blowing them up, and lighting you up so bright that you're easily visible from the other side of the solar system. Hey! I know! Let's put a tiny wormhole on our spaceships so we can just send all the heat someplace else. Great idea! That'll work, right?

Actually, yes. If you have small enough wormholes, it will work great. But there are consequences.

So you're running your coolant loops from your reactor through the wormhole to some cooling towers on your home planet. Much better than radiators. But – why do you still have that reactor on your spaceship? It's big and heavy and needs lots of big and heavy shielding and poses nasty radiation hazards. Surely you could leave it behind on your home planet as well, next to those cooling towers, and just run a power line through the wormhole? You'll be able to run a bigger, more powerful reactor that way anyway, and that extra weight won't be bogging your ship down.

And why does your spaceship have its lasers on board? You can have a much bigger laser on the planet that you don't need to lug around. Just shine the laser beam through the wormhole and re-direct it with mirrors.

And all of your missiles? Might as well just feed them through the wormhole when you need to fire them.

And then do you really need all that life support equipment? Just run an HVAC and plumbing that sends fresh air and water to the spacecraft and takes used air and water off. And remember to send the occasional snack, too.

Oh, but wait. Why is all your crew even on the spacecraft again? They have a wormhole to the spaceship. You can just have all your crew run the spacecraft remotely from a mission control station. Most of your equipment is on the planet anyway, so most of the engineering staff is already here. Now you don't actually need to put people in harm's way. If someone does need to fix something just put them in a space suit and send them through, let them do their job, then bring them back. Most of that can be done by a tele-operated robot, in any event. And being able to go back to their families when their shift is over is great for morale.

Hmmm, your sensors could just look through the wormhole instead of having all the vulnerable and heavy and expensive equipment actually on the spacecraft. And as long as you are doing that, you can just leave your laser beam pointer at home too, and aim the laser by pointing it through the wormhole.

So now what's actually left in your actual spacecraft? Oh, the rocket itself. But remember how we already talked about how we could just keep the rocket engine and propellant on the planet and shoot the rocket jet through the wormhole, and it will move just like a rocket? Yeah, so no rocket either.

Congratulations, you've successfully gotten rid of your spaceship. The wormhole itself is your spaceship. It's all you need, it's safer, it's more capable, it's stealthier, and just better all around than some contraption of steel and Mylar and fissioning uranium fuel rods and fragile people going through space.

===Adding mass to wormholes===

So you just made a pair of wormholes. Chances are, you pulled them apart as near Planck mass objects, with about 20 micrograms of mass. Well, that's not very useful! What if you want to send something through that's bigger than 20 micrograms? You're going to need to add mass to your wormhole mouths.

A similar issue comes up if you project a wormhole at some far away planet or star system using the wormhole-rocket trick already discussed, maybe by using powerful lasers to make a photon rocket. To be practical, your projected wormhole mouth will need to be very low mass. But once you get it to where you want to go, you'll need to give the mouth enough mass to let your explorers and colonists and equipment through.

A simple way to do this is to simply suck stuff into the wormhole. This adds mass to the wormhole mouth the stuff comes through. Of course, if you take it out from the other side, that other mouth will lose that much mass, and if it didn't have than much mass to begin with you'll likely have problems. So just leave that stuff in there! Shove it off to the side. Let it clutter up the unused areas out of the main traffic route in the way that basements and attics and the unused spaces of garages accumulate all that old junk that you never use any more but don't want to throw away. Build a wall inside of your wormhole, so travelers don't have to see all the asteroidal rock or super salty water or very surprised natives or whatever it was you scooped up on your alien world. Shove it into the undesirable high curvature places with nasty tides. This wall can also help keep the legitimate passengers away from the inner working of the wormhole, the strange energies that support it, the cables and pipes and crawlspaces that only your maintenance workers should access anyway, and the extreme space-time curvatures that would rip the passengers to shreds if they ever were directly exposed to them.

===Tiny wormholes===

Wormholes that are too small to fit people through could still be useful for sending information. You could use them to rapidly communicate over long distances. If the wormholes were light enough and cheap enough, they might even replace cell phones and laggy land lines with super high bit rate connections.

You could also send energy through tiny wormholes, in the form of light or electric current. Storms that knock down power lines no longer need to cut off your home's electricity. You could also use them to power machinery that need light weight and high power output – electric transportation like airplanes, shipping, or trucking; directed energy weapons; or high specific impulse rockets.

===Causality denial, causality attacks===

We have shown that moving wormholes around can lead to time travel, and that there are plausible physical mechanisms that can prevent time-travel allowing wormhole configurations from happening. People being people, you just know that if this is possible, that it will not be long before these effects are used in politics and warfare.

In a network of wormholes that is about to create a time machine, one likely resolution is that things break until a time machine is no longer possible. In any chain stressed to its point of failure, it is the weakest link that breaks. So in a path through wormholes and normal space-time that is about to form a time machine, it is the weakest wormhole that will collapse (or otherwise have something bad happen to it). So if your rival polity has a wormhole that allows then easy trade with a rich world, and you want to edge in on that trade and simultaneously deny that trade to your rival, you can just send a bigger, stronger wormhole in such a way that forces a time loop. Now your rival's wimpy wormhole breaks, and you get access to the trade opportunities with the world! This is the basics of a causality attack – use the causality-enforcing properties of wormholes to attack your enemy's wormhole infrastructure.

But what if your rival polity anticipated this? What if they have strong wormholes, with an arrangement in space-time that blocks off easy access of your wormholes to a profitable region of space? This is a causality denial action, or a causality fortress. By controlling the time displacement of the wormholes in their network, a polity can control what additional connections can be made. This can be used to enforce borders, prevent infiltration, or restrict civilian wormholes to legitimate uses.

===Moving one wormhole through another wormhole===

If you have a wormhole mouth, you can move it through another wormhole as long as the mouth (and any supporting structures) is small enough to fit through the throat of the wormhole it is going through. The geometry all works out. In fact, this has a rather nice side-effect. If you are looking through the wormhole being moved from back home, you only see it take as long to go through the other wormhole as a wormhole transit usually takes even if it got projected far into the future or past. So the moved wormhole acquires the same time differences as the wormhole it just went through (added on to any time differences it already had, it doesn't get reset or anything). This can let you easily build wormhole networks that connect to themselves in loops. It also makes the network more susceptible to breaking, because there is much less leeway for error with a closed loop as far as making a time machine and breaking the loop. So it can be done, but there are consequences.

===Dropping a wormhole into a black hole===

In principle, if you have a wormhole connection to the inside of a black hole, you can get information out from inside the event horizon. The theorem that prevents things from escaping a black hole was made under the assumption that the energy can't go negative. And as we've already seen, wormholes require regions of negative energy to work. You still wouldn't be able to see further toward the singularity at the center than the wormhole's location. And the wormhole would be inevitably dragged into the singularity in a finite amount of time. But, in principle, during that time you could see what is happening in the black hole.

But there's a big catch with this. You can't actually get the wormhole into the black hole. Not if you assume that time machines can't form, anyway. From the point of view of the wormhole mouth falling into the black hole, it will pass through the event horizon in a finite amount of time. From the point of view of someone watching from outside, the infalling mouth gets more and more time dilated to the point that it never crosses the horizon. This time dilation sets up the conditions for a time machine, and any time machine preventing physics happens on the wormhole before it can ever get in.

== Possible Networks ==
=== Acyclic wormhole networks ===

Recall that wormholes are space-time connections. They bridge space as well as time. Assuming we want to build this network in a stable manner, we want to avoid Closed Timelike Curves from appearing. Let us start with some simple anaylsis of consequences on this. Assume projected wormholes have one mouth sent at relativistic speeds to a distant target and the other mouth is kept at rest at home. From the perspective of the dispatching party, this will lead to the new connection opening very quickly. Done again and again, large volumes of space can be accessed in a short time, a powerful advantage.<br />

[[File:Wormholes PartialOrder.svg|400px|thumb|The partial order relationship of the wormhole connection]]

Every projected connection thus extends onto the future. Then we must avoid forming loops, or more generally "backtracking" back towards a dispatched node. This nets us a graph with no closed loops - an acyclical graph, also called a tree. Since connections may not return to previously established nodes if they are time-shifting from the past into the future, we also get a partial order: nodes <math display="inline">u</math> that dispatch wormholes have a higher order over the nodes <math display="inline">v</math> they establish: <math display="inline">u < v</math>. This lets us establish a direction on the whole graph. For a particular node <math display="inline">n</math>, its '''parent node''' established it, <math display="inline">parent(n) < n</math>, and it in turn establishes '''children nodes''', <math display="inline">n < child(n)</math>. The node that only has children nodes is the tree's '''root node'''. Nodes that have no children are '''leafs''' of the tree. The trees '''height''' is the longest outward path one can take from the root to a leaf.<br />

In terms of traversal, such nodes have some interesting and useful properties. Since wormhole traversals may be assumed to be effectively instantaneously bridging large distances, travel time is mostly defined by the travel distance between wormhole mouths. And as such, we want to minimize the amount of transits we have to make to get from a node <math display="inline">A</math> to a node <math display="inline">B</math> the fastest. It turns out, in order to achieve this, we want to have as large a choice of connections at each node. In the ideal case, any node except root can be reached in two transits: one transit to root, and from there to the other node. In the less ideal case we have to make one further stop-over, choosing from a number of connections at a second node to reach our destinations. The less ideal case from that sees us make two transits through such "switch yards". Assuming each node except the leafs in such a wormhole network has exactly <math display="inline">k</math> connections and there are <math display="inline">n</math> nodes in the whole network, we can reach any destination from root in <math display="inline">\log_k (n)</math> choices of wormhole transit. This is a powerful way to get to connections quickly through centralization. (In computer science, such k-nary trees are of great interest for quickly search-able data structures for this reason.) Configurations that are less "balanced" net appropriately lesser advantages and longer transit times.<br />

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Wormholes AcyclicGraphExample.svg |600px|frameless]]
<td>[[File:Wormholes Trees.svg|600px|frameless]]
<tr>
<td width=400>An example of an acyclic wormhole network in space-time.
<td width=400>The connected tree graph of the network exampke, and a k-nary tree example.
</table>

Also, traffic to most other nodes ''has'' to go through a root-wards node. This gives these nodes significant presence in the larger network. Routing through them is unavoidable if you want to reach most of the rest of the network. Root nodes can profit from taxation, services to travelers, control of routing infrastructure for information, and so on. But there is also a catch. Assuming leaf nodes each send the same amount of traffic inward. The amount of traffic load increases polynomial as you go towards root. Your wormhole infrastructure (and what transports cargo between wormholes, in space or on planets) has to cope with this scaling. Congestion might become an issue.<br />

=== Mesh wormhole networks ===

The acyclic network may have some bothersome problems. The central routing focuses power and can keep transit times down quite significantly depending on the specific network infrastructure, but it’s also congestion-prone, and some close locations may be widely separated on the network. Or maybe you want to avoid centralized routing dependencies for political reasons. Root might like being root; everyone else being subject to root’s whims, not so much. So, can we build networks where there are two routes between the same two nodes? We can, but it comes with some necessary adjustment.<br />

Recall why the acyclic network is acyclic: to avoid forming CTCs due to the time connection between future and past from the projected wormholes. But if we let both wormhole mouths experience the same time dilation, the problem that ultimately leads to CTCs - one connection routing displacing further into the future than the other connection, and thus allowing one to arrive at a node before ones departure - is avoided entirely, since there is no time displacement to be accumulated. Thus we get meshes, where there may be many connections and cycles between nodes. An observer at one node watching for his transition from another node 10 light years away would have to wait for ten years to watch his light arrive. Closed timelike loops are not possible so long as this network is kept balanced.<br />

[[File:Wormholes MeshNetwork.svg|thumb|center|700px|An example of such a mesh network]]

The disadvantage - this network must take great care to ensure that the time dilation that inenvitably occurs due to different star velocities and the galactic gravitational field don’t accidentally create a time shift and CTC in the network. And also we cannot exploit time dilation to open connections very quickly, from the home node’s perspective. One has to wait out the full years it takes the wormhole’s far mouth to get to a remote destination, before the connection can be used.<br />

=== Semi-privileged wormhole networks ===

The interesting thing is, it is very much possible to build networks that make use both of relativistic projection to seemingly accelerate the network construction time, ''and'' contain cycles, with alternate routes to get to another node. The acyclic construction is a strategy to avoid CTC formation, but we should remember the ultimate constraint reason: we avoid cycles so that there are no connections which, through a certain time shift on them, a CTC could be formed. Acyclic construction is one strategy. Only building cyclic connections with careful time shift balancing is another!<br />

The foundation for such networks is an acyclic "base graph" that may be isolated from the larger network graph. We also call this the '''red graph'''. The base graph maintains all the partially ordered properties of the simple acyclic graph. We can call this the "privileged" graph, since its nodes and their connections define the principle positions in space-time of the network, and we must avoid generating CTCs on this definition. How do we avoid forming CTCs but still create alternative paths and thus loops?<br />

Let’s assume we have two nodes <math display="inline">A</math> and <math display="inline">B</math> ten light years apart. They both had connections established from a parent node <math display="inline">P</math>. Those connections are time-shifted over 99 years and part of the privileged base graph.

We couldn’t build a time-shifted connection between <math display="inline">A</math> and <math display="inline">B</math> without causing a CTC to be formed. But we are entirely fine sending signals '''Fast-as-Light''' from <math display="inline">A</math> to <math display="inline">B</math> and back. This is not a violation. Now consider: how does a wormhole with no time shift across it behave? In the mesh network case, such connections don’t cause CTCs to appear. And it doesn’t cause one to appear here either. (A/N: this could be illustrated even better with a space-time lightcone illustration I am prototyping.) How can we build such connections? We can of course use projection with relativistic speeds at both ends, like how we would build Mesh networks. But there is an even faster solution. If we can move wormholes through wormholes, we can shift the new mouths through the existing network to create this connection!<br />

The main catch is that we have to tightly watch out for temporal drift on such new connections. Through adding such edges that overall form a "blue graph" on top of the red graph, we are creating loops on the red graph underpinning the network. If one end of a wormhole mouth drifts more in time than the other (which will inevitably happen because stars won’t have the same mass, experience the same galactic gravity, or have the same velocity relative to each other) the conditions for a CTC will start to appear. The only way to pre-empt this is with continuous maintenance, taking connections offline and adjusting their temporal position using synchrotons, and having "safety space-time" spacing between wormholes. In this way, there is some "room" where signals can arrive faster and faster, but not before they were sent, allowing the connection to be used most of the time with only short intervals of adjustment.<br />

This works even when both ends connected don’t have the same time distance from the root node - the important thing is that the new shortcut accomodates this difference with a time shift of its own. We can calculate this difference easily. For two non-root nodes <math display="inline">A,B</math>, for each node we sum the temporal difference over the path from root to the nodes along the red graph, then subtract the difference from each other. The resulting remaining temporal difference is that the new connection on the blue graphh must have to keep the network stable. This means we can even do things like connect systems far out directly to core systems, or bridge from one edge of the graph to the other - arbitrary routings are possible. These routings can allow faster signal transmission and mass transport than through root, mitigating bandwith issues. The main problem is to watch out for network stability. As nodes accumulate, CTC avoidance becomes more and more delicate, and safety spacing must be maintained diligently. This can add to the effective travel time through such alternate connections, and thus maintains some of the "privilege" of the acyclic base graph.<br />

Arguably this network configuration is the most powerful, marrying many advantages and removing some disadvantages. It is however also bound to the need for safety spacing. Depending on your infrastructure assumptions, this can pose issues. If you want to implement serious stretches of safety space, wormhole links not part of the acyclic base graph would likely have to be located at interplanetary distances away from the main links, with light hours to light days of safety spacing. If your main network runs on planetary trains, this is rather a bother. Adjusting the internal throat length of wormholes can help in this regard, but of course now poses the issues of how to safely navigate such a long wormhole. Maybe you could extend something like space elevator tethers through such stretchy throats. With no gravity inside the wormhole itself, such cables could probably stretch quite significant distances. If your culture is handling wormholes mostly in space, you still have to account for added travel times through the safety spacing. Communications links would probably be easier connected between arbitrary nodes than macroscopic, craft-traversable wormholes.

==Credit==
Authors: Luke Campbell and Sevoris

==References==

File:Wormholes MeshNetwork.svg

2022-08-21T08:49:10Z

Sevoris: An example of a mesh architecture wormhole network Created by Sevoris Doe for Galactic Library.

== Summary ==
An example of a mesh architecture wormhole network

Created by Sevoris Doe for Galactic Library.

Wormholes

2022-08-21T08:48:36Z

Sevoris: /* Acyclic wormhole networks */

Wormholes are hypothetical structures in space-time allowed by the general theory of relativity.
They provide short cuts through space-time that connect one region of space and time to another; potentially, these regions can be very distant from each other.
The symmetries in space-time of our universe mean that wormholes move like physical objects, and acquire the conserved (or in some cases approximately conserved) quantities of things that go into them and lose those quantities of things that come out – properties like energy, mass, momentum, angular momentum, and electric charge.

==What is a wormhole? The geometry of space-time and how to twist it into a pretzel==

We're going to have to start out with some pretty heady stuff. Like the very nature of existence. And what is reality, anyway?
Usually, space and time are so much an integral part of our existence that we don't even think of them at all. They simply form a backdrop on which all the interesting stuff plays out. But when we do think about them, we're so accustomed to living in a world where they exist and where distances are well defined and events can be uniquely identified based on <i>where</i> and <i>when</i> they happened (or are happening or will happen) that a lot of stuff seems so obvious that we don't stop to think about if it really has to be this way or if there are other possible ways of doing things.

===A toy model===

So lets consider a hypothetical scenario, where an ant can be in any of several states, represented by the squares in the pictures below. The ant can transition to any other state that has a numbered side with the same number as the state it is in. We will start on the left. If the ant chooses the "2" transition, we end up with the picture in the middle. If it then chooses the "10" transition, we end up with the state on the right.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Space_elements_1.png|frameless]]
<td width=400>[[File:Space_elements_2.png|frameless]]
<td width=400>[[File:Space_elements_3.png|frameless]]
</table>
But now notice that the behavior of this system is exactly the same as if we re-arrange the states into a grid, as shown below on the left. The disconnected states now become a space; an expanse of coordinates on which the ant can exist. Now, the ant's transition between states can be described as a trajectory as shown in the right picture below.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Space_elements_grid.png|frameless]]
<td width=255>[[File:Space_elements_grid_path.png|frameless]]
</table>

This gives us an inkling of how something like space-time might end up being assembled from individual events. But what if we don't have things match up into quite such a perfect grid? We
ll take our space elements and tile them into an infinite grid. Two sections of the grid are shown below. But now we remove two of those space-tiles, and we connect the edges adjacent to the removed tiles with a corresponding edge of the other removed tile (although, as shown in the figure, we don't have to keep the same orientation). Now the ant, entering side 1 (which we have color-coded red for convenience) comes out of side 1 of the other missing square. This is perhaps the most basic wormhole.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Space_elements_wormhole_grid_1.png|frameless]]
<td width=400>[[File:Space_elements_wormhole_grid_2.png|frameless]]
</table>

Of course, this simple assembly only stitches together a space. But the world we live in is one of space-time. As the ant transitions from one square to another, it is also transitioning further ahead in time. But when the ant passes through the wormhole, while it locally experiences no discontinuity that wormhole might connect to a different time as well as a different region of space. So the ant might emerge far in the past or future of when it stepped through.

===Manifolds and coordinate patches===

The toy model used above hopefully prepares you for some of the ideas to follow - one of how we can piece together a complete space and time by connecting together different pieces. But before we go further, lets define an important term: a <i>manifold</i> is a "shape" of a given number of dimensions where, if you zoom in close enough, you can always get to a scale where it looks like the shape is flat. We'll use this definition to help us describe how we can put space and time together to get a wormhole. But first, let's look at a few illustrative examples.

====One dimension====

The simplest one dimensional shape is a line. And a line is everywhere flat. So a line is a one dimensional manifold.

A one dimensional shape might seem to be a wiggly path if inscribed in two or more dimensions, but it has no <i>intrinsic</i> curvature. That is, if everything about your world is confined to that path, you can't tell that it is curved – all you can do is measure the distance you go along the path. This greatly limits the kinds of shapes that we can consider. So any curved path of infinite extent looks from the inside like a line, and so any infinite length path is also a manifold.

But there's another class of one dimensional shapes that are manifolds – loops. A loop is a path of finite length that connects back with itself. You can make one by drawing a closed curve in a higher dimensional space. Or you can take a straight line but then say that a given point <i>here</i> maps on to another point <i>there</i>, and all the points between <i>here</i> and <i>there</i> form a loop. When you are marching along the line starting at <i>there</i> and you reach <i>here</i> you are also at <i>there</i>, so if you go further you just end up back where you were. This last bit may seem like a pointless bit of pedantry that just confuses a simple problem, but it will be a very useful way of looking at things as we go on.

====Two dimensions====

The simplest two dimensional manifold is a plane. A plane is everywhere flat, and extends off to infinity. A small region around any point on a plane (naturally) locally looks like a plane, so it meets our definition of a manifold.

=====Curved manifolds=====

But in two dimensions, we can have something that we don't have in one dimension – intrinsic curvature. Consider a sphere. If you get close enough to the sphere, it will look like you are on a plane; we live on a world that is (approximately) spherical (if you neglect mountains and ocean basins and all that other stuff), but when you stand up and look around you it mostly looks flat. So a sphere is a manifold. But take a line on the surface of your sphere. Extend it out along the surface, drawing it as straight as possible. Eventually, it will go all the way around the sphere, dividing it in two, and meet itself. This path is called a great circle. The Earth's equator is (approximately) a great circle, as are the lines of longitude (but not lines of latitude, because they do not divide the Earth into two equal-sized halves). The straightest-possible path on a curved surface (like a great circle on a sphere) is called a <i>geodesic</i>. But to keep the jargon down, we'll just call them lines for this non-technical presentation, and hopefully this will keep things more intuitive and understandable to the layman.

In flat space, two lines that are initially parallel remain parallel, always remaining the same distance apart. But in a curved space, initially parallel lines do not remain the same distance apart. On a sphere, for example, any two great circles will intersect at two points. A sphere thus has intrinsic curvature.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Sphere_and_great_circles.png|frameless]]
<td>[[File:Hyperbolic_surface_and_lines.png|frameless]]
<td>[[File:Cylinder_and_lines.png|frameless]]
<tr>
<td width=400>Two great circles going around a sphere. Although initially parallel, they meet at two points.
<td width=400>Two initially parallel lines on a hyperbolic surface. As you get farther from the place where they are parallel, the lines diverge.
<td width=400>Lines that are initially parallel on a cylinder remain parallel everywhere.
</table>

On a sphere, parallel lines always end up converging. One example of a curved surface where parallel lines diverge is a hyperbolic surface. Like a sphere, a hyperbolic surface is a manifold. But it has the opposite sense of curvature.

Now consider a cylinder. Again, a cylinder is a manifold. A cylinder can be made by rolling up a plane. Like our construction of a loop, this can be done by simply taking a long rectangular section of a plane and saying that two of the parallel edges are actually the same (this is essentially what you do if you roll up a sheet of paper so that two of its parallel edges touch, and then pretend that they are merged together where they meet). This "rolling up" preserves distances, so any two parallel lines on the plane remain the same distance apart on the cylinder (think about that sheet of paper, and if you drew two parallel lines on it - they will stay the same distance apart when the paper is rolled up). A cylinder has <i>extrinsic</i> curvature – you can tell that it is curved when looking at it from three dimensions. But it has no <i>intrinsic</i> curvature – if you are confined to the surface of the cylinder it behaves locally as if it is flat.

A manifold that is flat obeys all the rules of Euclidean geometry (although you'll need to go to the three dimensional case for dealing with volumes), and the standard Cartesian coordinate system (representing position by two orthogonal coordinates, commonly called <math>x</math> and <math>y</math>) can be used to specify any point. A manifold that is not flat violates Euclid's fifth postulate; the geometry of the manifold will be <i>non-Euclidean</i> (somewhere, H. P. Lovecraft is quietly having a mental breakdown). It cannot be represented by Cartesian coordinates, but different sets of coordinates – or perhaps patches of multiple coordinate systems that apply in different places – can be defined instead to locate points.

=====Topology: simply and multiply connected manifolds=====
If you take a plane or a sphere, any closed loop on the manifold can be shrunk in a continuous manner until it vanishes. This is called <i>simply connected</i>. But not all manifolds are simply connected. Those which are not simply connected are called <i>multiply connected</i>. The most basic multiply connected manifold is a torus, like a bagel or doughnut or bicycle inner tube. While there are closed loops on a torus that can be shrunk to a point and made to vanish, those that go all the way around the torus tube cannot – they'll get caught up circling the tube. Likewise, loops that go all the way along the tube will get caught up on the central hole if they are shrunk and again cannot be made to go to a point.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:torus_and_loops.png|frameless]]
<td width=400>[[File:Topological_torus_and_loops.png|frameless]]
<tr>
<td colspan=2>Three classes of loops on a torus. On the left is a standard geometrical torus; on the right is a topological torus that is flat but has opposite edges identified so that they are connected. The blue loop doesn't go around the torus, it can be continuously deformed to shrink to a point and vanish. The red line goes around the torus the long way. If you shrink it, it will get caught up going around the hole in the middle and get stuck there; it cannot be continuously deformed to make it any smaller. The green line loops around the torus from the outside to the inside and back. Again, if you shrink it the line gets caught up circling the torus tube and can't be made any smaller.
</table>

And this brings us to another interesting kind of manifold. You can get a manifold that is flat but has the topology of a torus. You can do this by taking a rectangle and saying that the opposite sides are actually the same. This gives you the effect of a "wrap-around screen" like on that old arcade game Asteroids – if you go off the edge on one side, you just come back on the opposite side at the same height. in physics, this is known as "periodic boundary conditions". Closed loops that go all the way around one way or the other cannot be shrunk to a point. But this kind of surface <i>cannot</i> be embedded in three-dimensional space! We might have grown used to surfaces that we can envision existing in the space we live in, but there are manifolds, even fairly simple two-dimensional manifolds, that cannot be embedded in higher-dimensional spaces like that. Remember this when we get to actual wormholes.

You can have multiply connected manifolds that are not toruses, even though they might have the same topology; any manifold with one or more "loops" or "handles" is multiply connected.

=====Non-orientable manifolds=====

And there is one other interesting kind of manifold that is relevant to wormholes – manifolds that are <i>non-orientable</i>. An orientable manifold is one where you can define a consistent sense of clockwise and counter-clockwise everywhere on the manifold. If you start with a clock spinning clockwise, or a right-handed glove, or a copy of Lewis Carrol's "Through the Looking-Glass, and What Alice Found There", you can move it anywhere on the surface and if it meets back up with itself it will still be spinning clockwise, or right handed, or normally readable. But in a non-orientable surface, there are ways you can move the object so that when you get it back to where it originally was the clock is spinning counter-clockwise, or the glove is left-handed, or the book is written in mirror writing. The simplest example of such a surface is a Mobius strip. As shown below, if you move an oriented circle around the strip, it comes back with the opposite orientation.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Topological_Mobius_strip.png|frameless]]
<td width=400>[[File:Mobius_twist.png|frameless]]
<td width=400>[[File:Mobius_strip.png|frameless]]
<tr>
<td colspan=3>A Mobius strip … a strip of surface with a twist so the edges where it connects meet up "the wrong way around". On the right is a 2D strip where you connect along the short edge matching the 1's together and the 2's together. The center shows the same strip with a twist in it so you can line the edges up the way you are supposed to, demonstrating how the twist reverses the orientation of the spinning circles. An embedded image in 3D space with the edges connected together is shown on the left. You cannot define a consistent orientation to this surface – starting at the red counter-clockwise circle, if you move the circle continuously along the surface when you get back where you started you will find it is going clockwise!
</table>

====Three dimensions====

The most basic three-dimensional manifold is a flat space. This is, well, just a normal space. You can go off in any combination of three independent directions (left-right, up-down, front-back), and because it is flat, initially parallel lines stay parallel. It all behaves much as you would expect three dimensional spaces to behave.

But you can get variations on this. They are harder to visualize than two-dimensional cases, because we don't have the brainspace for processing curved spaces inside of a four-dimensional hyperspace. But you can do things like identifying two planes to be the same, such that if you go far enough in one direction you can end back up where you started again (akin to a cylinder), or make them curved such that distances are distorted and lines don't remain parallel and maybe you can put larger things inside of smaller surfaces. There are 3D equivalents of spherical and hyperbolic surfaces (called, naturally enough, spherical and hyperbolic spaces). There are even non-orientable spaces.

One way of trying to envision a curved space, and to gain some intuition about them, is to take a two-dimensional slice of that curved space and embed that in a flat three-dimensional space. This is called an embedding diagram. Remember that this is a reduced representation that throws away a lot of the information (unless there are special symmetries you can exploit) and is not actually the thing being described.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Space_with_embedded_plane.png|frameless]]
<td width=400>[[File:Embedded_plane.png|frameless]]
<td width=400>[[File:Embedding_diagram_example.png|frameless]]
<td width=400>[[File:Embedding_diagram_example_very_curved.png|frameless]]
<tr>
<td>A space with an embedded plane. We show a sphere in this space, with the plane bisecting the sphere.
<td>Just the embedded plane. Note how in a spherically symmetric manifold (like this space), a circle on the embedding diagram represents a complete 2-dimensional spherical surface.
<td>An embedding diagram of a "dimpled" space. Just as before, the closed circles represent closed spherical surfaces. It is just that in this case, the distance between two concentric spheres of radius <math>R_1</math> and <math>R_2</math> is longer than the usual <math>R_1 - R_2</math> because of the spatial curvature, as shown by the increasing distance along the dimple's surface.
<td>An embedding diagram of an extremely curved section of space. In this example, there are several places where we have a bigger sphere inside of a smaller one! Curved space is weird like that.
</table>

Remember that not every manifold can be embedded, so not every curved space can be represented by an embedding diagram. But you can take patches of the curved space and show them by embedding diagrams, and then describe how these patches stitch together to create a contiguous surface.

====Space-time: three space dimensions and one time dimension====

This is the world we live in. So this is the class of manifolds that we really need to use when making a wormhole. Sometimes it's abbreviated as 3+1D (as opposed to 1D, 2D, 3D, etc.).

The basic, infinite, flat space-time manifold is often called Minkowski space. I know, it's a big word using scary foreign-sounding names. But all it means is that its a 3+1D manifold that goes on forever and is flat. In flat space-time, everything obeys the rules of special relativity. Which means it really doesn't work when masses become significant enough to make gravity.

Our best model of how gravity works is general relativity. In this theory, the curvature of space-time is created by mass. At least, that's what they always say in the over-simplified pop-sci summaries of general relativity. Actually, space-time curvature is created by all the components of the stress-energy tensor. It's just that in almost all cases, mass is the most significant part of this. But momentum, stresses, pressures, and shear forces all have contributions as well. In general relativity, physical objects move along the straightest possible lines in this curved space-time. The way the curvature causes the lines to bend and converge is what leads to the appearance of gravity as a force if the curvature is sufficiently weak.<ref name="Misner_Thorne_and_Wheeler">Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler, "Gravitation", W. H. Freemann and Company, New York (1973)</ref>

The most basic curved space-time manifold is the Schwarzschild geometry, which is what you get when you have an uncharged, non-spinning spherical mass doing its distortion thing on space-time. The Schwarzschild geometry is <i>static</i> meaning it does not change in time, and is <i>spherically symmetric</i>, meaning that it has a definite center and it doesn't matter which direction you are relative to that center – it is independent of angle. If the mass becomes dense enough, space-time gets so distorted that an event horizon forms. Nothing that goes into the event horizon can ever come back out, and the inevitability of the geometry crushes the mass down to an infinitely dense point at the center called a singularity. Nothing on the inside of the horizon can ever escape. It forms a <i>black hole</i>, that forever traps any light or matter that enters it. For masses that don't exceed this critical density, the gravitational distortion outside of the object is the same as for a black hole of equal mass, but there is no event horizon. The object can still be seen and interacted with.<ref name="Misner_Thorne_and_Wheeler"></ref>

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Schwarzschild_star.png|frameless]]
<td width=400>[[File:Schwarzschild_black_hole.png|frameless]]
<tr>
<td>An embedding diagram of the Schwarzschild geometry of a dense, massive star. The physical extent of the star is shown in blue.
<td>An embedding diagram of the Schwarzschild geometry of a black hole of the same mass as the star to the left. The event horizon is shown in red.
</table>

There is one distinction between space-time manifolds that will become important later. Some manifolds have an isolated region of high curvature (corresponding to a concentration of mass, or at least of stress-energy) but as you go farther away from the strongly curved region the space-time becomes increasingly flat. If you can always go out far enough away from a curved region that space-time becomes flatter that whatever flatness criterion you choose, no matter what direction you choose to leave the curved region, the manifold is called <i>asymptotically flat</i>. For our purposes, if you can go far enough away that the Newtonian approximation to general relativity is accurate, and gravity can be described as a force rather than requiring a geometric description, then the geometry can be considered asymptotically flat. You can also allow gravitational waves in your asymptotically flat space-time – their very small space-time curvature is not enough to cause problems and they can still be described by a linearized version of gravity. Both the Schwarzschild and the Minkowski geometries are asymptotically flat. Spherical or hyperbolic geometries are not asymptotically flat. Planets, stars, galaxies, neutron stars, black holes, and galactic clusters are all approximately asymptotically flat; but the universe as a whole is <i>not</i>. As long as we confine our attention to merely galactic clusters, we can make the approximation of asymptotic flatness.

===Making a wormhole===

Now that we have covered some of the introductory material, we can get down to the main course. To make a wormhole, take two regions of space-time, usually far apart and usually asymptotically flat (although neither of these is strictly required) and connect them together. That's it. You're done. Sounds easy, doesn't it?

Note that if you are connecting two regions in the same universe, your new geometry will be multiply connected. A loop that goes through the wormhole and then connects back up outside the wormhole cannot be shrunk to a point. There will now be two equivalent ways to get between any two points – the path that goes through your usual space-time, and the path that goes through the wormhole. If you use a wormhole to connect two separate universes, the combination of the two universes and the wormhole can still be simply connected.

In general relativity, you can distort the geometry of space-time with stress-energy, but there is no way to change its topology; that is, no way to turn a simply connected universe into one that is multiply connected, or to add or remove more "loops" or "handles" to an already multiply connected universe. Such changes, if they are possible at all, would only be possible in the realm of quantum gravity and we do not know enough about quantum gravity to know if it would be possible or not. So, it might not be possible to make intra-universal wormholes at all. Or maybe quantum gravity does allow topology changes in space-time and there are ways to connect different parts of our universe together with wormholes. On the plus side, if you can't change the topology of space-time, you can always bud out a new universe from our own – perhaps by triggering the inflationary field that started our universe. It will be connected to ours by a wormhole. And because the topology can't change, that wormhole would never be able to break. We would have an eternal bridge to the new universe. So a system of physics that does not allow topology changes still allows wormholes to other universes.

==Common kinds of wormholes==

===Spherically symmetric wormholes===

General relativity involves a lot of non-trivial math. You need to deal with four-dimensional curvature tensors with up to 20 independent components, and figure out how they relate to a given distribution of matter, fields, energy, stresses, and material flows. If you can reduce the number of dimensions you need to keep track of, it can simplify things a lot. One way of doing this is assuming your space-time geometry does not change with time (it is static), and looks the same in all directions when viewed from a central point (it is spherically symmetric). This makes things a lot easier to analyze.

Astute readers will have noticed that these conditions are satisfied by the Schwarzschild geometry. In fact, if you play around with the Schwarzschild geometry enough, you will find that you can even get a wormhole-like solution. At the event horizon, you can connect your black hole solution to a time-reversed structure called a white hole. This is called an Einstein-Rosen bridge. Unfortunately, it has some issues. For one, it can't form naturally from infalling matter creating a black hole. For another, going through an Einstein-Rosen bridge involves passing through an event horizon, so the trip is necessarily one way. And finally, you can show that even the smallest amount of matter or radiation falling into the Einstein-Rosen bridge leads to it collapsing before the matter or radiation can pass through, trapping it inside the black hole. So this is not looking particularly useful.

So what are we going to do? Well, the Einstein-Rosen bridge, like a lot of early work in general relativity, was made by assuming a distribution of stress-energy (in this case, matter) that was assumed to be physically plausible, and finding out what the resulting geometry would be. But we want a wormhole, dangit! So what happens if we just demand a static tunnel connecting two asymptotically flat regions of space-time? If you do this, you get a Morris-Thorne wormhole<ref name="MorrisThorne">Michael S. Morris and Kip S. Thorne, "Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity", American Journal of Physics <b>56</b>(5), 395-412, May 1988</ref>. You still have the relation between stress-energy and geometry, but now it is your assumed geometry dictating the stress-energy rather than the other way around. One difficulty with this method is that the distribution of stress-energy you end up with might not end up seeming physically plausible. In particular, it can be shown that any wormhole must have regions in it with negative energy density<ref name="Visser_Lorentzian_wormholes">Matt Visser, "Lorentzian Wormholes: From Einstein to Hawking", Springer/AIP Press (1996)</ref>, and these spherically symmetric wormholes are no different. We will discuss the issues around and solutions to the negative energy density problem later.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Morris_Thorne_wormhole.png|frameless]]
<tr>
<td>An embedding diagram of a Morris-Thorne wormhole. For this wormhole, we just took two copies of the Schwarzschild geometry from above and connected the middle "dimples" with a cylindrical patch of space-time.
</table>

The places where you enter a wormhole are called it's <i>mouths</i>, and most wormholes have two. The space-time tunnel between the mouths is called the wormhole <i>throat</i>.

====Some common problems with spherically symmetric wormholes====

if you don't have your distribution of matter and energy going off to infinity, then eventually if you go far enough away your geometry will reduce to the Schwarzschild geometry. A good many published examples of such wormholes are actually Schwarzschild-like up until pretty close to where the event horizon would be. The first problem you might encounter, then, is the formation of a horizon. A single horizon makes any trip one-way. Going into it on the side with the horizon would be indistinguishable from entering a black hole - except that you pop out somewhere else instead of getting removed from existence at the singularity. And because of the horizon, you couldn't ever go back. If both sides have a horizon, you simply can't get out of it. Why do you even have this thing, anyway?

The strong curvature as you get close to the throat can also produce very strong tidal forces. Due to the gravitational forces, if you go in feet-first your feet could be pulled much harder than your head as you go in and you could end up getting pulled apart – spaghettified is the technical term. In addition to these pure spatial tides, it turns out that you also have velocity-dependent tides. These tides are transverse to the radial direction - if you dive in really fast, you will be squeezed even harder from side to size.

Early attempts to limit the tidal forces while still keeping a throat large enough to fit a person through resulted in very large masses. Like planetary scale masses or even solar scale masses. This would mean you would need to spend a lot of propellant de-orbiting into the wormhole and then rocketing your way back out. Some of these early wormholes also had issues like the extreme gravity blue-shifting the cosmic microwave background to blow-torch intensity x-rays. Sometimes, attempts to keep the tides down also lead to really large throat sizes. Like the distance from the Sun to the Earth sizes. Or even light year sizes. These gigantic wormholes would be rather inconvenient.

None of these are necessarily intrinsic limitations to spherical wormholes. But it does indicate that a perfectly reasonable assumption for the wormhole tech in your science fiction universe could involve wormholes with Jupiter-level masses, extreme tides, and other fun details.

===Thin shell wormholes===

Take a shape located somewhere in flat (Minkowski) space-time. The important part of this shape is that it must be a closed surface that completely encloses a given volume. Now go someplace else in space-time and take an identical shape. Remove all of the space-time from inside both shapes, and then say that the corresponding parts of the two surfaces of the shapes are actually the same points. Now you have a thin shell wormhole. The throat region is infinitesimally short – the section that corresponds to the wormhole is just that surface of your shape, hence the name "thin shell".

The first thin shell wormholes proposed were simple polyhedra <ref name="Polyhedral_wormholes">Matt Visser, "Traversable wormholes: Some simple examples", Physical Review D vol. 29 no. 10 pages 3182-3184 (1989)</ref>. These are attractive choices for a potential wormhole-user because the flat faces of a polyhedral thin-shell wormhole have no tidal effects whatsoever. Further, the flat faces do not have any stress-energy either. It is simply flat space-time, no different from any other area of flat space-time. Going through the face of a polyhedral wormhole is no different from going through the flat space-time of the doorway separating your hallway from your bedroom. The negative energy stuff is concentrated entirely in the edges and corners of the polyhedron. And hoo boy, do you need a lot of it. For a cubic wormhole, each edge requires approximately <math>-1.52 \times 10^{43}</math> J/m of length. This is <math>-1.69 \times 10^{26}</math> kg/m, or close to a tenth of a Jupiter mass per meter of edge length. It has been proposed that perhaps cosmic strings – defects of the primordial universe where things couldn't line up the right way and left a linear flaw in reality – with negative tension might be able to meet this requirement and hold open a polyhedral wormhole.

But there's a strange thing when you consider the mass that these wormholes require. They are embedded in flat space-time. Which means that outside of that thin shell they have, they induce no curvature on the space-time around them. Remember that mass causes space-time to curve, dimpling up as in the Schwarzschild geometry shown earlier. Because these wormholes induce no far-flung curvature to imprint on the surrounding space-time, they have zero mass (technically ADM mass). And because mass is equivalent to energy, it should technically be possible to make one of these for no energy input at all from those of us in flat space-time. Forces are the rate of change of energy with distance, pressures are the rate of change of energy with volume. But because the energy is always zero no matter what size they are (as measured from out in flat space-time), there would not be any force or pressure needed to hold them open. The extreme conditions of their interior contrasts with the relatively benign conditions once away from the thin shell (or, indeed, within any flat area within the shell).

A curious feature of polyhedral wormholes is that each face of the polyhedron might connect to a <i>different</i> polyhedron. You might, for example, have a collection of seven cubical wormhole mouths, and each of the six faces of the wormhole you are facing connect to a different one of the six remaining mouths.

For more general thin-shell wormholes, you just need to smear that negative mass out a bit. As long as you keep a flat area to go through, travelers will not experience any strange tides or encounter space-warping exotic energies. You could, for example, have your standard circular portal beloved of fiction. Or you could have any other arbitrary shape you wanted.

Thin shell wormholes are what you get when you take a mathematical limit; they would be an approximation of a more physical distribution of stress and energy. Any realizable version would have a finite, although possible small, thickness to its shell and hence a finite, although possible small, throat length. As we will discuss later, they would likely also have a positive, although possible small, mass. But as an approximation, and as a valid solution to Einstein's field equations in general relativity, they allow us to explore many aspects of wormhole physics using a relatively simple model that is easy to calculate, and can lead to a lot of interesting inspiration for fictional ideas of wormholes.

====Tidal distortion during transit====

Consider a thin shell wormhole whose shell-shape is everywhere convex - no flat areas this time. For convenience, let's make it a sphere. As a traveler passes through the shell, the sections of his body that pass the shell must suddenly "pop" from curving one way to curving the other way, as illustrated below:

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Velocity_dependent_tides_pre_transit.png|frameless]]
<td width=400>[[File:Velocity_dependent_tides_in_transit.png|frameless]]
<tr>
<td>A traveler (left rectangular shape) about to go through a spherical thin-shell wormhole (here represented as circles, with the interior volume blacked out to indicate that there is no space-time there and nothing can exist in that region).
<td>The traveler in the process of passing through the wormhole. Note that on the wormhole shell, the portion of the traveler to the left has a leftward-curvature, while the portion on the right has a rightward curvature. As the traveler passes through, each section that goes through the shell immediately jumps from left-curving to right-curving. This will induce stresses in the traveler as his interior is deformed.
</table>

it is clear that as the curvature of the wormhole becomes large compared to the size of the traveler, the distortion will decrease. In the limit of a flat section of the surface, no distortion is experienced at all.

A consequence of this is that something going through a curved section of a thin shell wormhole will experience stresses (distortions) and strains (forces). The elastic energy stored by a strained object is proportional to the product of the stress and the strain, added up over all of its volume. Assuming that the object going in to the wormhole is initially in its relaxed state, it will require energy to push it into the wormhole shell. If it enters with insufficient initial kinetic energy to get it through or external forces pushing on it, it will just bounce off.

===Cylindrical and toroidal wormholes===

In principle, you can have a wormhole with cylindrical symmetry
<ref>Kirill A. Bronnikov and José P. S. Lemos, "Cylindrical wormholes", arXiv:0902.2360 [gr-qc] 24 Feb 2009</ref>
. This forms a valid solution to Einsteins field equations in general relativity. However, a true cylindrical wormhole would be infinite in length. So they probably don't exist naturally, and you won't be able to build one. Unlike all the other wormholes discussed here, they are not asymptotically flat. Unlike wormholes of finite extent, not all cylindrical wormholes require negative energy regions. However, it has been shown that negative energy stuff is needed if the geometry radially far enough away from the wormhole approaches the flat space-time of our usual universe, so a universe with a cylindrical wormhole without negative energy would not look like our own.

So the thought is that maybe you can get close enough to a cylindrical wormhole by making a really long wormhole that bends ever so slightly and comes back and meets itself, forming an extremely stringy torus. The general case for this has not yet been worked out, but a thin shell approximation of a toroidal wormhole has been developed
<ref>Vladimir Dzhunushaliev, Vladimir Folomeev, Burkhard Kleihaus, and Jutta Kunz, "Thin-shell toroidal wormhole", arXiv:1901.07545v2 [gr-qc] 28 Jan 2019</ref>
.
These thin shell toriodal wormholes have been shown to be stable to perturbations, one nice benefit that has not been established for a lot of other kinds of wormholes. They do require some negative energy regions on their surface, but there will also be some regions that do not contain negative energy.

===Non-orientable wormholes===

In the discussion on non-orientable manifolds, we showed how in these manifolds you can't properly determine the clockwise - counter-clockwise orientation of objects or motions, or the orientation of objects with respect to their mirror reflections. If you move an object along certain paths, when it comes back it might come back the wrong way around. So it is probably no surprise that a non-orientable wormhole turns the universe into a non-orientable manifold. Something going through a non-orientable wormhole comes out looking like it has been mirror reflected.

Normally this seems like it would be rather boring and useless, but for an obscure constraint from quantum physics called the CPT theorem. This theorem, which holds for all physical phenomena, means that the product of the discrete symmetries of charge conjugation (C), parity transformation (P), and time reversal (T) always returns you to your original state. Huh? What does that mean? Well, parity transformation is equivalent to a mirror reflection; so passing through a non-orientable wormhole means parity is inverted. This means that exactly one of the other two symmetries must also be inverted for the CPT theorem to hold. The thing coming out on the other end is still manifestly going forward in time, so it must be charge conjugated, whatever that is.

Charge conjugation means you turn all particles into their antiparticles. So anything passing through a non-orientable wormhole emerges on the other end made entirely out of antimatter.

While this particular turn of events might discourage people from passing through, it does offer obvious application for energy generation, space propulsion, and weaponry.

==Exotic energy conditions==

It is fairly easy to see than any wormhole requires negative energy densities (or more generally, regions where the tension is higher than the energy in natural units) to exist. Take a look at any of the wormhole diagrams above – the spherically symmetric ones will probably make it the most clear. If you send a parallel ray of stuff – maybe particles or light or golf balls – through the wormhole, and trace out the straightest possible path of all of those parallel lines, you will see that after they go through the wormhole all those initially parallel trajectories will now be diverging. Positive energy (such as mass) causes attractive forces, drawing trajectories toward it and making the paths converging. To get diverging paths you need the opposite – repulsive gravity – which you can get from negative energy densities or very high tensions. Having the tension be higher than the energy density turns out to be equivalent to a negative energy density – in some frames of reference observers will measure a negative energy density there.

A lot of physicists used to be reflexively dismissive of negative energy densities. After all, they can't really exist; all energy in the real world is positive.

Except that it is not. There are certain odd cases that allow for negative energy densities. One of these is the Casimir vacuum. A conducting boundary cannot support an electric field parallel to that boundary (if there was such a field, charge would flow under the force of the field until the parallel component of the field is screened out). So if you have two parallel conductive plates, the only electromagnetic waves that you can get between them require their fields to vanish at the plates. This restricts the number of ways the electromagnetic field can vibrate, called modes. In quantum mechanics, merely the potential for a vibration is associated with a certain energy (called <i>zero point energy</i>) even if no quanta of vibration are actually present. By making these modes impossible, they cannot contribute their zero point energy to the region between the plates. This lowers the total energy of the space between the plates compared to the region outside the plates (or with the plates absent). Because the energy of empty space is zero, this means that the region between these Casimir plates has negative energy. The Casimir effect has been measured in the laboratory. It is fairly weak, but it is present. In fact, not long after traversable wormholes were first described, one proposal suggested holding open the wormhole using the Casimir effect<ref name=MirrosThorneYurtsever">Michael S. Morris, Kip S. Thorne, and Ulvi Turtsever, "Wormholes, Time Machines, and the Weak Energy Condition", Physical Review Letters, Volume 61, Number 13, 26 September 1988, pages 1446-1449</ref>

Another situation that gives rise to a negative energy density is a <i>squeezed vacuum</i>. A squeezed state is a way of manipulating a quantum oscillator (like a pulse of light) to get around the uncertainty relations of quantum mechanics to increase the precision in one variable by increasing the uncertainty in its conjugate variable. Applying this operator reduces the energy of the oscillator, and if this is applied to the vacuum state, which already has zero energy, you can end up with regions of negative energy.

It is worth noting that not just any old negative energy region of exotic space-time will work. In particular, dark energy is a condition of the universe we live in where even bare vacuum itself has a small amount of negative energy. The current best model we have for this is as a cosmological constant - where every part of the universe has an equal density of negative energy and a related amount of pressure. But the wormhole allowing contributions to space-time curvature of that negative energy are exactly cancelled by the wormhole-denying contributions of the pressure. So according to our best current models, dark energy will not support a wormhole.

Also, the Casimir vacuum described above is also problematic. The exotic vacuum between the plates does have the necessary properties (if not magnitudes) to support a wormhole, but this is generally far overwhelmed by the immediately adjacent positive mass matter making up the conductive plates. The technical term for the criterion that determines whether you can have a wormhole or not is the Averaged Null Energy Condition (ANEC). If you project a ray through space-time that follows the path that light (in a vacuum) could take, and do some math along that ray and add up all the contributions from all the places along that ray, then if your result is zero or positive then the ANEC is said to be satisfied. If this is the case, you can't get a wormhole. If the ANEC is less than zero, it is said to be violated and in principle you could use that weird exotic energy stuff to prop open a wormhole's throat. There are other so-called energy conditions, and if you have regions of space-time that also violate those energy conditions they can be used to hold open wormholes as well - but all of these also violate the ANEC, so the ANEC is all you need to consider.

===So just how do I get the ANEC-violating stuff I need?===

The short answer is, we don't know.

As we already said, a cosmological constant form of dark energy won't do it (although different ideas about dark energy, such as phantom energy, could conceivably work). The inflationary vacuum thought to have existed very briefly during the first instances of our universe's existence is also usually thought to be a cosmological constant (although one that stopped existing after a very brief period of time, so I guess it is less than constant), so the inflationary vacuum probably won't work either.

If you can get the Casimir vacuum without the massive conducting plates that bound it, you might be able to do something with that. For example, the twisted geometry of space-time itself in a wormhole throat imposes boundary conditions that restrict the modes of vibration inside of it similar to the Casimir effect. This is called the <i>topological Casimir effect</i>, potentially it could be used to help support a wormhole. Also, although the ANEC is satisfied if the probing space-time ray goes through the Casimir plates, a ray that goes between the plates but never touches them will violate the ANEC. Can you use this to hold open a wormhole? Maybe a sufficiently clever person will eventually figure out how to do so.

Even more serendipitous, it turns out that gravity itself naturally squeezes the vacuum
<ref name="gravitationally_squeezed_vacuum">David Hochberg and Thomas W. Kephart, "Lorentzian wormholes from the gravitationally squeezed vacuum", Physics Letters B 268, 377-383 (1991)</ref>
. If you have the curved space-time of gravity, all of the modes of oscillation in it become squeezed states. So it might be possible that the curved space-time of a wormhole can squeeze the vacuum enough to support itself. Or a combination of gravitational vacuum squeezing and the topological Casimir vacuum from the wormhole's geometry constraints might be all that is needed. At least one proposal has suggested a class of wormholes that can be self supporting in such a way
<ref>S. V. Krasnikov, "Toward a Traversable Wormhole", arXiv:gr-qc/0003092v1 22 Mar 2000</ref>
<ref>S. Krasnikov, "Traversible Wormhole", Physical Review D, volume 62, article 084028 (2000)</ref>,
although the throat of these wormholes is very "wrinkly" or "crumpled up" and it is unclear what the effects of that would be on anyone passing through.

====Quantum energy inequalities====

When you apply quantum field theory to what happens when you have a region where the quantum fields produce local negative energy densities, you end up with a limit of how much negative energy stuff you can have before it has to be balanced out by an even greater amount of nearby positive energy stuff
<ref>[https://web.archive.org/web/20060206055950/http://maths.york.ac.uk/www/PhysicsQIneq.htm|Quantum Energy Inequalities]</ref>
<ref>[http://www.phys.lsu.edu/mog/mog20/node16.html|Quantum field theory on curved spacetime at the Erwin Schrödinger Institute]</ref>
<ref name="Fewster_Roman_2005">Christopher J. Fewster and Thomas A. Roman, "On wormholes with arbitrarily small quantities of exotic matter", Physical Review D <b>72</b>, 044023 (2005)</ref>
.

====Matter with negative mass====

It might be tempting to try to solve the negative energy density requirement for wormholes by introducing some sort of matter that naturally has negative mass.
However, negative mass would introduce all kinds of problems to the smooth operation of how the world works. For one thing, if negative mass (or equivalently energy) can exist, why doesn't it just spontaneously pop out of empty space accompanied by an equal magnitude of positive mass (or energy)? For another, negative masses lead to all kinds of runaway instabilities. For example, imagine what happens if you push on an object with a negative mass. You exert a force on it with your hand. But by Newton's second law of motion, an object's acceleration is proportional to its mass. So the negative mass thing will accelerate backwards, against the direction of the force, and into your hand. Which means you will be pushing even harder on it, which makes it push back even harder than before. You have a runaway process that blows up until your hand breaks, and any fragments of flesh and bone and sinew that were in the way of the negative mass thing have been flung off.

Or consider a negative mass thing with an electric charge. If you put something with an electric charge in an electric field the field will exert a force on the charge. For a normal positive mass charged thing, it will move in the direction of the force. The work done is the force in the direction of the displacement in position, so because the force and motion of the object are in the same direction, the field does work on the charged object and energy flows from the field into the kinetic energy of the charged object. But if the charged object has negative mass it will begin to move opposite to the direction of the force. This will do work on the field instead of the object, increasing the energy of the field. The energy comes from the kinetic energy of the negative mass charged object – but because kinetic energy is proportional to mass and increases with increasing speed, as the negative mass object loses kinetic energy it gains speed, going ever faster. Both the field strength and the speed of the object increase without bound. If the electric field originates from the electric part of an electromagnetic wave, it will vary sinusoidally with time and space. As the negative mass object is forced into a sinusoidal trajectory opposing the electric force on it, it will amplify that wave making it ever more intense. A single, isolated charge exposed to an electromagnetic wave will eventually have its inertia keep it moving from the previously imposed force even after the force has switched direction, so if the negative mass is free with no other forces acting on it there will be no net amplification. But if there are any other damping forces, or internal arrangements of bound charge that can resonate with the wave, then instead of being absorbed (as would happen with a positive mass), the wave will be amplified indefinitely and the ever-increasing energy of the wave will come from the ever increasing temperature of the negative mass thing (because the mass is negative, the thermal motion of temperature gives it a negative energy whose magnitude only increases as the temperature rises).

And, of course, the quantum energy inequalities indicate that you can't just have isolated lumps of negative mass stuff floating around.

So just having lumps of stuff that have a negative mass is probably not realistic, and it were possible it would lead to all kinds of universe-shattering cataclysms. Hopefully, we can get wormholes to stay open just using the exotic vacuum states from the Casimir effect and squeezed vacuums.

===Reducing or eliminating ANEC violating stuff===

As soon as people figured out that the wormholes they want needed stuff they couldn't get, they set about trying to find ways to use as little of the stuff they couldn't get as possible. A number of attempts have been made to hammer out a geometry for spherical wormholes that are stable, meet the quantum energy inequalities, and allow a person to pass through them in a reasonable length of time without being shredded by tides
<ref name="Fewster_Roman_2005"></ref>
<ref name="Visser_Kar_Dadhich">Matt Visser, Sayan Kar, and Naresh Dadhich, "Traversable Wormholes with Arbitrarily Small Energy Condition Violations", Physical Review Letters Vol. 90 No. 20 article 201102 (2003)</ref>
<ref name="Kuhfittig">Peter K. F. Kuhfittig, "More on wormholes supported by small amounts of exotic matter", Physical Review D <b>73</b>, 084014 (2006); Peter K. F. Kuhfittig, "Wormholes supported by small amounts of exotic matter: some corrections", arXiv:gr-qc/0508060v1 15 Aug 2005</ref>
.
Often, these resulting structures end up having extreme dimensions, such as throat sizes on the order of light years
<ref name="Fewster_Roman_2005"></ref>
<ref>Remo Garattini, "Generalized Absurdly Benign Traversable Wormholes powered by Casimir Energy", arXic:2008.05901v1 [gr-qc] 12 Aug 2020</ref>

Thin shell wormholes have also been the subject of investigation. One study <ref name="Mazharimousavi_Halilsoy">S. Habib Mazharimousavi, M. Halilsoy, "3 + 1-dimensional thin shell wormhole with deformed throat can be supported by normal matter", Eur. Phys. J. C (2015) 75:271 DOI 10.1140/epjc/s10052-015-3506-6</ref> found a number of thin shell wormholes with sharp-edged shapes could be supported entirely without any exotic energy. However, this only applies if the edges are infinitely sharp. A finite radius of curvature at the edges would require negative energy at those edges.

If the wormhole is taken have the properties of a class of subatomic particles called a fermion, then if the ratio of the electric charge to the mass is sufficiently large you can get a wormhole that can remain open without any negative energy regions at all
<ref name="Blazquez-Salcedo_Knoll_Radu">Jose Luis Blázquez-Salcedo, Christian Knoll, and Eugen Radu, “Transversable wormholes in Einstein-Dirac-Maxwell theory”, Physical Review Letters <b>126</b>, 101102 (2021); ArXiv: 2010.07317v2 [gr-qc].</ref>
.

==Fields and conserved properties==

Our best theories of how the world works – quantum mechanics and general relativity – predict that certain things are [[Conservation_Laws:_Limits_to_Cheating|conserved]]. For asymptotically flat space-times, these quantities are energy, linear momentum (or just momentum), angular momentum, and electric charge. Furthermore, in the non-relativistic limit mass is conserved independently of non-mass energy.

And not only are these things conserved, but they are always conserved <i>locally</i>. This means you can't just get rid of, say, some energy in one place and expect things to balance out by having that energy appear somewhere else.

What does it mean for something to be conserved? Mathematically, it means that the thing obeys the continuity equation. There are various ways of writing the continuity equation, all equivalent, but basically it boils down to the following: if you have a region of space surrounded by a closed surface, then the amount of conserved stuff inside that surface can only increase if some of the stuff enters through the surface and it can only decrease if some of the stuff leaves through the surface. Makes sense, right? If you have three loaves of bread in a room, then if the deliveryman brings in another two loaves you will have five loaves. But unlike loaves of bread, which can be baked or eaten or grow moldy or burn up or be broken into crumbs, the only way to get energy or electric charge or the other conserved things into or out of the room is to have them come into or go out of the room.

Each of these conserved quantities has an associated field. Electric charges create an electric field. Energy and momentum and angular momentum each create a distribution of the curvature tensor of space-time. In the non-relativistic limit, this is the mass part of the energy creating the gravitational field.

There is a relationship between the amount of a conserved quantity in our hypothetical closed surface and the net amount of field that penetrates that surface. This is called [https://en.wikipedia.org/wiki/Gauss%27s_law <i>Gauss's law</i>]. Basically, it says that if you add up all of the amount of field that goes out of the surface, and subtract off all of the field that goes back into the surface, the resulting amount will be directly proportional to the amount of the conserved stuff in there. Do you start off with a kilogram of mass inside your room? Then you have a certain amount of gravitational field from that mass that leaks out of the room. Add another kilo for two kilos total, and the amount of gravitational field leaving is also doubled. The distribution of where that field comes out may change, but as long as you don't take that mass out of the room or add more mass in the room, the total amount of field leaking out won't change. And any gravitational field from masses outside of the room that leaks into the room must also leak back out again <i>somewhere</i>. Those masses outside the room will not affect the net amount of field leaking through the surface. What this means is that you can determine the amount of conserved stuff inside a surface if you know the amount of field going through that surface.

This can be visualized in the field line approximation. The field around a positive charge can be approximated as a series of directed lines that radiate away from the charge. The field around a negative charge in this approximation is a series of lines that converge on the charge (for non-relativistic gravitation, you only have positive charges because mass is always positive. We'll just ignore the fact that wormholes require negative energy density stuff in order to exist which will lead to negative mass – we're just trying to get across the basic idea here). Field lines can only ever start on a positive charge and only ever end on a negative charge. The number of lines radiating away or converging on the charge is proportional to the size of that charge (so if you double the charge, you double the number of lines that connect to it). Field lines propagating through space bend away from a positive charge and bend toward a negative charge.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=322>[[File:charges_positive.png|frameless]]
<td width=322>[[File:charges_negative.png|frameless]]
<td width=322>[[File:charges_dipole.png|frameless]]
<tr>
<td>The field lines radiating away from a positive electric charge.
<td>The field lines converging on a negative electric charge.
<td>The field lines emerging from a positive charge and curving around to go into a negative charge of the same magnitude. This configuration, with a positive and negative charge of the same magnitude, is called a <i>dipole</i>.
</table>

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=322>[[File:charges_positive_Gauss_law.png|frameless]]
<td width=322>[[File:charges_negative_Gauss_law.png|frameless]]
<td width=322>[[File:charges_dipole_Gauss_law.png|frameless]]
<td width=322>[[File:charges_exterior_dipole_Gauss_law.png|frameless]]
<tr>
<td>A closed surface encloses a positive charge. A net amount of field lines exit the surface.
<td>A closed surface encloses a negative charge. A net amount of field lines enter the surface.
<td>A closed surface encloses a dipole. The total amount of charge inside the surface is zero because the negative charge cancels the positive charge. No net amount of field goes through the surface - just as many field lines enter as leave.
<td>A closed surface that has no charge inside of it, but various charges outside of it (in this case a dipole). Again, no net amount of field goes through the surface - every field line that enters comes back out somewhere else.
</table>

Now consider the case of an isolated positive charge, like one one shown above, that is far away from anything else. We will bring up one mouth of a wormhole close to the charge. Because the field lines going out of the charge cannot break, they cannot go through the wormhole. Instead, they will curve around the wormhole mouth. As the charge enters the wormhole mouth and travels into its throat, it drags its field lines along with it. With all of its field lines entering the mouth, that mouth looks like it has a net positive charge. As the charge leaves the other mouth, it continues to drag ints field line out of the wormhole - the field lines that start on the charge now curve around to enter the wormhole mouth. The mouth that the charge exited from now looks like it has a negative charge.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Wormhole_with_charges_initial.png|frameless]]
<td width=400>[[File:Wormhole_with_charges_inside.png|frameless]]
<td width=400>[[File:Wormhole_with_charges_passed_through.png|frameless]]
<td width=400>[[File:Wormhole_with_charges_long_gone.png|frameless]]
<tr>
<td>The charge approaches the wormhole.
<td>The charge is inside the wormhole. The mouth of the wormhole that the charge entered looks like it has a positive charge.
<td>The charge has passed through the wormhole. The mouth that the charge entered still looks like it has a positive charge, but now the mouth that the charge came out of looks like it has a negative charge.
<td>The positive charge has now moved very far away. Both mouths of the wormhole still appear to be charged.
</table>

This can also be understood by the continuity equation. Picture an imaginary closed surface around the mouth of the wormhole. This is like the case of the room, before, except that <i>now there is a way out of the enclosed volume that does not pass through the surface</i> – by going through the wormhole! Any conserved quantity that goes into the surface adds its quantity to the stuff in the surface. If it goes through the wormhole, it does not come out through the surface. The conserved quantity inside that surface stays the same! To an outside observer, it looks like the wormhole mouth has that quantity.

Either way you look at it, the results are the same. A wormhole mouth acquires any conserved quantity of the things that enter it, and loses any conserved quantity of the stuff that exits from it.
<ul>
<li> A wormhole mouth acquires the <i>energy</i> of anything that goes into it. Its energy decreases by the energy of anything that goes out of it.
<li> In the non-relativistic limit, this means that a wormhole mouth acquires the <i>mass</i> of anything that goes into it. Its mass decreases by the mass of anything that goes out of it.
<li> A wormhole mouth acquires the <i>electric charge</i> of anything that goes into it. It gains negative the electric charge of anything that goes out of it.
<li> A wormhole mouth acquires the <i>momentum</i> vector of anything that goes into it. It gains the negative of the momentum vector of anything that goes out of it.
<li> A wormhole mouth acquires the <i>angular momentum</i> vector of anything that goes into it. It gains the negative of the angular momentum vector of anything that goes out of it.
</ul>

This perhaps should not be a surprising result – other general relativistic space-time distortions also exhibit this behavior. A black hole, for example, gains the mass, charge, momentum, and angular momentum of anything that goes into it.

You can use these same arguments to find that the same results apply to any asymptotically flat manifolds of space-time. They do not necessarily hold for space-time geometries that are not asymptotically flat. If the gravitational distortion of a wormhole extends infinitely far away, you will not be able to even define its total energy or momentum or angular momentum, so it will be impossible to say if they are conserved. But do note that all of the wormhole examples that we have discussed so far (except for the cylindrical wormhole of infinite extent) are asymptotically flat, and will obey the local conservation of all of these quantities.

===Wormhole dynamics===

A wormhole mouth has the gravitational field and conserved properties of mass, momentum, and angular momentum. By the equivalence principle of general relativity, this must mean that it behaves as if it had those properties. So a wormhole mouth will follow ballistic trajectories in vacuum through space-time as if it were an object in free fall. If it is around a concentrated mass, like a planet or sun, it will have a Keplerian orbit around that object (or, if the wormhole has more mass than the planet or sun, perhaps it would be more reasonable to say that the planet or sun would orbit it). The mass of the wormhole mouth will resist acceleration like any other mass, according to Newton's laws of motion. Force is the time rate of change of momentum, so as the wormhole mouth's momentum changes it will experience forces, and these will change its trajectory in the same way as if that momentum (or forces) acted on a material object with the same mass. And if the wormhole mouth has an electric charge, it will experience forces from electric and magnetic fields; again, these can change its trajectory.

In other words, a wormhole mouth acts like a physical object of the same mass, charge, energy, momentum, and angular momentum.

The motion of the wormhole mouths through our normal space does not affect their relative separation through the wormhole's throat. That can remain the same length regardless of the behavior of the mouths in our normal universe.

As we covered, if an object enters a wormhole mouth that mouth gains the object's momentum. This will give the wormhole a "kick" changing its motion in the direction that the entering object was originally moving. The dynamics work out the same as if the wormhole were a sticky blob and the object collided and stuck to it. In both cases, you get a final object with the combined masses of both initial things (the mouth and the entering object) that has been knocked off course a bit by the momentum of the entering object hitting it.

If that object comes out of the other mouth, then as that mouth gains the opposite of the objects momentum it gets a recoil kick in the opposite direction. Again, this results in dynamics that are similar to things we already know about – like a gun with a bullet loaded firing the bullet. As the bullet leaves, the gun + bullet system loses the mass of the bullet, and as the bullet shoots away the gun recoils in the opposite direction.

So now consider what happens if you direct a constant stream of material through a wormhole. As the jet of material exits its mouth, that mouth loses mass at the same rate as the mass flow rate of the jet. It also continually gains momentum by the recoil kick of the gas. These dynamics are identical to that of a rocket - as the propellant is ejected, the rocket loses the propellant's mass and the recoil momentum from the escaping gas jet pushes the rocket in the opposite direction. So by shooting stuff through a wormhole you can turn it into a rocket, with dynamics identical to a rocket, obeying the [https://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation Tsiolkovsky rocket equation], and everything else that happens with rockets.

===Does gravity go through a wormhole?===

The above diagrams of the electric field of a charge as it approaches a wormhole mouth also hold for the gravitational field of a mass that is near a wormhole mouth. In the same way that the electric field cannot enter the wormhole's interior or leak through to the other side unless the charge enters the wormhole or goes through to the other side, so to will the gravitational field of a nearby mass curve around the wormhole. The gravity of a nearby planet will fall to zero inside the wormhole throat and will not affect those near the opposite mouth.

===Can waves go through a wormhole?===

So if gravity won't go through a wormhole, and static electric and magnetic fields can't go through a wormhole, then what about light? That's made up of electricity and magnetism, right?

So, turns out that light can go through a wormhole. So can other electromagnetic waves like radio and x-rays. So can gravitational waves. The rules for drawing field lines only really work for static charges; for radiating waves you also need to include the parts about how the changing field generates more field. In the end, waves that are smaller in wavelength than the size of the wormhole can go through just fine. You'll get some [[Diffraction|diffraction]] around the mouths, but for wavelengths much smaller than the mouth size even that might not be noticable.

===Can negative mass wormholes exist?===

So a wormhole mouth loses the mass of anything that comes out of it. What happens if you send something through a wormhole with more mass than the mouth which it exits from? This rule says that the mass must become negative, but as we saw earlier this leads to all kinds of problems (especially because the wormhole can also be charged) and is likely to be forbidden by the laws of nature.

If a wormhole's mouth can reach negative mass, you can also get other interesting instabilities. The positive mass mouth will attract matter into it, which will pass through, increasing the mass of the positive mass mouth and making the mass of the negative mass mouth even more negative. Meanwhile the gravity of the negative mass mouth will repel matter, preventing any return flow. Both the negative and positive masses will gradually increase over time.

So what's an aspiring science fiction writer to do? If you have a particular desire for negative mass things in the fiction you create, go ahead and allow it. Just be aware of the consequences. Otherwise, it is probably easier to say that negative mass wormhole mouths can't exist, and justify that with the quantum energy inequalities. Which, of course, brings up the question of what happens when you try to push an object through a wormhole that would make the mass of one of its mouths go negative? Two obvious possibilities are that either you get a back-reaction force that makes it so you can't ever make something go through the mouth to make its mass negative (perhaps by the formation of a horizon as the mass approaches zero), or the wormhole collapses before the mass can ever reach zero. Perhaps inventive science fiction authors and fans can come up with other plausible options as well.

==Wormholes and time travel==

Wormholes connect across both space and time. So it is natural to consider the possibility of using wormholes for time travel.

Now maybe you don't want to have to deal with time travel in your fiction. It introduces all kinds of opportunities for paradoxes, and you have to keep track of plot lines in a self-consistent way that go back and forth through time, influencing themselves in complicated ways. Okay, no problem. Lets just have our wormholes always connect to the same time. Easy right? In fact, it is natural to justify this. Physics is local, so if you can make wormholes you probably have to create both ends right next to each other at the same time. You can then move the wormhole ends away from each other – maybe putting one end on an interstellar spacecraft or something – but they stay at the same time. Right?

Right?

Except ...

In relativity, motion not only affects how fast you go through space, but also how fast you go through time. When things move, they experience time dilation, so if you move one wormhole rapidly far away and then bring it back, the time dilation it experienced will naturally allow it to form a time machine so you can send things and people back to interfere with their own pasts. Even worse, time can also be slowed down by gravity, so if your wormholes are in areas affected differently by your galaxy's own gravity, or around stars of different mass, or just at different elevations on the same planet, the rate they experience time will be different. Temporal complications ahoy!

Let's look at how this works in more detail. We'll take our standard science fiction empire, called The Empire. Like all empires, it has a central area of authority, the metropole (which we will place on a planet called Metropole), which extracts wealth from various colonies and subjugated client states for its own enrichment in the form of taxes, tribute, and forced favorable trading opportunities. Let's give our Empire two colony worlds which we'll call Colony A and Colony B (these Empire folks sure aren't very creative in their naming, are they?). Just for convenience, we'll put Metropole, Colony A, and Colony B all 100 light years away from each other, in an equilateral triangle.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:WormholesTimeTravel graphic1 wb.svg|600px|frameless]]
<td>[[File:WormholesTimeTravel graphic2 wb.svg|600px|frameless]]
<tr>
<td width=400>Metropole and its two colonies A and B
<td width=400>Metropole sending a wormhole to Colony A.
</table>

So Metropole launches a wormhole mouth out to Colony A. They want to get to Colony A quickly, so they'll make the wormhole go really fast. Let's say they fling it out at 99.9999% the speed of light. At this speed, it takes 100.0001 years for the wormhole mouth to reach its destination in the reference frame of Metropole and Colony A. But due to relativistic time dilation, in the reference frame of the projected wormhole mouth it only takes 0.1414 years to go from Metropole to Colony A. Because the techs at Metropole can look through the wormhole, they will see that 0.1414 years after launch it arrives at its destination. At that point, brave explorers from Metropole can go through and set foot on Colony A – they only need wait about a month and a half to feel alien soil under their feet, rather than a century. In Metropole's reference frame, going through the wormhole takes you 100 light years away, and 99.8596 years into the future. Going the other way, from Colony A to Metropole, takes you 100 light years away and 99.8596 years into the past.

[[File:WormholesTimeTravel graphic3 wb.svg|800px|frameless]]

One might think that these time warps would let you engage in all kinds of time travel. It is easy to see that the Metropole—Colony A situation described here doesn't allow these kinds of shenanigans. For practical purposes, you only have a time machine when you can go back to the place you left at a time before you left. And you can't do that here. Go from Colony A to Metropole and you go back in time 99.8596 years. Go back to Colony A through the wormhole, and you go forward in time the same amount, plus any time you spent on Metropole, so you get back after you left. If you go back through flat space-time, it will always take at least 100 years since you can't go faster than the speed of light so you also get back after you left. No paradoxes for you!

However, it is easy to imagine situations where a wormhole, or a configuration of wormholes, does make a time machine. For example, what happens if we immediately turn around and send the wormhole mouth from Colony A back to Metropole? In the reference frame of Metropole, the wormhole comes back with its "Return to sender" sticker 200.0002 years after it was launched. But going through from the left-behind mouth to the round trip mouth will take you 199.7192 years into the future. And if you step into the round trip mouth, you will go back in time by 199.7192 years into your own past. You now have a real honest to goodness time machine, complete with paradoxes.

[[File:WormholesTimeTravel graphic6´4 wb.svg|800px|frameless]]

Or maybe the wormhole mouth stays at Colony A. And Metropole sends another one to Colony B, that also goes 99.8596 years into the future. Now Colony A sends a wormhole to Colony B. This wormhole also goes 99.8564 years into the future as a consequence of its trip. This means if a traveler at Colony B went through the Colony A wormhole he would go back in time 99.8564 years. Then going from Colony A to Metropole he would go back in time another 99.8564 years. Then he could go from Metropole to Colony B and go forward in time 99.8564 years. The net result is that he ended up back where he started nearly a century before he left.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:WormholesTimeTravel graphic5 wb.svg|600px|frameless]]
<td>[[File:WormholesTimeTravel graphic6 wb.svg|600px|frameless]]
</table>

And it is not just motion that causes time dilation. Being different depths into a gravitational potential does this as well. If Colony A is deeper into the gravity well of the galaxy, or orbits a heavier star, it will experience more gravitational time dilation than Metropole. So eventually it will build up more than 100 years of time slip between the two end of the wormhole and you can have a time machine.

So is time travel inevitable if you have wormholes? What a nuisance!

Well, maybe not. Here's why. Think about what happens when the Colony A – Colony B wormhole has gone just far enough that a light signal going through the wormholes can get back to where it left just as it is leaving. Now, since the propagating signal and the newly transmitted signal are both leaving at the same time, you have double the intensity. So this doubled intensity signal goes around and meets itself again, quadrupling its intensity. And so on. At this point, just as the configuration is on the verge of becoming a time machine, it becomes a perfect resonator for light signals, which then build up to arbitrarily high intensities until something breaks and you don't have an incipient time machine any more.

Now, this won't always work if you just consider light signals. Light can get defocused or sent off in odd directions or something so that you don't have a good path back until after you end up with a time machine. But if you look at what happens in semiclassical quantum gravity, will get a buildup in the amplitude of quantum fluctuations as soon as you start to get a time machine, called <i>vacuum polarization</i>. These fluctuations are expected to build up to sufficient amplitude as to destroy the time machine to be, at least for the few simple cases studied. Or perhaps it just causes the wormholes to bounce away from each other so as to prevent time machine formation. This has not yet been shown to occur in general. But to does seem to happen for all possible configurations of one or two wormholes about to become time machines
<ref name="Visser_Lorentzian_wormholes"></ref>
.

But let us assume for the moment that wormholes cannot form time machines in any configuration due to this vacuum polarization mechanism. Now there are limits on just how you can place and move your wormholes. You don't want to accidentally almost form a time machine and break your transportation network. There are several ways to do this:
<ul>
<li> Move your wormholes slowly, so as not to build up much time dilation (or <i>time lag</i>, if you will).
<li> Don't form closed loops in your wormhole transportation network. The less distance you need to travel to go all the way around a loop to get back where you started, the less time lag is needed to make a time machine.
<li> "Discharge" the time lag by periodically rapidly moving the mouth of a wormhole that is "behind" in time in such a way that it comes back to where it was. Perhaps you can put a charge on it by sending a [[Particle_Beam_Weapons|particle beam]] through it, and then accelerate it up to relativistic speeds in a synchrotron for a while.
<li> Send new wormholes that you are going to place through the existing wormhole network. This way they acquire the same time lag across their mouths as all of the wormholes they traverse. Until they build up some additional time lag of their own, you provide a perfect return trip without making a time machine.
</ul>

In conclusion, if you want a story or setting that involves time travel, you can plausibly still use wormholes somehow to allow that (although it might take loops of several of them). If you want a story that avoids the complications of time travel, you can still plausibly have wormholes in that setting (although you now need to beware of the limitations on your wormhole transportation network imposed by the no time machine rule).

==Practical implications==

===Getting places===

Usually in science fiction, people want wormholes because they want to be able to get to cool places far away without taking forever. Very often, this involves the technology (or the protagonist, with some kind of psychic power) "opening" a wormhole from the current location to the desired location when there wasn't one before.

Sadly, the ability to just create a wormhole to anywhere you want to seems unlikely. Physics tends to be local, which means that if you can make a wormhole both ends will emerge right next to each other. You will need to move the ends to where you want them to be using other methods (mailing them by post, putting them on a spacecraft, etc.). As was mentioned earlier, if you shoot a stream of stuff through a wormhole, the end that the stuff comes out of becomes a rocket. So one way to move an end of a wormhole is to shoot stuff through the wormhole. Because you can leave all the big heavy equipment at home (generators, rocket motors, cooling systems, and everything else), if you can make your wormhole light enough this could be a low cost and efficient method of getting it places. Remember that you can exploit time dilation to make it seem like you can get someplace in much less time than you would expect light to take going to that place (at the expense of taking you some time forward in the future as well when you step through). If you can get your wormhole mass down to a few grams, or even several kilograms, using powerful lasers to shoot through the wormhole to make a photon rocket could be a highly efficient way to go places at relativistic speeds. Perhaps it could even fuel itself by collecting mass from the interstellar medium (ISM) as it flies along, and it might be able to brake against the ISM or solar wind of the destination star system to slow down, reducing mass loss even further.

===Keeping time===

We have already shown that wormholes connect across both space and time, so that a trip between star systems could take you hundreds of years into the future, and the return trip takes you hundreds of years back in time. And this is even before we throw in how time slips between planets when considering relativistic time dilation due to different speeds and gravitational potentials.

Fortunately, all the weirdness of different time rates and going backward and forward in time can be ignored by the average person. This is because you never need to go from one world to another, or back, across the vast gulfs of interstellar space. You just take the wormhole between them. All you ever need to worry about is the coordinate frame that goes across the wormhole. When considering this reference frame, you're not hopping all over the place in time. If it takes ten minutes to cross the wormhole between the two planets, when you get to your destination world the clocks will read ten minutes later than they did when you left your departure world. By coordinating their time-keeping across the wormhole network, all of the worlds of the network can agree on a common time to coordinate their activities. This is all travelers ever need to worry about, and they can then ignore all the relativistic weirdness. Your network engineers will still need to keep track of relative time drift and how close a given configuration is getting to a time loop. But unless your protagonist is a network engineer, they can just ignore all that stuff. And, as an author, so can you! Assume your engineers are competent, you have good regulatory bodies and standards institutions, and don't worry about any of this "time travel" that doesn't actually let you cause paradoxes.

===Keeping wormholes away from planets===

By now, people expect science fiction to have spaceships. Wormholes offer a convenient excuse to have spaceships, because it provides a way for those spaceships to get to other worlds that are not in our solar system. But if you have wormholes, sometimes you need to ask why you still have spaceships? Why not just put one end on Earth and the other on an alien planet and step through to get where you are going?

Sometimes there are good reasons for this. If your wormhole has as much mass as Jupiter, you are not just going to keep it in Topeka, Kansas. That would kind of end the Earth. So you would put it in orbit around the Sun, and send spaceships from Earth out to the wormhole and through it to wherever it goes.

But all too often we see authors using special pleading to pigeonhole in their spaceships without thinking about good physics. Things like "it can't work in atmosphere" (use an airlock), "it can't work within a certain number of planetary radii" (why? There is no physical plausibility to such a restriction), "It can't work in a gravitational field of more than X" (violates the equivalence principle), and so on. But if you insist on having spaceships, and you also insist on wormholes that mass less than a planet, there are a few plausible ways to justify your preconceived ideas that at least plausibly align with physics.

For example, the closest you can plausibly get to "X number of planetary radii" or "gravity field less than X" is a criterion that relies on local measurements of space-time curvature only. And it turns out that this quantity is the tidal forces (curvature is a direct measurement of tides). Tidal forces are going to be proportional to the mass of the nearby thing causing the tides, and inversely proportional to the cube of the distance. If <math>R</math> is a measure of your space-time curvature, <math>M</math> is your object's mass, and <math>r</math> is the distance from the object to the wormhole, then

<div align=center><math>R \propto \frac{M}{r^3}</math>.</div>

So if you want your wormholes to be at least ten planetary radii from Earth (<math>r = 64 \times 10^6</math> m, <math>M = 6 \times 10^{24}</math> kg), then the curvature limit is <math>R = 23</math> kg/m³ in some convenient system of units where we don't have to include constants of proportionality. But now suppose you want to send a <math>26 \times 10^6</math> kg ore freighter through that wormhole. You'll only be able to get your freighter within about 100 meters of the wormhole before the freighter's tides collapse the wormhole. This <i>might</i> work if the freighter were really long and skinny, much longer than 100 meters, to reduce its transverse tides; or if the wormhole was much larger than 100 meters in diameter. But in general, note that the curvature and tides have units of density, so if the average density in any volume around the wormhole exceeds the critical density – oops, there it goes.

Also, having a limit on the geometric curvature of space-time similar to what you get around mere planets doesn't end up being very plausible – the curvatures in the wormhole itself exceed that by so very many orders of magnitude that having so stringent of a curvature tolerance seems very implausible, and just dancing on the edge of disaster. For example, if your wormhole is about 100 meters across, its curvature in these same units will be approximately 500 billion trillion kg/m³. Keeping the tides to within 23 kg/m³ is the equivalent of an engineering tolerance of one part in 25 billion trillion.

So if you don't really have a good way to limit where you can place a wormhole based on the nearby "gravity", are there other ways you can exclude them from the vicinity of planets?

Safety might be one good reason. We have already discussed various ways a wormhole might collapse (forming a time machine, having its mass go negative). What happens when a wormhole collapses? If there are any parts of the wormhole or things in the wormhole that are charged or made of charged particles (like ordinary matter) you could justify a considerable fraction of the wormhole's mass-energy being converted to electromagnetic radiation. In addition, if the wormhole collapses into a black hole it will evaporate away its mass-energy primarily in the form of electromagnetic waves as it undergoes Hawking radiation. Large black holes will last a long time, but if the black hole left behind has a mass of 500 tons or less it will last less than 10 seconds. And 500 tons worth of energy delivered in 10 seconds would do quite a number on a planet. These sorts of considerations could lead to a regulatory environment that require wormholes to be located several light seconds away from inhabited places (like planets).

===Wormholes on planets===

But sometimes you might want to have the wormhole on a planet. Perhaps you are envisioning one of those circular or oval fantasy portals, looking like a window to an exotic foreign land, and when you step through you are actually there. We already know that thin-shell wormholes can be very low mass, and can have a circular shape. So that should work, right?

There are at least two issues to consider here. One is that even if both wormhole ends are on the same planet, natural variations in atmospheric pressure due to weather will lead to very significant differences in air pressure from one side of the wormhole to another (unless the wormholes are so close that they are within a small distance of the extent of the same weather system – say, within a few km of each other). This pressure difference will drive severe winds through the wormhole, usually comparable to the winds of a hurricane (although there will be periods of relative calm when the pressure at both wormholes just happens to be at the same pressure). This effect will only be exacerbated if the wormhole mouths are at different altitudes, due to the decrease in air pressure with altitude. And if the wormhole connects between different worlds, the effect will be much worse. In addition to making travel inconvenient and possibly damaging the equipment, the winds will produce large uncontrollable mass flows across the wormhole (air on earth has a density of about 1 kg/m³, so if you have 50 m/s winds blowing through a 2 meter diameter portal you have a mass flow of about 160 kg/s through your wormhole). This will make it difficult for mass balance, and keeping both ends having high enough masses that neither gets so close to zero that the wormhole breaks or forms a horizon or goes negative mass and causes a runaway planetary catastrophe or whatever wormholes do when one end gets driven to near negative mass by the stuff leaving it.

As a consequence, planetary wormhole mouths will probably be kept in airlocks.

The other issue is that different latitudes on a planet will be rotating at different speeds; at the poles they won't be moving at all while at the equator they will be going a full planetary circumference every day. This means wormhole mouths at different latitudes will be building up different time dilations with respect to each other. In addition, wormholes at different altitudes will experience different gravitational time dilation rates. If you let this go on for long enough, the differences in time across the wormhole mouths will form a time machine. Whether this collapses the wormholes, makes them "bounce" apart from each other (likely ruining a lot of property and infrastructure in the process), or just forms a time loop that leads to various inconvenient paradoxes depends on the assumptions you made for the physics of your world. But it will probably be something you will want to avoid. Re-balancing the time differences between wormhole ends will probably need to be done every few decades.

===Wormholes on spaceships===

The necessity for radiators on spacecraft is really annoying. Dealing with all that waste heat is not fun, and these issues are not awesome kewl stuff like gigawatts and megatons and hundreds of g's of thrust and other things that let you show everyone how much your spaceships totally rule and everyone else's spacecraft totally drool. Instead you need these big fragile stupid-looking things hanging off the side of your ship, ruining your aesthetics and efficiency and being vulnerable to your enemies blowing them up, and lighting you up so bright that you're easily visible from the other side of the solar system. Hey! I know! Let's put a tiny wormhole on our spaceships so we can just send all the heat someplace else. Great idea! That'll work, right?

Actually, yes. If you have small enough wormholes, it will work great. But there are consequences.

So you're running your coolant loops from your reactor through the wormhole to some cooling towers on your home planet. Much better than radiators. But – why do you still have that reactor on your spaceship? It's big and heavy and needs lots of big and heavy shielding and poses nasty radiation hazards. Surely you could leave it behind on your home planet as well, next to those cooling towers, and just run a power line through the wormhole? You'll be able to run a bigger, more powerful reactor that way anyway, and that extra weight won't be bogging your ship down.

And why does your spaceship have its lasers on board? You can have a much bigger laser on the planet that you don't need to lug around. Just shine the laser beam through the wormhole and re-direct it with mirrors.

And all of your missiles? Might as well just feed them through the wormhole when you need to fire them.

And then do you really need all that life support equipment? Just run an HVAC and plumbing that sends fresh air and water to the spacecraft and takes used air and water off. And remember to send the occasional snack, too.

Oh, but wait. Why is all your crew even on the spacecraft again? They have a wormhole to the spaceship. You can just have all your crew run the spacecraft remotely from a mission control station. Most of your equipment is on the planet anyway, so most of the engineering staff is already here. Now you don't actually need to put people in harm's way. If someone does need to fix something just put them in a space suit and send them through, let them do their job, then bring them back. Most of that can be done by a tele-operated robot, in any event. And being able to go back to their families when their shift is over is great for morale.

Hmmm, your sensors could just look through the wormhole instead of having all the vulnerable and heavy and expensive equipment actually on the spacecraft. And as long as you are doing that, you can just leave your laser beam pointer at home too, and aim the laser by pointing it through the wormhole.

So now what's actually left in your actual spacecraft? Oh, the rocket itself. But remember how we already talked about how we could just keep the rocket engine and propellant on the planet and shoot the rocket jet through the wormhole, and it will move just like a rocket? Yeah, so no rocket either.

Congratulations, you've successfully gotten rid of your spaceship. The wormhole itself is your spaceship. It's all you need, it's safer, it's more capable, it's stealthier, and just better all around than some contraption of steel and Mylar and fissioning uranium fuel rods and fragile people going through space.

===Adding mass to wormholes===

So you just made a pair of wormholes. Chances are, you pulled them apart as near Planck mass objects, with about 20 micrograms of mass. Well, that's not very useful! What if you want to send something through that's bigger than 20 micrograms? You're going to need to add mass to your wormhole mouths.

A similar issue comes up if you project a wormhole at some far away planet or star system using the wormhole-rocket trick already discussed, maybe by using powerful lasers to make a photon rocket. To be practical, your projected wormhole mouth will need to be very low mass. But once you get it to where you want to go, you'll need to give the mouth enough mass to let your explorers and colonists and equipment through.

A simple way to do this is to simply suck stuff into the wormhole. This adds mass to the wormhole mouth the stuff comes through. Of course, if you take it out from the other side, that other mouth will lose that much mass, and if it didn't have than much mass to begin with you'll likely have problems. So just leave that stuff in there! Shove it off to the side. Let it clutter up the unused areas out of the main traffic route in the way that basements and attics and the unused spaces of garages accumulate all that old junk that you never use any more but don't want to throw away. Build a wall inside of your wormhole, so travelers don't have to see all the asteroidal rock or super salty water or very surprised natives or whatever it was you scooped up on your alien world. Shove it into the undesirable high curvature places with nasty tides. This wall can also help keep the legitimate passengers away from the inner working of the wormhole, the strange energies that support it, the cables and pipes and crawlspaces that only your maintenance workers should access anyway, and the extreme space-time curvatures that would rip the passengers to shreds if they ever were directly exposed to them.

===Tiny wormholes===

Wormholes that are too small to fit people through could still be useful for sending information. You could use them to rapidly communicate over long distances. If the wormholes were light enough and cheap enough, they might even replace cell phones and laggy land lines with super high bit rate connections.

You could also send energy through tiny wormholes, in the form of light or electric current. Storms that knock down power lines no longer need to cut off your home's electricity. You could also use them to power machinery that need light weight and high power output – electric transportation like airplanes, shipping, or trucking; directed energy weapons; or high specific impulse rockets.

===Causality denial, causality attacks===

We have shown that moving wormholes around can lead to time travel, and that there are plausible physical mechanisms that can prevent time-travel allowing wormhole configurations from happening. People being people, you just know that if this is possible, that it will not be long before these effects are used in politics and warfare.

In a network of wormholes that is about to create a time machine, one likely resolution is that things break until a time machine is no longer possible. In any chain stressed to its point of failure, it is the weakest link that breaks. So in a path through wormholes and normal space-time that is about to form a time machine, it is the weakest wormhole that will collapse (or otherwise have something bad happen to it). So if your rival polity has a wormhole that allows then easy trade with a rich world, and you want to edge in on that trade and simultaneously deny that trade to your rival, you can just send a bigger, stronger wormhole in such a way that forces a time loop. Now your rival's wimpy wormhole breaks, and you get access to the trade opportunities with the world! This is the basics of a causality attack – use the causality-enforcing properties of wormholes to attack your enemy's wormhole infrastructure.

But what if your rival polity anticipated this? What if they have strong wormholes, with an arrangement in space-time that blocks off easy access of your wormholes to a profitable region of space? This is a causality denial action, or a causality fortress. By controlling the time displacement of the wormholes in their network, a polity can control what additional connections can be made. This can be used to enforce borders, prevent infiltration, or restrict civilian wormholes to legitimate uses.

===Moving one wormhole through another wormhole===

If you have a wormhole mouth, you can move it through another wormhole as long as the mouth (and any supporting structures) is small enough to fit through the throat of the wormhole it is going through. The geometry all works out. In fact, this has a rather nice side-effect. If you are looking through the wormhole being moved from back home, you only see it take as long to go through the other wormhole as a wormhole transit usually takes even if it got projected far into the future or past. So the moved wormhole acquires the same time differences as the wormhole it just went through (added on to any time differences it already had, it doesn't get reset or anything). This can let you easily build wormhole networks that connect to themselves in loops. It also makes the network more susceptible to breaking, because there is much less leeway for error with a closed loop as far as making a time machine and breaking the loop. So it can be done, but there are consequences.

===Dropping a wormhole into a black hole===

In principle, if you have a wormhole connection to the inside of a black hole, you can get information out from inside the event horizon. The theorem that prevents things from escaping a black hole was made under the assumption that the energy can't go negative. And as we've already seen, wormholes require regions of negative energy to work. You still wouldn't be able to see further toward the singularity at the center than the wormhole's location. And the wormhole would be inevitably dragged into the singularity in a finite amount of time. But, in principle, during that time you could see what is happening in the black hole.

But there's a big catch with this. You can't actually get the wormhole into the black hole. Not if you assume that time machines can't form, anyway. From the point of view of the wormhole mouth falling into the black hole, it will pass through the event horizon in a finite amount of time. From the point of view of someone watching from outside, the infalling mouth gets more and more time dilated to the point that it never crosses the horizon. This time dilation sets up the conditions for a time machine, and any time machine preventing physics happens on the wormhole before it can ever get in.

== Possible Networks ==
=== Acyclic wormhole networks ===

Recall that wormholes are space-time connections. They bridge space as well as time. Assuming we want to build this network in a stable manner, we want to avoid Closed Timelike Curves from appearing. Let us start with some simple anaylsis of consequences on this. Assume projected wormholes have one mouth sent at relativistic speeds to a distant target and the other mouth is kept at rest at home. From the perspective of the dispatching party, this will lead to the new connection opening very quickly. Done again and again, large volumes of space can be accessed in a short time, a powerful advantage.<br />

[[File:Wormholes PartialOrder.svg|400px|thumb|The partial order relationship of the wormhole connection]]

Every projected connection thus extends onto the future. Then we must avoid forming loops, or more generally "backtracking" back towards a dispatched node. This nets us a graph with no closed loops - an acyclical graph, also called a tree. Since connections may not return to previously established nodes if they are time-shifting from the past into the future, we also get a partial order: nodes <math display="inline">u</math> that dispatch wormholes have a higher order over the nodes <math display="inline">v</math> they establish: <math display="inline">u < v</math>. This lets us establish a direction on the whole graph. For a particular node <math display="inline">n</math>, its '''parent node''' established it, <math display="inline">parent(n) < n</math>, and it in turn establishes '''children nodes''', <math display="inline">n < child(n)</math>. The node that only has children nodes is the tree's '''root node'''. Nodes that have no children are '''leafs''' of the tree. The trees '''height''' is the longest outward path one can take from the root to a leaf.<br />

In terms of traversal, such nodes have some interesting and useful properties. Since wormhole traversals may be assumed to be effectively instantaneously bridging large distances, travel time is mostly defined by the travel distance between wormhole mouths. And as such, we want to minimize the amount of transits we have to make to get from a node <math display="inline">A</math> to a node <math display="inline">B</math> the fastest. It turns out, in order to achieve this, we want to have as large a choice of connections at each node. In the ideal case, any node except root can be reached in two transits: one transit to root, and from there to the other node. In the less ideal case we have to make one further stop-over, choosing from a number of connections at a second node to reach our destinations. The less ideal case from that sees us make two transits through such "switch yards". Assuming each node except the leafs in such a wormhole network has exactly <math display="inline">k</math> connections and there are <math display="inline">n</math> nodes in the whole network, we can reach any destination from root in <math display="inline">\log_k (n)</math> choices of wormhole transit. This is a powerful way to get to connections quickly through centralization. (In computer science, such k-nary trees are of great interest for quickly search-able data structures for this reason.) Configurations that are less "balanced" net appropriately lesser advantages and longer transit times.<br />

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Wormholes AcyclicGraphExample.svg |600px|frameless]]
<td>[[File:Wormholes Trees.svg|600px|frameless]]
<tr>
<td width=400>An example of an acyclic wormhole network in space-time.
<td width=400>The connected tree graph of the network exampke, and a k-nary tree example.
</table>

Also, traffic to most other nodes ''has'' to go through a root-wards node. This gives these nodes significant presence in the larger network. Routing through them is unavoidable if you want to reach most of the rest of the network. Root nodes can profit from taxation, services to travelers, control of routing infrastructure for information, and so on. But there is also a catch. Assuming leaf nodes each send the same amount of traffic inward. The amount of traffic load increases polynomial as you go towards root. Your wormhole infrastructure (and what transports cargo between wormholes, in space or on planets) has to cope with this scaling. Congestion might become an issue.<br />

=== Mesh wormhole networks ===

The acyclic network may have some bothersome problems. The central routing focuses power and can keep transit times down quite significantly depending on the specific network infrastructure, but it’s also congestion-prone, and some close locations may be widely separated on the network. Or maybe you want to avoid centralized routing dependencies for political reasons. Root might like being root; everyone else being subject to root’s whims, not so much. So, can we build networks where there are two routes between the same two nodes? We can, but it comes with some necessary adjustment.<br />

Recall why the acyclic network is acyclic: to avoid forming CTCs due to the time connection between future and past from the projected wormholes. But if we let both wormhole mouths experience the same time dilation, the problem that ultimately leads to CTCs - one connection routing displacing further into the future than the other connection, and thus allowing one to arrive at a node before ones departure - is avoided entirely, since there is no time displacement to be accumulated. Thus we get meshes, where there may be many connections and cycles between nodes. An observer at one node watching for his transition from another node 10 light years away would have to wait for ten years to watch his light arrive. Closed timelike loops are not possible so long as this network is kept balanced.<br />

The disadvantage - this network must take great care to ensure that the time dilation that inenvitably occurs due to different star velocities and the galactic gravitational field don’t accidentally create a time shift and CTC in the network. And also we cannot exploit time dilation to open connections very quickly, from the home node’s perspective. One has to wait out the full years it takes the wormhole’s far mouth to get to a remote destination, before the connection can be used.<br />

=== Semi-privileged wormhole networks ===

The interesting thing is, it is very much possible to build networks that make use both of relativistic projection to seemingly accelerate the network construction time, ''and'' contain cycles, with alternate routes to get to another node. The acyclic construction is a strategy to avoid CTC formation, but we should remember the ultimate constraint reason: we avoid cycles so that there are no connections which, through a certain time shift on them, a CTC could be formed. Acyclic construction is one strategy. Only building cyclic connections with careful time shift balancing is another!<br />

The foundation for such networks is an acyclic "base graph" that may be isolated from the larger network graph. We also call this the '''red graph'''. The base graph maintains all the partially ordered properties of the simple acyclic graph. We can call this the "privileged" graph, since its nodes and their connections define the principle positions in space-time of the network, and we must avoid generating CTCs on this definition. How do we avoid forming CTCs but still create alternative paths and thus loops?<br />

Let’s assume we have two nodes <math display="inline">A</math> and <math display="inline">B</math> ten light years apart. They both had connections established from a parent node <math display="inline">P</math>. Those connections are time-shifted over 99 years and part of the privileged base graph.

We couldn’t build a time-shifted connection between <math display="inline">A</math> and <math display="inline">B</math> without causing a CTC to be formed. But we are entirely fine sending signals '''Fast-as-Light''' from <math display="inline">A</math> to <math display="inline">B</math> and back. This is not a violation. Now consider: how does a wormhole with no time shift across it behave? In the mesh network case, such connections don’t cause CTCs to appear. And it doesn’t cause one to appear here either. (A/N: this could be illustrated even better with a space-time lightcone illustration I am prototyping.) How can we build such connections? We can of course use projection with relativistic speeds at both ends, like how we would build Mesh networks. But there is an even faster solution. If we can move wormholes through wormholes, we can shift the new mouths through the existing network to create this connection!<br />

The main catch is that we have to tightly watch out for temporal drift on such new connections. Through adding such edges that overall form a "blue graph" on top of the red graph, we are creating loops on the red graph underpinning the network. If one end of a wormhole mouth drifts more in time than the other (which will inevitably happen because stars won’t have the same mass, experience the same galactic gravity, or have the same velocity relative to each other) the conditions for a CTC will start to appear. The only way to pre-empt this is with continuous maintenance, taking connections offline and adjusting their temporal position using synchrotons, and having "safety space-time" spacing between wormholes. In this way, there is some "room" where signals can arrive faster and faster, but not before they were sent, allowing the connection to be used most of the time with only short intervals of adjustment.<br />

This works even when both ends connected don’t have the same time distance from the root node - the important thing is that the new shortcut accomodates this difference with a time shift of its own. We can calculate this difference easily. For two non-root nodes <math display="inline">A,B</math>, for each node we sum the temporal difference over the path from root to the nodes along the red graph, then subtract the difference from each other. The resulting remaining temporal difference is that the new connection on the blue graphh must have to keep the network stable. This means we can even do things like connect systems far out directly to core systems, or bridge from one edge of the graph to the other - arbitrary routings are possible. These routings can allow faster signal transmission and mass transport than through root, mitigating bandwith issues. The main problem is to watch out for network stability. As nodes accumulate, CTC avoidance becomes more and more delicate, and safety spacing must be maintained diligently. This can add to the effective travel time through such alternate connections, and thus maintains some of the "privilege" of the acyclic base graph.<br />

Arguably this network configuration is the most powerful, marrying many advantages and removing some disadvantages. It is however also bound to the need for safety spacing. Depending on your infrastructure assumptions, this can pose issues. If you want to implement serious stretches of safety space, wormhole links not part of the acyclic base graph would likely have to be located at interplanetary distances away from the main links, with light hours to light days of safety spacing. If your main network runs on planetary trains, this is rather a bother. Adjusting the internal throat length of wormholes can help in this regard, but of course now poses the issues of how to safely navigate such a long wormhole. Maybe you could extend something like space elevator tethers through such stretchy throats. With no gravity inside the wormhole itself, such cables could probably stretch quite significant distances. If your culture is handling wormholes mostly in space, you still have to account for added travel times through the safety spacing. Communications links would probably be easier connected between arbitrary nodes than macroscopic, craft-traversable wormholes.

==Credit==
Authors: Luke Campbell and Sevoris

==References==

File:Wormholes Trees.svg

2022-08-21T08:42:25Z

Sevoris: The trees of the acyclic graph example, and a 4-nary graph example. Created by Sevoris Doe for Galactic Library.

== Summary ==
The trees of the acyclic graph example, and a 4-nary graph example.

Created by Sevoris Doe for Galactic Library.

Wormholes

2022-08-21T08:41:00Z

Sevoris: /* Acyclic wormhole networks */

Wormholes are hypothetical structures in space-time allowed by the general theory of relativity.
They provide short cuts through space-time that connect one region of space and time to another; potentially, these regions can be very distant from each other.
The symmetries in space-time of our universe mean that wormholes move like physical objects, and acquire the conserved (or in some cases approximately conserved) quantities of things that go into them and lose those quantities of things that come out – properties like energy, mass, momentum, angular momentum, and electric charge.

==What is a wormhole? The geometry of space-time and how to twist it into a pretzel==

We're going to have to start out with some pretty heady stuff. Like the very nature of existence. And what is reality, anyway?
Usually, space and time are so much an integral part of our existence that we don't even think of them at all. They simply form a backdrop on which all the interesting stuff plays out. But when we do think about them, we're so accustomed to living in a world where they exist and where distances are well defined and events can be uniquely identified based on <i>where</i> and <i>when</i> they happened (or are happening or will happen) that a lot of stuff seems so obvious that we don't stop to think about if it really has to be this way or if there are other possible ways of doing things.

===A toy model===

So lets consider a hypothetical scenario, where an ant can be in any of several states, represented by the squares in the pictures below. The ant can transition to any other state that has a numbered side with the same number as the state it is in. We will start on the left. If the ant chooses the "2" transition, we end up with the picture in the middle. If it then chooses the "10" transition, we end up with the state on the right.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Space_elements_1.png|frameless]]
<td width=400>[[File:Space_elements_2.png|frameless]]
<td width=400>[[File:Space_elements_3.png|frameless]]
</table>
But now notice that the behavior of this system is exactly the same as if we re-arrange the states into a grid, as shown below on the left. The disconnected states now become a space; an expanse of coordinates on which the ant can exist. Now, the ant's transition between states can be described as a trajectory as shown in the right picture below.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Space_elements_grid.png|frameless]]
<td width=255>[[File:Space_elements_grid_path.png|frameless]]
</table>

This gives us an inkling of how something like space-time might end up being assembled from individual events. But what if we don't have things match up into quite such a perfect grid? We
ll take our space elements and tile them into an infinite grid. Two sections of the grid are shown below. But now we remove two of those space-tiles, and we connect the edges adjacent to the removed tiles with a corresponding edge of the other removed tile (although, as shown in the figure, we don't have to keep the same orientation). Now the ant, entering side 1 (which we have color-coded red for convenience) comes out of side 1 of the other missing square. This is perhaps the most basic wormhole.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Space_elements_wormhole_grid_1.png|frameless]]
<td width=400>[[File:Space_elements_wormhole_grid_2.png|frameless]]
</table>

Of course, this simple assembly only stitches together a space. But the world we live in is one of space-time. As the ant transitions from one square to another, it is also transitioning further ahead in time. But when the ant passes through the wormhole, while it locally experiences no discontinuity that wormhole might connect to a different time as well as a different region of space. So the ant might emerge far in the past or future of when it stepped through.

===Manifolds and coordinate patches===

The toy model used above hopefully prepares you for some of the ideas to follow - one of how we can piece together a complete space and time by connecting together different pieces. But before we go further, lets define an important term: a <i>manifold</i> is a "shape" of a given number of dimensions where, if you zoom in close enough, you can always get to a scale where it looks like the shape is flat. We'll use this definition to help us describe how we can put space and time together to get a wormhole. But first, let's look at a few illustrative examples.

====One dimension====

The simplest one dimensional shape is a line. And a line is everywhere flat. So a line is a one dimensional manifold.

A one dimensional shape might seem to be a wiggly path if inscribed in two or more dimensions, but it has no <i>intrinsic</i> curvature. That is, if everything about your world is confined to that path, you can't tell that it is curved – all you can do is measure the distance you go along the path. This greatly limits the kinds of shapes that we can consider. So any curved path of infinite extent looks from the inside like a line, and so any infinite length path is also a manifold.

But there's another class of one dimensional shapes that are manifolds – loops. A loop is a path of finite length that connects back with itself. You can make one by drawing a closed curve in a higher dimensional space. Or you can take a straight line but then say that a given point <i>here</i> maps on to another point <i>there</i>, and all the points between <i>here</i> and <i>there</i> form a loop. When you are marching along the line starting at <i>there</i> and you reach <i>here</i> you are also at <i>there</i>, so if you go further you just end up back where you were. This last bit may seem like a pointless bit of pedantry that just confuses a simple problem, but it will be a very useful way of looking at things as we go on.

====Two dimensions====

The simplest two dimensional manifold is a plane. A plane is everywhere flat, and extends off to infinity. A small region around any point on a plane (naturally) locally looks like a plane, so it meets our definition of a manifold.

=====Curved manifolds=====

But in two dimensions, we can have something that we don't have in one dimension – intrinsic curvature. Consider a sphere. If you get close enough to the sphere, it will look like you are on a plane; we live on a world that is (approximately) spherical (if you neglect mountains and ocean basins and all that other stuff), but when you stand up and look around you it mostly looks flat. So a sphere is a manifold. But take a line on the surface of your sphere. Extend it out along the surface, drawing it as straight as possible. Eventually, it will go all the way around the sphere, dividing it in two, and meet itself. This path is called a great circle. The Earth's equator is (approximately) a great circle, as are the lines of longitude (but not lines of latitude, because they do not divide the Earth into two equal-sized halves). The straightest-possible path on a curved surface (like a great circle on a sphere) is called a <i>geodesic</i>. But to keep the jargon down, we'll just call them lines for this non-technical presentation, and hopefully this will keep things more intuitive and understandable to the layman.

In flat space, two lines that are initially parallel remain parallel, always remaining the same distance apart. But in a curved space, initially parallel lines do not remain the same distance apart. On a sphere, for example, any two great circles will intersect at two points. A sphere thus has intrinsic curvature.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Sphere_and_great_circles.png|frameless]]
<td>[[File:Hyperbolic_surface_and_lines.png|frameless]]
<td>[[File:Cylinder_and_lines.png|frameless]]
<tr>
<td width=400>Two great circles going around a sphere. Although initially parallel, they meet at two points.
<td width=400>Two initially parallel lines on a hyperbolic surface. As you get farther from the place where they are parallel, the lines diverge.
<td width=400>Lines that are initially parallel on a cylinder remain parallel everywhere.
</table>

On a sphere, parallel lines always end up converging. One example of a curved surface where parallel lines diverge is a hyperbolic surface. Like a sphere, a hyperbolic surface is a manifold. But it has the opposite sense of curvature.

Now consider a cylinder. Again, a cylinder is a manifold. A cylinder can be made by rolling up a plane. Like our construction of a loop, this can be done by simply taking a long rectangular section of a plane and saying that two of the parallel edges are actually the same (this is essentially what you do if you roll up a sheet of paper so that two of its parallel edges touch, and then pretend that they are merged together where they meet). This "rolling up" preserves distances, so any two parallel lines on the plane remain the same distance apart on the cylinder (think about that sheet of paper, and if you drew two parallel lines on it - they will stay the same distance apart when the paper is rolled up). A cylinder has <i>extrinsic</i> curvature – you can tell that it is curved when looking at it from three dimensions. But it has no <i>intrinsic</i> curvature – if you are confined to the surface of the cylinder it behaves locally as if it is flat.

A manifold that is flat obeys all the rules of Euclidean geometry (although you'll need to go to the three dimensional case for dealing with volumes), and the standard Cartesian coordinate system (representing position by two orthogonal coordinates, commonly called <math>x</math> and <math>y</math>) can be used to specify any point. A manifold that is not flat violates Euclid's fifth postulate; the geometry of the manifold will be <i>non-Euclidean</i> (somewhere, H. P. Lovecraft is quietly having a mental breakdown). It cannot be represented by Cartesian coordinates, but different sets of coordinates – or perhaps patches of multiple coordinate systems that apply in different places – can be defined instead to locate points.

=====Topology: simply and multiply connected manifolds=====
If you take a plane or a sphere, any closed loop on the manifold can be shrunk in a continuous manner until it vanishes. This is called <i>simply connected</i>. But not all manifolds are simply connected. Those which are not simply connected are called <i>multiply connected</i>. The most basic multiply connected manifold is a torus, like a bagel or doughnut or bicycle inner tube. While there are closed loops on a torus that can be shrunk to a point and made to vanish, those that go all the way around the torus tube cannot – they'll get caught up circling the tube. Likewise, loops that go all the way along the tube will get caught up on the central hole if they are shrunk and again cannot be made to go to a point.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:torus_and_loops.png|frameless]]
<td width=400>[[File:Topological_torus_and_loops.png|frameless]]
<tr>
<td colspan=2>Three classes of loops on a torus. On the left is a standard geometrical torus; on the right is a topological torus that is flat but has opposite edges identified so that they are connected. The blue loop doesn't go around the torus, it can be continuously deformed to shrink to a point and vanish. The red line goes around the torus the long way. If you shrink it, it will get caught up going around the hole in the middle and get stuck there; it cannot be continuously deformed to make it any smaller. The green line loops around the torus from the outside to the inside and back. Again, if you shrink it the line gets caught up circling the torus tube and can't be made any smaller.
</table>

And this brings us to another interesting kind of manifold. You can get a manifold that is flat but has the topology of a torus. You can do this by taking a rectangle and saying that the opposite sides are actually the same. This gives you the effect of a "wrap-around screen" like on that old arcade game Asteroids – if you go off the edge on one side, you just come back on the opposite side at the same height. in physics, this is known as "periodic boundary conditions". Closed loops that go all the way around one way or the other cannot be shrunk to a point. But this kind of surface <i>cannot</i> be embedded in three-dimensional space! We might have grown used to surfaces that we can envision existing in the space we live in, but there are manifolds, even fairly simple two-dimensional manifolds, that cannot be embedded in higher-dimensional spaces like that. Remember this when we get to actual wormholes.

You can have multiply connected manifolds that are not toruses, even though they might have the same topology; any manifold with one or more "loops" or "handles" is multiply connected.

=====Non-orientable manifolds=====

And there is one other interesting kind of manifold that is relevant to wormholes – manifolds that are <i>non-orientable</i>. An orientable manifold is one where you can define a consistent sense of clockwise and counter-clockwise everywhere on the manifold. If you start with a clock spinning clockwise, or a right-handed glove, or a copy of Lewis Carrol's "Through the Looking-Glass, and What Alice Found There", you can move it anywhere on the surface and if it meets back up with itself it will still be spinning clockwise, or right handed, or normally readable. But in a non-orientable surface, there are ways you can move the object so that when you get it back to where it originally was the clock is spinning counter-clockwise, or the glove is left-handed, or the book is written in mirror writing. The simplest example of such a surface is a Mobius strip. As shown below, if you move an oriented circle around the strip, it comes back with the opposite orientation.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Topological_Mobius_strip.png|frameless]]
<td width=400>[[File:Mobius_twist.png|frameless]]
<td width=400>[[File:Mobius_strip.png|frameless]]
<tr>
<td colspan=3>A Mobius strip … a strip of surface with a twist so the edges where it connects meet up "the wrong way around". On the right is a 2D strip where you connect along the short edge matching the 1's together and the 2's together. The center shows the same strip with a twist in it so you can line the edges up the way you are supposed to, demonstrating how the twist reverses the orientation of the spinning circles. An embedded image in 3D space with the edges connected together is shown on the left. You cannot define a consistent orientation to this surface – starting at the red counter-clockwise circle, if you move the circle continuously along the surface when you get back where you started you will find it is going clockwise!
</table>

====Three dimensions====

The most basic three-dimensional manifold is a flat space. This is, well, just a normal space. You can go off in any combination of three independent directions (left-right, up-down, front-back), and because it is flat, initially parallel lines stay parallel. It all behaves much as you would expect three dimensional spaces to behave.

But you can get variations on this. They are harder to visualize than two-dimensional cases, because we don't have the brainspace for processing curved spaces inside of a four-dimensional hyperspace. But you can do things like identifying two planes to be the same, such that if you go far enough in one direction you can end back up where you started again (akin to a cylinder), or make them curved such that distances are distorted and lines don't remain parallel and maybe you can put larger things inside of smaller surfaces. There are 3D equivalents of spherical and hyperbolic surfaces (called, naturally enough, spherical and hyperbolic spaces). There are even non-orientable spaces.

One way of trying to envision a curved space, and to gain some intuition about them, is to take a two-dimensional slice of that curved space and embed that in a flat three-dimensional space. This is called an embedding diagram. Remember that this is a reduced representation that throws away a lot of the information (unless there are special symmetries you can exploit) and is not actually the thing being described.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Space_with_embedded_plane.png|frameless]]
<td width=400>[[File:Embedded_plane.png|frameless]]
<td width=400>[[File:Embedding_diagram_example.png|frameless]]
<td width=400>[[File:Embedding_diagram_example_very_curved.png|frameless]]
<tr>
<td>A space with an embedded plane. We show a sphere in this space, with the plane bisecting the sphere.
<td>Just the embedded plane. Note how in a spherically symmetric manifold (like this space), a circle on the embedding diagram represents a complete 2-dimensional spherical surface.
<td>An embedding diagram of a "dimpled" space. Just as before, the closed circles represent closed spherical surfaces. It is just that in this case, the distance between two concentric spheres of radius <math>R_1</math> and <math>R_2</math> is longer than the usual <math>R_1 - R_2</math> because of the spatial curvature, as shown by the increasing distance along the dimple's surface.
<td>An embedding diagram of an extremely curved section of space. In this example, there are several places where we have a bigger sphere inside of a smaller one! Curved space is weird like that.
</table>

Remember that not every manifold can be embedded, so not every curved space can be represented by an embedding diagram. But you can take patches of the curved space and show them by embedding diagrams, and then describe how these patches stitch together to create a contiguous surface.

====Space-time: three space dimensions and one time dimension====

This is the world we live in. So this is the class of manifolds that we really need to use when making a wormhole. Sometimes it's abbreviated as 3+1D (as opposed to 1D, 2D, 3D, etc.).

The basic, infinite, flat space-time manifold is often called Minkowski space. I know, it's a big word using scary foreign-sounding names. But all it means is that its a 3+1D manifold that goes on forever and is flat. In flat space-time, everything obeys the rules of special relativity. Which means it really doesn't work when masses become significant enough to make gravity.

Our best model of how gravity works is general relativity. In this theory, the curvature of space-time is created by mass. At least, that's what they always say in the over-simplified pop-sci summaries of general relativity. Actually, space-time curvature is created by all the components of the stress-energy tensor. It's just that in almost all cases, mass is the most significant part of this. But momentum, stresses, pressures, and shear forces all have contributions as well. In general relativity, physical objects move along the straightest possible lines in this curved space-time. The way the curvature causes the lines to bend and converge is what leads to the appearance of gravity as a force if the curvature is sufficiently weak.<ref name="Misner_Thorne_and_Wheeler">Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler, "Gravitation", W. H. Freemann and Company, New York (1973)</ref>

The most basic curved space-time manifold is the Schwarzschild geometry, which is what you get when you have an uncharged, non-spinning spherical mass doing its distortion thing on space-time. The Schwarzschild geometry is <i>static</i> meaning it does not change in time, and is <i>spherically symmetric</i>, meaning that it has a definite center and it doesn't matter which direction you are relative to that center – it is independent of angle. If the mass becomes dense enough, space-time gets so distorted that an event horizon forms. Nothing that goes into the event horizon can ever come back out, and the inevitability of the geometry crushes the mass down to an infinitely dense point at the center called a singularity. Nothing on the inside of the horizon can ever escape. It forms a <i>black hole</i>, that forever traps any light or matter that enters it. For masses that don't exceed this critical density, the gravitational distortion outside of the object is the same as for a black hole of equal mass, but there is no event horizon. The object can still be seen and interacted with.<ref name="Misner_Thorne_and_Wheeler"></ref>

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Schwarzschild_star.png|frameless]]
<td width=400>[[File:Schwarzschild_black_hole.png|frameless]]
<tr>
<td>An embedding diagram of the Schwarzschild geometry of a dense, massive star. The physical extent of the star is shown in blue.
<td>An embedding diagram of the Schwarzschild geometry of a black hole of the same mass as the star to the left. The event horizon is shown in red.
</table>

There is one distinction between space-time manifolds that will become important later. Some manifolds have an isolated region of high curvature (corresponding to a concentration of mass, or at least of stress-energy) but as you go farther away from the strongly curved region the space-time becomes increasingly flat. If you can always go out far enough away from a curved region that space-time becomes flatter that whatever flatness criterion you choose, no matter what direction you choose to leave the curved region, the manifold is called <i>asymptotically flat</i>. For our purposes, if you can go far enough away that the Newtonian approximation to general relativity is accurate, and gravity can be described as a force rather than requiring a geometric description, then the geometry can be considered asymptotically flat. You can also allow gravitational waves in your asymptotically flat space-time – their very small space-time curvature is not enough to cause problems and they can still be described by a linearized version of gravity. Both the Schwarzschild and the Minkowski geometries are asymptotically flat. Spherical or hyperbolic geometries are not asymptotically flat. Planets, stars, galaxies, neutron stars, black holes, and galactic clusters are all approximately asymptotically flat; but the universe as a whole is <i>not</i>. As long as we confine our attention to merely galactic clusters, we can make the approximation of asymptotic flatness.

===Making a wormhole===

Now that we have covered some of the introductory material, we can get down to the main course. To make a wormhole, take two regions of space-time, usually far apart and usually asymptotically flat (although neither of these is strictly required) and connect them together. That's it. You're done. Sounds easy, doesn't it?

Note that if you are connecting two regions in the same universe, your new geometry will be multiply connected. A loop that goes through the wormhole and then connects back up outside the wormhole cannot be shrunk to a point. There will now be two equivalent ways to get between any two points – the path that goes through your usual space-time, and the path that goes through the wormhole. If you use a wormhole to connect two separate universes, the combination of the two universes and the wormhole can still be simply connected.

In general relativity, you can distort the geometry of space-time with stress-energy, but there is no way to change its topology; that is, no way to turn a simply connected universe into one that is multiply connected, or to add or remove more "loops" or "handles" to an already multiply connected universe. Such changes, if they are possible at all, would only be possible in the realm of quantum gravity and we do not know enough about quantum gravity to know if it would be possible or not. So, it might not be possible to make intra-universal wormholes at all. Or maybe quantum gravity does allow topology changes in space-time and there are ways to connect different parts of our universe together with wormholes. On the plus side, if you can't change the topology of space-time, you can always bud out a new universe from our own – perhaps by triggering the inflationary field that started our universe. It will be connected to ours by a wormhole. And because the topology can't change, that wormhole would never be able to break. We would have an eternal bridge to the new universe. So a system of physics that does not allow topology changes still allows wormholes to other universes.

==Common kinds of wormholes==

===Spherically symmetric wormholes===

General relativity involves a lot of non-trivial math. You need to deal with four-dimensional curvature tensors with up to 20 independent components, and figure out how they relate to a given distribution of matter, fields, energy, stresses, and material flows. If you can reduce the number of dimensions you need to keep track of, it can simplify things a lot. One way of doing this is assuming your space-time geometry does not change with time (it is static), and looks the same in all directions when viewed from a central point (it is spherically symmetric). This makes things a lot easier to analyze.

Astute readers will have noticed that these conditions are satisfied by the Schwarzschild geometry. In fact, if you play around with the Schwarzschild geometry enough, you will find that you can even get a wormhole-like solution. At the event horizon, you can connect your black hole solution to a time-reversed structure called a white hole. This is called an Einstein-Rosen bridge. Unfortunately, it has some issues. For one, it can't form naturally from infalling matter creating a black hole. For another, going through an Einstein-Rosen bridge involves passing through an event horizon, so the trip is necessarily one way. And finally, you can show that even the smallest amount of matter or radiation falling into the Einstein-Rosen bridge leads to it collapsing before the matter or radiation can pass through, trapping it inside the black hole. So this is not looking particularly useful.

So what are we going to do? Well, the Einstein-Rosen bridge, like a lot of early work in general relativity, was made by assuming a distribution of stress-energy (in this case, matter) that was assumed to be physically plausible, and finding out what the resulting geometry would be. But we want a wormhole, dangit! So what happens if we just demand a static tunnel connecting two asymptotically flat regions of space-time? If you do this, you get a Morris-Thorne wormhole<ref name="MorrisThorne">Michael S. Morris and Kip S. Thorne, "Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity", American Journal of Physics <b>56</b>(5), 395-412, May 1988</ref>. You still have the relation between stress-energy and geometry, but now it is your assumed geometry dictating the stress-energy rather than the other way around. One difficulty with this method is that the distribution of stress-energy you end up with might not end up seeming physically plausible. In particular, it can be shown that any wormhole must have regions in it with negative energy density<ref name="Visser_Lorentzian_wormholes">Matt Visser, "Lorentzian Wormholes: From Einstein to Hawking", Springer/AIP Press (1996)</ref>, and these spherically symmetric wormholes are no different. We will discuss the issues around and solutions to the negative energy density problem later.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Morris_Thorne_wormhole.png|frameless]]
<tr>
<td>An embedding diagram of a Morris-Thorne wormhole. For this wormhole, we just took two copies of the Schwarzschild geometry from above and connected the middle "dimples" with a cylindrical patch of space-time.
</table>

The places where you enter a wormhole are called it's <i>mouths</i>, and most wormholes have two. The space-time tunnel between the mouths is called the wormhole <i>throat</i>.

====Some common problems with spherically symmetric wormholes====

if you don't have your distribution of matter and energy going off to infinity, then eventually if you go far enough away your geometry will reduce to the Schwarzschild geometry. A good many published examples of such wormholes are actually Schwarzschild-like up until pretty close to where the event horizon would be. The first problem you might encounter, then, is the formation of a horizon. A single horizon makes any trip one-way. Going into it on the side with the horizon would be indistinguishable from entering a black hole - except that you pop out somewhere else instead of getting removed from existence at the singularity. And because of the horizon, you couldn't ever go back. If both sides have a horizon, you simply can't get out of it. Why do you even have this thing, anyway?

The strong curvature as you get close to the throat can also produce very strong tidal forces. Due to the gravitational forces, if you go in feet-first your feet could be pulled much harder than your head as you go in and you could end up getting pulled apart – spaghettified is the technical term. In addition to these pure spatial tides, it turns out that you also have velocity-dependent tides. These tides are transverse to the radial direction - if you dive in really fast, you will be squeezed even harder from side to size.

Early attempts to limit the tidal forces while still keeping a throat large enough to fit a person through resulted in very large masses. Like planetary scale masses or even solar scale masses. This would mean you would need to spend a lot of propellant de-orbiting into the wormhole and then rocketing your way back out. Some of these early wormholes also had issues like the extreme gravity blue-shifting the cosmic microwave background to blow-torch intensity x-rays. Sometimes, attempts to keep the tides down also lead to really large throat sizes. Like the distance from the Sun to the Earth sizes. Or even light year sizes. These gigantic wormholes would be rather inconvenient.

None of these are necessarily intrinsic limitations to spherical wormholes. But it does indicate that a perfectly reasonable assumption for the wormhole tech in your science fiction universe could involve wormholes with Jupiter-level masses, extreme tides, and other fun details.

===Thin shell wormholes===

Take a shape located somewhere in flat (Minkowski) space-time. The important part of this shape is that it must be a closed surface that completely encloses a given volume. Now go someplace else in space-time and take an identical shape. Remove all of the space-time from inside both shapes, and then say that the corresponding parts of the two surfaces of the shapes are actually the same points. Now you have a thin shell wormhole. The throat region is infinitesimally short – the section that corresponds to the wormhole is just that surface of your shape, hence the name "thin shell".

The first thin shell wormholes proposed were simple polyhedra <ref name="Polyhedral_wormholes">Matt Visser, "Traversable wormholes: Some simple examples", Physical Review D vol. 29 no. 10 pages 3182-3184 (1989)</ref>. These are attractive choices for a potential wormhole-user because the flat faces of a polyhedral thin-shell wormhole have no tidal effects whatsoever. Further, the flat faces do not have any stress-energy either. It is simply flat space-time, no different from any other area of flat space-time. Going through the face of a polyhedral wormhole is no different from going through the flat space-time of the doorway separating your hallway from your bedroom. The negative energy stuff is concentrated entirely in the edges and corners of the polyhedron. And hoo boy, do you need a lot of it. For a cubic wormhole, each edge requires approximately <math>-1.52 \times 10^{43}</math> J/m of length. This is <math>-1.69 \times 10^{26}</math> kg/m, or close to a tenth of a Jupiter mass per meter of edge length. It has been proposed that perhaps cosmic strings – defects of the primordial universe where things couldn't line up the right way and left a linear flaw in reality – with negative tension might be able to meet this requirement and hold open a polyhedral wormhole.

But there's a strange thing when you consider the mass that these wormholes require. They are embedded in flat space-time. Which means that outside of that thin shell they have, they induce no curvature on the space-time around them. Remember that mass causes space-time to curve, dimpling up as in the Schwarzschild geometry shown earlier. Because these wormholes induce no far-flung curvature to imprint on the surrounding space-time, they have zero mass (technically ADM mass). And because mass is equivalent to energy, it should technically be possible to make one of these for no energy input at all from those of us in flat space-time. Forces are the rate of change of energy with distance, pressures are the rate of change of energy with volume. But because the energy is always zero no matter what size they are (as measured from out in flat space-time), there would not be any force or pressure needed to hold them open. The extreme conditions of their interior contrasts with the relatively benign conditions once away from the thin shell (or, indeed, within any flat area within the shell).

A curious feature of polyhedral wormholes is that each face of the polyhedron might connect to a <i>different</i> polyhedron. You might, for example, have a collection of seven cubical wormhole mouths, and each of the six faces of the wormhole you are facing connect to a different one of the six remaining mouths.

For more general thin-shell wormholes, you just need to smear that negative mass out a bit. As long as you keep a flat area to go through, travelers will not experience any strange tides or encounter space-warping exotic energies. You could, for example, have your standard circular portal beloved of fiction. Or you could have any other arbitrary shape you wanted.

Thin shell wormholes are what you get when you take a mathematical limit; they would be an approximation of a more physical distribution of stress and energy. Any realizable version would have a finite, although possible small, thickness to its shell and hence a finite, although possible small, throat length. As we will discuss later, they would likely also have a positive, although possible small, mass. But as an approximation, and as a valid solution to Einstein's field equations in general relativity, they allow us to explore many aspects of wormhole physics using a relatively simple model that is easy to calculate, and can lead to a lot of interesting inspiration for fictional ideas of wormholes.

====Tidal distortion during transit====

Consider a thin shell wormhole whose shell-shape is everywhere convex - no flat areas this time. For convenience, let's make it a sphere. As a traveler passes through the shell, the sections of his body that pass the shell must suddenly "pop" from curving one way to curving the other way, as illustrated below:

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Velocity_dependent_tides_pre_transit.png|frameless]]
<td width=400>[[File:Velocity_dependent_tides_in_transit.png|frameless]]
<tr>
<td>A traveler (left rectangular shape) about to go through a spherical thin-shell wormhole (here represented as circles, with the interior volume blacked out to indicate that there is no space-time there and nothing can exist in that region).
<td>The traveler in the process of passing through the wormhole. Note that on the wormhole shell, the portion of the traveler to the left has a leftward-curvature, while the portion on the right has a rightward curvature. As the traveler passes through, each section that goes through the shell immediately jumps from left-curving to right-curving. This will induce stresses in the traveler as his interior is deformed.
</table>

it is clear that as the curvature of the wormhole becomes large compared to the size of the traveler, the distortion will decrease. In the limit of a flat section of the surface, no distortion is experienced at all.

A consequence of this is that something going through a curved section of a thin shell wormhole will experience stresses (distortions) and strains (forces). The elastic energy stored by a strained object is proportional to the product of the stress and the strain, added up over all of its volume. Assuming that the object going in to the wormhole is initially in its relaxed state, it will require energy to push it into the wormhole shell. If it enters with insufficient initial kinetic energy to get it through or external forces pushing on it, it will just bounce off.

===Cylindrical and toroidal wormholes===

In principle, you can have a wormhole with cylindrical symmetry
<ref>Kirill A. Bronnikov and José P. S. Lemos, "Cylindrical wormholes", arXiv:0902.2360 [gr-qc] 24 Feb 2009</ref>
. This forms a valid solution to Einsteins field equations in general relativity. However, a true cylindrical wormhole would be infinite in length. So they probably don't exist naturally, and you won't be able to build one. Unlike all the other wormholes discussed here, they are not asymptotically flat. Unlike wormholes of finite extent, not all cylindrical wormholes require negative energy regions. However, it has been shown that negative energy stuff is needed if the geometry radially far enough away from the wormhole approaches the flat space-time of our usual universe, so a universe with a cylindrical wormhole without negative energy would not look like our own.

So the thought is that maybe you can get close enough to a cylindrical wormhole by making a really long wormhole that bends ever so slightly and comes back and meets itself, forming an extremely stringy torus. The general case for this has not yet been worked out, but a thin shell approximation of a toroidal wormhole has been developed
<ref>Vladimir Dzhunushaliev, Vladimir Folomeev, Burkhard Kleihaus, and Jutta Kunz, "Thin-shell toroidal wormhole", arXiv:1901.07545v2 [gr-qc] 28 Jan 2019</ref>
.
These thin shell toriodal wormholes have been shown to be stable to perturbations, one nice benefit that has not been established for a lot of other kinds of wormholes. They do require some negative energy regions on their surface, but there will also be some regions that do not contain negative energy.

===Non-orientable wormholes===

In the discussion on non-orientable manifolds, we showed how in these manifolds you can't properly determine the clockwise - counter-clockwise orientation of objects or motions, or the orientation of objects with respect to their mirror reflections. If you move an object along certain paths, when it comes back it might come back the wrong way around. So it is probably no surprise that a non-orientable wormhole turns the universe into a non-orientable manifold. Something going through a non-orientable wormhole comes out looking like it has been mirror reflected.

Normally this seems like it would be rather boring and useless, but for an obscure constraint from quantum physics called the CPT theorem. This theorem, which holds for all physical phenomena, means that the product of the discrete symmetries of charge conjugation (C), parity transformation (P), and time reversal (T) always returns you to your original state. Huh? What does that mean? Well, parity transformation is equivalent to a mirror reflection; so passing through a non-orientable wormhole means parity is inverted. This means that exactly one of the other two symmetries must also be inverted for the CPT theorem to hold. The thing coming out on the other end is still manifestly going forward in time, so it must be charge conjugated, whatever that is.

Charge conjugation means you turn all particles into their antiparticles. So anything passing through a non-orientable wormhole emerges on the other end made entirely out of antimatter.

While this particular turn of events might discourage people from passing through, it does offer obvious application for energy generation, space propulsion, and weaponry.

==Exotic energy conditions==

It is fairly easy to see than any wormhole requires negative energy densities (or more generally, regions where the tension is higher than the energy in natural units) to exist. Take a look at any of the wormhole diagrams above – the spherically symmetric ones will probably make it the most clear. If you send a parallel ray of stuff – maybe particles or light or golf balls – through the wormhole, and trace out the straightest possible path of all of those parallel lines, you will see that after they go through the wormhole all those initially parallel trajectories will now be diverging. Positive energy (such as mass) causes attractive forces, drawing trajectories toward it and making the paths converging. To get diverging paths you need the opposite – repulsive gravity – which you can get from negative energy densities or very high tensions. Having the tension be higher than the energy density turns out to be equivalent to a negative energy density – in some frames of reference observers will measure a negative energy density there.

A lot of physicists used to be reflexively dismissive of negative energy densities. After all, they can't really exist; all energy in the real world is positive.

Except that it is not. There are certain odd cases that allow for negative energy densities. One of these is the Casimir vacuum. A conducting boundary cannot support an electric field parallel to that boundary (if there was such a field, charge would flow under the force of the field until the parallel component of the field is screened out). So if you have two parallel conductive plates, the only electromagnetic waves that you can get between them require their fields to vanish at the plates. This restricts the number of ways the electromagnetic field can vibrate, called modes. In quantum mechanics, merely the potential for a vibration is associated with a certain energy (called <i>zero point energy</i>) even if no quanta of vibration are actually present. By making these modes impossible, they cannot contribute their zero point energy to the region between the plates. This lowers the total energy of the space between the plates compared to the region outside the plates (or with the plates absent). Because the energy of empty space is zero, this means that the region between these Casimir plates has negative energy. The Casimir effect has been measured in the laboratory. It is fairly weak, but it is present. In fact, not long after traversable wormholes were first described, one proposal suggested holding open the wormhole using the Casimir effect<ref name=MirrosThorneYurtsever">Michael S. Morris, Kip S. Thorne, and Ulvi Turtsever, "Wormholes, Time Machines, and the Weak Energy Condition", Physical Review Letters, Volume 61, Number 13, 26 September 1988, pages 1446-1449</ref>

Another situation that gives rise to a negative energy density is a <i>squeezed vacuum</i>. A squeezed state is a way of manipulating a quantum oscillator (like a pulse of light) to get around the uncertainty relations of quantum mechanics to increase the precision in one variable by increasing the uncertainty in its conjugate variable. Applying this operator reduces the energy of the oscillator, and if this is applied to the vacuum state, which already has zero energy, you can end up with regions of negative energy.

It is worth noting that not just any old negative energy region of exotic space-time will work. In particular, dark energy is a condition of the universe we live in where even bare vacuum itself has a small amount of negative energy. The current best model we have for this is as a cosmological constant - where every part of the universe has an equal density of negative energy and a related amount of pressure. But the wormhole allowing contributions to space-time curvature of that negative energy are exactly cancelled by the wormhole-denying contributions of the pressure. So according to our best current models, dark energy will not support a wormhole.

Also, the Casimir vacuum described above is also problematic. The exotic vacuum between the plates does have the necessary properties (if not magnitudes) to support a wormhole, but this is generally far overwhelmed by the immediately adjacent positive mass matter making up the conductive plates. The technical term for the criterion that determines whether you can have a wormhole or not is the Averaged Null Energy Condition (ANEC). If you project a ray through space-time that follows the path that light (in a vacuum) could take, and do some math along that ray and add up all the contributions from all the places along that ray, then if your result is zero or positive then the ANEC is said to be satisfied. If this is the case, you can't get a wormhole. If the ANEC is less than zero, it is said to be violated and in principle you could use that weird exotic energy stuff to prop open a wormhole's throat. There are other so-called energy conditions, and if you have regions of space-time that also violate those energy conditions they can be used to hold open wormholes as well - but all of these also violate the ANEC, so the ANEC is all you need to consider.

===So just how do I get the ANEC-violating stuff I need?===

The short answer is, we don't know.

As we already said, a cosmological constant form of dark energy won't do it (although different ideas about dark energy, such as phantom energy, could conceivably work). The inflationary vacuum thought to have existed very briefly during the first instances of our universe's existence is also usually thought to be a cosmological constant (although one that stopped existing after a very brief period of time, so I guess it is less than constant), so the inflationary vacuum probably won't work either.

If you can get the Casimir vacuum without the massive conducting plates that bound it, you might be able to do something with that. For example, the twisted geometry of space-time itself in a wormhole throat imposes boundary conditions that restrict the modes of vibration inside of it similar to the Casimir effect. This is called the <i>topological Casimir effect</i>, potentially it could be used to help support a wormhole. Also, although the ANEC is satisfied if the probing space-time ray goes through the Casimir plates, a ray that goes between the plates but never touches them will violate the ANEC. Can you use this to hold open a wormhole? Maybe a sufficiently clever person will eventually figure out how to do so.

Even more serendipitous, it turns out that gravity itself naturally squeezes the vacuum
<ref name="gravitationally_squeezed_vacuum">David Hochberg and Thomas W. Kephart, "Lorentzian wormholes from the gravitationally squeezed vacuum", Physics Letters B 268, 377-383 (1991)</ref>
. If you have the curved space-time of gravity, all of the modes of oscillation in it become squeezed states. So it might be possible that the curved space-time of a wormhole can squeeze the vacuum enough to support itself. Or a combination of gravitational vacuum squeezing and the topological Casimir vacuum from the wormhole's geometry constraints might be all that is needed. At least one proposal has suggested a class of wormholes that can be self supporting in such a way
<ref>S. V. Krasnikov, "Toward a Traversable Wormhole", arXiv:gr-qc/0003092v1 22 Mar 2000</ref>
<ref>S. Krasnikov, "Traversible Wormhole", Physical Review D, volume 62, article 084028 (2000)</ref>,
although the throat of these wormholes is very "wrinkly" or "crumpled up" and it is unclear what the effects of that would be on anyone passing through.

====Quantum energy inequalities====

When you apply quantum field theory to what happens when you have a region where the quantum fields produce local negative energy densities, you end up with a limit of how much negative energy stuff you can have before it has to be balanced out by an even greater amount of nearby positive energy stuff
<ref>[https://web.archive.org/web/20060206055950/http://maths.york.ac.uk/www/PhysicsQIneq.htm|Quantum Energy Inequalities]</ref>
<ref>[http://www.phys.lsu.edu/mog/mog20/node16.html|Quantum field theory on curved spacetime at the Erwin Schrödinger Institute]</ref>
<ref name="Fewster_Roman_2005">Christopher J. Fewster and Thomas A. Roman, "On wormholes with arbitrarily small quantities of exotic matter", Physical Review D <b>72</b>, 044023 (2005)</ref>
.

====Matter with negative mass====

It might be tempting to try to solve the negative energy density requirement for wormholes by introducing some sort of matter that naturally has negative mass.
However, negative mass would introduce all kinds of problems to the smooth operation of how the world works. For one thing, if negative mass (or equivalently energy) can exist, why doesn't it just spontaneously pop out of empty space accompanied by an equal magnitude of positive mass (or energy)? For another, negative masses lead to all kinds of runaway instabilities. For example, imagine what happens if you push on an object with a negative mass. You exert a force on it with your hand. But by Newton's second law of motion, an object's acceleration is proportional to its mass. So the negative mass thing will accelerate backwards, against the direction of the force, and into your hand. Which means you will be pushing even harder on it, which makes it push back even harder than before. You have a runaway process that blows up until your hand breaks, and any fragments of flesh and bone and sinew that were in the way of the negative mass thing have been flung off.

Or consider a negative mass thing with an electric charge. If you put something with an electric charge in an electric field the field will exert a force on the charge. For a normal positive mass charged thing, it will move in the direction of the force. The work done is the force in the direction of the displacement in position, so because the force and motion of the object are in the same direction, the field does work on the charged object and energy flows from the field into the kinetic energy of the charged object. But if the charged object has negative mass it will begin to move opposite to the direction of the force. This will do work on the field instead of the object, increasing the energy of the field. The energy comes from the kinetic energy of the negative mass charged object – but because kinetic energy is proportional to mass and increases with increasing speed, as the negative mass object loses kinetic energy it gains speed, going ever faster. Both the field strength and the speed of the object increase without bound. If the electric field originates from the electric part of an electromagnetic wave, it will vary sinusoidally with time and space. As the negative mass object is forced into a sinusoidal trajectory opposing the electric force on it, it will amplify that wave making it ever more intense. A single, isolated charge exposed to an electromagnetic wave will eventually have its inertia keep it moving from the previously imposed force even after the force has switched direction, so if the negative mass is free with no other forces acting on it there will be no net amplification. But if there are any other damping forces, or internal arrangements of bound charge that can resonate with the wave, then instead of being absorbed (as would happen with a positive mass), the wave will be amplified indefinitely and the ever-increasing energy of the wave will come from the ever increasing temperature of the negative mass thing (because the mass is negative, the thermal motion of temperature gives it a negative energy whose magnitude only increases as the temperature rises).

And, of course, the quantum energy inequalities indicate that you can't just have isolated lumps of negative mass stuff floating around.

So just having lumps of stuff that have a negative mass is probably not realistic, and it were possible it would lead to all kinds of universe-shattering cataclysms. Hopefully, we can get wormholes to stay open just using the exotic vacuum states from the Casimir effect and squeezed vacuums.

===Reducing or eliminating ANEC violating stuff===

As soon as people figured out that the wormholes they want needed stuff they couldn't get, they set about trying to find ways to use as little of the stuff they couldn't get as possible. A number of attempts have been made to hammer out a geometry for spherical wormholes that are stable, meet the quantum energy inequalities, and allow a person to pass through them in a reasonable length of time without being shredded by tides
<ref name="Fewster_Roman_2005"></ref>
<ref name="Visser_Kar_Dadhich">Matt Visser, Sayan Kar, and Naresh Dadhich, "Traversable Wormholes with Arbitrarily Small Energy Condition Violations", Physical Review Letters Vol. 90 No. 20 article 201102 (2003)</ref>
<ref name="Kuhfittig">Peter K. F. Kuhfittig, "More on wormholes supported by small amounts of exotic matter", Physical Review D <b>73</b>, 084014 (2006); Peter K. F. Kuhfittig, "Wormholes supported by small amounts of exotic matter: some corrections", arXiv:gr-qc/0508060v1 15 Aug 2005</ref>
.
Often, these resulting structures end up having extreme dimensions, such as throat sizes on the order of light years
<ref name="Fewster_Roman_2005"></ref>
<ref>Remo Garattini, "Generalized Absurdly Benign Traversable Wormholes powered by Casimir Energy", arXic:2008.05901v1 [gr-qc] 12 Aug 2020</ref>

Thin shell wormholes have also been the subject of investigation. One study <ref name="Mazharimousavi_Halilsoy">S. Habib Mazharimousavi, M. Halilsoy, "3 + 1-dimensional thin shell wormhole with deformed throat can be supported by normal matter", Eur. Phys. J. C (2015) 75:271 DOI 10.1140/epjc/s10052-015-3506-6</ref> found a number of thin shell wormholes with sharp-edged shapes could be supported entirely without any exotic energy. However, this only applies if the edges are infinitely sharp. A finite radius of curvature at the edges would require negative energy at those edges.

If the wormhole is taken have the properties of a class of subatomic particles called a fermion, then if the ratio of the electric charge to the mass is sufficiently large you can get a wormhole that can remain open without any negative energy regions at all
<ref name="Blazquez-Salcedo_Knoll_Radu">Jose Luis Blázquez-Salcedo, Christian Knoll, and Eugen Radu, “Transversable wormholes in Einstein-Dirac-Maxwell theory”, Physical Review Letters <b>126</b>, 101102 (2021); ArXiv: 2010.07317v2 [gr-qc].</ref>
.

==Fields and conserved properties==

Our best theories of how the world works – quantum mechanics and general relativity – predict that certain things are [[Conservation_Laws:_Limits_to_Cheating|conserved]]. For asymptotically flat space-times, these quantities are energy, linear momentum (or just momentum), angular momentum, and electric charge. Furthermore, in the non-relativistic limit mass is conserved independently of non-mass energy.

And not only are these things conserved, but they are always conserved <i>locally</i>. This means you can't just get rid of, say, some energy in one place and expect things to balance out by having that energy appear somewhere else.

What does it mean for something to be conserved? Mathematically, it means that the thing obeys the continuity equation. There are various ways of writing the continuity equation, all equivalent, but basically it boils down to the following: if you have a region of space surrounded by a closed surface, then the amount of conserved stuff inside that surface can only increase if some of the stuff enters through the surface and it can only decrease if some of the stuff leaves through the surface. Makes sense, right? If you have three loaves of bread in a room, then if the deliveryman brings in another two loaves you will have five loaves. But unlike loaves of bread, which can be baked or eaten or grow moldy or burn up or be broken into crumbs, the only way to get energy or electric charge or the other conserved things into or out of the room is to have them come into or go out of the room.

Each of these conserved quantities has an associated field. Electric charges create an electric field. Energy and momentum and angular momentum each create a distribution of the curvature tensor of space-time. In the non-relativistic limit, this is the mass part of the energy creating the gravitational field.

There is a relationship between the amount of a conserved quantity in our hypothetical closed surface and the net amount of field that penetrates that surface. This is called [https://en.wikipedia.org/wiki/Gauss%27s_law <i>Gauss's law</i>]. Basically, it says that if you add up all of the amount of field that goes out of the surface, and subtract off all of the field that goes back into the surface, the resulting amount will be directly proportional to the amount of the conserved stuff in there. Do you start off with a kilogram of mass inside your room? Then you have a certain amount of gravitational field from that mass that leaks out of the room. Add another kilo for two kilos total, and the amount of gravitational field leaving is also doubled. The distribution of where that field comes out may change, but as long as you don't take that mass out of the room or add more mass in the room, the total amount of field leaking out won't change. And any gravitational field from masses outside of the room that leaks into the room must also leak back out again <i>somewhere</i>. Those masses outside the room will not affect the net amount of field leaking through the surface. What this means is that you can determine the amount of conserved stuff inside a surface if you know the amount of field going through that surface.

This can be visualized in the field line approximation. The field around a positive charge can be approximated as a series of directed lines that radiate away from the charge. The field around a negative charge in this approximation is a series of lines that converge on the charge (for non-relativistic gravitation, you only have positive charges because mass is always positive. We'll just ignore the fact that wormholes require negative energy density stuff in order to exist which will lead to negative mass – we're just trying to get across the basic idea here). Field lines can only ever start on a positive charge and only ever end on a negative charge. The number of lines radiating away or converging on the charge is proportional to the size of that charge (so if you double the charge, you double the number of lines that connect to it). Field lines propagating through space bend away from a positive charge and bend toward a negative charge.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=322>[[File:charges_positive.png|frameless]]
<td width=322>[[File:charges_negative.png|frameless]]
<td width=322>[[File:charges_dipole.png|frameless]]
<tr>
<td>The field lines radiating away from a positive electric charge.
<td>The field lines converging on a negative electric charge.
<td>The field lines emerging from a positive charge and curving around to go into a negative charge of the same magnitude. This configuration, with a positive and negative charge of the same magnitude, is called a <i>dipole</i>.
</table>

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=322>[[File:charges_positive_Gauss_law.png|frameless]]
<td width=322>[[File:charges_negative_Gauss_law.png|frameless]]
<td width=322>[[File:charges_dipole_Gauss_law.png|frameless]]
<td width=322>[[File:charges_exterior_dipole_Gauss_law.png|frameless]]
<tr>
<td>A closed surface encloses a positive charge. A net amount of field lines exit the surface.
<td>A closed surface encloses a negative charge. A net amount of field lines enter the surface.
<td>A closed surface encloses a dipole. The total amount of charge inside the surface is zero because the negative charge cancels the positive charge. No net amount of field goes through the surface - just as many field lines enter as leave.
<td>A closed surface that has no charge inside of it, but various charges outside of it (in this case a dipole). Again, no net amount of field goes through the surface - every field line that enters comes back out somewhere else.
</table>

Now consider the case of an isolated positive charge, like one one shown above, that is far away from anything else. We will bring up one mouth of a wormhole close to the charge. Because the field lines going out of the charge cannot break, they cannot go through the wormhole. Instead, they will curve around the wormhole mouth. As the charge enters the wormhole mouth and travels into its throat, it drags its field lines along with it. With all of its field lines entering the mouth, that mouth looks like it has a net positive charge. As the charge leaves the other mouth, it continues to drag ints field line out of the wormhole - the field lines that start on the charge now curve around to enter the wormhole mouth. The mouth that the charge exited from now looks like it has a negative charge.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Wormhole_with_charges_initial.png|frameless]]
<td width=400>[[File:Wormhole_with_charges_inside.png|frameless]]
<td width=400>[[File:Wormhole_with_charges_passed_through.png|frameless]]
<td width=400>[[File:Wormhole_with_charges_long_gone.png|frameless]]
<tr>
<td>The charge approaches the wormhole.
<td>The charge is inside the wormhole. The mouth of the wormhole that the charge entered looks like it has a positive charge.
<td>The charge has passed through the wormhole. The mouth that the charge entered still looks like it has a positive charge, but now the mouth that the charge came out of looks like it has a negative charge.
<td>The positive charge has now moved very far away. Both mouths of the wormhole still appear to be charged.
</table>

This can also be understood by the continuity equation. Picture an imaginary closed surface around the mouth of the wormhole. This is like the case of the room, before, except that <i>now there is a way out of the enclosed volume that does not pass through the surface</i> – by going through the wormhole! Any conserved quantity that goes into the surface adds its quantity to the stuff in the surface. If it goes through the wormhole, it does not come out through the surface. The conserved quantity inside that surface stays the same! To an outside observer, it looks like the wormhole mouth has that quantity.

Either way you look at it, the results are the same. A wormhole mouth acquires any conserved quantity of the things that enter it, and loses any conserved quantity of the stuff that exits from it.
<ul>
<li> A wormhole mouth acquires the <i>energy</i> of anything that goes into it. Its energy decreases by the energy of anything that goes out of it.
<li> In the non-relativistic limit, this means that a wormhole mouth acquires the <i>mass</i> of anything that goes into it. Its mass decreases by the mass of anything that goes out of it.
<li> A wormhole mouth acquires the <i>electric charge</i> of anything that goes into it. It gains negative the electric charge of anything that goes out of it.
<li> A wormhole mouth acquires the <i>momentum</i> vector of anything that goes into it. It gains the negative of the momentum vector of anything that goes out of it.
<li> A wormhole mouth acquires the <i>angular momentum</i> vector of anything that goes into it. It gains the negative of the angular momentum vector of anything that goes out of it.
</ul>

This perhaps should not be a surprising result – other general relativistic space-time distortions also exhibit this behavior. A black hole, for example, gains the mass, charge, momentum, and angular momentum of anything that goes into it.

You can use these same arguments to find that the same results apply to any asymptotically flat manifolds of space-time. They do not necessarily hold for space-time geometries that are not asymptotically flat. If the gravitational distortion of a wormhole extends infinitely far away, you will not be able to even define its total energy or momentum or angular momentum, so it will be impossible to say if they are conserved. But do note that all of the wormhole examples that we have discussed so far (except for the cylindrical wormhole of infinite extent) are asymptotically flat, and will obey the local conservation of all of these quantities.

===Wormhole dynamics===

A wormhole mouth has the gravitational field and conserved properties of mass, momentum, and angular momentum. By the equivalence principle of general relativity, this must mean that it behaves as if it had those properties. So a wormhole mouth will follow ballistic trajectories in vacuum through space-time as if it were an object in free fall. If it is around a concentrated mass, like a planet or sun, it will have a Keplerian orbit around that object (or, if the wormhole has more mass than the planet or sun, perhaps it would be more reasonable to say that the planet or sun would orbit it). The mass of the wormhole mouth will resist acceleration like any other mass, according to Newton's laws of motion. Force is the time rate of change of momentum, so as the wormhole mouth's momentum changes it will experience forces, and these will change its trajectory in the same way as if that momentum (or forces) acted on a material object with the same mass. And if the wormhole mouth has an electric charge, it will experience forces from electric and magnetic fields; again, these can change its trajectory.

In other words, a wormhole mouth acts like a physical object of the same mass, charge, energy, momentum, and angular momentum.

The motion of the wormhole mouths through our normal space does not affect their relative separation through the wormhole's throat. That can remain the same length regardless of the behavior of the mouths in our normal universe.

As we covered, if an object enters a wormhole mouth that mouth gains the object's momentum. This will give the wormhole a "kick" changing its motion in the direction that the entering object was originally moving. The dynamics work out the same as if the wormhole were a sticky blob and the object collided and stuck to it. In both cases, you get a final object with the combined masses of both initial things (the mouth and the entering object) that has been knocked off course a bit by the momentum of the entering object hitting it.

If that object comes out of the other mouth, then as that mouth gains the opposite of the objects momentum it gets a recoil kick in the opposite direction. Again, this results in dynamics that are similar to things we already know about – like a gun with a bullet loaded firing the bullet. As the bullet leaves, the gun + bullet system loses the mass of the bullet, and as the bullet shoots away the gun recoils in the opposite direction.

So now consider what happens if you direct a constant stream of material through a wormhole. As the jet of material exits its mouth, that mouth loses mass at the same rate as the mass flow rate of the jet. It also continually gains momentum by the recoil kick of the gas. These dynamics are identical to that of a rocket - as the propellant is ejected, the rocket loses the propellant's mass and the recoil momentum from the escaping gas jet pushes the rocket in the opposite direction. So by shooting stuff through a wormhole you can turn it into a rocket, with dynamics identical to a rocket, obeying the [https://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation Tsiolkovsky rocket equation], and everything else that happens with rockets.

===Does gravity go through a wormhole?===

The above diagrams of the electric field of a charge as it approaches a wormhole mouth also hold for the gravitational field of a mass that is near a wormhole mouth. In the same way that the electric field cannot enter the wormhole's interior or leak through to the other side unless the charge enters the wormhole or goes through to the other side, so to will the gravitational field of a nearby mass curve around the wormhole. The gravity of a nearby planet will fall to zero inside the wormhole throat and will not affect those near the opposite mouth.

===Can waves go through a wormhole?===

So if gravity won't go through a wormhole, and static electric and magnetic fields can't go through a wormhole, then what about light? That's made up of electricity and magnetism, right?

So, turns out that light can go through a wormhole. So can other electromagnetic waves like radio and x-rays. So can gravitational waves. The rules for drawing field lines only really work for static charges; for radiating waves you also need to include the parts about how the changing field generates more field. In the end, waves that are smaller in wavelength than the size of the wormhole can go through just fine. You'll get some [[Diffraction|diffraction]] around the mouths, but for wavelengths much smaller than the mouth size even that might not be noticable.

===Can negative mass wormholes exist?===

So a wormhole mouth loses the mass of anything that comes out of it. What happens if you send something through a wormhole with more mass than the mouth which it exits from? This rule says that the mass must become negative, but as we saw earlier this leads to all kinds of problems (especially because the wormhole can also be charged) and is likely to be forbidden by the laws of nature.

If a wormhole's mouth can reach negative mass, you can also get other interesting instabilities. The positive mass mouth will attract matter into it, which will pass through, increasing the mass of the positive mass mouth and making the mass of the negative mass mouth even more negative. Meanwhile the gravity of the negative mass mouth will repel matter, preventing any return flow. Both the negative and positive masses will gradually increase over time.

So what's an aspiring science fiction writer to do? If you have a particular desire for negative mass things in the fiction you create, go ahead and allow it. Just be aware of the consequences. Otherwise, it is probably easier to say that negative mass wormhole mouths can't exist, and justify that with the quantum energy inequalities. Which, of course, brings up the question of what happens when you try to push an object through a wormhole that would make the mass of one of its mouths go negative? Two obvious possibilities are that either you get a back-reaction force that makes it so you can't ever make something go through the mouth to make its mass negative (perhaps by the formation of a horizon as the mass approaches zero), or the wormhole collapses before the mass can ever reach zero. Perhaps inventive science fiction authors and fans can come up with other plausible options as well.

==Wormholes and time travel==

Wormholes connect across both space and time. So it is natural to consider the possibility of using wormholes for time travel.

Now maybe you don't want to have to deal with time travel in your fiction. It introduces all kinds of opportunities for paradoxes, and you have to keep track of plot lines in a self-consistent way that go back and forth through time, influencing themselves in complicated ways. Okay, no problem. Lets just have our wormholes always connect to the same time. Easy right? In fact, it is natural to justify this. Physics is local, so if you can make wormholes you probably have to create both ends right next to each other at the same time. You can then move the wormhole ends away from each other – maybe putting one end on an interstellar spacecraft or something – but they stay at the same time. Right?

Right?

Except ...

In relativity, motion not only affects how fast you go through space, but also how fast you go through time. When things move, they experience time dilation, so if you move one wormhole rapidly far away and then bring it back, the time dilation it experienced will naturally allow it to form a time machine so you can send things and people back to interfere with their own pasts. Even worse, time can also be slowed down by gravity, so if your wormholes are in areas affected differently by your galaxy's own gravity, or around stars of different mass, or just at different elevations on the same planet, the rate they experience time will be different. Temporal complications ahoy!

Let's look at how this works in more detail. We'll take our standard science fiction empire, called The Empire. Like all empires, it has a central area of authority, the metropole (which we will place on a planet called Metropole), which extracts wealth from various colonies and subjugated client states for its own enrichment in the form of taxes, tribute, and forced favorable trading opportunities. Let's give our Empire two colony worlds which we'll call Colony A and Colony B (these Empire folks sure aren't very creative in their naming, are they?). Just for convenience, we'll put Metropole, Colony A, and Colony B all 100 light years away from each other, in an equilateral triangle.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:WormholesTimeTravel graphic1 wb.svg|600px|frameless]]
<td>[[File:WormholesTimeTravel graphic2 wb.svg|600px|frameless]]
<tr>
<td width=400>Metropole and its two colonies A and B
<td width=400>Metropole sending a wormhole to Colony A.
</table>

So Metropole launches a wormhole mouth out to Colony A. They want to get to Colony A quickly, so they'll make the wormhole go really fast. Let's say they fling it out at 99.9999% the speed of light. At this speed, it takes 100.0001 years for the wormhole mouth to reach its destination in the reference frame of Metropole and Colony A. But due to relativistic time dilation, in the reference frame of the projected wormhole mouth it only takes 0.1414 years to go from Metropole to Colony A. Because the techs at Metropole can look through the wormhole, they will see that 0.1414 years after launch it arrives at its destination. At that point, brave explorers from Metropole can go through and set foot on Colony A – they only need wait about a month and a half to feel alien soil under their feet, rather than a century. In Metropole's reference frame, going through the wormhole takes you 100 light years away, and 99.8596 years into the future. Going the other way, from Colony A to Metropole, takes you 100 light years away and 99.8596 years into the past.

[[File:WormholesTimeTravel graphic3 wb.svg|800px|frameless]]

One might think that these time warps would let you engage in all kinds of time travel. It is easy to see that the Metropole—Colony A situation described here doesn't allow these kinds of shenanigans. For practical purposes, you only have a time machine when you can go back to the place you left at a time before you left. And you can't do that here. Go from Colony A to Metropole and you go back in time 99.8596 years. Go back to Colony A through the wormhole, and you go forward in time the same amount, plus any time you spent on Metropole, so you get back after you left. If you go back through flat space-time, it will always take at least 100 years since you can't go faster than the speed of light so you also get back after you left. No paradoxes for you!

However, it is easy to imagine situations where a wormhole, or a configuration of wormholes, does make a time machine. For example, what happens if we immediately turn around and send the wormhole mouth from Colony A back to Metropole? In the reference frame of Metropole, the wormhole comes back with its "Return to sender" sticker 200.0002 years after it was launched. But going through from the left-behind mouth to the round trip mouth will take you 199.7192 years into the future. And if you step into the round trip mouth, you will go back in time by 199.7192 years into your own past. You now have a real honest to goodness time machine, complete with paradoxes.

[[File:WormholesTimeTravel graphic6´4 wb.svg|800px|frameless]]

Or maybe the wormhole mouth stays at Colony A. And Metropole sends another one to Colony B, that also goes 99.8596 years into the future. Now Colony A sends a wormhole to Colony B. This wormhole also goes 99.8564 years into the future as a consequence of its trip. This means if a traveler at Colony B went through the Colony A wormhole he would go back in time 99.8564 years. Then going from Colony A to Metropole he would go back in time another 99.8564 years. Then he could go from Metropole to Colony B and go forward in time 99.8564 years. The net result is that he ended up back where he started nearly a century before he left.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:WormholesTimeTravel graphic5 wb.svg|600px|frameless]]
<td>[[File:WormholesTimeTravel graphic6 wb.svg|600px|frameless]]
</table>

And it is not just motion that causes time dilation. Being different depths into a gravitational potential does this as well. If Colony A is deeper into the gravity well of the galaxy, or orbits a heavier star, it will experience more gravitational time dilation than Metropole. So eventually it will build up more than 100 years of time slip between the two end of the wormhole and you can have a time machine.

So is time travel inevitable if you have wormholes? What a nuisance!

Well, maybe not. Here's why. Think about what happens when the Colony A – Colony B wormhole has gone just far enough that a light signal going through the wormholes can get back to where it left just as it is leaving. Now, since the propagating signal and the newly transmitted signal are both leaving at the same time, you have double the intensity. So this doubled intensity signal goes around and meets itself again, quadrupling its intensity. And so on. At this point, just as the configuration is on the verge of becoming a time machine, it becomes a perfect resonator for light signals, which then build up to arbitrarily high intensities until something breaks and you don't have an incipient time machine any more.

Now, this won't always work if you just consider light signals. Light can get defocused or sent off in odd directions or something so that you don't have a good path back until after you end up with a time machine. But if you look at what happens in semiclassical quantum gravity, will get a buildup in the amplitude of quantum fluctuations as soon as you start to get a time machine, called <i>vacuum polarization</i>. These fluctuations are expected to build up to sufficient amplitude as to destroy the time machine to be, at least for the few simple cases studied. Or perhaps it just causes the wormholes to bounce away from each other so as to prevent time machine formation. This has not yet been shown to occur in general. But to does seem to happen for all possible configurations of one or two wormholes about to become time machines
<ref name="Visser_Lorentzian_wormholes"></ref>
.

But let us assume for the moment that wormholes cannot form time machines in any configuration due to this vacuum polarization mechanism. Now there are limits on just how you can place and move your wormholes. You don't want to accidentally almost form a time machine and break your transportation network. There are several ways to do this:
<ul>
<li> Move your wormholes slowly, so as not to build up much time dilation (or <i>time lag</i>, if you will).
<li> Don't form closed loops in your wormhole transportation network. The less distance you need to travel to go all the way around a loop to get back where you started, the less time lag is needed to make a time machine.
<li> "Discharge" the time lag by periodically rapidly moving the mouth of a wormhole that is "behind" in time in such a way that it comes back to where it was. Perhaps you can put a charge on it by sending a [[Particle_Beam_Weapons|particle beam]] through it, and then accelerate it up to relativistic speeds in a synchrotron for a while.
<li> Send new wormholes that you are going to place through the existing wormhole network. This way they acquire the same time lag across their mouths as all of the wormholes they traverse. Until they build up some additional time lag of their own, you provide a perfect return trip without making a time machine.
</ul>

In conclusion, if you want a story or setting that involves time travel, you can plausibly still use wormholes somehow to allow that (although it might take loops of several of them). If you want a story that avoids the complications of time travel, you can still plausibly have wormholes in that setting (although you now need to beware of the limitations on your wormhole transportation network imposed by the no time machine rule).

==Practical implications==

===Getting places===

Usually in science fiction, people want wormholes because they want to be able to get to cool places far away without taking forever. Very often, this involves the technology (or the protagonist, with some kind of psychic power) "opening" a wormhole from the current location to the desired location when there wasn't one before.

Sadly, the ability to just create a wormhole to anywhere you want to seems unlikely. Physics tends to be local, which means that if you can make a wormhole both ends will emerge right next to each other. You will need to move the ends to where you want them to be using other methods (mailing them by post, putting them on a spacecraft, etc.). As was mentioned earlier, if you shoot a stream of stuff through a wormhole, the end that the stuff comes out of becomes a rocket. So one way to move an end of a wormhole is to shoot stuff through the wormhole. Because you can leave all the big heavy equipment at home (generators, rocket motors, cooling systems, and everything else), if you can make your wormhole light enough this could be a low cost and efficient method of getting it places. Remember that you can exploit time dilation to make it seem like you can get someplace in much less time than you would expect light to take going to that place (at the expense of taking you some time forward in the future as well when you step through). If you can get your wormhole mass down to a few grams, or even several kilograms, using powerful lasers to shoot through the wormhole to make a photon rocket could be a highly efficient way to go places at relativistic speeds. Perhaps it could even fuel itself by collecting mass from the interstellar medium (ISM) as it flies along, and it might be able to brake against the ISM or solar wind of the destination star system to slow down, reducing mass loss even further.

===Keeping time===

We have already shown that wormholes connect across both space and time, so that a trip between star systems could take you hundreds of years into the future, and the return trip takes you hundreds of years back in time. And this is even before we throw in how time slips between planets when considering relativistic time dilation due to different speeds and gravitational potentials.

Fortunately, all the weirdness of different time rates and going backward and forward in time can be ignored by the average person. This is because you never need to go from one world to another, or back, across the vast gulfs of interstellar space. You just take the wormhole between them. All you ever need to worry about is the coordinate frame that goes across the wormhole. When considering this reference frame, you're not hopping all over the place in time. If it takes ten minutes to cross the wormhole between the two planets, when you get to your destination world the clocks will read ten minutes later than they did when you left your departure world. By coordinating their time-keeping across the wormhole network, all of the worlds of the network can agree on a common time to coordinate their activities. This is all travelers ever need to worry about, and they can then ignore all the relativistic weirdness. Your network engineers will still need to keep track of relative time drift and how close a given configuration is getting to a time loop. But unless your protagonist is a network engineer, they can just ignore all that stuff. And, as an author, so can you! Assume your engineers are competent, you have good regulatory bodies and standards institutions, and don't worry about any of this "time travel" that doesn't actually let you cause paradoxes.

===Keeping wormholes away from planets===

By now, people expect science fiction to have spaceships. Wormholes offer a convenient excuse to have spaceships, because it provides a way for those spaceships to get to other worlds that are not in our solar system. But if you have wormholes, sometimes you need to ask why you still have spaceships? Why not just put one end on Earth and the other on an alien planet and step through to get where you are going?

Sometimes there are good reasons for this. If your wormhole has as much mass as Jupiter, you are not just going to keep it in Topeka, Kansas. That would kind of end the Earth. So you would put it in orbit around the Sun, and send spaceships from Earth out to the wormhole and through it to wherever it goes.

But all too often we see authors using special pleading to pigeonhole in their spaceships without thinking about good physics. Things like "it can't work in atmosphere" (use an airlock), "it can't work within a certain number of planetary radii" (why? There is no physical plausibility to such a restriction), "It can't work in a gravitational field of more than X" (violates the equivalence principle), and so on. But if you insist on having spaceships, and you also insist on wormholes that mass less than a planet, there are a few plausible ways to justify your preconceived ideas that at least plausibly align with physics.

For example, the closest you can plausibly get to "X number of planetary radii" or "gravity field less than X" is a criterion that relies on local measurements of space-time curvature only. And it turns out that this quantity is the tidal forces (curvature is a direct measurement of tides). Tidal forces are going to be proportional to the mass of the nearby thing causing the tides, and inversely proportional to the cube of the distance. If <math>R</math> is a measure of your space-time curvature, <math>M</math> is your object's mass, and <math>r</math> is the distance from the object to the wormhole, then

<div align=center><math>R \propto \frac{M}{r^3}</math>.</div>

So if you want your wormholes to be at least ten planetary radii from Earth (<math>r = 64 \times 10^6</math> m, <math>M = 6 \times 10^{24}</math> kg), then the curvature limit is <math>R = 23</math> kg/m³ in some convenient system of units where we don't have to include constants of proportionality. But now suppose you want to send a <math>26 \times 10^6</math> kg ore freighter through that wormhole. You'll only be able to get your freighter within about 100 meters of the wormhole before the freighter's tides collapse the wormhole. This <i>might</i> work if the freighter were really long and skinny, much longer than 100 meters, to reduce its transverse tides; or if the wormhole was much larger than 100 meters in diameter. But in general, note that the curvature and tides have units of density, so if the average density in any volume around the wormhole exceeds the critical density – oops, there it goes.

Also, having a limit on the geometric curvature of space-time similar to what you get around mere planets doesn't end up being very plausible – the curvatures in the wormhole itself exceed that by so very many orders of magnitude that having so stringent of a curvature tolerance seems very implausible, and just dancing on the edge of disaster. For example, if your wormhole is about 100 meters across, its curvature in these same units will be approximately 500 billion trillion kg/m³. Keeping the tides to within 23 kg/m³ is the equivalent of an engineering tolerance of one part in 25 billion trillion.

So if you don't really have a good way to limit where you can place a wormhole based on the nearby "gravity", are there other ways you can exclude them from the vicinity of planets?

Safety might be one good reason. We have already discussed various ways a wormhole might collapse (forming a time machine, having its mass go negative). What happens when a wormhole collapses? If there are any parts of the wormhole or things in the wormhole that are charged or made of charged particles (like ordinary matter) you could justify a considerable fraction of the wormhole's mass-energy being converted to electromagnetic radiation. In addition, if the wormhole collapses into a black hole it will evaporate away its mass-energy primarily in the form of electromagnetic waves as it undergoes Hawking radiation. Large black holes will last a long time, but if the black hole left behind has a mass of 500 tons or less it will last less than 10 seconds. And 500 tons worth of energy delivered in 10 seconds would do quite a number on a planet. These sorts of considerations could lead to a regulatory environment that require wormholes to be located several light seconds away from inhabited places (like planets).

===Wormholes on planets===

But sometimes you might want to have the wormhole on a planet. Perhaps you are envisioning one of those circular or oval fantasy portals, looking like a window to an exotic foreign land, and when you step through you are actually there. We already know that thin-shell wormholes can be very low mass, and can have a circular shape. So that should work, right?

There are at least two issues to consider here. One is that even if both wormhole ends are on the same planet, natural variations in atmospheric pressure due to weather will lead to very significant differences in air pressure from one side of the wormhole to another (unless the wormholes are so close that they are within a small distance of the extent of the same weather system – say, within a few km of each other). This pressure difference will drive severe winds through the wormhole, usually comparable to the winds of a hurricane (although there will be periods of relative calm when the pressure at both wormholes just happens to be at the same pressure). This effect will only be exacerbated if the wormhole mouths are at different altitudes, due to the decrease in air pressure with altitude. And if the wormhole connects between different worlds, the effect will be much worse. In addition to making travel inconvenient and possibly damaging the equipment, the winds will produce large uncontrollable mass flows across the wormhole (air on earth has a density of about 1 kg/m³, so if you have 50 m/s winds blowing through a 2 meter diameter portal you have a mass flow of about 160 kg/s through your wormhole). This will make it difficult for mass balance, and keeping both ends having high enough masses that neither gets so close to zero that the wormhole breaks or forms a horizon or goes negative mass and causes a runaway planetary catastrophe or whatever wormholes do when one end gets driven to near negative mass by the stuff leaving it.

As a consequence, planetary wormhole mouths will probably be kept in airlocks.

The other issue is that different latitudes on a planet will be rotating at different speeds; at the poles they won't be moving at all while at the equator they will be going a full planetary circumference every day. This means wormhole mouths at different latitudes will be building up different time dilations with respect to each other. In addition, wormholes at different altitudes will experience different gravitational time dilation rates. If you let this go on for long enough, the differences in time across the wormhole mouths will form a time machine. Whether this collapses the wormholes, makes them "bounce" apart from each other (likely ruining a lot of property and infrastructure in the process), or just forms a time loop that leads to various inconvenient paradoxes depends on the assumptions you made for the physics of your world. But it will probably be something you will want to avoid. Re-balancing the time differences between wormhole ends will probably need to be done every few decades.

===Wormholes on spaceships===

The necessity for radiators on spacecraft is really annoying. Dealing with all that waste heat is not fun, and these issues are not awesome kewl stuff like gigawatts and megatons and hundreds of g's of thrust and other things that let you show everyone how much your spaceships totally rule and everyone else's spacecraft totally drool. Instead you need these big fragile stupid-looking things hanging off the side of your ship, ruining your aesthetics and efficiency and being vulnerable to your enemies blowing them up, and lighting you up so bright that you're easily visible from the other side of the solar system. Hey! I know! Let's put a tiny wormhole on our spaceships so we can just send all the heat someplace else. Great idea! That'll work, right?

Actually, yes. If you have small enough wormholes, it will work great. But there are consequences.

So you're running your coolant loops from your reactor through the wormhole to some cooling towers on your home planet. Much better than radiators. But – why do you still have that reactor on your spaceship? It's big and heavy and needs lots of big and heavy shielding and poses nasty radiation hazards. Surely you could leave it behind on your home planet as well, next to those cooling towers, and just run a power line through the wormhole? You'll be able to run a bigger, more powerful reactor that way anyway, and that extra weight won't be bogging your ship down.

And why does your spaceship have its lasers on board? You can have a much bigger laser on the planet that you don't need to lug around. Just shine the laser beam through the wormhole and re-direct it with mirrors.

And all of your missiles? Might as well just feed them through the wormhole when you need to fire them.

And then do you really need all that life support equipment? Just run an HVAC and plumbing that sends fresh air and water to the spacecraft and takes used air and water off. And remember to send the occasional snack, too.

Oh, but wait. Why is all your crew even on the spacecraft again? They have a wormhole to the spaceship. You can just have all your crew run the spacecraft remotely from a mission control station. Most of your equipment is on the planet anyway, so most of the engineering staff is already here. Now you don't actually need to put people in harm's way. If someone does need to fix something just put them in a space suit and send them through, let them do their job, then bring them back. Most of that can be done by a tele-operated robot, in any event. And being able to go back to their families when their shift is over is great for morale.

Hmmm, your sensors could just look through the wormhole instead of having all the vulnerable and heavy and expensive equipment actually on the spacecraft. And as long as you are doing that, you can just leave your laser beam pointer at home too, and aim the laser by pointing it through the wormhole.

So now what's actually left in your actual spacecraft? Oh, the rocket itself. But remember how we already talked about how we could just keep the rocket engine and propellant on the planet and shoot the rocket jet through the wormhole, and it will move just like a rocket? Yeah, so no rocket either.

Congratulations, you've successfully gotten rid of your spaceship. The wormhole itself is your spaceship. It's all you need, it's safer, it's more capable, it's stealthier, and just better all around than some contraption of steel and Mylar and fissioning uranium fuel rods and fragile people going through space.

===Adding mass to wormholes===

So you just made a pair of wormholes. Chances are, you pulled them apart as near Planck mass objects, with about 20 micrograms of mass. Well, that's not very useful! What if you want to send something through that's bigger than 20 micrograms? You're going to need to add mass to your wormhole mouths.

A similar issue comes up if you project a wormhole at some far away planet or star system using the wormhole-rocket trick already discussed, maybe by using powerful lasers to make a photon rocket. To be practical, your projected wormhole mouth will need to be very low mass. But once you get it to where you want to go, you'll need to give the mouth enough mass to let your explorers and colonists and equipment through.

A simple way to do this is to simply suck stuff into the wormhole. This adds mass to the wormhole mouth the stuff comes through. Of course, if you take it out from the other side, that other mouth will lose that much mass, and if it didn't have than much mass to begin with you'll likely have problems. So just leave that stuff in there! Shove it off to the side. Let it clutter up the unused areas out of the main traffic route in the way that basements and attics and the unused spaces of garages accumulate all that old junk that you never use any more but don't want to throw away. Build a wall inside of your wormhole, so travelers don't have to see all the asteroidal rock or super salty water or very surprised natives or whatever it was you scooped up on your alien world. Shove it into the undesirable high curvature places with nasty tides. This wall can also help keep the legitimate passengers away from the inner working of the wormhole, the strange energies that support it, the cables and pipes and crawlspaces that only your maintenance workers should access anyway, and the extreme space-time curvatures that would rip the passengers to shreds if they ever were directly exposed to them.

===Tiny wormholes===

Wormholes that are too small to fit people through could still be useful for sending information. You could use them to rapidly communicate over long distances. If the wormholes were light enough and cheap enough, they might even replace cell phones and laggy land lines with super high bit rate connections.

You could also send energy through tiny wormholes, in the form of light or electric current. Storms that knock down power lines no longer need to cut off your home's electricity. You could also use them to power machinery that need light weight and high power output – electric transportation like airplanes, shipping, or trucking; directed energy weapons; or high specific impulse rockets.

===Causality denial, causality attacks===

We have shown that moving wormholes around can lead to time travel, and that there are plausible physical mechanisms that can prevent time-travel allowing wormhole configurations from happening. People being people, you just know that if this is possible, that it will not be long before these effects are used in politics and warfare.

In a network of wormholes that is about to create a time machine, one likely resolution is that things break until a time machine is no longer possible. In any chain stressed to its point of failure, it is the weakest link that breaks. So in a path through wormholes and normal space-time that is about to form a time machine, it is the weakest wormhole that will collapse (or otherwise have something bad happen to it). So if your rival polity has a wormhole that allows then easy trade with a rich world, and you want to edge in on that trade and simultaneously deny that trade to your rival, you can just send a bigger, stronger wormhole in such a way that forces a time loop. Now your rival's wimpy wormhole breaks, and you get access to the trade opportunities with the world! This is the basics of a causality attack – use the causality-enforcing properties of wormholes to attack your enemy's wormhole infrastructure.

But what if your rival polity anticipated this? What if they have strong wormholes, with an arrangement in space-time that blocks off easy access of your wormholes to a profitable region of space? This is a causality denial action, or a causality fortress. By controlling the time displacement of the wormholes in their network, a polity can control what additional connections can be made. This can be used to enforce borders, prevent infiltration, or restrict civilian wormholes to legitimate uses.

===Moving one wormhole through another wormhole===

If you have a wormhole mouth, you can move it through another wormhole as long as the mouth (and any supporting structures) is small enough to fit through the throat of the wormhole it is going through. The geometry all works out. In fact, this has a rather nice side-effect. If you are looking through the wormhole being moved from back home, you only see it take as long to go through the other wormhole as a wormhole transit usually takes even if it got projected far into the future or past. So the moved wormhole acquires the same time differences as the wormhole it just went through (added on to any time differences it already had, it doesn't get reset or anything). This can let you easily build wormhole networks that connect to themselves in loops. It also makes the network more susceptible to breaking, because there is much less leeway for error with a closed loop as far as making a time machine and breaking the loop. So it can be done, but there are consequences.

===Dropping a wormhole into a black hole===

In principle, if you have a wormhole connection to the inside of a black hole, you can get information out from inside the event horizon. The theorem that prevents things from escaping a black hole was made under the assumption that the energy can't go negative. And as we've already seen, wormholes require regions of negative energy to work. You still wouldn't be able to see further toward the singularity at the center than the wormhole's location. And the wormhole would be inevitably dragged into the singularity in a finite amount of time. But, in principle, during that time you could see what is happening in the black hole.

But there's a big catch with this. You can't actually get the wormhole into the black hole. Not if you assume that time machines can't form, anyway. From the point of view of the wormhole mouth falling into the black hole, it will pass through the event horizon in a finite amount of time. From the point of view of someone watching from outside, the infalling mouth gets more and more time dilated to the point that it never crosses the horizon. This time dilation sets up the conditions for a time machine, and any time machine preventing physics happens on the wormhole before it can ever get in.

== Possible Networks ==
=== Acyclic wormhole networks ===

Recall that wormholes are space-time connections. They bridge space as well as time. Assuming we want to build this network in a stable manner, we want to avoid Closed Timelike Curves from appearing. Let us start with some simple anaylsis of consequences on this. Assume projected wormholes have one mouth sent at relativistic speeds to a distant target and the other mouth is kept at rest at home. From the perspective of the dispatching party, this will lead to the new connection opening very quickly. Done again and again, large volumes of space can be accessed in a short time, a powerful advantage.<br />

[[File:Wormholes PartialOrder.svg|400px|thumb|The partial order relationship of the wormhole connection]]

Every projected connection thus extends onto the future. Then we must avoid forming loops, or more generally "backtracking" back towards a dispatched node. This nets us a graph with no closed loops - an acyclical graph, also called a tree. Since connections may not return to previously established nodes if they are time-shifting from the past into the future, we also get a partial order: nodes <math display="inline">u</math> that dispatch wormholes have a higher order over the nodes <math display="inline">v</math> they establish: <math display="inline">u < v</math>. This lets us establish a direction on the whole graph. For a particular node <math display="inline">n</math>, its '''parent node''' established it, <math display="inline">parent(n) < n</math>, and it in turn establishes '''children nodes''', <math display="inline">n < child(n)</math>. The node that only has children nodes is the tree's '''root node'''. Nodes that have no children are '''leafs''' of the tree. The trees '''height''' is the longest outward path one can take from the root to a leaf.<br />

In terms of traversal, such nodes have some interesting and useful properties. Since wormhole traversals may be assumed to be effectively instantaneously bridging large distances, travel time is mostly defined by the travel distance between wormhole mouths. And as such, we want to minimize the amount of transits we have to make to get from a node <math display="inline">A</math> to a node <math display="inline">B</math> the fastest. It turns out, in order to achieve this, we want to have as large a choice of connections at each node. In the ideal case, any node except root can be reached in two transits: one transit to root, and from there to the other node. In the less ideal case we have to make one further stop-over, choosing from a number of connections at a second node to reach our destinations. The less ideal case from that sees us make two transits through such "switch yards". Assuming each node except the leafs in such a wormhole network has exactly <math display="inline">k</math> connections and there are <math display="inline">n</math> nodes in the whole network, we can reach any destination from root in <math display="inline">\log_k (n)</math> choices of wormhole transit. This is a powerful way to get to connections quickly through centralization. (In computer science, such k-nary trees are of great interest for quickly search-able data structures for this reason.) Configurations that are less "balanced" net appropriately lesser advantages and longer transit times.<br />

Also, traffic to most other nodes ''has'' to go through a root-wards node. This gives these nodes significant presence in the larger network. Routing through them is unavoidable if you want to reach most of the rest of the network. Root nodes can profit from taxation, services to travelers, control of routing infrastructure for information, and so on. But there is also a catch. Assuming leaf nodes each send the same amount of traffic inward. The amount of traffic load increases polynomial as you go towards root. Your wormhole infrastructure (and what transports cargo between wormholes, in space or on planets) has to cope with this scaling. Congestion might become an issue.<br />

=== Mesh wormhole networks ===

The acyclic network may have some bothersome problems. The central routing focuses power and can keep transit times down quite significantly depending on the specific network infrastructure, but it’s also congestion-prone, and some close locations may be widely separated on the network. Or maybe you want to avoid centralized routing dependencies for political reasons. Root might like being root; everyone else being subject to root’s whims, not so much. So, can we build networks where there are two routes between the same two nodes? We can, but it comes with some necessary adjustment.<br />

Recall why the acyclic network is acyclic: to avoid forming CTCs due to the time connection between future and past from the projected wormholes. But if we let both wormhole mouths experience the same time dilation, the problem that ultimately leads to CTCs - one connection routing displacing further into the future than the other connection, and thus allowing one to arrive at a node before ones departure - is avoided entirely, since there is no time displacement to be accumulated. Thus we get meshes, where there may be many connections and cycles between nodes. An observer at one node watching for his transition from another node 10 light years away would have to wait for ten years to watch his light arrive. Closed timelike loops are not possible so long as this network is kept balanced.<br />

The disadvantage - this network must take great care to ensure that the time dilation that inenvitably occurs due to different star velocities and the galactic gravitational field don’t accidentally create a time shift and CTC in the network. And also we cannot exploit time dilation to open connections very quickly, from the home node’s perspective. One has to wait out the full years it takes the wormhole’s far mouth to get to a remote destination, before the connection can be used.<br />

=== Semi-privileged wormhole networks ===

The interesting thing is, it is very much possible to build networks that make use both of relativistic projection to seemingly accelerate the network construction time, ''and'' contain cycles, with alternate routes to get to another node. The acyclic construction is a strategy to avoid CTC formation, but we should remember the ultimate constraint reason: we avoid cycles so that there are no connections which, through a certain time shift on them, a CTC could be formed. Acyclic construction is one strategy. Only building cyclic connections with careful time shift balancing is another!<br />

The foundation for such networks is an acyclic "base graph" that may be isolated from the larger network graph. We also call this the '''red graph'''. The base graph maintains all the partially ordered properties of the simple acyclic graph. We can call this the "privileged" graph, since its nodes and their connections define the principle positions in space-time of the network, and we must avoid generating CTCs on this definition. How do we avoid forming CTCs but still create alternative paths and thus loops?<br />

Let’s assume we have two nodes <math display="inline">A</math> and <math display="inline">B</math> ten light years apart. They both had connections established from a parent node <math display="inline">P</math>. Those connections are time-shifted over 99 years and part of the privileged base graph.

We couldn’t build a time-shifted connection between <math display="inline">A</math> and <math display="inline">B</math> without causing a CTC to be formed. But we are entirely fine sending signals '''Fast-as-Light''' from <math display="inline">A</math> to <math display="inline">B</math> and back. This is not a violation. Now consider: how does a wormhole with no time shift across it behave? In the mesh network case, such connections don’t cause CTCs to appear. And it doesn’t cause one to appear here either. (A/N: this could be illustrated even better with a space-time lightcone illustration I am prototyping.) How can we build such connections? We can of course use projection with relativistic speeds at both ends, like how we would build Mesh networks. But there is an even faster solution. If we can move wormholes through wormholes, we can shift the new mouths through the existing network to create this connection!<br />

The main catch is that we have to tightly watch out for temporal drift on such new connections. Through adding such edges that overall form a "blue graph" on top of the red graph, we are creating loops on the red graph underpinning the network. If one end of a wormhole mouth drifts more in time than the other (which will inevitably happen because stars won’t have the same mass, experience the same galactic gravity, or have the same velocity relative to each other) the conditions for a CTC will start to appear. The only way to pre-empt this is with continuous maintenance, taking connections offline and adjusting their temporal position using synchrotons, and having "safety space-time" spacing between wormholes. In this way, there is some "room" where signals can arrive faster and faster, but not before they were sent, allowing the connection to be used most of the time with only short intervals of adjustment.<br />

This works even when both ends connected don’t have the same time distance from the root node - the important thing is that the new shortcut accomodates this difference with a time shift of its own. We can calculate this difference easily. For two non-root nodes <math display="inline">A,B</math>, for each node we sum the temporal difference over the path from root to the nodes along the red graph, then subtract the difference from each other. The resulting remaining temporal difference is that the new connection on the blue graphh must have to keep the network stable. This means we can even do things like connect systems far out directly to core systems, or bridge from one edge of the graph to the other - arbitrary routings are possible. These routings can allow faster signal transmission and mass transport than through root, mitigating bandwith issues. The main problem is to watch out for network stability. As nodes accumulate, CTC avoidance becomes more and more delicate, and safety spacing must be maintained diligently. This can add to the effective travel time through such alternate connections, and thus maintains some of the "privilege" of the acyclic base graph.<br />

Arguably this network configuration is the most powerful, marrying many advantages and removing some disadvantages. It is however also bound to the need for safety spacing. Depending on your infrastructure assumptions, this can pose issues. If you want to implement serious stretches of safety space, wormhole links not part of the acyclic base graph would likely have to be located at interplanetary distances away from the main links, with light hours to light days of safety spacing. If your main network runs on planetary trains, this is rather a bother. Adjusting the internal throat length of wormholes can help in this regard, but of course now poses the issues of how to safely navigate such a long wormhole. Maybe you could extend something like space elevator tethers through such stretchy throats. With no gravity inside the wormhole itself, such cables could probably stretch quite significant distances. If your culture is handling wormholes mostly in space, you still have to account for added travel times through the safety spacing. Communications links would probably be easier connected between arbitrary nodes than macroscopic, craft-traversable wormholes.

==Credit==
Authors: Luke Campbell and Sevoris

==References==

File:Wormholes PartialOrder.svg

2022-08-21T08:38:22Z

Sevoris: Partial order relation on wormholes in space and time. Created by Sevoris Doe for Galactic Library.

== Summary ==
Partial order relation on wormholes in space and time.

Created by Sevoris Doe for Galactic Library.

File:Wormholes AcyclicGraphExample.svg

2022-08-21T08:36:19Z

Sevoris: An example of an acyclic wormhole network. Created by Sevoris Doe for Galactic Library.

== Summary ==
An example of an acyclic wormhole network.

Created by Sevoris Doe for Galactic Library.

Particle Accelerators

2022-08-20T17:07:11Z

Sevoris: Removed the under construction notice. Article is considered complete at 1.0 state.

Particle accelerators are an incredibly useful piece of technology and a diverse set of artifacts produced for many purposes. While particle accelerators are often invoked as weapons of war in science fiction, and have captured the public attention as giant research systems the size of cities, they have an incredible widespread range of applications. There are also many, many ways to build a particle accelerator, depending on what effects you need to achieve and what circumstances you work under.

This article aims to be a general primer on particle accelerators, their physics and a selection of technologies and construction styles. In detail, all particle accelerators are high-tech engineering customized to their circumstances in many, many aspects.

Further pages covering application fields are in the work.

= Accelerator physics =
The simple explanation is that we use electric fields to accelerate charged particles. But from this basic principle of leveraging the electromagnetic forces springs an entire set of accelerator technologies with different performances, trade-offs and characteristics. The most basic particle accelerators are just a big high-voltage source and can still be found in old CRT TV’s and dentist x-ray machines around the world. Things rapidly get more complex from there, but first we need to lay out some terms so we can all have a common conversation.

The first thing to keep in mind with particle accelerators is that we are talking about charged beams here, even if we neutralize them later. So no neutrons, photons, or other innately neutral particles. Your choices range from the simple electron up to anti-uranium and more, but they all must be charged. The particle of choice determines just about everything else about the accelerator and any potential uses. Really heavy particles (like large ions) don’t accelerate easily and take a lot of energy to get relativistic enough for time dilation tricks to take into place. Really light particles (like electrons) accelerate super easily, but can have other radiation problems to worry about.

The second most important thing to know about a given accelerator is the particle energy. This is a measurement of the kinetic energy of any individual electron, ion, whatever in the beam and is measured in a unit called the electron-volt or eV for short. The lower bound for any worthwhile particle accelerator is in the MeV (or million eV) range and we typically want a lot more! Inside of the accelerator we tend to talk about the electric field doing the accelerating (measured in Volts/meter). For singly charged particles (like electrons) this gets really to figure out the energy because of how the eV is defined. We just take the electric field (also called the accelerating gradient) and multiply by the length of the machine! Boom, now you have particle energy! For more complex particles you have to do some scaling based on mass and charge state.

Also a quick trick to help you with deciding if something is relativistic or not is to divide the particle energy <math>E</math> by the mass energy <math>M</math> of the accelerated particle (0.511 MeV for electrons/positrons and about 1 GeV for a proton) and add one. The resulting number is called the Lorentz gamma value and is used all over in accelerator engineering.
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>\gamma = 1+\frac{E}{M}</math></div>
Once you get a value much larger than 1, you have relativistic particles. Once that number is in the hundreds you get an ultra-relativistic beam. The speed as a fraction of the speed of light <math>\beta</math> is a function of <math>\gamma</math>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
\beta = \sqrt{1 - \frac{1}{\gamma^2}}
</math></div>
and conversely
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
\gamma = \sqrt{\frac{1}{1 - \beta^2}}
</math></div>
Figure 1 helps show the relationship between Lorentz gamma and speed of the particle.

[[File:LorentzGvsSped.svg|900px|thumb|center|Lorentz gamma versus fraction of the speed of light]]

The third most important feature of a particle accelerator is the emittance which is just a fancy term for divergence of the beam and can be thought of similar to the wavelength of lasers. No beam is made perfect and they will naturally want to expand outwards. The smaller the emittance, the longer the beam will stay in a tight spot. Emittance can be altered with a wide variety of beam engineering techniques that take entire books to discuss, but one other cheating thing you can do is drive the beam energy higher. This naturally lowers the emittance by a factor of Lorentz gamma due to the cheating effect of special relativity! Thank god for Einstein!

== Acceleration forces ==
Every accelerator tries to build an electric gradient appropriate for the charge of the particles it is accelerating. The greater the electric gradient, the bigger the amount of work that can be done, the more energy is imparted on a particle as it travels through the gradient, and thus the higher the acceleration. The simplest way to build up a gradient is simply two charged electrodes with a strong potential between them. Many other accelerator methods use electromagnetic fields, in the form of waves confined in electrically conductive structures. In this case we exploit that electromagnetic fields have both an electric and a magnetic component. The magnetic component cannot be used to accelerate particles (only bend their trajectory - which can be exploited for effects like deflection or focusing) but as the electromagnetic waves oscillate, so does an electric field. This naturally creates a charge difference across half of the phase of an electromagnetic wave - this electrical gradient can accelerate particles. Both of these technologies encounter problems with the physical materials asked to contain the electric or electromagnetic energy. Eventually the fields get so intense that solid materials get broken down and an arc is formed. As such, various advanced concepts use clouds of charged particles and plasmas, which are not subject to the material limitations of solid matter.

The strong and weak nuclear force as well as gravity (which are all of the forces we know of at the time of writing) are not suitable for accelerating particles. This is related to the power and range of these forces. The strong and weak nuclear force have extremely short ranges - even if we had a means to artificially shape them like with electromagnets, their range would be too short. Gravity has the range but is incredibly weak. To generate a measly 1 G of acceleration, 9.81 m/s², requires the mass-energy of the entire earth, some 5.97e24 kg of mass, to do the job! The acceleration involved in particle accelerators is much, much, much larger (on the order of 10^13 times larger or more).

{{MascotSays|Interestingly, one source of very high-energy particles are giant gravity wells. Black holes generate conditions around them that can eject particles with enormous energies. The jets of particles that shoot from the north and south poles of many such stellar entities are particle accelerators.}}

If you want to use a force other than electromagnetism to accelerate particles, it’ll have to be a fictional one. The forcefields of various sci-fi properties could be considered, since they can often apply forces to particles. If you propagate a wave of such forcefields down a tube, you could have something quite a lot like many particle accelerator systems!

== The particle bunch ==
Essentially all high energy particle beams are not constant streams of particles, but a time sequence of discrete groups of particles, commonly referred to as “bunches”. These bunches can be described both in their distribution in lateral space to the traveling direction (x and y axis) and longitude (z axis) along the traveling direction, anchored on an origin point (0,0,0) that moves in time.

[[File:ParticleBunch.png|center|900px]]

Bunches can have various profiles. Generally they are somewhat distributed in the lateral direction just because the charged particles have slightly different starting positions and lateral velocities (also called transverse emittance). In longitudinal space, differing acceleration experienced by individual particles as well as dynamics within the bunch scatter them out, with “noses” and “tails” that contain fewer particles. The end results is that most beam profiles follow a gaussian distribution, though there are exceptions.

This bunching behavior allows us to analyze other properties of particle beams very similar to lasers. There is a “pulse frequency” given in Hertz. Each bunch contains a certain energy, which can be large enough that we give it in Joules and not electronvolts. There is an average output energy over a given timespan, given in Watts. We have a beam peak power, given by how quickly a bunch arrives at a target and deposits its energy, which we give in Watts. Related to this we also have a related peak intensity depending on how much area/volume the beam is deposited into, with units of either W/m² or W/m³. If we have charged particles being accelerated, we get a beam voltage, the same way as in a wire. Beams can also have a temperature, the same way as any other group of particles with random motion from their energy, given in Kelvin or eV.

The systems that generate particle beams of course have efficiencies, with the wall plug efficiency (power inserted into the entire system versus power deposited in the particle beam) being the most interesting for first-order engineering. They also have an operating temperature.

== Particles for acceleration ==
With the constraint that our particles must have an electric charge in order to be accelerated with anything but fantastic technology, we can discuss quite a large family of particles, from the conventional to the exotic. They differ in the applications they can be interesting for. Some can be similar, but no two particles behave entirely alike.

=== Electrons ===
The ubiquitous electron is the most common particle to be accelerated and the first one ever pushed up near the speed of light by humans. Due to its excellent charge to mass ratio (the best possible due to the extremely low mass and charge of 1e) electrons are very easy to get up to relativistic speeds. Electron accelerators for radiography and sterilization are often portable by a single person and not much larger than a laptop in size (although they can be much bigger for power output reasons). The ease in which electrons reach relativistic speeds means that electron RF accelerators can be designed with no speed changes assumed (since the electrons are already moving at effectively light speed) and thus tend to be highly efficient and have the highest acceleration gradients achievable. For similar reasons electrons are also able to be accelerated in plasma based accelerators, allowing for even higher acceleration gradients and smaller machines for a given energy. The major downside of electrons is that due to their large charge to mass ratio they will radiate energy like crazy when bent by a magnetic field. Thus ring shaped accelerators are of limited use for electrons since you have to make the ring extremely large to not just lose all of your energy as beams of x-rays coming out of the machine! This very trick is used to make high quality x-ray sources for industrial and research applications.

When electrons hit matter they tend to release their energy as photons in a process called bremsstrahlung, or braking radiation. These extremely high energy photons (roughly ~⅓ the peak energy of the electron on average) then proceed to make electron/positron pairs that then make photons that then make pairs that then make photons… This whole thing is a massive cascade of radiation that ionizes, heats and can even damage on a nuclear level! At high enough energies you will generate every particle known and might even make some new ones!

=== Proton ===
Protons are the next most common particle to be accelerated and are currently used by the absolute largest particle accelerator ever made (the Large Hadron Collider or LHC) as the primary particle of choice. Since they have ~2000X lower charge to mass ratio than electrons, protons are about ~2000X harder to get up to speed. Unlike electrons they are not promptly at light speed so we have to send protons through a series of various accelerators depending on the energy of choice. For example the protons in the LHC the protons go through 5 different particle accelerators plus the extremely low energy stage that first makes the beam of ions! This difficulty in reaching highly relativistic speeds has two other bad side effects. Firstly, if we want to transmit our protons through empty space we will probably need to neutralize them with a beam of electrons.

{{Note|''When shooting extremely high current (kA), relatively low energy (not highly relativistic or in some cases relativistic at all) beams in the atmosphere the higher mass and positive charge of protons gives them many beam propagation advantages over electrons if the beams can be made. In this case no neutralization is required as the atmosphere itself becomes a plasma and neutralizes the beam.''}}
Luckily this is not very hard and will add almost nothing to your comparatively large proton accelerator, but may increase your beam spread a little bit via some neutralization physics. This does mean your beam is technically plasma, which is cool! Second, we effectively cannot use those neat plasma based accelerators for protons since we would need nearly TeV class proton beams to inject into the plasma accelerator and by the time you reach those energies you have basically already done all of the work. Maybe in the future there will be ways around this, but as of right now we are limited! There is one major benefit to this poor charge to mass ratio though, and that is the lack of synchrotron radiation! Ring shaped accelerators are ideal for protons, and basically all ions.

When protons hit matter they can interact with all of the four fundamental forces, although only two of them (Strong Nuclear and Electromagnetism) tend to play big roles. Protons barrel on in causing huge ionization cascades and can also release bremsstrahlung photons at high enough energies. They are also ideal at knocking neutrons, protons, deuterons and more out of the target nuclei which just adds to the fun! Protons also come with a neat trick known as the Bragg peak, where they deposit a large portion of their energy right at the end of their trajectory through a target material. This is a very useful trick for medicine and means you can “tune” the beam to almost ignore large portions of material and primarily heat, ionize, and destroy the area of interest. You can imagine how useful this is for cancer therapy or more nefarious things!

=== Light ions ===
Light ions are a rather fuzzy term but here we will use the definition of any ion heavier than a proton but lighter cobalt. The fuzzy reasoning here is that you get on the other end of the nuclear binding curve (i.e. can’t get net energy from fusion) at iron, so cobalt will then no longer be “light” and has to be made in a different process. Feel free to pick your own cut off! Historically beams up to neon have been experimented with for nuclear fission, nuclear fusion and medical therapy applications. Light ions have even worse charge to mass ratios than protons (Z/A times worse) and typically you will find light ions have about ½ the charge to mass ratio of a proton on average, although the isotope 3He is a notable exception. This means they are even slower to accelerate, although often the exact same machines as protons can be used with minor tweaks. Light ions have similar requirements for neutralization as protons, although their heavier mass can help fight beam bloom for a similar beam speed (not energy since the increased mass means the particles are higher energy for a given speed). The decreased charge to mass ratio also means that synchrotron radiation is even lower so beams can be driven to higher speeds before radiating. Light ions have similar target interaction physics as protons, although the ion itself will break up at very low speeds and add to the radiation environment.

=== Heavy ions ===
Heavy ions are being defined here as cobalt up to whatever unstable element you can jam into your accelerator! Typically heavy ion beams are made of elements like lead (used in some runs on the LHC), gold (used in the Relativistic Heavy Ion Collider) and even californium (used in some runs on the Argonne Tandem Linac Accelerator System). These beams tend to be for pure nuclear physics research, but there has been a long standing interest in heavy ion beams for inertial confinement fusion research and as potential methods of propelling spacecraft. These extremely heavy ions will be even harder to accelerate to ultra high speeds than light ions and much slower for a given energy, but they do provide a lot of momentum to make up for that! They have the same neutralization requirements and same potential upsides for beam bloom as light ions, but will most likely be orders of magnitude slower reaching our targets than the lighter beams. Heavy ions can also be used in the same machines as protons with small modifications and generate almost no synchrotron radiation as a result of their much smaller charge to mass ratio. They also have similar interaction physics to protons and light ions, but will stop in much thinner layers of materials. While electron, proton and light ion beams tend to both mechanically destroy and irradiate targets through-out, heavy ion beams will act much more like lasers and burn in from the outside of the target on in. There will still be a spray of ionizing radiation, but far less of one than an equivalent energy electron, proton, or light ion beam. The much higher momentum per particle, lower radiation concerns and ease of deflection makes heavy ion beams very interesting for beamed propulsion purposes.

=== Macrons ===
Macrons are typically defined as macroscopic particles, such as extremely large molecules or even grains of sand, that are charged up and accelerated via similar technologies as traditional particle accelerators. There are many difficulties with getting macrons up to appreciable speeds due to the extremely low charge to mass ratio, engineering difficulties with getting consistent and good charging, phase matching with accelerating cavities and lack of effective focusing optics. There are so called “dust accelerators” in existence that utilize the first stage of ion accelerators to generate their “beams” and proposals have been made to create better versions using updated technology. Macrons will interact just like a very small and very fast bullet unless they are designed to carry a nuclear payload to boost this output. Antimatter is an assured option for getting energy gain from a macron impacting a target, although there are potentially options for fusion, fission and fission/fusion given the correct sizes (mm scale or bigger), clever target design and the right impact speeds (most likely >>100 km/sec). The various energy gains versus speeds can be seen here:

[[File:IvKmacron.svg|900px|thumb|center]]

It can be seen that antimatter proves to be worth it up near the speed of light, but both fission and fusion stop providing much gain well below the relativistic regime. This also helps explain why heavy ion uranium beams are not going to provide some incredible advantage over any other ion.

=== Antimatter ===

Antimatter is the evil cousin of normal matter and whenever the two meet a release of energy equivalent to their combined mass energy follows. Now this does sound like an amazing option for beams but there are two downsides. First, antimatter is energetically expensive to make! Positrons (antimatter electrons) are the “cheapest” at 1.02 MeV of input energy at a theoretical best, but they only give you that much energy back when hitting a target. Since a typical positron beam can be expected to be equivalent to an electron beam, there is little gain in adding 1.02 MeV per particle to a multiple GeV (or even TeV) beam. You can see this explicitly:

[[File:IvKhBarPostron.svg|900px|thumb|center]]

For anti-protons the math becomes more favorable, but creating these particles is much more difficult and energy consuming. At best anti-protons require ~2 GeV of input energy to create, but in practice >400 GeV per anti-proton has been needed. Luckily significant energy gain is had up to several GeV of beam energy, so if anti-protons can be made at a central facility of some kind and easily stored (neither being easy things) then there may be some advantage to such a beam. Heavier ions are significantly harder to make since fusion with antimatter must be performed to generate those ions. Barring a naturally occurring source being found, antimatter heavier than a proton is most likely not worth using.

{{Note|'' Positrons can potentially fill an interesting niche either as a co-propagating neutralizing beam for electrons or perhaps as a higher performance atmospheric beam than electrons due to the positive charge. In this case the annihilation properties are no what is of interest, rather the positive charge and relative ease of acceleration compared to ions. Since positrons share all properties with electrons except for charge, they can be accelerated just as easily and even co-accelerated with electrons in the same machine.}}

=== Muons ===
Muons are an interesting cousin of electrons and positrons. They are unstable and heavy cousins of electrons, massing about 200X more than electrons and living about 2 microseconds in the lab frame before decaying into an electron (or positron) and a neutrino. If we accelerate them up to relativistic speeds (about 10X easier than protons and the muons are typically born relativistically thanks to how we make them) then the muons can be made to “live” far longer. Muons have long been of interest to accelerator physicists due to their position as an almost ideal intermediary between electrons and protons. The requirement to make them on site has always been an impediment though and muons are currently only used for certain niche experiments. During the planning stages for a potential muon collider it was also noted that a muon collider would also have a serious issue with neutrino radiation! This is due to how neutrinos are more likely to interact with matter at high energies and a muon collider would be generating a LOT of really high energy neutrinos in a ring around the machine. This has been proposed as a sort of super weapon even!

Muons should interact with matter in a similar fashion to electron beams and the beams will probably be treated in a similar manner. They can propagate through the atmosphere quite well in theory and should outperform electron and proton beams for a given beam energy, although at quite a cost… Muons are typically generated with proton beams impacting targets and will create an equal number of positive and negative muons at high energies that must be carefully captured. This would allow for the easy creation of a neutral muon beam though, which may make for a great giant doom beam! Do note that negative muons also can help catalyze nuclear fusion so a very high power beam hitting a fusion fuel tank could produce some rather… Interesting results!

=== Other exotic options ===
There are a wide variety of odd particles that could also be chosen for use in beams given a specific need for them, although generation and utilization of these particles will be quite difficult! Extremely short lived particles like pions, kaons and taus can be generated and accelerated up to extreme speeds to help propagate them far enough to matter, although that will need some very extreme speeds. At the upper end of extreme there are even proposals for accelerating tiny charged blackholes!

= Accelerator technologies =
There are a wide variety of accelerator technologies and here we try to summarize them as best as possible. Different accelerator types may be sequenced one after another. Some particle accelerator technologies depend on the particles fulfilling certain requirements of speed (thus energy) and hence different systems may be employed in sequence. Accelerator systems are described in terms of their efficiency of turning input power into particle energy, their acceleration gradient measured in electric field which translates to volts per meter, and the mass per length meter. An interesting secondary parameter is their operating temperature. Some accelerators can operate well at high temperatures. Others require cold or cryogenic conditions to work well.

== Core accelerator assembly ==
We define the core accelerator assembly as the structure where energy is imparted onto the particles.

=== DC Accelerators ===
For very low beam energies we can utilize large DC electric fields between plates or grids to accelerate particles up to speed. This is the same technique as is used in ion drives and is the most common industrial accelerator used. Every dentist's office and old CRT television uses this acceleration technique. Ion accelerators often also use a DC accelerator to generate the initial beam of ions, while electron beams utilize other techniques. These machines are limited by material breakdown to about 0.5 MV/m accelerating gradient and typically no more than a couple of MeV total particle energy for extremely large machines. The highest energy DC accelerators can reach 40 MeV for proton beams, but this is for machines massing in the 100s of tons and at low total beam powers. Very large DC accelerators are currently the only tested method for accelerating macrons up to 10’s of km/sec.These accelerators are limited by the risk of arcs jumping between the electrodes at too high voltage differentials.

=== Radiofrequency Quadrupole ===
A more advanced first stage accelerator for ion beams is the radiofrequency quadrupole or RFQ. These machines utilize low frequency radiowaves and carefully designed conductive bar structures to efficiently generate high current ion beams up to energies of ~7 MeV in a couple of meters. RFQs were a revolution in ion accelerator physics and caused quite a stir when the USSR first showed them off as they provided a much higher power and higher quality ion source in a smaller package than anything the USA had built! RFQs have also been proposed as methods of efficiently accelerating macrons up to >100 km/sec. These machines can be normal or superconducting and can operate with efficiencies as high as RF LINACs in theory.

=== Cyclotrons and Betatrons ===
One of the earliest ways to accelerate particles up to relativistic speeds is with circular accelerators known as cyclotrons (for ions) and betatrons (for electrons). In Cyclotrons the accelerator is a round chamber which houses two half-circle electrodes roughly shaped like a D (and thus called “dees”) separated by a gap, sandwiched between the poles of a large electromagnet. The magnet creates a static magnetic field which bends the path of charged particles into a circular motion. An oscillating alternative voltage is applied to the electrodes (what we often call RF or radiofrequency) to accelerate the ions. This radiofrequency field’s oscillation is matched so that one cycle of the voltage matches one circuit of the particles. As a consequence when crossing the gap from one electrode to the other, the charged ions experience a voltage gradient and are accelerated. As the particles gain energy and speed, their radius in the magnetic field gets larger. The result is an overall spiraling trajectory from an injection point in the center to an exit point at the rim where the charged particles are bent out of the cyclotron. Cyclotrons use the same voltage gradient many times over, defeating the electrostatic breakdown that limits the single-time electrostatic accelerators. As a consequence, proton energies of up to 70 MeV may be reached. Cyclotrons energies can also be increased by switching from normal magnets and RF resonators to superconductive components. This eliminates various losses from electric resistance and absorption and also allows for much stronger bending magnets, which allows the cyclotron to contain particles up to a higher energy if desired. Superconducting cyclotrons have achieved up to 520 MeV. Still, cyclotrons suffer from poor beam current, pulse rate, and beam quality issues that make them a less than ideal choice. The large magnet required (measured in meters) also does not lend itself to a lightweight weapon and as such, cyclotrons are regulated to niche medical and physics uses.

A close cousin is the betatron, which is used to accelerate electrons. In this case the magnetic field strength is increased as the electrons speed up, which prevents any spiraling trajectories of the particles. Betatrons have been operated up to electrons energies of 300 MeV, but are more typically in the ~10 MeV range. In principle a betatron could also be made superconducting, although magnet ramping may prove to be difficult. Other types of ring shaped accelerators may prove to be more useful, although electrons are heavily limited by synchrotron radiation and it is often better to use linear accelerators for them. For now betatrons remain limited to small (sometimes portable) accelerators for nuclear physics, industry and security work.

=== Radiofrequency Linear Accelerator ===
The real work-horse of the particle accelerator world is the RF linear accelerator. These babies use the same alternating field trick that cyclotrons use, but in a straight line rather than needing to bend the beam around in a circle. This provides a couple of nice advantages over the more traditional cyclotron. First and foremost you get to leave that large, bulky magnet behind which saves you a ton of mass, cooling and power consumption! You also get to now have much higher RF frequencies (1-10 GHz right now compared to the MHz used in cyclotrons), which means higher accelerating gradients. To generate these powerful, high frequency RF waves (also known as microwaves) that drive the accelerator we typically use military radar power supplies or (in more advanced cases) even lower energy particle accelerators! State of the art RF LINAC technology is well into the 100 MV/m range and have no upper limit on beam energy! The Stanford Linear Accelerator reaches 50 GeV and was built back in the 1960’s! There have been traditional RF LINAC designs for machines reaching up to 1 TeV, with the only limit being how much space you have to put these monsters. They do operate best (i.e. with high accelerating gradients and efficiencies) with particles already moving at relativistic speeds, so any ions fed in need to use a series of lower gradients LINACs or other pre-accelerators. Electrons and positrons can naturally be used in LINACs since they accelerate up to relativistic speeds so quickly.

LINACs also have been shown to operate at high average powers (MW scale is now in use) and high pulse energy (10’s of kJ pulses fired at high rep rate) while also maintaining a high quality beam that stays well focused. The two newest advancements have been to push the frequency ever higher for higher acceleration gradients (300+ MV/m has been shown and there are plans for even more) and to switch to superconducting RF accelerating cavities (the name of the actual piece of equipment that does the accelerating) which then dramatically increases the electrical to beam efficiency (80-90% has been shown and even copper waveguides have been operated at 70%). All in all, LINACs beat the tar out of cyclotrons which is why we don’t build cyclotrons anymore. To quote an old Fermilab engineer “cyclotrons are an expensive and bulky way to get a really crappy beam or a cheap way to get a giant chunk of lightly radioactive steel that can never leave your lab”.

LINACs work in detail by shaping the radiofrequency energy in a confining waveguide. The oscillations of the RF energy may be held in space, or allowed to travel down the waveguide over time. Radiofrequency waves have connected magnetic and electric components, which oscillate.

[[File:2022 08 16 GalLib ElectromagneticWave.svg|center|800px]]

Inside the RF accelerator, the electromagnetic radiation is oriented 90 degrees to the particle travel direction, which points the associated electrical field in the travel direction of the particles.The radiofrequency field naturally oscilates along its wave, and the connected electrical and magnetic fields oscilate as well: building up, reaching a peak, and diminishing, inversing, building up, reaching a peak, and diminishing, and the cycle repeats. The detailed behavior depends on the specific cavity.

[[File:CavityWithEM.png|600px|thumb|A resonant RF acceleration cavity and the electric field inside of it, created by resonating RF fields, at one moment in time.]]

Particles of the right charge in the right space at the right time experience an acceleration from the electrical field, being drawn to the opposite electric charge. If they land in the other region, they are instead decelerated. Inbetween, particles experience less to no acceleration. This behavior naturally separates a stream of particles into discrete '''bunches'''.

[[File:CavityWithEMAndParticles.svg|600px|thumb|Bunches being accelerated inside the electric field. ]]

It also creates an implicit limit to what can be effectively accelerated using a specific radiofrequency accelerator design. The particles driven through it have to keep pace with the oscillations of the electric field, reaching the next correct region inside the cavity at that time when the electric field is oriented correctly. If they are too slow to keep inside the accelerating regions, the accelerating action will be ineffective. This requires RF cavities to be matched to the particle's velocity gain under energy imparture and their velocity when entering, being inside, and exiting the accelerator. For electrons and positrons, this isn’t that complicated. Since these lightweight particles accelerate readily, they have few problems “staying in the pocket”. Particles that are slower to gain speed, such as ions and even larger particles, must however be accomodated. This in turn impacts how much acceleration and energy per meter can be fitted into the accelerator in the first place. The acceleration of heavy particles requires specifically designed and sequenced RF accelerators for specific regions of particle speed. This in turn complicates the design of the larger accelerator infrastructure, with multiple discrete stages getting the particles to the energy required so that the next stage in sequence can take over.

RF cavity accelerators are limited in their eV/m capability by the critical current at the containing waveguide. To increase the acceleration provided, more intense electrical fields must be built up. Eventually the electrical charge of the RF waves becomes so strong it can rip electrons out of the cavity walls, causing an arc to form and the cavity material to break down physically.

=== Advanced Accelerator technologies ===
Present accelerator technology caps out at around 250 MeV/m for the most overdriven designs. This poses issues for a variety of applications, where higher energies in shorter distances are desired. In science, "desktop" and "chipset" accelerators are of interest. When militarizing particle beams, some effects only become available at very high energies - but you want something compact enough to turret.

urrent and even near future accelerator isn’t really ready to supply that sort of scale quite yet, so we have to move on to the cutting edge of accelerator physics. There are three big options with sub technologies within these broad fields. We have extreme RF cavities using much higher frequencies, plasma accelerators using lasers for power and plasma accelerators using particle beams for power. Each technology provides a unique series of advantages and disadvantages, but fundamentally they are all a way to make linear accelerators that are much smaller.

==== Dielectric Wakefield Accelerator ====
LINACs using much higher frequencies than traditional LINACs have two fundamental problems to fight. The first is electrical breakdown and the second is generation of those mm-wave (10-100 GHz) or even THz (100-1000+ GHz) sources at sufficient power and efficiency to power a particle accelerator. As the frequency of the driving power goes up the electric field climbs up, which means the accelerating gradient gets larger (which is good) but also means that we run a higher risk of a huge arc forming in the accelerator and breaking it (which is bad). This is often fought by changing the accelerator cavity material from a metal (or superconductor) to a dielectric material like ceramic, or by using a coating of dielectric, which can sometimes earn them the moniker of Dielectric Wake Field Accelerator (DWFA). The issue of creating powerful and efficient enough sources of this higher frequency drive power is being actively pursued for a variety of fields (5G cell signals, beamed power, etc) but is still quite immature compared to the microwave RF sources of traditional LINACs. There are options using essentially "over-clocked" microwave sources, low energy particle beams, and even laser-plasmas. There is still a lot to be done here, but these sorts of problems are being actively worked on there are working accelerators using this technology. There is potential for GV/m or greater acceleration gradiatens and all of the same advantages of traditional RF LINACs (i.e. high powers, high fire rates, high energy pulses, quality beams) should stay the same. The wall plug efficiency is liable to be a bit lower, but that will be highly dependent on the efficiency of the source of these high frequency waves.

==== Laser-Plasma Accelerator ====
Plasma accelerators using lasers for power are a whole world in and of themselves. There are a wide variety of ways to accelerate particles with lasers and plasmas, but we will focus on the most studied technique known as Laser Plasma Accelerators (LPA). LPA accelerators utilize ultrashort (<1psec) and ultraintense (>10^18 W/cm^2) lasers to drive a plasma wake (thus sometimes earning them the name laser wakefield accelerator) and accelerate particles through the resulting charge differential. They have only shown the ability to accelerate electrons and positrons, but any ions that could be injected at the right time and at extremely relativistic speeds could conceivably be accelerated this way. In the operation of a typical LPA the ultraintense laser is fired into a jet of gas (turning it into plasma) or a channel of lightly pre-ionized plasma. The laser then blasts through this plasma like a boat through water and leaves a void behind it (the wake). This void ends up having a HUGE accelerating gradient (100 GV/m) and suffers no risk of electrical breakdown since the plasma is already broken down! A pre-accelerated bunch of particles can be injected into that void at the right time (called the witness bunch) for a BIG acceleration or you can rely on the wake itself spewing electrons out into the void and make a beam out of the plasma itself. The gas or plasma then quickly is reformed and the laser fires as soon as its cooling and charging allows. Keep in mind that even though the beams can hit incredible energies in short distances, the laser itself is not a pocket sized device. These lasers are room sized at minimum and not easy to shrink. The pulse lengths required are also much shorter than most of our hypothetical weapons-grade pulse lasers that may operate in the nanoseconds regime - dual-use may prove difficult. The intensities reached also pose significant issues for the laser hardware. If you don’t spread the energies out, it becomes quite possible you will break parts of your laser from the produced intensities.

Current LPA systems suffer from a couple of problems that are being actively worked on. First and foremost they are incredibly efficiency limited by the laser tech being used. There are plans to move on to much more efficient fiber lasers and eventually make LPA machines that operate in the 25% or greater wall plug efficiency, but that is not currently doable. These lasers also need to be fired at higher rates than the current ~1 Hz, but that is much more of a limit of laser engineering than accelerating engineering. LPA’s also have yet to link more than one accelerator together and the lasers are limited in just how much plasma they can accelerate through right now (~20 cm max) so the beam energy runs into a wall quite quickly. Aside from R&D on linking LPA’s together in a way similar to RF LINACs there are two other options. The acceleration gradient is higher the denser the plasma, so a move to denser plasmas and shorter wavelength lasers (to penetrate the plasma) could provide higher beam energies in similar distances. The other option is to use clever optical techniques like the so-called “flying focus” to allow for the laser to go through much longer plasmas. This could in theory allow for more efficient LWFA’s and single stage machines that provide ~1 TeV electron beams in meters! The final issue is beam quality. Even the best LPAs will not make beams that have as tight of a beam spot, energy spread, or emittance as a normal RF LINAC. This means they will end up needing some beam “cleaning” if you want to get a lot of range out of them.

LPAs are of engineering interest for very short wavelength X-ray free-electron lasers in a compact format. Currently these facilities require linear electron accelerators many kilometers in length. With an LPA as the electron accelerator, an XFEL able to image molecular structures and chemical reactions as they happen might be shrunken down to a much more compact format. (Sometimes this is discussed under the ultimate promise of the “benchtop” or even “chip-format”(sic!) XFEL)

==== Beam-Plasma Accelerator ====
Plasma accelerators using particle beams for power are very similar to LPAs in principle but use a particle beam rather than a laser. We typically just call these things Beam Plasma Accelerators (BPA) for short. The concept is very similar to the LPA in that a high intensity beam comes in and blasts a void into a plasma, which particles can then be accelerated in. BPA provides some interesting advantages over LPA in that you can use highly efficient superconducting RF LINACs (or synchrotrons if you want) to provide the particles at very high efficiency. They also have little in the way of plasma density limits and the accelerated beam takes on the characteristics of the beam driving the BPA. Thus little if any beam cleaning will be needed after acceleration. The big downside is the need for a relativistic particle beam to drive the effect, and typically one in the 100’s of MeV to low GeV range, depending on the particle used. Thus your final accelerator may be short, but you still won’t be making a handheld particle beam device.

== Accelerator Shapes ==
The accelerator technologies discussed above are the means by which we create situations for particles to gain energy. The accelerators become embedded in a larger structure that produces, directs, accelerates, even stores the particles until they are directed towards their final use.

There are two principle shapes for accelerator assemblies. Ones where the beampath is not closed onto itself, and designs where the beam path is closed on itself in part of the assembly, forming a loop.

=== Linear accelerator ===
Everything from electron generation to acceleration to conditioning is built in a straight line, one after another. This is the Linear Accelerator or LINAC. The main catch here is that there is only one single passage for the particles through the entire accelerator. Re-using accelerator hardware to drive the same particle to higher energies by repeated passage is out. The particles have to reach their desired energy in a single pass. Practically this puts limitations on the energy achievable with heavy particles in a LINAC. The length required gets impractically large. Even for electrons, lengths of hundreds of meters to kilometers may be required, unless advanced accelerators with high gradients are used. This said, every accelerator contains linear sections. Since so much high-performance accelerator technology depends on particles having a minimum velocity, ions must be “pre-conditioned” in a set of linear accelerators before the re-use of an accelerator through looped beam paths can be considered.

LINACs of low energy may be used in many applications. Medicinal they find use in radiation therapy and radiosurgery.

=== Synchrotrons ===
The other shape which has some interesting properties are accelerators where beampaths are looped onto themselves. These are called synchrotrons. Since beam paths are closed onto themselves, particles can pass through the same accelerator unit repeatedly, gaining more energy on every passage. This is advantageous for heavy particles which do not gain much speed on any single passage. They can pass through the same accelerator thousands or even millions of times instead. This is obviously advantageous for facility bulk and mass and allows us to reach energies that would otherwise be impractical. The accelerated beam bunches can also be held in storage, looping like the rotating mass of a flywheel, holding energy. The LHC can store some 700 Megajoules of energy in the recirculating beam after full speed has been reached.

That said, the devil is in the details. When charged particles are bent in circles they emit electromagnetic radiation which scales very rapidly (to the fourth power!) with relativistic gamma factor and with bending radius (inversely to the second power). This phenomenon is called synchrotron radiation. Due to the scaling, high energy particles emit more synchrotron radiation than lower energy ones and lighter particles (like electrons) at those high energies emit much more than heavier particles. In addition, the bigger you can make the bending radius of the synchrotron (often called “the ring”) the less synchrotron radiation you get.

[[File:SynchrVsE.svg|800px|center|thumb|Synchrotron radiation per particle versus particle energy for electrons and protons in a 50 meter in radius ring.]]

[[File:SynchroVsRing.svg|800px|center|thumb|Synchrotron radiation emission per particle for different particles at different energies, over varying bending radii. ]]

Although these numbers look small, keep in mind that this is PER PARTICLE. A typical pulse in an electron accelerator will have more than more than a billion electrons in it! Large synchrotrons like the LHC will have over 300 trillion ions circulating through it! Synchrotron radiation necessarily has to be managed to reduce radiation shielding, the radiation load (and possibility of hardware failure) on the accelerator, and to reach high energies in the first place - you might lose all the energy your accelerators are adding to synchrotron radiation!

Deflecting the beam also requires the application of magnetic fields, and higher degrees of deflection, in a smaller length, at higher particle energies, require stronger and stronger magnetic fields. Eventually multi-Tesla fields from superconductors will be required. And due to action and reaction, those magnets need to be braced against the pressure the beam exerts as it is deflected. Ultimately this leads to the same limitations as with flywheels - the mechanical strength of the braces and static connections that keep the magnets together as a larger structure is the accelerator’s limitation on J/kg held in storage. A further complication: The magnet strength has to be matched to particle energy. As a beam is accelerated, the deflectors have to ramp up their power to keep the degree of deflection constant.

Sometimes, synchrotron radiation from electrons is intentionally induced in a ring-shaped accelerator. Since synchrotron radiation of high-energy electrons tends to be hard x-rays emitted along narrow vectors, it’s a way to get x-rays used in various imaging techniques in material science and molecular physics.

Synchrotrons used for ions also need to respect the limitations of the involved accelerators. Synchrotrons often use radiofrequency accelerators. Since radiofrequency accelerators must match the speed and speed gain of their particles, an ion synchrotron cannot be made up of a single accelerator loop. Instead there are sequential loops connected in sequence, each designed to boost the particle through a specific velocity and energy region so that the next accelerator takes over. A designed example is the LHC. Particles headed for the main LHC actually circulate in two other synchrotrons beforehand, gaining sufficient energy so that the LHCs RF cavities can take over. (Also, before being injected into the first synchrotron ring, the ions must first go through multiple linear accelerators using different technologies to reach a base energy.)

An interesting design consideration: Synchrotrons don’t necessarily have to take the form of a (near-ideal) ring. Funamentally any dipole represents a single point of deflection of a straight path - they're polygons! Any geometric shape that contains a loop, and where any single bending curve keeps within the limits of the synchrotron radiation you are willing to tolerate, works. As such, complicated two- or three-dimensional “folded-up” shapes are also possible formats for a synchrotron.

=== Storage Rings ===
Since synchrotrons have to match their deflection magnet strength to the energy of the particles being accelerated, a single synchrotron track cannot hold more than one small section of beam bunches in storage at a time, and becomes unavailable to accelerate further bunches until it discharges the stored beam. This is disadvantageous for various applications where we may want to store a fully accelerated particle beam “bolt” for a longer time.

Storage rings are separate synchrotrons designed to hold multiple sets of bunches in circulation at a target energy at a time. The bunches are delivered from an accelerating synchrotron (sometimes called a booster ring) and held in circulation in the storage ring. Since there are synchrotron radiation losses, a few accelerator sections must compensate for these.

Storage rings can also be used to condition a stored beam. A common thing to do in storage rings is to “cool” a beam, draining unwanted movement energy out of the particles. Beam cooling may take significant time, which makes it ideal to conduct in a storage ring. The result is a beam with less tendency to drift apart, advantageous for many applications.

== Other parts ==
=== Particle sources ===
Particle sources are assemblies which supply particles to accelerators. Here we can group them into three categories: electron sources, ion sources, and exotic particle sources. The later involve the production of particles by means of gathering them from collision reactions (or the handwavium blackbox of your choice) and feeding them into another accelerator for further use. Positrons for example may be manufactured in beam collisions.

Electron sources require the production of a free stream of electrons. The oldest method here are thermionic sources. A hot piece of material ejects free electrons into a vacuum, where they can be drawn in using electric differentials and forwarded to the accelerator.

Field emission electron sources instead use a strong potential between a needle-like cathode and an anode plate with a hole. The electrons are drawn from the needle and through the anode due to the high charge differential. This method produces a “cooler” beam with less random divergence than thermionic sources.

A further improvement still are photoemission sources. Here the precise application of laser light onto a photosensitive material energizes electrons in the photocathode until they break free. This process is thermodynamically very precise and due to the nature of the laser light, also very controllable. The result is a low-emittance stream of electrons. These may be accelerated using electrostatic attraction with an anode at one end of the vacuum chamber, or directly using an RF field.

Ion beam sources are more complicated. The physics of ion formation are complex, which opens many design choices when creating an ion beam. Generally a plasma must be produced, from which the ions can be extracted.

The plasma can be formed in a variety of ways. Electrons from a local cathode-anode array, plasma frequency heating, the injection of electron beams, the application of laser light. The end results can be as diverse. Some methods only produce weakly charged ions and on many vectors, so with high emittance. Other methods offer greater ion charges or reduced emittance.

=== Magnets ===
While acceleration of charged particles is effected with electric fields, magnetic fields can be used to bend the trajectories of particle beams. As such, electromagnets can be used to bend the trajectories of charged particles in various ways, and in a controllable fashion.

==== Dipole Magnets ====
Dipole electromagnet are used to steer the entire particle beam in one axis. (To steer in two axes, we must either rotate one dipole about the beam axis, or couple two dipoles in sequence.) They can be used to direct a beam inside a particle accelerator, inject and extract beams out of looping constructs by switching on and off at the right moment, and point a beam at a target.

In a dipole magnet, a positive and negative coil oppose each other, embedded in a “choke” assembly often made of steel. Charged particles moving through the magnetic field between the coils experience a deflection in one direction specified by the polarization of the field, and a degree of deflection specific to the field strength.

==== Quadrupole Magnet ====
Dipoles can steer, but they influence all particles equally. As such, they cannot be used to manipulate the beam's shape. The first type of magnet able to influence a particle beam’s shape are quadrupole magnets.

They are assembled from four coils at 90 degree angles, with coils of the same polarity opposing each other. The passing beam is compressed on one axis, and expanded on the other. To focus the beam overall, at least two quadpoles must be arrayed in sequence.

Quadrupoles are the basic and essential “magnetic lens” of particle beams. Without them, charged beams would inevitably defocus and run havoc.

==== Hexapole Magnets ====
Hexapole magnets exist to address one problem of quadrupoles - they focus differently depending on the energy of the particle passing through the magnetic field. The effect is likened by particle accelerator physicists to the different diffraction of light in a prism - and hence the effect is called “chromatic aberration”. Hexapoles are used to correct for this, stronger deflecting higher-energy beams.

==== Integrated optics ====
As the field of particle accelerator science and engineering advances, more complex designs are proposed to handle more complex phenomena inside accelerators. An upcoming field is “integrated optics” - complex magnetic systems with nonlinear behaviors. These could provide more advanced manipulation of electron beams, building upon existing capabilities and adding new ones for shaping a beam.

=== Plasma lenses ===
Plasma lenses are an up-and-coming technology, where very high currents in plasma are used to create very strong magnetic fields. Similar to plasma-based accelerator technology, the fact that charges are already free in plasma is used to circumvent conventional charge breakdown and critical current limitations on solid-matter devices. The resulting magnetic fields can be much stronger and bend beams over shorter distances. Furthermore, by creating complicated profiles in the plasma, elaborate beam manipulations could be performed, more akin to sophisticated optical elements. As well, the charges in the plasma neutralize the particle beam in the region of the plasma lense - affecting its propagation behavior further and in a positive manner, as self-repulsion is neutralized.

The catch is that this is plasma - that means you are dealing with all the complications of plasma physics. Heating of the plasma has been an early challenge to successfully forming and maintaining plasma lenses.

Plasma lenses require an appropriate chamber, possibly active containment to keep the plasma from losing energy to the walls or leaking into other accelerator parts, and the necessary plasma sources, maintenance heating devices and then lense shapers and control sensors. In high-powered plasma lenses there will also be a need to cool the plasma as it heats up from waste heat. Potentially we may manage this by open-cycle cooling - venting the plasma controlled in one direction into open space, rather than have a closed-cycle solution for reconditioning the plasma. This could cause some very cool visuals!

Plasma lenses for particle beams should not be confused with optical plasma lenses for lasers! Those exist as well, but are their own topic.

=== Chicanes ===
Chicanes are assemblies of multiple dipole magnets used to manage the longitudinal dispersion of a particle beam due to different particle energies. Since particles with higher energy move faster, they drift ahead in the beam distribution over time; and particles with less energy increasingly lag behind.

Chicanes deflect high-energy particles onto an outward curving trajectory. Since traversing the curve takes longer, the high-energy particles are effectively delayed, allowing lower-energy particles to catch up. By the end of the chicane assembly, the bunch has been compressed in the longitudinal direction.

Chicanes however, do not remove energy from too-fast particles. That is the responsibility of other structures.

=== Undulators and wigglers ===
Undulators and wigglers are magnetic assemblies used to transform particle energy into electromagnetic emissions, with different properties from the specific implementation of a shared basic principle.

Assembled from stacks of generally dipole magnets (other configurations exist), both undulators and wigglers force charged particles onto “wiggling” trajectories by bending their path left and right. Wigglers force very strong arcs; undulators less so. In both cases, the rapid deflection of the charged particles causes them to shed parts of their energy as electromagnetic emissions. Due to the energy of the particles, these photons are generally highly energetic x-rays.

Undulators may be used to build free-electron laser systems.

They can also be used as a means of cooling the beam. The removal of energy from particles can also suppress unwanted movement modes, allowing accelerator structures to re-add energy only to desired movement modes. Of course, this means the particle beam actively loses energy, and produces x-rays that have to be absorbed (and cooled away.)

=== Beam coolers ===
Beam coolers are a collection of systems and strategies that are used to remove unwanted movements from a particle beam. Undulators and wigglers are one way to do this.

One method set of beam cooling relies on finding particles who are very “hot” with random motion, and either removing the particles from the beam using a kick device, or applying a precise counter-force that acts against the unwanted motion.

If one is dealing with ions, one clever trick is to inject a beam of electrons. Due to the thermodynamics within the particle population, the lighter electrons absorb heat from the heavier ions, taking this unwanted motion energy with them when the electron beam is bent out.

As well, lasers can be used (what can’t lasers be used for!) to apply counter-force on ions and remove unwanted movement in this manner.

Beam coolers are generally not instantaneous devices. They work over time to make a beam better. Often they involve the removal of energy or particles from the stream, requiring accelerators and sources to make up the loss and creating waste heat.

=== Beam recovery ===
Some accelerator systems can actually be run in reverse - they can turn beam energy back into some other energy format and then electricity. This can be used to recover some to most of the energy from a particle beam before disposing of it. This is advantageous both for your energy bills and making a very high-energy beam liable to spray radiation everywhere, easier to absorb in a beam dump.

=== Beam dump ===
A beam dump is a construction where a particle beam can be routed in case it has to be disposed. Beam dumps are designed to safely absorb the particle's energy without leakage of dangeorus radiation or activated materials to the outside enviroment. Common materials are carbon and metals, embedded in an actively cooled assembly.

= Propagation physics =
==Magnetic fields==

Magnetic fields are things of motion. They are created by moving electric charges. And they only affect electric charges that are moving. An electric charge at rest in the presence of only a strong magnetic field will just sit there. But when it starts moving, the magnetic field interacts with it, starts deflecting it.

But not along the field. Oh no. The magnetic field does not point in the direction it will move the charge. No, the charge experiences a force that is perpendicular to the direction of the field. And also that is perpendicular to its own velocity. A charged particle moving straight down a magnetic field line will not experience any force.

On the other hand, if the particle is moving directly perpendicular to the field, the magnetic force will always push it to the side, even as it keeps turning. This results in the particle moving in a circle. The radius of this circle is called the <b>gyroradius</b>, and the rate at which the particle makes circles is called the <b>gyrofrequency</b> (or sometimes <b>cyclotron frequency</b>).
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Circular_motion.png|frameless]]
<td width=255>[[File:magnetic_forces.png|frameless]]
<td width=255>[[File:spiral_around_field_lines.png|frameless]]
<tr>
<td>To get circular motion, you need a force (blue) that is always the same amount but always pushing perpendicular to the direction of motion, toward the center of the circle. A charged particle in a constant magnetic field moving perpendicular to the magnetic field meets these conditions, and moves in a circular path.
<td>A diagram of how a magnetic field (magenta) exerts forces (blue) on a moving particle (with velocity shown in green).
<td>If a particle is moving partially along the magnetic field (magenta), it will drift down the field lines while spiraling around it (black path). Where the magnetic field is stronger (indicated by having the field lines closer together) it will execute tighter, faster circles in its spiral and will drift more slowly than when the field is weaker.
</table>

It can be convenient to split up a particle's velocity into a part parallel to the magnetic field (the <b>longitudinal velocity</b>) and a part perpendicular to the field (the <b>transverse velocity</b>). The longitudinal velocity describes how fast the particle drifts along the magnetic field line. The transverse velocity determines how tightly and how quickly the particle circles around the field lines.

If the gyroradius is much smaller than the extent of the magnetic field, you get a case where the particle approximately cannot cross magnetic field lines. It just spirals around them. On the other hand, if the magnetic field ends before the particle can make a significant portion of a full revolution, then the particle ends up being somewhat deflected from its original trajectory. If the gydroradius is enogh to deflect the particle into a region where the magnetic field has a different strength or direction, then things get complicated!

When a particle is spiraling around the magnetic field while drifting down the field lines into regions where the magnetic field is stronger, it will circle around the field lines in a faster, smaller circle. This increases the kinetic energy in the spiraling motion; this kinetic energy must come from the kinetic energy of the longitudinal motion. If the field becomes too strong, the particle will slow down all the way and come to a stop. Then it will be reflected back, pushed the other way down the field lines toward the region where the field is not as strong. This phenomenon is called magnetic mirroring, and is responsible for the way radiation gets trapped in the Van Allen belts around Earth with the particles bouncing back and forth along the Earth's field lines and getting reflected back and forth at the poles where the field is strongest.

If you want to calculate for yourself what the gyroradius <math>r_c</math> and gyrofrequency <math>f_c</math> are
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
r_c = \frac{\gamma \, m \, v_\perp}{q \, B}
</math></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
f_c = \frac{q \, B}{\gamma \, m}
</math></div>
where <math>B</math> is the strength of the magnetic field, <math>q</math> is the amount of electric charge of the particle, <math>m</math> is the mass of the particle, <math>\gamma</math> is the Lorentz factor, and <math>v_\perp</math> is the part of the velocity that is perpendicular to the magnetic field. To get these formula to work, you need to use a consistent set of units. So for example if you measure the magnetic field in tesla you need to measure the charge in coulombs (not fundamental charges) and the mass in kilograms (not eV).

One consequence of this is that you can bend the beam coming out of your accelerator by placing a uniform magnetic field across the end. So you can steer your beam to where it needs to go. You can also bend the beam inside of your accelerator – to make it go around in a circle, for example.

Another consequence is that if your beam has to pass through regions with magnetic fields in it, it will get deflected. If you want the beam to end up at the right spot, you will need to account for this. If the exact strength or direction of the field are not known, you will get unpredictable deflections to your beam.

If your beam has been neutralized, the magnetic fields will work to bend the positively charged particles one way and the negatively charged particles the other way. If you have enough beam current, these separated charges will attract each other and counteract the magnetic force. If you don't, the beam will split and you'll get positive charges zipping off in one direction and negative charges going another direction. If you needed to neutralize your beam so that it doesn't electrostatically blow itself apart, well now it will be doing just that.

Even perfectly neutral atoms can be torn apart if the atoms are going fast enough and if the magnetic field is strong enough. Atoms are pretty resistant to this, but you can get conditions where it happens.

Magnetic fields affect the path of beams. But beams are made of charged particles and thus can have currents in them. Currents create magnetic fields themselves, which can influence the surrounding fields. Even a neutralized beam forming a plasma, when it encounters a magnetic field, will get currents induced in the plasma that helps to screen its interior from the field. A beam with enough punch can deform the fields around itself rather than having the fields bending it. As a rough rule of thumb, if the energy density of the beam over a regions larger than a gyroradius is more than the energy density of the field over that region, then the beam gets to have its way with the field, and vice versa. Highly focused, high current beams can thus be expected to better deal with ambient magnetic fields than those with less focus or lower current.

===Synchrotron radiation===

A charged particle is surrounded by an electromagnetic field. Information can only move through this field at the speed of light. If the particle is accelerated, the information doesn't catch up to all of the field before it has moved on, and bits of field can detach themselves from the particle and go off on their own as electromagnetic radiation. A particle being bent by a magnetic field is being accelerated to the side, and this produces radiation. It is called <b>synchrotron radiation</b>. The energy that goes into synchrotron radiation comes from the kinetic energy of the particle. Depending on the details of the particle, its speed, and how much it is bending, the energy loss can be significant.

The amount of energy loss for a full turn of radius <math>r</math> is<ref name="Jackson">J. D. Jackson, "Classical Electrodynamics, Second Edition", John Wiley & sons (1975). Note that this text is written entirely in cgs units, I have taken the liberty of converting to SI for this reference.</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
\delta E = \frac{1}{4 \, \pi \, \varepsilon_0} \frac{4 \, \pi}{3} \frac{e^2}{r} \beta^2 \gamma^4 \ \ \ \ \ \ \ \ \ \ \mbox{energy loss for a full revolution}
</math></div>
where <math>\gamma</math> is the Lorentz factor, <math>\beta</math> is the speed as a fraction of the speed of light, <math>e = 1.602176634 \times 10^{-19}</math> C is the fundamental charge unit, and <math>\frac{1}{4 \, \pi \, \epsilon_0} = 8.9875517923 \times 10^9</math> kg m²/C² is the Coulomb constant.
For highly relativistic electrons with <math>\beta \to 1</math>, an expression in convenient units is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
\delta E \mbox{(MeV)} = 8.85 \times 10^{-2} \frac{E \mbox{(GeV)}^4}{r \mbox{(m)}} \ \ \ \ \ \ \ \ \ \ \mbox{energy loss for a full revolution, electrons, } \ \beta \to 1.
</math></div>
If your particle only gets deflected a bit and does not go a full revolution, but is only deflected by an angle <math>\theta</math>, multiply the energy loss per revolution by <math>\theta/360^{\circ}</math> (degrees) or <math>\theta/(2 \pi)</math> (radians) to find the energy loss for being bent by that amount.

If your don't know the radius of revolution <math>r</math>, but you do know the angle of deflection <math>\theta</math> and the distance over which the deflection happens <math>d</math>, then
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
r = \frac{d}{\sin (\theta)}.
</math></div>
For small <math>\theta</math> measured in radians, this simplifies to <math>r = d/\theta</math>.

<blockquote>
<i>Example</i>: We have a 50 GeV electron beam coming out of a plasma accelerator with a beam radius of 0.1 mm. We want to expand the beam radius out to 10 cm so we can focus it better on a distant target. We will use one plasma mirror where the beam comes out to expand it, and a second plasma mirror 10 m away to finish the focusing. In that 10 m, the beam has to drift 5 cm in each direction before it hits the second lens, so that is an angle of approximately <math>0.05/10 = 0.005</math> radians in the small angle limit. If the plasma mirrors are 1 meter long each, then plugging in <math>d = 1</math> m into the formula above, we get a radius of curvature of <math>r = 200</math> m. Putting this in to our energy loss formula, we get an energy loss per revolution of approximately 2770 MeV. We only go through a fraction <math>0.005/(2 \pi)</math> of a revolution, though, so our energy loss is 2.2 MeV. But we then lose another 2.2 MeV as our diverging beam is re-focused at the second plasma lens 10 m away. So this focusing procedure loses roughly 5 MeV out of 50 GeV, or 0.01% of the beam energy, which is deemed by your engineering supervisor to be an acceptable loss.
</blockquote>

<blockquote>
<i>Example</i>: You are part of the planetary defense force in low orbit around Earth. Your space warcraft are equipped with the electron beams in the above example. An invading alien force is attacking! You aim your electron beam at an alien spacecraft 100 km away and begin zapping it.

The magnetic field around Earth is approximately 0.0001 tesla. For the shot you need to take, your beam will be moving perpendicular to the magnetic field. At 50 GeV, your electrons have a <math>\gamma</math> of nearly 100,000. With <math>v_\perp = 300 000</math> m/s (so close to the speed of light as to make no difference), an electron mass of <math>9.11 \times 10^{-31}</math> kg, and an electron charge of <math>1.6 \times 10^{-19}</math> C the gyroradius of your beam in the Earth's field is 1,700 km. The angle of deflection of your beam will be very close to <math>\theta = 100 \mbox{km} / 1700 \mbox{km} = 0.059</math> radians, so the fraction of a full revolution your electrons will take en route to the target is 0.0094. The energy loss for a full revolution at 1700 km gyroradius and 50 GeV energy is 0.33 MeV; because your beam only goes through 0.0094 of a revolution your beam only loses 0.003 MeV on its way to deliver hot radioactive death to your enemy.
</blockquote>

==Focusing==

===Focusing equipment===
In the previous section, we discussed how a particle beam can be steered by directing it through a uniform magnetic field. We can make this kind of field with a magnetic north pole directly opposite a magnetic south pole (with additional shimming magnets to keep things uniform). Because it has two poles, one north and one south, this kind of magnet is called a <b>dipole magnet</b>.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:dipole_beam_steering.png|frameless]]
</table>

But we want to do more than just steer the beam. we want to be able to expand it, contract it, and focus it to a point. To do this, we need a different kind of configuration. We will put two north poles opposite each other and, and 90 degrees, two south poles opposite each other. With four poles, this is called a <b>quadrupole magnet</b>, or sometimes just a "quad".
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_magnet.png|frameless]]
</table>
At the center of the quad arrangement, the magnetic fields from each side cancel out. there is no field there. A particle going through the center will travel straight through without deflection. However, the farther particles get from the center, the higher the field they encounter and the more they get deflected. A particle slightly off to one side may experience a force that curves it toward the center-line. If the quads are properly shimmed, a particle twice as far away will experience twice the force and twice the deflection from its trajectory back toward the center. This curves all the particles so they meet in one spot.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_focusing.png|frameless]]
</table>
However, there is a catch. Those particles that are offset from the center-line at 90 degrees from these focused trajectories are bent away from the center-line rather than towards it! A quad focuses in one direction and de-focuses in the other.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_defocusing.png|frameless]]
</table>

Is there anything e can do to get an arrangement of magnets that purely focuses the beam? Well yes. We can put another quad arrangement downstream of the first, rotated by 90 degrees. Now consider a set of particles in an arrangement that is de-focused by the first quad. When they get to the second quad, they are further from the center than they were when they went through the first quad. Because the second quad is aligned to focus anything the first quad de-focuses, these particles will be focused; and because they are farther from the center, the focusing deflection will be larger than their previous de-focusing deflection. So the net result is that the particles are focused.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Dual_quad_focusing.png|frameless]]
</table>
For particles with the opposite orientation that are first focused, these will be de-focused by the second quad but because they will have converged they will be going through a region of lower field and will be defocused with a smaller deflection than they were initially focused. This dual quad arrangement will focus any particle that goes through it, regardless of orientation.

A more speculative method of focusing particle beams uses a plasma lens<ref>[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.121.194801 C. A. Lindstrøm <i>et al.</i>, "Emittance Preservation in an Aberration-Free Active Plasma Lens", Phys. Rev. Lett. <b>121</b>, 194801 (2018)]</ref>. A long tube of gas is flashed to plasma by a strong current pulse. A uniform current tube has no magnetic field at its center, but the magnetic field strength increases linearly as you travel radially out from the center.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=800>[[File:Plasma_lens_perspective.png|800 px|frameless]]
<td width=400>[[File:Plasma_lens_end_on_view.png|400 px|frameless]]
<tr>
<td>A cutaway perspective view of a plasma lens tube, showing the axial current (magneta) and the circulating azimuthal field (cyan).
<td>An end-on view of the plasma lens tube, showing the circulating magnetic field (cyan). The field is zero at the center, but rises in strength as you approach the edges.
</table>

A particle beam traveling down the length of the tube will be crossing perpendicular to the magnetic field lines. As we have discussed, this gives a force perpendicular to both the magnetic field and the particle direction – which in this case is either inward or outward (depending on the direction the particles are moving and on the direction of the magnetic field, which in turn depends on the direction of current flow). Because the field strength increases uniformly the farther you go from the center it produces a focusing effect with particles moving down the center not deflected at all and the farther out you go the more the particle gets deflected.

A plasma lens has a number of drawbacks that need to be taken into account. First, it needs to be very long compared to its width. In order to get a uniform current you need a very long path for the current to flow on compared to the distance across the tube. Second, you need some way to confine the gas (and the momentary plasma), such as windows. A material window might not deal well with very high intensity particle beams that might damage the window. A potential alternative are plasma windows, with a plasma at the ends held in place by a strong magnetic field, but this confining field will deflect the particle beam and this will need to be taken into account. Third, all plasma lenses made so far have been really tiny, on the order of a millimeter across. This is not much of an issue if you are dealing with various plamsa wakefield accelerators (which generally produce beams about 0.1 mm across, so they can easily fit through these lenses), but if you want to focus your particle beam to small spot sizes at long distances you will want to put in a lot of engineering to get wider plasma lens channels, which will in turn mean a much longer apparatus and high energy costs involved in making the current discharge for the lens.

===Emittance===

For small deflection angles, a dual quad set-up acts like an ideal lens for particles. An ideal lens can focus all the particles diverging from a point onto another point, or into perfectly parallel lines. It can focus particles on perfectly parallel lines onto a point. So now we can use the particle beam to deliver focused particles on our target.

Or, at least we could if all the particles came from a point, or if they traveled on perfectly parallel lines. Unfortunately, they don't. When particles are made, they come from a source with a finite spot size and a finite temperature. The random motion of the particles moving around from their initial thermal motion means that they are never moving perfectly parallel to each other. And when you try to focus them, instead of all converging onto a single point they make a spot of finite size. And if your focusing and beam bending elements are not perfect, you add additional imperfections to the beam that further hider your ability to focus it.

Your ability to focus can be quantified by a measure called <b>emittance</b>. Emittance is measured in units of a length scale times an angle - nominally meters-radians (m<math>\cdot</math>rad) but the technical literature more commonly gives units of millimeters-milliradians (or mm<math>\cdot</math>mrad) or even 2 <math>\pi</math> mm<math>\cdot</math>mrad. The divergence angle of a beam that you try to make parallel will the the emittance divided by the width of the beam. If you focus a beam down so it converges at an angle, the minimum spot size you can get at the focal point is the emittance divided by the beam angle. If you try to focus a beam with emittance <math>\epsilon</math> on a spot a distance <math>R</math> away with an initial beam diameter of <math>D</math>, the spot size <math>S</math> on the target you can achieve is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
S = R \frac{\epsilon}{D}
</math></div>
It is interesting to note that this is very nearly the same spot size you get from a laser due to [[Diffraction#Spot_Size|diffraction]] with a wavelength equal to the particle beam's enissivity. And, like lasers, the more you can expand the beam to large diameters at the focusing equipment, the tighter the spot you can achieve on the target.

So to get good focusing, we want low emittance. Is there a way we can lower the emittance beyond what we get from our source? Engineers are pretty bright folks, and they have found some ways to do this. You can, for example, pass a beam through a thin foil. This causes some stopping, slowing the beam down. You can then accelerate the beam back up to its original speed. However, the stopping is along the direction of the beam particle's trajectory, even if that trajectory is at an angle. Meanwhile, all the acceleration is in the direction of the beam axis. So the net result is to reduce the random motion perpendicular to the beam, and thus reducing its emittance. Similar tricks can be played with magnet set-ups that force particles going through them to emit radiation.

Another method can be used with a relatively high emittance beam, say a beam of protons or ions or muons. Electron beams can be made with substantially lower emittances. So you can send a low emittance electron beam to co-propagate along with your primary particle beam. The beams will interact, with the particles scattering off each other. This exchanges heat between the two beams and so the temperatures, and thus emittances, of the two beams get closer to each other (and, if given enough time, would become equal). If you repeat this often enough, you can get the emittance down to the emittance of the electron beams. This is more practical in beams going around circular accelerators, because they will have time to get cooled over and over.

A final method that has been used is to install sensors to measure the beam fluctuations. For beams going in circles, the time to get a light-speed signal directly across the diameter to the opposite side wil always be less than the time it takes for the particles to go around the curve to that spot. So you can send signals to give electromagnetic kicks to the beam particles to knock them back into alignment when they get to that spot.

It is intriguing to note that these beam cooling techniques can take <i>hours</i> to work!<ref name="RHIC emittance">[https://www.bnl.gov/isd/documents/86783.pdf M. Minty, R. Connolly, C. Liu,T. Summers, and S. Tepikian, "Absolute beam emittance measurements at RHIC using ionization profile monitors", Brookhaven National Laboratory formal report BNL-105970-2014-IR]</ref> However, without beam cooling methods like this, the emittance of ions or protons in storage rings can slowly increase on a time scale of hours.<ref>[https://arxiv.org/abs/1204.6022 Vladimir Shiltsev and Alvin Tollestrup. "Emittance growth mechanisms in the Tevatron beams", arXiv:1204.6022 [physics.acc-ph]]</ref>

But perhaps the simplest method to reduce emittance is just to speed the particles up. For particles moving at a fraction of light speed <math>\beta</math> and Lorentz factor <math>\gamma</math>, the emittance of particles from the same source with the same temperature and same random motion will be proportional to <math>1/(\beta \, \gamma)</math>. To reflect this, it is convenient to define a normalized emittance <math>\epsilon_n</math> such that for the usual geometric emittance <math>\epsilon</math> that we have been talking about
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
\epsilon_n = \beta \, \gamma \, \epsilon.
</math></div>

Normalized emittance measures makes it convenient to get basic estimates of particle beam performance and compare emittance of different sources and beams. For example, various high-performing electron beams around the world have managed to push their normalized emittances down to about <math>2 \times 10^{-6}</math> m<math>\cdot</math>rad to <math>3 \times 10^{-6}</math> m<math>\cdot</math>rad. Facilities with proton or ion beams seem to have about an order of magnitude higher normalized emittance.
<table>
<tr><td width=200>Machine <td width=300>normalized emittance (m<math>\cdot</math>rad) <td width=100>particles<td>reference
<tr><td>European XFEL <td><math>2.3\times10^{-6}</math> <td> electrons <td><ref>[https://journals.aps.org/prab/pdf/10.1103/PhysRevAccelBeams.24.110702 T. Hara <i>et al.</i>, "Low-emittance beam injection for a synchrotron radiation source using an X-ray free-electron laser linear accelerator", Physical Review Accelerators and Beams <b>24</b>, 110702 (2021)]</ref>, using reported <math>\epsilon \approx 1.5\times 10^{-10}</math> m<math>\cdot</math>rad and electrons at 8 GeV for <math>\beta \to 1</math> and <math>\gamma = 15700</math>.
<tr><td>DESY PETRA III<td><math>0.93\times10^{-6}</math> <td> electrons <td><ref>[https://www.desy.de/news/news_search/index_eng.html?openDirectAnchor=444&two_columns=0 New emittance world record at PETRA III]</ref>, using reported <math>\epsilon = 1.6\times 10^{-10}</math> m<math>\cdot</math>rad and electrons at 3 GeV for <math>\beta \to 1</math> and <math>\gamma = 5870</math>.
<tr><td>SLAC <td><math>3\times 10^{-6}</math> <td> electrons <td><ref>[https://accelconf.web.cern.ch/pac97/papers/pdf/4W009.PDF M. Hernandez <i>et al.</i>, "Emittance Measurements for the SLAC Gun Test Facility", 1998 IEEE]</ref>
<tr><td>LHC <td><math>3.75\times 10^{-5}</math> <td> protons <td><ref>[https://www.lhc-closer.es/taking_a_closer_look_at_lhc/0.beta___emittance Beta & Emittance; Taking a closer look at LHC]</ref>
<tr><td>Fermilab Tevatron <td><math>\approx 5\times 10^{-5}</math> <td> protons <td><ref>[https://accelconf.web.cern.ch/p03/PAPERS/TPPB070.PDF J. Slaughter, J. Estrada, K. Genser, A. Jansson, P. Lebrun, and J. C. Yun, "Tevatron Run II Luminosity, Emittance and Collision Point Size", IEEE Proceedings of the 2003 Particle Accelerator Conference, pages 1763-1765]</ref>
<tr><td>RHIC (with protons) <td><math>\approx 5\times 10^{-5}</math> <td> protons <td><ref>[http://www.osti.gov/servlets/purl/910428-6sKfP0/ Zhang, S, and Ptitsyn, V., "Proton Beam Emittance Growth at RHIC", Brookhaven National Laboratory formal report BNL-77563-2007-CP]</ref>
<tr><td>RHIC (on startup) <td><math>\approx 1.5\times 10^{-5}</math> <td> Au-197 <td><ref name="RHIC emittance"></ref>, using reported <math>\epsilon \approx 12\times 10^{-6}</math> m<math>\cdot</math>rad and Au-197 ions at 100 GeV for <math>\beta = 0.76</math> and <math>\gamma = 1.545</math>.
<tr><td>RHIC (after several hours cooling) <td><math>\approx 5\times 10^{-6}</math> <td> Au-197 <td><ref name="RHIC emittance"></ref>, using reported <math>\epsilon \approx 4\times 10^{-6}</math> m<math>\cdot</math>rad and Au-197 ions at 100 GeV for <math>\beta = 0.76</math> and <math>\gamma = 1.545</math>.
</table>
So as a plausible first estimate for your sci-fi electron guns, you can take <math>\epsilon_n \approx 10^{-6}</math> m<math>\cdot</math>rad and for proton guns <math>\epsilon_n \approx 10^{-5}</math> m<math>\cdot</math>rad. Modify for any improved future-tech that can bring the emittance down. Then divide by <math>\beta \, \gamma</math> for your beam to find the geometrical emittance. Which in turn can let you figure out how well you can focus your beam.

However, we should add one caution here. Unlike a laser, which can get a beam diameter nearly as wide as the width of the optical element (be it a lens or mirror) focusing it, a particle beam needs to go through a small hole inside much larger focusing elements (although this constraint might be somewhat decreased with a plasma lens, at the expense of making your lens much longer). Thus, a particle beam width (what goes into the spot size calculation) is likely to be much smaller than the physical size of the focusing equipment. You may be able to overcome this limitation by making the geometrical emittance very low, but it will be an issue to keep in mind.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Laser_focus.png|400 px|frameless]]
<td width=150>
<td width=556>[[File:P-beam_focus.png|556 px|frameless]]
<tr>
<td>Laser focused with a mirror
<td>
<td>Particle beam focused with quadrupole magnets
</table>

==Beam self-forces==

particle beams are composed of moving charged particles. Moving charged particles are electric currents. And currents and charges exert forces on each other.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Charges_and_forces_pn.png|frameless]]
<td width=255>Opposite electric charges exert an attractive force on each other
<tr>
<td>[[File:Charges_and_forces_pp.png|frameless]]
<td rowspan=2>Electric charges of the same type exert repulsive forces on each other
<tr>
<td width=349>[[File:Charges_and_forces_nn.png|frameless]]
</table>
<td>
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Currents_and_forces_opposite.png|frameless]]
<td>[[File:Currents_and_forces_same.png|frameless]]
<tr>
<td width=200>Electric currents in opposite directions repel each other
<td width=200>Electric currents in the same direction attract each other
</table>
</table>

In a particle beam, the collection of like-charged particles and their mutual repulsion is called <b>space charge</b>, and can have various effects ranging from making beams expand and trumpet out if not properly neutralized to even making particle beams stop, turn around, and come back the way they came! The currents created by all the particles, on the other hand, are moving in the same direction so they attract each other. This attraction can be called a <b>z-pinch</b>, or sometimes a <b>Bennett pinch</b> after one of the first guys to study it.

===Looking at the same thing two different ways===

Imagine that you are sitting calmly, watching a collection of positively charged particles (as you do). Because the particles all have the same electric charge, they repel each other. You observe the particles pushing away each other and flying apart.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Charges_at_rest.png|478 px|frameless]]
</table>

Ah, but what we didn't tell you was that these particles were just shot out of a particle accelerator at high speed. You just happen to be moving at the same velocity as all the particles, so to you they appear to be at rest. But your colleague running the accelerator sees the particles (and, presumably, you) flying past at high speed. She sees not just particles of the same charge pushing out against each other, but also all the currents of these charges. And because the currents are all in the same direction, there is an attractive force pulling the particles together.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Charges_in_motion.png|750 px|frameless]]
</table>

With the extra forces from the current squeezing the particles together, your colleague should see the particles expanding less fast than you do. Maybe she even sees them collapse in on themselves if the current-current forces are higher than the charge-charge forces. What gives? Who is right?

It turns out, you both are. Because, as it turns out, magnetic fields are just a consequence of relativity balancing things out when you look at moving things from a different coordinate system. You may be familiar with the idea that in the theory of relativity, moving things experience time at a slower rate than things at rest (as observed by someone who is "at rest" in that frame of reference). So your colleague sees the things happening in your frame of reference happening slower than you do. Maybe only just a tiny bit slower, if you are not moving very fast at all. Maybe much, much slower if you are nearly relativistic with respect to her. But some degree slower, in any event. And so she won't see the particles expanding as fast as you do. Each of the particles has the same electric charge for both of you, so the electric forces will be the same. So your colleague must observe some other force in addition to the electric forces that act to partially compensate for them. And these are the magnetic forces, which are present in her frame of reference but not yours. And now you can see that, for bare isolated charges, the magnetic self-pinching forces can never quite overcome the electric self-repulsion charges – although if the particle bunch is going very fast, they might come very close.

===Beam propagation in vacuum with self-forces===

To quantify the effects of the charge and current self-forces on the beam, you can calculate a number called the <b>perveance</b> <math>K</math>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
K = \frac{2 \,q \, I}{4 \, \pi \, \varepsilon_0 \, m \, (\beta \, \gamma \, c)^3}
</math></div>
where <math>q</math> is the charge per particle, <math>I</math> is the electric current in the beam, <math>1 / (4 \, \pi \, \varepsilon_0) = 8.9875517923 \times 10^{9}</math> N m²/C² is the Coulomb constant, <math>m</math> is the particle mass, <math>\beta</math> is the speed as a fraction of the speed of light, <math>\gamma = 1/\sqrt{1-\beta^2}</math> is the Lorentz factor, <math>c</math> is the speed of light, and <math>R</math> is the radius of the beam. If <math>K</math> is much larger than <math>\epsilon^2/R^2</math> the beam will be dominated by self-force expansion; for <math>K</math> much smaller than <math>\epsilon^2/R^2</math> it will be dominated by emittance (and if <math>K \approx \epsilon^2/R^2</math> you get a mix of both).

(Useful values: the charge of an electron is <math>q = -1.602176634 \times 10^{-19}</math> C and that of the proton is <math>q = +1.602176634 \times 10^{-19}</math> C. The electron mass <math>m = 9.1093837015 \times 10^{-31}</math> kg and for a proton <math>m = 1.67262192369 \times 10^{-27}</math> kg. Ions have a mass of <math>m = 1.66053906660 \times 10^{-27}</math> kg times their standard atomic weight. The speed of light is <math>c = 299792458</math> m/s. If you know the total Power <math>P</math> in watts delivered by your beam and you know the energy of each particle <math>V</math> in eV, then the beam current in amperes is <math>I = P/V</math>.)

<blockquote>
<i>Example</i>: Let's consider an electron beam with <math>P = 1</math> MW of power and <math>V = 10</math> GeV energy particles. It needs to have a current of <math>I = 0.0001</math> A to get this power with this particle energy. The beam will initially be 1 cm wide (<math>R = 0.01</math> m). At this particle energy, <math>\gamma = 19600</math>, and <math>\beta</math> is so close to one as to make no difference. For a reasonable normalized emittance of <math>\epsilon_n = 1\times 10^{-6}</math> m<math>\cdot</math>rad, we end up with a geometrical emittance of <math>\epsilon = 5.1 \times 10^{-11}</math> m<math>\cdot</math>rad.

Plugging these numbers in, we find that <math>K = 1.56 \times 10^{-21}</math>. The quantity <math>(\epsilon/R)^2 = 2.6 \times 10^{-17}</math>. So for this particle beam, emittance will dominate over self-forces and we only really need to worry about emittance. Only if you try to focus it down to about a tenth of a millimeter or less will <math>(\epsilon/R)^2</math> be comparable to <math>K</math> such that you would need to take self-forces into account.
</blockquote>

<blockquote>
<i>Example</i>: Now consider an electron beam with the same <math>P = 1</math> MW of power but particles with only <math>V = 10</math> MeV energy. This requires a current of <math>I = 0.1</math> A. we'll use the same 1 cm wide (<math>R = 0.01</math> m) beam width. At this particle energy, <math>\gamma = 20.57</math>, and <math>\beta = 0.999</math>. For the same normalized emittance of <math>\epsilon_n = 1\times 10^{-6}</math> m<math>\cdot</math>rad, we end up with a geometrical emittance of <math>\epsilon = 4.86 \times 10^{-8}</math> m<math>\cdot</math>rad.

Plugging these numbers in, we find that <math>K = 1.37 \times 10^{-9}</math>. The quantity <math>(\epsilon/R)^2 = 2.36 \times 10^{-11}</math>. For this particle beam, self-forces will have a much greater effect than emittance.
</blockquote>

To get actual numbers, take a look at the picture below. It shows the shape of a beam emitted from an accelerator. The beam reaches a minimum width of <math>R_m</math> at a position <math>z=0</math>. At any distance <math>z</math> along the beam from this minimum, the beam width will be <math>R(z)</math>.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=600>[[File:Self-charge_shaped_beam.png|600 px|frameless]]
</table>
It is useful to define <math>\chi</math> as the fractional amount by which the beam expands
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
\chi = \frac{R(z)}{R_m}.
</math></div>
So if the beam doubles in radius, then <math>\chi = 2</math>.

We also need to introduce a function <math>F(\chi)</math>, which doesn't have any simple expression (it is defined as an integral over the reciprocal of a logarithm), so it is easiest just to give it in tables and figures.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:F_of_chi_close.png|frameless]]
<td>[[File:F_of_chi_over_chi_close.png|frameless]]
<tr>
<td>[[File:F_of_chi_far.png|frameless]]
<td>[[File:F_of_chi_over_chi_far.png|frameless]]
</table>
To use these to get the beam size at any distance from the beam minimum, we can use<ref>Stanley Humphries, Jr., "Charged Particle Beams", Originally published in 1990 by John Wiley and Sons (QC786.H86 1990, ISBN 0-471-60014-8)</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
F(\chi) = \sqrt{2 K} \, \frac{z}{R_m}
</math></div>
and
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
\frac{F(\chi)}{\chi} = \sqrt{2 K} \, \frac{z}{R(z)}.
</math></div>

<blockquote>
<i>Example</i>: Let's use our 1 MW, 10 MeV electron accelerator from before, with its 1 cm aperture and <math>K = 1.37 \times 10^{-9}</math>. Suppose we are using this as an electron cannon, and shooting at a target 2 km away. We want to know how small of a spot we can direct on to our target. We use <math>R_m = 0.01</math> m (the beam radius at the aperture) and <math>z = 2,000</math> m for the distance to the target. This gives us <math>F(\chi) = 10.46</math>. Looking at the graph, <math>F(\chi)</math> is about 10 where <math>/chi</math> is about 12. So we know that at the target the beam spot at the target is about 12 times larger than at the aperture, or about 12 cm across.
</blockquote>

===Beam propagation in matter with self-forces===

The propagation of high intensity particle beams in matter soon turns into the propagation of high intensity particle beams in plasma, as the matter is ionized by the beam passage (if the beam intensity is not high enough to do this, see the next section). The electric charge of the beam particles will then attract particles in the plasma of the opposite charge and repel particles in the plasma of the same charge. This phenomenon is called <b>polarization</b>, and it acts to partially screen the electric charge of the beam. As a consequence, the electric self-repulsion is reduced. In many cases, however, there is no equivalent compensation for the magnetic self-attraction due to the beam's current. Unlike the case in vacuum, the attractive magnetic self-forces of the current can now overcome the repulsive self-forces of the charge, and the beam will self-pinch. This keeps the beam tightly focused as it propagates, potentially allowing the beam to go long distances and still maintain a tight focus when it gets there. The details of particle beam in plasma are very complex, well beyond the scope of this work. However, it is worth noting that experimentally particle beams have been made that can reach through hundreds of meters of air<ref name="Gsponer">[https://arxiv.org/abs/physics/0409157 Andre Gsponer, "The Pysics of high-intensity high-energy Particle Beam Propagation in open Air and outer-space Plasmas", arXiv:physics/0409157v3 11 Jan 2009]</ref>.

==Energy loss in matter==

Sometimes, a particle beam needs to go through stuff to get it where you want it to go. Stuff like air, for example. Or the tissue in the way of getting the beam to a tumor. Also, sometimes, you want your particle beam to interact with stuff. Like the material you are trying to machine with an ion beam, or the bacteria you are trying to kill on medical equipment. All of this falls under the general realm of the interaction of the beam particles with matter. This is an enormously complex subject, but we can cover many of the basics here.

The primary effect on the beam of any interactions with matter are loss of energy of the beam particles (<b>stopping</b>), random changes to the particle directions that make the beam spread out (<b>straggling</b>), and knocking particles clean out of the beam altogether (<b>scattering</b>).

===Ionization===

The particles coming out of your particle accelerator are charged. When a fast moving charged particle goes through matter, the electric field from the charge pushes and pulls on the various charged particles that make up that matter – mainly electrons and the atomic nuclei around which the electrons orbit. At speeds above approximately 1% of the speed of light, it is primarily light and nimble electrons being affected while the ponderous nuclei just sit there barely perturbed by the passage of the particle. When the passing particle yanks on an electron hard enough, it can pop it clean off of the atom it was attached to, leaving a positively charged ion behind and giving you a free electron that can go and glom on to a previously neutral atom to make a negative ion. This process, unsurprisingly, is called <b>ionization</b>. For most particle beams, ionization is the dominant interaction with matter; although when you get very fast or very slow particles other effects can become more important.

if your particle was not initially charged when it goes into matter, it soon will be. The electric charges of the electrons and nuclei in the matter will do the same thing to the particle that a charged particle would do to them. This will quickly strip off any electrons around a high speed atom, leaving an ion and bare electrons to go through the matter. Only particles that are fundamentally neutral (like neutrons or photons) will not be immediately ionized.

It takes about 1 to 10 eV to ionize an atom. This energy must come from the kinetic energy of the beam particle. Particles from an accelerator have millions to billions of eV. So it takes a lot of ionizations to meaningfully affect the particles. However, one thing about matter is that it has lots of atoms that can be ionized so you do get those lots of ionizations. Because each ionization is so small compared to the total energy this is often approximated as a continuous slowing down of the beam particles. So many individual ionizations tend to average out, leading to all the beam particles being slowed down uniformly.

When the particle is yanking on nearby electrons to tear them out of atoms, those electrons are yanking back. This jitters the beam particle slightly from side to side. Again, these sideways yanks tend to average out. The statistical variations from the average slowing down and side-to-size jerking are what lead to straggling. Protons and ions are heavy, and will be deflected little from their original paths - like bowling balls smashing through matchsticks. These particles tend to go in mostly straight lines. Electrons, on the other hand, have the same mass as the electrons they are knocking about. So electron tracks through matter tend to zig-zag around a lot more.

As the particle whizzes past, it leaves a trail of atomic destruction in its wake - a trail of ions where the atoms are literally torn apart (not the nuclei, just the atoms). These particle tracks can sometimes be seen in detectors like cloud chambers. The faster the particle is going, the less time it has to yank on nearby electrons and so the less ionization it leaves behind. As a result, fast particles are not slowed down much but as they start to slow down they continue to be slowed down faster and faster. A consequence of this is that the particle dumps most of its energy into the material at the end of its track, just before it comes to a stop. This sudden peak in energy deposition is called the <b>Bragg peak</b>.

The electrons that a particle knocks out can end up going pretty fast. So these secondary electrons can produce further ionization in what is called an <b>electron cascade</b>. if an inner core electron is knocked out of an atom, other electrons falling in to fill that empty spot in the atom can release x-rays (<b>x-ray fluorescence</b>) or cause shake-ups in the outer electrons that knock other electrons off the atom (<b>Auger effect</b>).

===Brehmsstrahlung===

A charged particle is surrounded by an electric field. If it is moving, it is also surrounded by a circulating magnetic field. These fields move along with the particle.

If the particle knocks into something, it will slow down and probably change direction. The information about this bump only goes through the electromagnetic field at the speed of light. So a lot of that field that was moving along with the particle might not get the memo in time and keep right on going. Stray electromagnetic fields like this get this self-inductance effect going on where they start making themselves over and over again, allowing them to propagate as freely moving waves of electromagnetic radiation. This EM radiation from a slowing down particle is given the delightful name of <b>bremsstrahlung</b>.

Particles from an accelerator can be moving so fast that they make bremsstrahlung in the x-ray or gamma ray part of the spectrum. If you get very high energy bremsstrahlung gamma rays, the gamma rays can interact with matter by boiling an electron and its antimatter counterpart, a positron, out of the vacuum (using the nearby matter as the necessary momentum sink for the process). These positrons and electrons can themselves be very high energy and can in turn create more bremsstrahlung, and the process can repeat until the particles are no longer high enough energy to continue making each other. This is called an <b>electron-gamma shower</b>.

The x-ray machines used by dentists and radiologists use both bremsstrahlung and x-ray fluorescence that you get when shining an electron beam onto tungsten to make their x-rays.

At very high speeds, bremsstrahlung can be more a important source of energy loss than ionization.

===Electron capture and screening===

When an ion has slowed down to about 1% of the speed of light or less, it is moving slower than the electrons zipping around their atoms in the matter the ion is going through. This allows the electrons to stick on to the ion, now becoming the electrons of the atom that is moving through the material rather than of the material itself. The ion begins to lose its charge, and the rate of ionization goes down. Soon, ionization becomes negligible compared to the beam atom bumping in to other atoms.

===Phonons and displacement===

When an atom or ion moving through a material bumps into another atom, it will cause that other atom to move. This can have one of two primary effects.

First, if the atoms whack into each other hard enough, the atom in the material can be knocked clean out of place. Like a pool ball being thwacked by the cue ball, the struck atom will go zinging through the material. Atoms getting knocked out of their place like this is called <b>displacement</b>. Displacement creates chemical changes to molecules as atoms are removed, and creates defects called <b>vacancies</b> in crystals (which, as you might imagine, is just a missing atom in the crystal lattice). When the atom comes to rest, it can chemically attack other molecules, again changing the chemistry, or create an extra vacancy called an <b>interstitial</b> jammed into the crystal lattice where it is not supposed to be. If the displaced atom is going fast enough, it can whack other atoms and displace them, leading to a <b>displacement cascade</b> like tenpins knocking each other down. As a result, at the end of its track, an ion can leave a region of disordered material and messed up chemistry where the atoms have all crashed into each other like a freeway pileup.

If an atom is displaced near the surface of a material, the displaced atom can shoot out of the surface and escape the material altogether. This is called <b>sputtering</b>.

Second, the beam atom might only deliver a glancing blow. Or the beam particle might be an electron, which is so light that it is usually incapable of displacing atoms. In this case, the struck atom gets kicked to the side a bit and runs into other atoms; maybe it is also pulled back by the chemical bonds of other atoms that it is attached to. This transfers the motion of the atom to other atoms around it in the form of pressure waves (and, in a solid, also shear waves). This adds heat to the material without causing direct damage – although as you might imagine, enough heat can also cause localized melting, evaporation, or decomposition. The individual quanta of vibration – fundamental particles of sound, if you want to think of them like that – are called phonons in analogy with the photons that are the fundamental particles of light. Much like ionization, a particle that excites pressure waves slows down gradually and continuously, unlike the sudden changes in speed and direction you get from knock-out displacement. In a very real sense, these pressure waves are the sonic boom of the particle passing through the material faster than the speed of sound in that material. Sadly, however, they are far to weak for us to actually hear.

===Nuclear collisions===

So far, we have only discussed the things that happen when beam particles interact with matter particles by electromagnetic forces. Even atoms knocking into each other to produce phonons or displacement happens through the mutual repulsion of the electrons around the atoms (although this is dressed by other weird effects like how two electrons can't be in the same place at the same time, or how the fields of one electron can affect other nearby electrons that in turn affect the original electrons, leading to odd correlation effects and charge density waves and other fun stuff that keeps condensed matter physicists and quantum chemists busy).

But electromagnetism isn't the only thing you can get going on. Below the level of structure of the atom, you have the atomic nucleus. And a direct hit on the nucleus can end up giving very different effects.

The simplest thing that can happen is an <b>elastic collision</b>, where no energy goes in to any internal nuclear state or into breaking up the nucleus. An elastic collision just ends up giving you displacement (or possibly phonons, but if you hit the nucleus you usually hit it hard enough to send it flying).

But beyond that you can cause any number of nuclear reactions. You can leave the nucleus in an excited nuclear state, from which it usually relaxes back down to its original state by emitting gamma rays. You can knock out nuclear particles, such as neutrons, protons, deuterons (a proton stuck to a neutron), tritons (a proton stuck to two neutrons), helions (a neutron stuck to two protons), alpha particles (two protons stuck to two neutrons), or potentially even larger nuclear fragments. If you hit a heavy nucleus like uranium, you can cause it to split apart by fission. And if you hit the nucleus really hard you might just shatter it into many tiny fragments of those protons, neutrons, deuterons, tritons, helions, alpha particles, and heavier nuclear fragments mentioned earlier. To get any of these <b>inelastic collisions</b>, you need to hit the nucleus hard enough. If your beam particle doesn't have enough energy to excite the nucleus or knock particles, these things simply can't happen. As it gets enough energy to excite a particular interaction channel (as they are called), that process becomes possible and as the energy increases the process becomes more and more likely.

A nucleus is only about a femtometer across (<math>1 \times 10^{-15}</math> m). Which makes it only about <math>1 \times 10^{-30}</math> m² in cross sectional area. In the meantime, chemical bonds between atoms are usually about 1/10th of a nanometer (<math>1 \times 10^{-10}</math> m) long. In condensed matter (liquids and solids), this means that you have somewhere about <math>1 \times 10^{30}</math> atoms per cubic meter. A meter thickness of atoms thus presents just about about enough atoms that their nuclei can cover the projected area – in other words, you can expect a particle to go through (very roughly) about a meter of condensed matter before it whacks into a nucleus.

This assumes that a particle that goes through a nucleus interacts with it. This is usually a good assumption for particles that interact by the strong nuclear force, such as protons, neutrons, mesons, and the atomic nuclei of ions. But many particles cannot interact by the strong nuclear force, such as electrons or muons. These are much less likely to interact with an atomic nucleus even if they pass through it, so they will end up going considerably farther between hitting nuclei than the above assumption. But nuclei can interact electromagnetically (they are made up of charged protons, after all, and at a smaller level of charged quarks), so electrons and muons can produce some nuclear interactions; just far less than actual nuclear particles like protons and ions.

A nuclear strike is a discrete event that will remove the impacting particle from the beam and will (except for elastic collisions) probably produce a spray of radiation.

===High energy physics===

As you get to higher and higher energies, you unlock all sorts of new and weird and interesting processes that can happen. Boiling exotic particles out of the vacuum, unifying forces that had been divorced since the big bang, and producing bizarre unstable resonances in the quantum fields. This is what physics build accelerators to measure. From the point of view of getting a beam to its target, or what happens to the target, however, the main effect is mostly that the interacting particle is removed from the beam and a bunch of additional radiation is produced.

===Beam evacuation===

As we have seen, it can be a challenge to get a particle beam through matter, as it loses energy to the matter via various processes. One potential method around this is to use the beam itself to heat up a channel in the air for as far as it can go. This hot gas or plasma will be at higher pressure than its surroundings, and will expand to lower density. A second pulse can then be sent down this channel through the lower density air. The second pulse will experience much less energy loss and, in addition, will be guided by any ionization channel left by the first pulse<ref name="Gsponer"></ref>. When the second pulse reaches the end of the evacuation tunnel, it can then pass into fresh air and extend the evacuation tunnel, allowing a longer distance for the passage of subsequent pulses. This process can be repeated to obtain a long enough channel for the beam to reach its target.

===Rough calculation methods===

Although the detailed physics of particle beam interactions with matter can be complicated, if you can get it to work you can roughly expect the beam to go through about the same areal density of stuff no matter what the stuff is made of. A beam that can punch a hole through a meter of concrete, with a density of 2.4 g/cm³, will be able to punch through an areal density of 240 g/cm². So it could also be expected to blast through about 2.4 meters of water or high density polyethylene or living tissue (density of 1 g/cm³. Note that 2.4 meters of tissue is roughly 10 people standing next to each other in a line), 30 cm of steel (density 7.8 g/cm³) regardless of the mechanical strength of the steel, or 1.8 km of air (density 0.00129 g/cm³). If the beam has to first go through a kilometer of air to get to its target, the areal density of the air it passes through will be subtracted from the areal density of what it can penetrate at that point.

= Operational Considerations =
== Logistics ==
The logistics of particle beam can be a mixed bag. By itself, any particle accelerator is a complicated piece of machinery full of delicate parts that have to deliver high performance. This may require cryogenic temperatures with associated challenges on the materials and mountings used (insulation, and distortions under temperature changes). When working, particle accelerators invariably generate some radiation across the whole assembly, which irradiates and activates the structure. Some parts may be exposed to significantly higher radiation flux due to their working principles. Often the particle beam is a dynamic system that must be managed with active, closed control loops, involving sensors, modeling and control software, actuators, and the computing and communication hardware that brings it all together. This poses engineering challenges across all disciplines.

A practical particle beam is inherently a counterintuitively sturdy device. If it was too delicate, it would break too much to be useful, though of course this is a sliding scale. Still, while sturdy, this doesn’t eliminate complexity, it manages complexity. The result is a system that is the product of a high-tech civilization. A superior machine shop and well-trained engineers are a must to maintain a particle beam system. Some parts of a particle accelerator may be mundane - others may be custom-machined parts that require their own dedicated industries to produce. That said, the manifold utility of particle accelerators is on your side here. Since there are so many practical applications, it is economic to produce parts for industrial demand. (that said, these may not be the kind of components you want in specific high-performance applications.)

The main thing a particle accelerator requires the most is electricity. Any mature power production facility can supply the power.

However, all particle beams require at the least a source of neutralizing mass. The projection of a charged beam will leave the projecting object oppositely charged. In an atmosphere the natural plasma formed by the beam will ground everything out, but in space the diffuse vacuum will not do the job.

Furthermore particle beams using ions also require a feedstock for their ions to produce from, and in an appropriate format. This can be a gas, a fluid or an easily ionized solid. It can be the feedstock species in its pure form or in an easily split molecule, though every complexity adds to the amount of preparation work that has to be done before the feedstock enters the particle beam.

Don’t worry about the mass of your source or neutralizing ions though. Large particle accelerators, like the LHC in Switzerland, only consume tiny gas bottles worth of feed gas every couple years. The mass of a single 5.56 NATO bullet worth of hydrogen could feed a 10 GW, 10 GeV ion beam for nearly 4 days of firing! You will not be running out of beam feedstock easily.

== Cooling ==
Cooling a particle accelerator can be a serious challenge. Various bits of a particle accelerator love to be supercooled for efficiency or basic working principle - any bit of superconductor used in the accelerator, or magnets, or EM generators, will require cooling to its generally supercool operating temperature.

Even if Standard Temperature and Pressure Superconductors (273 Kelvin, 100 Kilopascal) are available, present superconductors have demonstrated superior performance by being supercooled. That means to achieve even stronger fields, even STP superconductors may be cooled to cryogenic conditions.

Cooling cryogenic systems sucks. Thermodynamics shows itself from its worst side. You can forget running cryogenic radiators - the Stefan-Boltzman Law penalizes the direct radiative cooling of anything cryogenic severely as much as it empowers very high-temperature radiators. This only leaves stages of heat pumps to provide cryogenic fluids that can in turn, cool cryogenic systems. Heat pumps draw power, and for such a giant temperature gradient as we can have here, they will draw a lot of power.

However, we have other options. Cryogenic systems also tend to be highly efficient, so while they do heat up, they do so very slowly and there is little overall energy absorbed. This provides the option of running off of bunkered cryogenic fluid in dewars for the duration of operating the equipment and not running any heat pumps all the time.

Other bits of a particle accelerator system aren’t as required, but really nothing is a fan of high temperatures. Going higher than room temperature can be treated as an oddity. High-performance lasers may not require cryogenic conditions, but they certainly like it cool. Radiofrequency producers of any kind generally fare worse with rising temperatures. That means cooling overall remains a challenge - 300 Kelvin is still not that great of a hot loop temperature for radiators.

== Safety <ref>Thomas Otto, "Safety for particle accelerators", Springer 2021, Available online at: https://library.oapen.org/handle/20.500.12657/46120</ref>==
=== Ionizing Radiation ===
There are surprisingly many sources of ionizing radiation in a particle accelerator, besides the obvious threat of the output beam itself. Not all particles are accelerated correctly or in the correct direction. When such particles interact with the structure of the accelerator, they release a shower of secondary ionizing radiation. In electron accelerators and sources, stray electrons may be accelerated and release x-rays from bremsstrahlung as they collide with surrounding matter. The resulting doses can be significant and dangerous. Also, such collisions leave behind residual activation, making the accelerator a radiation hazard for some time even after it has been shut down. Generally, people should not work near the particle accelerator while it is functioning, and radiation barriers isolate the core accelerator from adjacent facilities. (In real-life facilities, you will often see accelerators inside larger buildings being cocooned inside doped concrete plates to shield the rest of the room.)

=== Radiofrequency ===
RF energy is used in many kinds of particle accelerators. This energy may leak from the structures it is generated, contained and used in due to damages or incorrect assembly. As microwave radiation and above is used, the radiation can be readily absorbed by bodies, hydrogen-rich materials or other objects. The end result of this can be rapid heating and thermal effects. Especially in accelerators designed to throughput a lot of energy (say a GW or more), this may become a serious issue if a waveguide is damaged. It’s not recommendable to stand in front of a multi-MW radar either!

=== High-powered electrical and electronic systems ===
The operation of the magnetic systems and accelerators often involves high-powered electrical and electronic systems. High voltages and/or amperes are involved. This poses risks like shock hazards, electrical arcs, and electrical fires. Some electrical systems such as capacitors or inductors may have combustive or explosive failure modes.

=== Flammables ===
Parts of an accelerator or its infrastructure may be manufactured from flammable substances such as hydrocarbons. In the case of an ignition source such as a high-powered beam cutting into a physical substance or an electrical fault, these may catch fire.

=== Lasers ===
Some accelerator systems use high-powered lasers. Laser plasma accelerators use very high-powered pulse lasers to provide pumping power. The safety concerns of lasers for blinding, physical damage, and possible exposure to ionizing radiation as a secondary effect of the beam impact, apply. Some high-powered lasers can also drive electrons which generate strong EMPs, which may damage or destroy other systems in turn. Parts of a laser assembly or laser beam guide may shatter if exposed to overloading conditions.

=== Magnetic systems ===
Parts of a particle accelerator may use strong magnets, up to superconductive magnets with multi-Tesla fields. While the magnets are integrated into chokes that shape the magnetic fields, fringe fields invariably extend beyond the core magnet installation.

Multi-Tesla transient electromagnetic field can affect the human nervous system temporarily, inducing nausea, vertigo, metallic taste and optical phenomena, if the person moves in relation to the magnetic field. Otherwise, the risk of exposure is low. That said, electronic systems that are not shielded properly, such as say, implants, could be adversely affected by the magnetic fields.

The biggest physical risk are reactions of other magnetic objects with the magnets. You can look up cases where metallic objects interacted with the multi-Tesla fields of MRI scanners. Such parts may fly across the room towards the magnet at increasing speeds, and damage anything or injure anyone in their path, possibly ending with the object or person squashed (or impaled!) to the magnet.

Also, large superconductive magnets store not insignificant amounts of energy. If the superconductive property breaks down, the magnet quenches and becomes a resistor. The resistance site heats up as current tries to flow through it, expanding the quench and possibly being heated until it sublimates into gas. This could possibly happen explosively. In such a case, up to many GJ of energy might be unleashed in a conflagration or explosion (subsonic or supersonic), damaging surrounding systems and ejecting shrapnell.

=== Cryogenic cooling systems ===
Particle accelerators often involve cryogenically cooled systems. This means cryogenic fluids like nitrogen and helium are present in the accelerator infrastructure. Such gasses have both a very low temperature and a high ratio of gaseous volume at high temperatures to their liquid volume.

Exposure to the leaked fluid can lead to cryogenic cold damage, injury, and death. Furthermore such gasses are efficient at displacing ambient breathing gasses such as oxygen, so they pose an invisible, and rapidly spreading, suffocation hazard.

Also, if insulation is damaged, ambient heat may leak into the cryogenic circuits, and cryogenic fluid may begin evaporating. This can lead to dramatic rises in internal pressure. Relief valves should trigger to safely direct such pressure spikes away from important places and keep the system from failing catastrophically, but otherwise, pipes and cryogenic storage devices may fail, up to explosively.

=== Beam excursions ===
If the magnets responsible for manipulating the beam break down, the beam may excurse from its designed paths. The beam may become defocused, or maintain its focus. The beam begins cutting into the accelerator structure, evaporating material and depositing energy. This will definitely foul the ultra-high vacuum inside the accelerator, leading to secondary radiation production as the beam collides with gas inside the accelerator. Some sensitive facilities requiring ultra-high vacuum such as some RF cavities may “crash” and become ruined. Further systems may become damaged or destroyed from either physical stress or the radiation pulse. If the beam bunches contain a lot of energy and maybe are heavy particles, the deposition may lead to an energetic local explosion. Alternatively the beam may escape and cut into structure beyond the particle accelerator itself, damaging and destroying equipment and seriously hurting or killing personnel.

In the worst cases, a beam excursion on a very powerful beam may lead to an explosion that destroys the accelerator utterly. Even in a less bad case, the radiation pulse might damage or destroy critical equipment around the accelerator and take the entire facility offline.

=== Beam dumping ===
If any other systems fail, the accelerator probably has to be shut down. This poses a problem - what do you do with the beam energy that is still in the accelerator system?

Sometimes, the beam is designed to leave the accelerator immediately, and shutting down the beam is easy. But in the case of cyclotrons and storage rings, the beam may circulate for a while. What do you do then? You have to dump the beam from the accelerator. Special beam stops are used on research and industrial accelerators, large blocks of dense material able to absorb the beam. During this process the blocks both heat up and become radioactive. For particle beams storing very high energies, this may become an increasing challenge. How do you safely dumb a beam containing many GJ or more? In some cases you may have no other choice but to dump the beam out into the environment. Where and when can you do this safely without threatening third parties?

= Application areas =
Particle accelerators have many, many, many areas of application.

The public examples of particle accelerators are the big research accelerators used to create conditions with which one can interrogate fundamental physics. These accelerators are after specific particles under specific, as finely controlled as possible, conditions.

However, particle accelerators are also used in imaging. The electron microscope is a classical particle accelerator-based imaging system, using the fact that electron are very tiny particles to bypass classical imaging limitations of photons. Thanks to them, the biological and material science world can be interrogated in great detail, down to single atoms, and advances in the sensors and processing software keep pushing the boundaries.

On a bigger scale, particle accelerators can be used to produce high-energy x-rays with very small wavelengths. The x-rays are projected at prepared crystaline targets. The x-rays interact with the electrons in the crystals, giving information about bond structures. Such crystallography has great importance in biochemistry, but also material science.

Similiarly electrons, ions and neutrons can be used to probe at the structure of materials of interest. Their different behaviors allow for surface or in-depth scanning with particular properties.

In manufacturing, particle beams can be used both to shape materials and to image them for quality control. Particle beams can deliver thermal energy to cut apart materials or weld them. In additive manufacture, particle beams can provide fine precision melting or even meld together volumetric composites with penetrating beams. At fine scales, precision beams can break down gasous feedswtock to deposit materials in micrometer-sized structures. Directed ion guns implant traces of other elements into surfaces for manufacture.

Industries have a certain demand for specific isotopes. Accelerator-driven manufacture of isotopes covers these needs.

And in medicine, particle accelerators of ever-greater sophistication find increasing use in cancer treatment and beyond, destroying life-threatening tissues with ever-reduced doses to surrounding healthy tissue.

For military purposes, particle beams have been probed for a long time, especially for use as ground- or space-based weapons to shoot down incoming intercontinental balistic missiles. Particle beams can be made to propagate well enough both in air and space with various methods. They can deliver ionizing radiation that damages electronics, cause fission and breakdown in nuclear materials, or deliver sufficient thermal energy in a tight spot to destroy hard materials. With modern and future technologies, particle beams can offer equal and superior performance characteristics for effective range as lasers, and with various target interactions paradigmas of interest. They can also play a role as part of sensors, producing high-energy radiation from their target interactions that sensors can pick up and analyze.

= Further material =
* [[Particle Beamed Propulsion]]
* [[Plasma Guns]]
* [[Particle Beam Weapons]]

== External reading ==
The US Particle Accelerator School posts material of its courses publically at times: https://uspas.fnal.gov/materials/materials-table.shtml

= Sources =

== Authors ==
Sevoris Doe, Gerrit Bruhaug, Luke Campbell.

== References ==

[[Category:Beams]]

Particle Accelerators

2022-08-18T12:53:30Z

Sevoris: /* Application areas */

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Particle accelerators are an incredibly useful piece of technology and a diverse set of artifacts produced for many purposes. While particle accelerators are often invoked as weapons of war in science fiction, and have captured the public attention as giant research systems the size of cities, they have an incredible widespread range of applications. There are also many, many ways to build a particle accelerator, depending on what effects you need to achieve and what circumstances you work under.

This article aims to be a general primer on particle accelerators, their physics and a selection of technologies and construction styles. In detail, all particle accelerators are high-tech engineering customized to their circumstances in many, many aspects.

Further pages covering application fields are in the work.

= Accelerator physics =
The simple explanation is that we use electric fields to accelerate charged particles. But from this basic principle of leveraging the electromagnetic forces springs an entire set of accelerator technologies with different performances, trade-offs and characteristics. The most basic particle accelerators are just a big high-voltage source and can still be found in old CRT TV’s and dentist x-ray machines around the world. Things rapidly get more complex from there, but first we need to lay out some terms so we can all have a common conversation.

The first thing to keep in mind with particle accelerators is that we are talking about charged beams here, even if we neutralize them later. So no neutrons, photons, or other innately neutral particles. Your choices range from the simple electron up to anti-uranium and more, but they all must be charged. The particle of choice determines just about everything else about the accelerator and any potential uses. Really heavy particles (like large ions) don’t accelerate easily and take a lot of energy to get relativistic enough for time dilation tricks to take into place. Really light particles (like electrons) accelerate super easily, but can have other radiation problems to worry about.

The second most important thing to know about a given accelerator is the particle energy. This is a measurement of the kinetic energy of any individual electron, ion, whatever in the beam and is measured in a unit called the electron-volt or eV for short. The lower bound for any worthwhile particle accelerator is in the MeV (or million eV) range and we typically want a lot more! Inside of the accelerator we tend to talk about the electric field doing the accelerating (measured in Volts/meter). For singly charged particles (like electrons) this gets really to figure out the energy because of how the eV is defined. We just take the electric field (also called the accelerating gradient) and multiply by the length of the machine! Boom, now you have particle energy! For more complex particles you have to do some scaling based on mass and charge state.

Also a quick trick to help you with deciding if something is relativistic or not is to divide the particle energy <math>E</math> by the mass energy <math>M</math> of the accelerated particle (0.511 MeV for electrons/positrons and about 1 GeV for a proton) and add one. The resulting number is called the Lorentz gamma value and is used all over in accelerator engineering.
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>\gamma = 1+\frac{E}{M}</math></div>
Once you get a value much larger than 1, you have relativistic particles. Once that number is in the hundreds you get an ultra-relativistic beam. Figure 1 helps show the relationship between Lorentz gamma and speed of the particle.

[[File:LorentzGvsSped.svg|900px|thumb|center|Lorentz gamma versus fraction of the speed of light]]

The third most important feature of a particle accelerator is the emittance which is just a fancy term for divergence of the beam and can be thought of similar to the wavelength of lasers. No beam is made perfect and they will naturally want to expand outwards. The smaller the emittance, the longer the beam will stay in a tight spot. Emittance can be altered with a wide variety of beam engineering techniques that take entire books to discuss, but one other cheating thing you can do is drive the beam energy higher. This naturally lowers the emittance by a factor of Lorentz gamma due to the cheating effect of special relativity! Thank god for Einstein!

== Acceleration forces ==
Every accelerator tries to build an electric gradient appropriate for the charge of the particles it is accelerating. The greater the electric gradient, the bigger the amount of work that can be done, the more energy is imparted on a particle as it travels through the gradient, and thus the higher the acceleration. The simplest way to build up a gradient is simply two charged electrodes with a strong potential between them. Many other accelerator methods use electromagnetic fields, in the form of waves confined in electrically conductive structures. In this case we exploit that electromagnetic fields have both an electric and a magnetic component. The magnetic component cannot be used to accelerate particles (only bend their trajectory - which can be exploited for effects like deflection or focusing) but as the electromagnetic waves oscillate, so does an electric field. This naturally creates a charge difference across half of the phase of an electromagnetic wave - this electrical gradient can accelerate particles. Both of these technologies encounter problems with the physical materials asked to contain the electric or electromagnetic energy. Eventually the fields get so intense that solid materials get broken down and an arc is formed. As such, various advanced concepts use clouds of charged particles and plasmas, which are not subject to the material limitations of solid matter.

The strong and weak nuclear force as well as gravity (which are all of the forces we know of at the time of writing) are not suitable for accelerating particles. This is related to the power and range of these forces. The strong and weak nuclear force have extremely short ranges - even if we had a means to artificially shape them like with electromagnets, their range would be too short. Gravity has the range but is incredibly weak. To generate a measly 1 G of acceleration, 9.81 m/s², requires the mass-energy of the entire earth, some 5.97e24 kg of mass, to do the job! The acceleration involved in particle accelerators is much, much, much larger (on the order of 10^13 times larger or more).

{{MascotSays|Interestingly, one source of very high-energy particles are giant gravity wells. Black holes generate conditions around them that can eject particles with enormous energies. The jets of particles that shoot from the north and south poles of many such stellar entities are particle accelerators.}}

If you want to use a force other than electromagnetism to accelerate particles, it’ll have to be a fictional one. The forcefields of various sci-fi properties could be considered, since they can often apply forces to particles. If you propagate a wave of such forcefields down a tube, you could have something quite a lot like many particle accelerator systems!

== The particle bunch ==
Essentially all high energy particle beams are not constant streams of particles, but a time sequence of discrete groups of particles, commonly referred to as “bunches”. These bunches can be described both in their distribution in lateral space to the traveling direction (x and y axis) and longitude (z axis) along the traveling direction, anchored on an origin point (0,0,0) that moves in time.

[[File:ParticleBunch.png|center|900px]]

Bunches can have various profiles. Generally they are somewhat distributed in the lateral direction just because the charged particles have slightly different starting positions and lateral velocities (also called transverse emittance). In longitudinal space, differing acceleration experienced by individual particles as well as dynamics within the bunch scatter them out, with “noses” and “tails” that contain fewer particles. The end results is that most beam profiles follow a gaussian distribution, though there are exceptions.

This bunching behavior allows us to analyze other properties of particle beams very similar to lasers. There is a “pulse frequency” given in Hertz. Each bunch contains a certain energy, which can be large enough that we give it in Joules and not electronvolts. There is an average output energy over a given timespan, given in Watts. We have a beam peak power, given by how quickly a bunch arrives at a target and deposits its energy, which we give in Watts. Related to this we also have a related peak intensity depending on how much area/volume the beam is deposited into, with units of either W/m² or W/m³. If we have charged particles being accelerated, we get a beam voltage, the same way as in a wire. Beams can also have a temperature, the same way as any other group of particles with random motion from their energy, given in Kelvin or eV.

The systems that generate particle beams of course have efficiencies, with the wall plug efficiency (power inserted into the entire system versus power deposited in the particle beam) being the most interesting for first-order engineering. They also have an operating temperature.

== Particles for acceleration ==
With the constraint that our particles must have an electric charge in order to be accelerated with anything but fantastic technology, we can discuss quite a large family of particles, from the conventional to the exotic. They differ in the applications they can be interesting for. Some can be similar, but no two particles behave entirely alike.

=== Electrons ===
The ubiquitous electron is the most common particle to be accelerated and the first one ever pushed up near the speed of light by humans. Due to its excellent charge to mass ratio (the best possible due to the extremely low mass and charge of 1e) electrons are very easy to get up to relativistic speeds. Electron accelerators for radiography and sterilization are often portable by a single person and not much larger than a laptop in size (although they can be much bigger for power output reasons). The ease in which electrons reach relativistic speeds means that electron RF accelerators can be designed with no speed changes assumed (since the electrons are already moving at effectively light speed) and thus tend to be highly efficient and have the highest acceleration gradients achievable. For similar reasons electrons are also able to be accelerated in plasma based accelerators, allowing for even higher acceleration gradients and smaller machines for a given energy. The major downside of electrons is that due to their large charge to mass ratio they will radiate energy like crazy when bent by a magnetic field. Thus ring shaped accelerators are of limited use for electrons since you have to make the ring extremely large to not just lose all of your energy as beams of x-rays coming out of the machine! This very trick is used to make high quality x-ray sources for industrial and research applications.

When electrons hit matter they tend to release their energy as photons in a process called bremsstrahlung, or braking radiation. These extremely high energy photons (roughly ~⅓ the peak energy of the electron on average) then proceed to make electron/positron pairs that then make photons that then make pairs that then make photons… This whole thing is a massive cascade of radiation that ionizes, heats and can even damage on a nuclear level! At high enough energies you will generate every particle known and might even make some new ones!

=== Proton ===
Protons are the next most common particle to be accelerated and are currently used by the absolute largest particle accelerator ever made (the Large Hadron Collider or LHC) as the primary particle of choice. Since they have ~2000X lower charge to mass ratio than electrons, protons are about ~2000X harder to get up to speed. Unlike electrons they are not promptly at light speed so we have to send protons through a series of various accelerators depending on the energy of choice. For example the protons in the LHC the protons go through 5 different particle accelerators plus the extremely low energy stage that first makes the beam of ions! This difficulty in reaching highly relativistic speeds has two other bad side effects. Firstly, if we want to transmit our protons through empty space we will probably need to neutralize them with a beam of electrons.

{{Note|''When shooting extremely high current (kA), relatively low energy (not highly relativistic or in some cases relativistic at all) beams in the atmosphere the higher mass and positive charge of protons gives them many beam propagation advantages over electrons if the beams can be made. In this case no neutralization is required as the atmosphere itself becomes a plasma and neutralizes the beam.''}}
Luckily this is not very hard and will add almost nothing to your comparatively large proton accelerator, but may increase your beam spread a little bit via some neutralization physics. This does mean your beam is technically plasma, which is cool! Second, we effectively cannot use those neat plasma based accelerators for protons since we would need nearly TeV class proton beams to inject into the plasma accelerator and by the time you reach those energies you have basically already done all of the work. Maybe in the future there will be ways around this, but as of right now we are limited! There is one major benefit to this poor charge to mass ratio though, and that is the lack of synchrotron radiation! Ring shaped accelerators are ideal for protons, and basically all ions.

When protons hit matter they can interact with all of the four fundamental forces, although only two of them (Strong Nuclear and Electromagnetism) tend to play big roles. Protons barrel on in causing huge ionization cascades and can also release bremsstrahlung photons at high enough energies. They are also ideal at knocking neutrons, protons, deuterons and more out of the target nuclei which just adds to the fun! Protons also come with a neat trick known as the Bragg peak, where they deposit a large portion of their energy right at the end of their trajectory through a target material. This is a very useful trick for medicine and means you can “tune” the beam to almost ignore large portions of material and primarily heat, ionize, and destroy the area of interest. You can imagine how useful this is for cancer therapy or more nefarious things!

=== Light ions ===
Light ions are a rather fuzzy term but here we will use the definition of any ion heavier than a proton but lighter cobalt. The fuzzy reasoning here is that you get on the other end of the nuclear binding curve (i.e. can’t get net energy from fusion) at iron, so cobalt will then no longer be “light” and has to be made in a different process. Feel free to pick your own cut off! Historically beams up to neon have been experimented with for nuclear fission, nuclear fusion and medical therapy applications. Light ions have even worse charge to mass ratios than protons (Z/A times worse) and typically you will find light ions have about ½ the charge to mass ratio of a proton on average, although the isotope 3He is a notable exception. This means they are even slower to accelerate, although often the exact same machines as protons can be used with minor tweaks. Light ions have similar requirements for neutralization as protons, although their heavier mass can help fight beam bloom for a similar beam speed (not energy since the increased mass means the particles are higher energy for a given speed). The decreased charge to mass ratio also means that synchrotron radiation is even lower so beams can be driven to higher speeds before radiating. Light ions have similar target interaction physics as protons, although the ion itself will break up at very low speeds and add to the radiation environment.

=== Heavy ions ===
Heavy ions are being defined here as cobalt up to whatever unstable element you can jam into your accelerator! Typically heavy ion beams are made of elements like lead (used in some runs on the LHC), gold (used in the Relativistic Heavy Ion Collider) and even californium (used in some runs on the Argonne Tandem Linac Accelerator System). These beams tend to be for pure nuclear physics research, but there has been a long standing interest in heavy ion beams for inertial confinement fusion research and as potential methods of propelling spacecraft. These extremely heavy ions will be even harder to accelerate to ultra high speeds than light ions and much slower for a given energy, but they do provide a lot of momentum to make up for that! They have the same neutralization requirements and same potential upsides for beam bloom as light ions, but will most likely be orders of magnitude slower reaching our targets than the lighter beams. Heavy ions can also be used in the same machines as protons with small modifications and generate almost no synchrotron radiation as a result of their much smaller charge to mass ratio. They also have similar interaction physics to protons and light ions, but will stop in much thinner layers of materials. While electron, proton and light ion beams tend to both mechanically destroy and irradiate targets through-out, heavy ion beams will act much more like lasers and burn in from the outside of the target on in. There will still be a spray of ionizing radiation, but far less of one than an equivalent energy electron, proton, or light ion beam. The much higher momentum per particle, lower radiation concerns and ease of deflection makes heavy ion beams very interesting for beamed propulsion purposes.

=== Macrons ===
Macrons are typically defined as macroscopic particles, such as extremely large molecules or even grains of sand, that are charged up and accelerated via similar technologies as traditional particle accelerators. There are many difficulties with getting macrons up to appreciable speeds due to the extremely low charge to mass ratio, engineering difficulties with getting consistent and good charging, phase matching with accelerating cavities and lack of effective focusing optics. There are so called “dust accelerators” in existence that utilize the first stage of ion accelerators to generate their “beams” and proposals have been made to create better versions using updated technology. Macrons will interact just like a very small and very fast bullet unless they are designed to carry a nuclear payload to boost this output. Antimatter is an assured option for getting energy gain from a macron impacting a target, although there are potentially options for fusion, fission and fission/fusion given the correct sizes (mm scale or bigger), clever target design and the right impact speeds (most likely >>100 km/sec). The various energy gains versus speeds can be seen here:

[[File:IvKmacron.svg|900px|thumb|center]]

It can be seen that antimatter proves to be worth it up near the speed of light, but both fission and fusion stop providing much gain well below the relativistic regime. This also helps explain why heavy ion uranium beams are not going to provide some incredible advantage over any other ion.

=== Antimatter ===

Antimatter is the evil cousin of normal matter and whenever the two meet a release of energy equivalent to their combined mass energy follows. Now this does sound like an amazing option for beams but there are two downsides. First, antimatter is energetically expensive to make! Positrons (antimatter electrons) are the “cheapest” at 1.02 MeV of input energy at a theoretical best, but they only give you that much energy back when hitting a target. Since a typical positron beam can be expected to be equivalent to an electron beam, there is little gain in adding 1.02 MeV per particle to a multiple GeV (or even TeV) beam. You can see this explicitly:

[[File:IvKhBarPostron.svg|900px|thumb|center]]

For anti-protons the math becomes more favorable, but creating these particles is much more difficult and energy consuming. At best anti-protons require ~2 GeV of input energy to create, but in practice >400 GeV per anti-proton has been needed. Luckily significant energy gain is had up to several GeV of beam energy, so if anti-protons can be made at a central facility of some kind and easily stored (neither being easy things) then there may be some advantage to such a beam. Heavier ions are significantly harder to make since fusion with antimatter must be performed to generate those ions. Barring a naturally occurring source being found, antimatter heavier than a proton is most likely not worth using.

{{Note|'' Positrons can potentially fill an interesting niche either as a co-propagating neutralizing beam for electrons or perhaps as a higher performance atmospheric beam than electrons due to the positive charge. In this case the annihilation properties are no what is of interest, rather the positive charge and relative ease of acceleration compared to ions. Since positrons share all properties with electrons except for charge, they can be accelerated just as easily and even co-accelerated with electrons in the same machine.}}

=== Muons ===
Muons are an interesting cousin of electrons and positrons. They are unstable and heavy cousins of electrons, massing about 200X more than electrons and living about 2 microseconds in the lab frame before decaying into an electron (or positron) and a neutrino. If we accelerate them up to relativistic speeds (about 10X easier than protons and the muons are typically born relativistically thanks to how we make them) then the muons can be made to “live” far longer. Muons have long been of interest to accelerator physicists due to their position as an almost ideal intermediary between electrons and protons. The requirement to make them on site has always been an impediment though and muons are currently only used for certain niche experiments. During the planning stages for a potential muon collider it was also noted that a muon collider would also have a serious issue with neutrino radiation! This is due to how neutrinos are more likely to interact with matter at high energies and a muon collider would be generating a LOT of really high energy neutrinos in a ring around the machine. This has been proposed as a sort of super weapon even!

Muons should interact with matter in a similar fashion to electron beams and the beams will probably be treated in a similar manner. They can propagate through the atmosphere quite well in theory and should outperform electron and proton beams for a given beam energy, although at quite a cost… Muons are typically generated with proton beams impacting targets and will create an equal number of positive and negative muons at high energies that must be carefully captured. This would allow for the easy creation of a neutral muon beam though, which may make for a great giant doom beam! Do note that negative muons also can help catalyze nuclear fusion so a very high power beam hitting a fusion fuel tank could produce some rather… Interesting results!

=== Other exotic options ===
There are a wide variety of odd particles that could also be chosen for use in beams given a specific need for them, although generation and utilization of these particles will be quite difficult! Extremely short lived particles like pions, kaons and taus can be generated and accelerated up to extreme speeds to help propagate them far enough to matter, although that will need some very extreme speeds. At the upper end of extreme there are even proposals for accelerating tiny charged blackholes!

= Accelerator technologies =
There are a wide variety of accelerator technologies and here we try to summarize them as best as possible. Different accelerator types may be sequenced one after another. Some particle accelerator technologies depend on the particles fulfilling certain requirements of speed (thus energy) and hence different systems may be employed in sequence. Accelerator systems are described in terms of their efficiency of turning input power into particle energy, their acceleration gradient measured in electric field which translates to volts per meter, and the mass per length meter. An interesting secondary parameter is their operating temperature. Some accelerators can operate well at high temperatures. Others require cold or cryogenic conditions to work well.

== Core accelerator assembly ==
We define the core accelerator assembly as the structure where energy is imparted onto the particles.

=== DC Accelerators ===
For very low beam energies we can utilize large DC electric fields between plates or grids to accelerate particles up to speed. This is the same technique as is used in ion drives and is the most common industrial accelerator used. Every dentist's office and old CRT television uses this acceleration technique. Ion accelerators often also use a DC accelerator to generate the initial beam of ions, while electron beams utilize other techniques. These machines are limited by material breakdown to about 0.5 MV/m accelerating gradient and typically no more than a couple of MeV total particle energy for extremely large machines. The highest energy DC accelerators can reach 40 MeV for proton beams, but this is for machines massing in the 100s of tons and at low total beam powers. Very large DC accelerators are currently the only tested method for accelerating macrons up to 10’s of km/sec.These accelerators are limited by the risk of arcs jumping between the electrodes at too high voltage differentials.

=== Radiofrequency Quadrupole ===
A more advanced first stage accelerator for ion beams is the radiofrequency quadrupole or RFQ. These machines utilize low frequency radiowaves and carefully designed conductive bar structures to efficiently generate high current ion beams up to energies of ~7 MeV in a couple of meters. RFQs were a revolution in ion accelerator physics and caused quite a stir when the USSR first showed them off as they provided a much higher power and higher quality ion source in a smaller package than anything the USA had built! RFQs have also been proposed as methods of efficiently accelerating macrons up to >100 km/sec. These machines can be normal or superconducting and can operate with efficiencies as high as RF LINACs in theory.

=== Cyclotrons and Betatrons ===
One of the earliest ways to accelerate particles up to relativistic speeds is with circular accelerators known as cyclotrons (for ions) and betatrons (for electrons). In Cyclotrons the accelerator is a round chamber which houses two half-circle electrodes roughly shaped like a D (and thus called “dees”) separated by a gap, sandwiched between the poles of a large electromagnet. The magnet creates a static magnetic field which bends the path of charged particles into a circular motion. An oscillating alternative voltage is applied to the electrodes (what we often call RF or radiofrequency) to accelerate the ions. This radiofrequency field’s oscillation is matched so that one cycle of the voltage matches one circuit of the particles. As a consequence when crossing the gap from one electrode to the other, the charged ions experience a voltage gradient and are accelerated. As the particles gain energy and speed, their radius in the magnetic field gets larger. The result is an overall spiraling trajectory from an injection point in the center to an exit point at the rim where the charged particles are bent out of the cyclotron. Cyclotrons use the same voltage gradient many times over, defeating the electrostatic breakdown that limits the single-time electrostatic accelerators. As a consequence, proton energies of up to 70 MeV may be reached. Cyclotrons energies can also be increased by switching from normal magnets and RF resonators to superconductive components. This eliminates various losses from electric resistance and absorption and also allows for much stronger bending magnets, which allows the cyclotron to contain particles up to a higher energy if desired. Superconducting cyclotrons have achieved up to 520 MeV. Still, cyclotrons suffer from poor beam current, pulse rate, and beam quality issues that make them a less than ideal choice. The large magnet required (measured in meters) also does not lend itself to a lightweight weapon and as such, cyclotrons are regulated to niche medical and physics uses.

A close cousin is the betatron, which is used to accelerate electrons. In this case the magnetic field strength is increased as the electrons speed up, which prevents any spiraling trajectories of the particles. Betatrons have been operated up to electrons energies of 300 MeV, but are more typically in the ~10 MeV range. In principle a betatron could also be made superconducting, although magnet ramping may prove to be difficult. Other types of ring shaped accelerators may prove to be more useful, although electrons are heavily limited by synchrotron radiation and it is often better to use linear accelerators for them. For now betatrons remain limited to small (sometimes portable) accelerators for nuclear physics, industry and security work.

=== Radiofrequency Linear Accelerator ===
The real work-horse of the particle accelerator world is the RF linear accelerator. These babies use the same alternating field trick that cyclotrons use, but in a straight line rather than needing to bend the beam around in a circle. This provides a couple of nice advantages over the more traditional cyclotron. First and foremost you get to leave that large, bulky magnet behind which saves you a ton of mass, cooling and power consumption! You also get to now have much higher RF frequencies (1-10 GHz right now compared to the MHz used in cyclotrons), which means higher accelerating gradients. To generate these powerful, high frequency RF waves (also known as microwaves) that drive the accelerator we typically use military radar power supplies or (in more advanced cases) even lower energy particle accelerators! State of the art RF LINAC technology is well into the 100 MV/m range and have no upper limit on beam energy! The Stanford Linear Accelerator reaches 50 GeV and was built back in the 1960’s! There have been traditional RF LINAC designs for machines reaching up to 1 TeV, with the only limit being how much space you have to put these monsters. They do operate best (i.e. with high accelerating gradients and efficiencies) with particles already moving at relativistic speeds, so any ions fed in need to use a series of lower gradients LINACs or other pre-accelerators. Electrons and positrons can naturally be used in LINACs since they accelerate up to relativistic speeds so quickly.

LINACs also have been shown to operate at high average powers (MW scale is now in use) and high pulse energy (10’s of kJ pulses fired at high rep rate) while also maintaining a high quality beam that stays well focused. The two newest advancements have been to push the frequency ever higher for higher acceleration gradients (300+ MV/m has been shown and there are plans for even more) and to switch to superconducting RF accelerating cavities (the name of the actual piece of equipment that does the accelerating) which then dramatically increases the electrical to beam efficiency (80-90% has been shown and even copper waveguides have been operated at 70%). All in all, LINACs beat the tar out of cyclotrons which is why we don’t build cyclotrons anymore. To quote an old Fermilab engineer “cyclotrons are an expensive and bulky way to get a really crappy beam or a cheap way to get a giant chunk of lightly radioactive steel that can never leave your lab”.

LINACs work in detail by shaping the radiofrequency energy in a confining waveguide. The oscillations of the RF energy may be held in space, or allowed to travel down the waveguide over time. Radiofrequency waves have connected magnetic and electric components, which oscillate.

[[File:2022 08 16 GalLib ElectromagneticWave.svg|center|800px]]

Inside the RF accelerator, the electromagnetic radiation is oriented 90 degrees to the particle travel direction, which points the associated electrical field in the travel direction of the particles.The radiofrequency field naturally oscilates along its wave, and the connected electrical and magnetic fields oscilate as well: building up, reaching a peak, and diminishing, inversing, building up, reaching a peak, and diminishing, and the cycle repeats. The detailed behavior depends on the specific cavity.

[[File:CavityWithEM.png|600px|thumb|A resonant RF acceleration cavity and the electric field inside of it, created by resonating RF fields, at one moment in time.]]

Particles of the right charge in the right space at the right time experience an acceleration from the electrical field, being drawn to the opposite electric charge. If they land in the other region, they are instead decelerated. Inbetween, particles experience less to no acceleration. This behavior naturally separates a stream of particles into discrete '''bunches'''.

[[File:CavityWithEMAndParticles.svg|600px|thumb|Bunches being accelerated inside the electric field. ]]

It also creates an implicit limit to what can be effectively accelerated using a specific radiofrequency accelerator design. The particles driven through it have to keep pace with the oscillations of the electric field, reaching the next correct region inside the cavity at that time when the electric field is oriented correctly. If they are too slow to keep inside the accelerating regions, the accelerating action will be ineffective. This requires RF cavities to be matched to the particle's velocity gain under energy imparture and their velocity when entering, being inside, and exiting the accelerator. For electrons and positrons, this isn’t that complicated. Since these lightweight particles accelerate readily, they have few problems “staying in the pocket”. Particles that are slower to gain speed, such as ions and even larger particles, must however be accomodated. This in turn impacts how much acceleration and energy per meter can be fitted into the accelerator in the first place. The acceleration of heavy particles requires specifically designed and sequenced RF accelerators for specific regions of particle speed. This in turn complicates the design of the larger accelerator infrastructure, with multiple discrete stages getting the particles to the energy required so that the next stage in sequence can take over.

RF cavity accelerators are limited in their eV/m capability by the critical current at the containing waveguide. To increase the acceleration provided, more intense electrical fields must be built up. Eventually the electrical charge of the RF waves becomes so strong it can rip electrons out of the cavity walls, causing an arc to form and the cavity material to break down physically.

=== Advanced Accelerator technologies ===
Present accelerator technology caps out at around 250 MeV/m for the most overdriven designs. This poses issues for a variety of applications, where higher energies in shorter distances are desired. In science, "desktop" and "chipset" accelerators are of interest. When militarizing particle beams, some effects only become available at very high energies - but you want something compact enough to turret.

urrent and even near future accelerator isn’t really ready to supply that sort of scale quite yet, so we have to move on to the cutting edge of accelerator physics. There are three big options with sub technologies within these broad fields. We have extreme RF cavities using much higher frequencies, plasma accelerators using lasers for power and plasma accelerators using particle beams for power. Each technology provides a unique series of advantages and disadvantages, but fundamentally they are all a way to make linear accelerators that are much smaller.

==== Dielectric Wakefield Accelerator ====
LINACs using much higher frequencies than traditional LINACs have two fundamental problems to fight. The first is electrical breakdown and the second is generation of those mm-wave (10-100 GHz) or even THz (100-1000+ GHz) sources at sufficient power and efficiency to power a particle accelerator. As the frequency of the driving power goes up the electric field climbs up, which means the accelerating gradient gets larger (which is good) but also means that we run a higher risk of a huge arc forming in the accelerator and breaking it (which is bad). This is often fought by changing the accelerator cavity material from a metal (or superconductor) to a dielectric material like ceramic, or by using a coating of dielectric, which can sometimes earn them the moniker of Dielectric Wake Field Accelerator (DWFA). The issue of creating powerful and efficient enough sources of this higher frequency drive power is being actively pursued for a variety of fields (5G cell signals, beamed power, etc) but is still quite immature compared to the microwave RF sources of traditional LINACs. There are options using essentially "over-clocked" microwave sources, low energy particle beams, and even laser-plasmas. There is still a lot to be done here, but these sorts of problems are being actively worked on there are working accelerators using this technology. There is potential for GV/m or greater acceleration gradiatens and all of the same advantages of traditional RF LINACs (i.e. high powers, high fire rates, high energy pulses, quality beams) should stay the same. The wall plug efficiency is liable to be a bit lower, but that will be highly dependent on the efficiency of the source of these high frequency waves.

==== Laser-Plasma Accelerator ====
Plasma accelerators using lasers for power are a whole world in and of themselves. There are a wide variety of ways to accelerate particles with lasers and plasmas, but we will focus on the most studied technique known as Laser Plasma Accelerators (LPA). LPA accelerators utilize ultrashort (<1psec) and ultraintense (>10^18 W/cm^2) lasers to drive a plasma wake (thus sometimes earning them the name laser wakefield accelerator) and accelerate particles through the resulting charge differential. They have only shown the ability to accelerate electrons and positrons, but any ions that could be injected at the right time and at extremely relativistic speeds could conceivably be accelerated this way. In the operation of a typical LPA the ultraintense laser is fired into a jet of gas (turning it into plasma) or a channel of lightly pre-ionized plasma. The laser then blasts through this plasma like a boat through water and leaves a void behind it (the wake). This void ends up having a HUGE accelerating gradient (100 GV/m) and suffers no risk of electrical breakdown since the plasma is already broken down! A pre-accelerated bunch of particles can be injected into that void at the right time (called the witness bunch) for a BIG acceleration or you can rely on the wake itself spewing electrons out into the void and make a beam out of the plasma itself. The gas or plasma then quickly is reformed and the laser fires as soon as its cooling and charging allows. Keep in mind that even though the beams can hit incredible energies in short distances, the laser itself is not a pocket sized device. These lasers are room sized at minimum and not easy to shrink. The pulse lengths required are also much shorter than most of our hypothetical weapons-grade pulse lasers that may operate in the nanoseconds regime - dual-use may prove difficult. The intensities reached also pose significant issues for the laser hardware. If you don’t spread the energies out, it becomes quite possible you will break parts of your laser from the produced intensities.

Current LPA systems suffer from a couple of problems that are being actively worked on. First and foremost they are incredibly efficiency limited by the laser tech being used. There are plans to move on to much more efficient fiber lasers and eventually make LPA machines that operate in the 25% or greater wall plug efficiency, but that is not currently doable. These lasers also need to be fired at higher rates than the current ~1 Hz, but that is much more of a limit of laser engineering than accelerating engineering. LPA’s also have yet to link more than one accelerator together and the lasers are limited in just how much plasma they can accelerate through right now (~20 cm max) so the beam energy runs into a wall quite quickly. Aside from R&D on linking LPA’s together in a way similar to RF LINACs there are two other options. The acceleration gradient is higher the denser the plasma, so a move to denser plasmas and shorter wavelength lasers (to penetrate the plasma) could provide higher beam energies in similar distances. The other option is to use clever optical techniques like the so-called “flying focus” to allow for the laser to go through much longer plasmas. This could in theory allow for more efficient LWFA’s and single stage machines that provide ~1 TeV electron beams in meters! The final issue is beam quality. Even the best LPAs will not make beams that have as tight of a beam spot, energy spread, or emittance as a normal RF LINAC. This means they will end up needing some beam “cleaning” if you want to get a lot of range out of them.

LPAs are of engineering interest for very short wavelength X-ray free-electron lasers in a compact format. Currently these facilities require linear electron accelerators many kilometers in length. With an LPA as the electron accelerator, an XFEL able to image molecular structures and chemical reactions as they happen might be shrunken down to a much more compact format. (Sometimes this is discussed under the ultimate promise of the “benchtop” or even “chip-format”(sic!) XFEL)

==== Beam-Plasma Accelerator ====
Plasma accelerators using particle beams for power are very similar to LPAs in principle but use a particle beam rather than a laser. We typically just call these things Beam Plasma Accelerators (BPA) for short. The concept is very similar to the LPA in that a high intensity beam comes in and blasts a void into a plasma, which particles can then be accelerated in. BPA provides some interesting advantages over LPA in that you can use highly efficient superconducting RF LINACs (or synchrotrons if you want) to provide the particles at very high efficiency. They also have little in the way of plasma density limits and the accelerated beam takes on the characteristics of the beam driving the BPA. Thus little if any beam cleaning will be needed after acceleration. The big downside is the need for a relativistic particle beam to drive the effect, and typically one in the 100’s of MeV to low GeV range, depending on the particle used. Thus your final accelerator may be short, but you still won’t be making a handheld particle beam device.

== Accelerator Shapes ==
The accelerator technologies discussed above are the means by which we create situations for particles to gain energy. The accelerators become embedded in a larger structure that produces, directs, accelerates, even stores the particles until they are directed towards their final use.

There are two principle shapes for accelerator assemblies. Ones where the beampath is not closed onto itself, and designs where the beam path is closed on itself in part of the assembly, forming a loop.

=== Linear accelerator ===
Everything from electron generation to acceleration to conditioning is built in a straight line, one after another. This is the Linear Accelerator or LINAC. The main catch here is that there is only one single passage for the particles through the entire accelerator. Re-using accelerator hardware to drive the same particle to higher energies by repeated passage is out. The particles have to reach their desired energy in a single pass. Practically this puts limitations on the energy achievable with heavy particles in a LINAC. The length required gets impractically large. Even for electrons, lengths of hundreds of meters to kilometers may be required, unless advanced accelerators with high gradients are used. This said, every accelerator contains linear sections. Since so much high-performance accelerator technology depends on particles having a minimum velocity, ions must be “pre-conditioned” in a set of linear accelerators before the re-use of an accelerator through looped beam paths can be considered.

LINACs of low energy may be used in many applications. Medicinal they find use in radiation therapy and radiosurgery.

=== Synchrotrons ===
The other shape which has some interesting properties are accelerators where beampaths are looped onto themselves. These are called synchrotrons. Since beam paths are closed onto themselves, particles can pass through the same accelerator unit repeatedly, gaining more energy on every passage. This is advantageous for heavy particles which do not gain much speed on any single passage. They can pass through the same accelerator thousands or even millions of times instead. This is obviously advantageous for facility bulk and mass and allows us to reach energies that would otherwise be impractical. The accelerated beam bunches can also be held in storage, looping like the rotating mass of a flywheel, holding energy. The LHC can store some 700 Megajoules of energy in the recirculating beam after full speed has been reached.

That said, the devil is in the details. When charged particles are bent in circles they emit electromagnetic radiation which scales very rapidly (to the fourth power!) with relativistic gamma factor and with bending radius (inversely to the second power). This phenomenon is called synchrotron radiation. Due to the scaling, high energy particles emit more synchrotron radiation than lower energy ones and lighter particles (like electrons) at those high energies emit much more than heavier particles. In addition, the bigger you can make the bending radius of the synchrotron (often called “the ring”) the less synchrotron radiation you get.

[[File:SynchrVsE.svg|800px|center|thumb|Synchrotron radiation per particle versus particle energy for electrons and protons in a 50 meter in radius ring.]]

[[File:SynchroVsRing.svg|800px|center|thumb|Synchrotron radiation emission per particle for different particles at different energies, over varying bending radii. ]]

Although these numbers look small, keep in mind that this is PER PARTICLE. A typical pulse in an electron accelerator will have more than more than a billion electrons in it! Large synchrotrons like the LHC will have over 300 trillion ions circulating through it! Synchrotron radiation necessarily has to be managed to reduce radiation shielding, the radiation load (and possibility of hardware failure) on the accelerator, and to reach high energies in the first place - you might lose all the energy your accelerators are adding to synchrotron radiation!

Deflecting the beam also requires the application of magnetic fields, and higher degrees of deflection, in a smaller length, at higher particle energies, require stronger and stronger magnetic fields. Eventually multi-Tesla fields from superconductors will be required. And due to action and reaction, those magnets need to be braced against the pressure the beam exerts as it is deflected. Ultimately this leads to the same limitations as with flywheels - the mechanical strength of the braces and static connections that keep the magnets together as a larger structure is the accelerator’s limitation on J/kg held in storage. A further complication: The magnet strength has to be matched to particle energy. As a beam is accelerated, the deflectors have to ramp up their power to keep the degree of deflection constant.

Sometimes, synchrotron radiation from electrons is intentionally induced in a ring-shaped accelerator. Since synchrotron radiation of high-energy electrons tends to be hard x-rays emitted along narrow vectors, it’s a way to get x-rays used in various imaging techniques in material science and molecular physics.

Synchrotrons used for ions also need to respect the limitations of the involved accelerators. Synchrotrons often use radiofrequency accelerators. Since radiofrequency accelerators must match the speed and speed gain of their particles, an ion synchrotron cannot be made up of a single accelerator loop. Instead there are sequential loops connected in sequence, each designed to boost the particle through a specific velocity and energy region so that the next accelerator takes over. A designed example is the LHC. Particles headed for the main LHC actually circulate in two other synchrotrons beforehand, gaining sufficient energy so that the LHCs RF cavities can take over. (Also, before being injected into the first synchrotron ring, the ions must first go through multiple linear accelerators using different technologies to reach a base energy.)

An interesting design consideration: Synchrotrons don’t necessarily have to take the form of a (near-ideal) ring. Funamentally any dipole represents a single point of deflection of a straight path - they're polygons! Any geometric shape that contains a loop, and where any single bending curve keeps within the limits of the synchrotron radiation you are willing to tolerate, works. As such, complicated two- or three-dimensional “folded-up” shapes are also possible formats for a synchrotron.

=== Storage Rings ===
Since synchrotrons have to match their deflection magnet strength to the energy of the particles being accelerated, a single synchrotron track cannot hold more than one small section of beam bunches in storage at a time, and becomes unavailable to accelerate further bunches until it discharges the stored beam. This is disadvantageous for various applications where we may want to store a fully accelerated particle beam “bolt” for a longer time.

Storage rings are separate synchrotrons designed to hold multiple sets of bunches in circulation at a target energy at a time. The bunches are delivered from an accelerating synchrotron (sometimes called a booster ring) and held in circulation in the storage ring. Since there are synchrotron radiation losses, a few accelerator sections must compensate for these.

Storage rings can also be used to condition a stored beam. A common thing to do in storage rings is to “cool” a beam, draining unwanted movement energy out of the particles. Beam cooling may take significant time, which makes it ideal to conduct in a storage ring. The result is a beam with less tendency to drift apart, advantageous for many applications.

== Other parts ==
=== Particle sources ===
Particle sources are assemblies which supply particles to accelerators. Here we can group them into three categories: electron sources, ion sources, and exotic particle sources. The later involve the production of particles by means of gathering them from collision reactions (or the handwavium blackbox of your choice) and feeding them into another accelerator for further use. Positrons for example may be manufactured in beam collisions.

Electron sources require the production of a free stream of electrons. The oldest method here are thermionic sources. A hot piece of material ejects free electrons into a vacuum, where they can be drawn in using electric differentials and forwarded to the accelerator.

Field emission electron sources instead use a strong potential between a needle-like cathode and an anode plate with a hole. The electrons are drawn from the needle and through the anode due to the high charge differential. This method produces a “cooler” beam with less random divergence than thermionic sources.

A further improvement still are photoemission sources. Here the precise application of laser light onto a photosensitive material energizes electrons in the photocathode until they break free. This process is thermodynamically very precise and due to the nature of the laser light, also very controllable. The result is a low-emittance stream of electrons. These may be accelerated using electrostatic attraction with an anode at one end of the vacuum chamber, or directly using an RF field.

Ion beam sources are more complicated. The physics of ion formation are complex, which opens many design choices when creating an ion beam. Generally a plasma must be produced, from which the ions can be extracted.

The plasma can be formed in a variety of ways. Electrons from a local cathode-anode array, plasma frequency heating, the injection of electron beams, the application of laser light. The end results can be as diverse. Some methods only produce weakly charged ions and on many vectors, so with high emittance. Other methods offer greater ion charges or reduced emittance.

=== Magnets ===
While acceleration of charged particles is effected with electric fields, magnetic fields can be used to bend the trajectories of particle beams. As such, electromagnets can be used to bend the trajectories of charged particles in various ways, and in a controllable fashion.

==== Dipole Magnets ====
Dipole electromagnet are used to steer the entire particle beam in one axis. (To steer in two axes, we must either rotate one dipole about the beam axis, or couple two dipoles in sequence.) They can be used to direct a beam inside a particle accelerator, inject and extract beams out of looping constructs by switching on and off at the right moment, and point a beam at a target.

In a dipole magnet, a positive and negative coil oppose each other, embedded in a “choke” assembly often made of steel. Charged particles moving through the magnetic field between the coils experience a deflection in one direction specified by the polarization of the field, and a degree of deflection specific to the field strength.

==== Quadrupole Magnet ====
Dipoles can steer, but they influence all particles equally. As such, they cannot be used to manipulate the beam's shape. The first type of magnet able to influence a particle beam’s shape are quadrupole magnets.

They are assembled from four coils at 90 degree angles, with coils of the same polarity opposing each other. The passing beam is compressed on one axis, and expanded on the other. To focus the beam overall, at least two quadpoles must be arrayed in sequence.

Quadrupoles are the basic and essential “magnetic lens” of particle beams. Without them, charged beams would inevitably defocus and run havoc.

==== Hexapole Magnets ====
Hexapole magnets exist to address one problem of quadrupoles - they focus differently depending on the energy of the particle passing through the magnetic field. The effect is likened by particle accelerator physicists to the different diffraction of light in a prism - and hence the effect is called “chromatic aberration”. Hexapoles are used to correct for this, stronger deflecting higher-energy beams.

==== Integrated optics ====
As the field of particle accelerator science and engineering advances, more complex designs are proposed to handle more complex phenomena inside accelerators. An upcoming field is “integrated optics” - complex magnetic systems with nonlinear behaviors. These could provide more advanced manipulation of electron beams, building upon existing capabilities and adding new ones for shaping a beam.

=== Plasma lenses ===
Plasma lenses are an up-and-coming technology, where very high currents in plasma are used to create very strong magnetic fields. Similar to plasma-based accelerator technology, the fact that charges are already free in plasma is used to circumvent conventional charge breakdown and critical current limitations on solid-matter devices. The resulting magnetic fields can be much stronger and bend beams over shorter distances. Furthermore, by creating complicated profiles in the plasma, elaborate beam manipulations could be performed, more akin to sophisticated optical elements. As well, the charges in the plasma neutralize the particle beam in the region of the plasma lense - affecting its propagation behavior further and in a positive manner, as self-repulsion is neutralized.

The catch is that this is plasma - that means you are dealing with all the complications of plasma physics. Heating of the plasma has been an early challenge to successfully forming and maintaining plasma lenses.

Plasma lenses require an appropriate chamber, possibly active containment to keep the plasma from losing energy to the walls or leaking into other accelerator parts, and the necessary plasma sources, maintenance heating devices and then lense shapers and control sensors. In high-powered plasma lenses there will also be a need to cool the plasma as it heats up from waste heat. Potentially we may manage this by open-cycle cooling - venting the plasma controlled in one direction into open space, rather than have a closed-cycle solution for reconditioning the plasma. This could cause some very cool visuals!

Plasma lenses for particle beams should not be confused with optical plasma lenses for lasers! Those exist as well, but are their own topic.

=== Chicanes ===
Chicanes are assemblies of multiple dipole magnets used to manage the longitudinal dispersion of a particle beam due to different particle energies. Since particles with higher energy move faster, they drift ahead in the beam distribution over time; and particles with less energy increasingly lag behind.

Chicanes deflect high-energy particles onto an outward curving trajectory. Since traversing the curve takes longer, the high-energy particles are effectively delayed, allowing lower-energy particles to catch up. By the end of the chicane assembly, the bunch has been compressed in the longitudinal direction.

Chicanes however, do not remove energy from too-fast particles. That is the responsibility of other structures.

=== Undulators and wigglers ===
Undulators and wigglers are magnetic assemblies used to transform particle energy into electromagnetic emissions, with different properties from the specific implementation of a shared basic principle.

Assembled from stacks of generally dipole magnets (other configurations exist), both undulators and wigglers force charged particles onto “wiggling” trajectories by bending their path left and right. Wigglers force very strong arcs; undulators less so. In both cases, the rapid deflection of the charged particles causes them to shed parts of their energy as electromagnetic emissions. Due to the energy of the particles, these photons are generally highly energetic x-rays.

Undulators may be used to build free-electron laser systems.

They can also be used as a means of cooling the beam. The removal of energy from particles can also suppress unwanted movement modes, allowing accelerator structures to re-add energy only to desired movement modes. Of course, this means the particle beam actively loses energy, and produces x-rays that have to be absorbed (and cooled away.)

=== Beam coolers ===
Beam coolers are a collection of systems and strategies that are used to remove unwanted movements from a particle beam. Undulators and wigglers are one way to do this.

One method set of beam cooling relies on finding particles who are very “hot” with random motion, and either removing the particles from the beam using a kick device, or applying a precise counter-force that acts against the unwanted motion.

If one is dealing with ions, one clever trick is to inject a beam of electrons. Due to the thermodynamics within the particle population, the lighter electrons absorb heat from the heavier ions, taking this unwanted motion energy with them when the electron beam is bent out.

As well, lasers can be used (what can’t lasers be used for!) to apply counter-force on ions and remove unwanted movement in this manner.

Beam coolers are generally not instantaneous devices. They work over time to make a beam better. Often they involve the removal of energy or particles from the stream, requiring accelerators and sources to make up the loss and creating waste heat.

=== Beam recovery ===
Some accelerator systems can actually be run in reverse - they can turn beam energy back into some other energy format and then electricity. This can be used to recover some to most of the energy from a particle beam before disposing of it. This is advantageous both for your energy bills and making a very high-energy beam liable to spray radiation everywhere, easier to absorb in a beam dump.

=== Beam dump ===
A beam dump is a construction where a particle beam can be routed in case it has to be disposed. Beam dumps are designed to safely absorb the particle's energy without leakage of dangeorus radiation or activated materials to the outside enviroment. Common materials are carbon and metals, embedded in an actively cooled assembly.

= Propagation physics =
==Magnetic fields==

Magnetic fields are things of motion. They are created by moving electric charges. And they only affect electric charges that are moving. An electric charge at rest in the presence of only a strong magnetic field will just sit there. But when it starts moving, the magnetic field interacts with it, starts deflecting it.

But not along the field. Oh no. The magnetic field does not point in the direction it will move the charge. No, the charge experiences a force that is perpendicular to the direction of the field. And also that is perpendicular to its own velocity. A charged particle moving straight down a magnetic field line will not experience any force.

On the other hand, if the particle is moving directly perpendicular to the field, the magnetic force will always push it to the side, even as it keeps turning. This results in the particle moving in a circle. The radius of this circle is called the <b>gyroradius</b>, and the rate at which the particle makes circles is called the <b>gyrofrequency</b> (or sometimes <b>cyclotron frequency</b>).
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Circular_motion.png|frameless]]
<td width=255>[[File:magnetic_forces.png|frameless]]
<td width=255>[[File:spiral_around_field_lines.png|frameless]]
<tr>
<td>To get circular motion, you need a force (blue) that is always the same amount but always pushing perpendicular to the direction of motion, toward the center of the circle. A charged particle in a constant magnetic field moving perpendicular to the magnetic field meets these conditions, and moves in a circular path.
<td>A diagram of how a magnetic field (magenta) exerts forces (blue) on a moving particle (with velocity shown in green).
<td>If a particle is moving partially along the magnetic field (magenta), it will drift down the field lines while spiraling around it (black path). Where the magnetic field is stronger (indicated by having the field lines closer together) it will execute tighter, faster circles in its spiral and will drift more slowly than when the field is weaker.
</table>

It can be convenient to split up a particle's velocity into a part parallel to the magnetic field (the <b>longitudinal velocity</b>) and a part perpendicular to the field (the <b>transverse velocity</b>). The longitudinal velocity describes how fast the particle drifts along the magnetic field line. The transverse velocity determines how tightly and how quickly the particle circles around the field lines.

If the gyroradius is much smaller than the extent of the magnetic field, you get a case where the particle approximately cannot cross magnetic field lines. It just spirals around them. On the other hand, if the magnetic field ends before the particle can make a significant portion of a full revolution, then the particle ends up being somewhat deflected from its original trajectory. If the gydroradius is enogh to deflect the particle into a region where the magnetic field has a different strength or direction, then things get complicated!

When a particle is spiraling around the magnetic field while drifting down the field lines into regions where the magnetic field is stronger, it will circle around the field lines in a faster, smaller circle. This increases the kinetic energy in the spiraling motion; this kinetic energy must come from the kinetic energy of the longitudinal motion. If the field becomes too strong, the particle will slow down all the way and come to a stop. Then it will be reflected back, pushed the other way down the field lines toward the region where the field is not as strong. This phenomenon is called magnetic mirroring, and is responsible for the way radiation gets trapped in the Van Allen belts around Earth with the particles bouncing back and forth along the Earth's field lines and getting reflected back and forth at the poles where the field is strongest.

If you want to calculate for yourself what the gyroradius <math>r_c</math> and gyrofrequency <math>f_c</math> are
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
r_c = \frac{\gamma \, m \, v_\perp}{q \, B}
</math></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
f_c = \frac{q \, B}{\gamma \, m}
</math></div>
where <math>B</math> is the strength of the magnetic field, <math>q</math> is the amount of electric charge of the particle, <math>m</math> is the mass of the particle, <math>\gamma</math> is the Lorentz factor, and <math>v_\perp</math> is the part of the velocity that is perpendicular to the magnetic field. To get these formula to work, you need to use a consistent set of units. So for example if you measure the magnetic field in tesla you need to measure the charge in coulombs (not fundamental charges) and the mass in kilograms (not eV).

One consequence of this is that you can bend the beam coming out of your accelerator by placing a uniform magnetic field across the end. So you can steer your beam to where it needs to go. You can also bend the beam inside of your accelerator – to make it go around in a circle, for example.

Another consequence is that if your beam has to pass through regions with magnetic fields in it, it will get deflected. If you want the beam to end up at the right spot, you will need to account for this. If the exact strength or direction of the field are not known, you will get unpredictable deflections to your beam.

If your beam has been neutralized, the magnetic fields will work to bend the positively charged particles one way and the negatively charged particles the other way. If you have enough beam current, these separated charges will attract each other and counteract the magnetic force. If you don't, the beam will split and you'll get positive charges zipping off in one direction and negative charges going another direction. If you needed to neutralize your beam so that it doesn't electrostatically blow itself apart, well now it will be doing just that.

Even perfectly neutral atoms can be torn apart if the atoms are going fast enough and if the magnetic field is strong enough. Atoms are pretty resistant to this, but you can get conditions where it happens.

Magnetic fields affect the path of beams. But beams are made of charged particles and thus can have currents in them. Currents create magnetic fields themselves, which can influence the surrounding fields. Even a neutralized beam forming a plasma, when it encounters a magnetic field, will get currents induced in the plasma that helps to screen its interior from the field. A beam with enough punch can deform the fields around itself rather than having the fields bending it. As a rough rule of thumb, if the energy density of the beam over a regions larger than a gyroradius is more than the energy density of the field over that region, then the beam gets to have its way with the field, and vice versa. Highly focused, high current beams can thus be expected to better deal with ambient magnetic fields than those with less focus or lower current.

==Focusing==

===Focusing equipment===
In the previous section, we discussed how a particle beam can be steered by directing it through a uniform magnetic field. We can make this kind of field with a magnetic north pole directly opposite a magnetic south pole (with additional shimming magnets to keep things uniform). Because it has two poles, one north and one south, this kind of magnet is called a <b>dipole magnet</b>.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:dipole_beam_steering.png|frameless]]
</table>

But we want to do more than just steer the beam. we want to be able to expand it, contract it, and focus it to a point. To do this, we need a different kind of configuration. We will put two north poles opposite each other and, and 90 degrees, two south poles opposite each other. With four poles, this is called a <b>quadrupole magnet</b>, or sometimes just a "quad".
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_magnet.png|frameless]]
</table>
At the center of the quad arrangement, the magnetic fields from each side cancel out. there is no field there. A particle going through the center will travel straight through without deflection. However, the farther particles get from the center, the higher the field they encounter and the more they get deflected. A particle slightly off to one side may experience a force that curves it toward the center-line. If the quads are properly shimmed, a particle twice as far away will experience twice the force and twice the deflection from its trajectory back toward the center. This curves all the particles so they meet in one spot.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_focusing.png|frameless]]
</table>
However, there is a catch. Those particles that are offset from the center-line at 90 degrees from these focused trajectories are bent away from the center-line rather than towards it! A quad focuses in one direction and de-focuses in the other.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_defocusing.png|frameless]]
</table>

Is there anything e can do to get an arrangement of magnets that purely focuses the beam? Well yes. We can put another quad arrangement downstream of the first, rotated by 90 degrees. Now consider a set of particles in an arrangement that is de-focused by the first quad. When they get to the second quad, they are further from the center than they were when they went through the first quad. Because the second quad is aligned to focus anything the first quad de-focuses, these particles will be focused; and because they are farther from the center, the focusing deflection will be larger than their previous de-focusing deflection. So the net result is that the particles are focused.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Dual_quad_focusing.png|frameless]]
</table>
For particles with the opposite orientation that are first focused, these will be de-focused by the second quad but because they will have converged they will be going through a region of lower field and will be defocused with a smaller deflection than they were initially focused. This dual quad arrangement will focus any particle that goes through it, regardless of orientation.

===Emittance===

For small deflection angles, a dual quad set-up acts like an ideal lens for particles. An ideal lens can focus all the particles diverging from a point onto another point, or into perfectly parallel lines. It can focus particles on perfectly parallel lines onto a point. So now we can use the particle beam to deliver focused particles on our target.

Or, at least we could if all the particles came from a point, or if they traveled on perfectly parallel lines. Unfortunately, they don't. When particles are made, they come from a source with a finite spot size and a finite temperature. The random motion of the particles moving around from their initial thermal motion means that they are never moving perfectly parallel to each other. And when you try to focus them, instead of all converging onto a single point they make a spot of finite size. And if your focusing and beam bending elements are not perfect, you add additional imperfections to the beam that further hider your ability to focus it.

Your ability to focus can be quantified by a measure called <b>emittance</b>. Emittance is measured in units of a length scale times an angle - nominally meters-radians (m<math>\cdot</math>rad) but the technical literature more commonly gives units of millimeters-milliradians (or mm<math>\cdot</math>mrad) or even 2 <math>\pi</math> mm<math>\cdot</math>mrad. The divergence angle of a beam that you try to make parallel will the the emittance divided by the width of the beam. If you focus a beam down so it converges at an angle, the minimum spot size you can get at the focal point is the emittance divided by the beam angle. If you try to focus a beam with emittance <math>\epsilon</math> on a spot a distance <math>R</math> away with an initial beam diameter of <math>D</math>, the spot size <math>S</math> on the target you can achieve is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
S = R \frac{\epsilon}{D}
</math></div>
It is interesting to note that this is very nearly the same spot size you get from a laser due to [[Diffraction#Spot_Size|diffraction]] with a wavelength equal to the particle beam's enissivity. And, like lasers, the more you can expand the beam to large diameters at the focusing equipment, the tighter the spot you can achieve on the target.

So to get good focusing, we want low emittance. Is there a way we can lower the emittance beyond what we get from our source? Engineers are pretty bright folks, and they have found some ways to do this. You can, for example, pass a beam through a thin foil. This causes some stopping, slowing the beam down. You can then accelerate the beam back up to its original speed. However, the stopping is along the direction of the beam particle's trajectory, even if that trajectory is at an angle. Meanwhile, all the acceleration is in the direction of the beam axis. So the net result is to reduce the random motion perpendicular to the beam, and thus reducing its emittance. Similar tricks can be played with magnet set-ups that force particles going through them to emit radiation.

Another method can be used with a relatively high emittance beam, say a beam of protons or ions or muons. Electron beams can be made with substantially lower emittances. So you can send a low emittance electron beam to co-propagate along with your primary particle beam. The beams will interact, with the particles scattering off each other. This exchanges heat between the two beams and so the temperatures, and thus emittances, of the two beams get closer to each other (and, if given enough time, would become equal). If you repeat this often enough, you can get the emittance down to the emittance of the electron beams. This is more practical in beams going around circular accelerators, because they will have time to get cooled over and over.

A final method that has been used is to install sensors to measure the beam fluctuations. For beams going in circles, the time to get a light-speed signal directly across the diameter to the opposite side wil always be less than the time it takes for the particles to go around the curve to that spot. So you can send signals to give electromagnetic kicks to the beam particles to knock them back into alignment when they get to that spot.

It is intriguing to note that these beam cooling techniques can take <i>hours</i> to work!<ref name="RHIC emittance">[https://www.bnl.gov/isd/documents/86783.pdf M. Minty, R. Connolly, C. Liu,T. Summers, and S. Tepikian, "Absolute beam emittance measurements at RHIC using ionization profile monitors", Brookhaven National Laboratory formal report BNL-105970-2014-IR]</ref> However, without beam cooling methods like this, the emittance of ions or protons in storage rings can slowly increase on a time scale of hours.<ref>[https://arxiv.org/abs/1204.6022 Vladimir Shiltsev and Alvin Tollestrup. "Emittance growth mechanisms in the Tevatron beams", arXiv:1204.6022 [physics.acc-ph]]</ref>

But perhaps the simplest method to reduce emittance is just to speed the particles up. For particles moving at a fraction of light speed <math>\beta</math> and Lorentz factor <math>\gamma</math>, the emittance of particles from the same source with the same temperature and same random motion will be proportional to <math>1/(\beta \, \gamma)</math>. To reflect this, it is convenient to define a normalized emittance <math>\epsilon_n</math> such that for the usual geometric emittance <math>\epsilon</math> that we have been talking about
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
\epsilon_n = \beta \, \gamma \, \epsilon.
</math></div>

Normalized emittance measures makes it convenient to get basic estimates of particle beam performance and compare emittance of different sources and beams. For example, various high-performing electron beams around the world have managed to push their normalized emittances down to about <math>2 \times 10^{-6}</math> m<math>\cdot</math>rad to <math>3 \times 10^{-6}</math> m<math>\cdot</math>rad. Facilities with proton or ion beams seem to have about an order of magnitude higher normalized emittance.
<table>
<tr><td width=200>Machine <td width=300>normalized emittance (m<math>\cdot</math>rad) <td width=100>particles<td>reference
<tr><td>European XFEL <td><math>2.3\times10^{-6}</math> <td> electrons <td><ref>[https://journals.aps.org/prab/pdf/10.1103/PhysRevAccelBeams.24.110702 T. Hara <i>et al.</i>, "Low-emittance beam injection for a synchrotron radiation source using an X-ray free-electron laser linear accelerator", Physical Review Accelerators and Beams <b>24</b>, 110702 (2021)]</ref>, using reported <math>\epsilon \approx 1.5\times 10^{-10}</math> m<math>\cdot</math>rad and electrons at 8 GeV for <math>\beta \to 1</math> and <math>\gamma = 15700</math>.
<tr><td>DESY PETRA III<td><math>0.93\times10^{-6}</math> <td> electrons <td><ref>[https://www.desy.de/news/news_search/index_eng.html?openDirectAnchor=444&two_columns=0 New emittance world record at PETRA III]</ref>, using reported <math>\epsilon = 1.6\times 10^{-10}</math> m<math>\cdot</math>rad and electrons at 3 GeV for <math>\beta \to 1</math> and <math>\gamma = 5870</math>.
<tr><td>SLAC <td><math>3\times 10^{-6}</math> <td> electrons <td><ref>[https://accelconf.web.cern.ch/pac97/papers/pdf/4W009.PDF M. Hernandez <i>et al.</i>, "Emittance Measurements for the SLAC Gun Test Facility", 1998 IEEE]</ref>
<tr><td>LHC <td><math>3.75\times 10^{-5}</math> <td> protons <td><ref>[https://www.lhc-closer.es/taking_a_closer_look_at_lhc/0.beta___emittance Beta & Emittance; Taking a closer look at LHC]</ref>
<tr><td>Fermilab Tevatron <td><math>\approx 5\times 10^{-5}</math> <td> protons <td><ref>[https://accelconf.web.cern.ch/p03/PAPERS/TPPB070.PDF J. Slaughter, J. Estrada, K. Genser, A. Jansson, P. Lebrun, and J. C. Yun, "Tevatron Run II Luminosity, Emittance and Collision Point Size", IEEE Proceedings of the 2003 Particle Accelerator Conference, pages 1763-1765]</ref>
<tr><td>RHIC (with protons) <td><math>\approx 5\times 10^{-5}</math> <td> protons <td><ref>[http://www.osti.gov/servlets/purl/910428-6sKfP0/ Zhang, S, and Ptitsyn, V., "Proton Beam Emittance Growth at RHIC", Brookhaven National Laboratory formal report BNL-77563-2007-CP]</ref>
<tr><td>RHIC (on startup) <td><math>\approx 1.5\times 10^{-5}</math> <td> Au-197 <td><ref name="RHIC emittance"></ref>, using reported <math>\epsilon \approx 12\times 10^{-6}</math> m<math>\cdot</math>rad and Au-197 ions at 100 GeV for <math>\beta = 0.76</math> and <math>\gamma = 1.545</math>.
<tr><td>RHIC (after several hours cooling) <td><math>\approx 5\times 10^{-6}</math> <td> Au-197 <td><ref name="RHIC emittance"></ref>, using reported <math>\epsilon \approx 4\times 10^{-6}</math> m<math>\cdot</math>rad and Au-197 ions at 100 GeV for <math>\beta = 0.76</math> and <math>\gamma = 1.545</math>.
</table>
So as a plausible first estimate for your sci-fi electron guns, you can take <math>\epsilon_n \approx 10^{-6}</math> m<math>\cdot</math>rad and for proton guns <math>\epsilon_n \approx 10^{-5}</math> m<math>\cdot</math>rad. Modify for any improved future-tech that can bring the emittance down. Then divide by <math>\beta \, \gamma</math> for your beam to find the geometrical emittance. Which in turn can let you figure out how well you can focus your beam.

However, we should add one caution here. Unlike a laser, which can get a beam diameter nearly as wide as the width of the optical element (be it a lens or mirror) focusing it, a particle beam needs to go through a small hole inside much larger focusing elements. Thus, a particle beam width (what goes into the spot size calculation) is likely to be much smaller than the physical size of the focusing equipment. You may be able to overcome this limitation by making the geometrical emittance very low, but it will be an issue to keep in mind.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Laser_focus.png|400 px|frameless]]
<td width=150>
<td width=556>[[File:P-beam_focus.png|556 px|frameless]]
<tr>
<td>Laser focused with a mirror
<td>
<td>Particle beam focused with quadrupole magnets
</table>

==Beam self-forces==

particle beams are composed of moving charged particles. Moving charged particles are electric currents. And currents and charges exert forces on each other.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Charges_and_forces_pn.png|frameless]]
<td width=255>Opposite electric charges exert an attractive force on each other
<tr>
<td>[[File:Charges_and_forces_pp.png|frameless]]
<td rowspan=2>Electric charges of the same type exert repulsive forces on each other
<tr>
<td width=349>[[File:Charges_and_forces_nn.png|frameless]]
</table>
<td>
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Currents_and_forces_opposite.png|frameless]]
<td>[[File:Currents_and_forces_same.png|frameless]]
<tr>
<td width=200>Electric currents in opposite directions repel each other
<td width=200>Electric currents in the same direction attract each other
</table>
</table>

In a particle beam, the collection of like-charged particles and their mutual repulsion is called <b>space charge</b>, and can have various effects ranging from making beams expand and trumpet out if not properly neutralized to even making particle beams stop, turn around, and come back the way they came! The currents created by all the particles, on the other hand, are moving in the same direction so they attract each other. This attraction can be called a <b>z-pinch</b>, or sometimes a <b>Bennett pinch</b> after one of the first guys to study it.

===Looking at the same thing two different ways===

Imagine that you are sitting calmly, watching a collection of positively charged particles (as you do). Because the particles all have the same electric charge, they repel each other. You observe the particles pushing away each other and flying apart.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Charges_at_rest.png|478 px|frameless]]
</table>

Ah, but what we didn't tell you was that these particles were just shot out of a particle accelerator at high speed. You just happen to be moving at the same velocity as all the particles, so to you they appear to be at rest. But your colleague running the accelerator sees the particles (and, presumably, you) flying past at high speed. She sees not just particles of the same charge pushing out against each other, but also all the currents of these charges. And because the currents are all in the same direction, there is an attractive force pulling the particles together.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Charges_in_motion.png|750 px|frameless]]
</table>

With the extra forces from the current squeezing the particles together, your colleague should see the particles expanding less fast than you do. Maybe she even sees them collapse in on themselves if the current-current forces are higher than the charge-charge forces. What gives? Who is right?

It turns out, you both are. Because, as it turns out, magnetic fields are just a consequence of relativity balancing things out when you look at moving things from a different coordinate system. You may be familiar with the idea that in the theory of relativity, moving things experience time at a slower rate than things at rest (as observed by someone who is "at rest" in that frame of reference). So your colleague sees the things happening in your frame of reference happening slower than you do. Maybe only just a tiny bit slower, if you are not moving very fast at all. Maybe much, much slower if you are nearly relativistic with respect to her. But some degree slower, in any event. And so she won't see the particles expanding as fast as you do. Each of the particles has the same electric charge for both of you, so the electric forces will be the same. So your colleague must observe some other force in addition to the electric forces that act to partially compensate for them. And these are the magnetic forces, which are present in her frame of reference but not yours. And now you can see that, for bare isolated charges, the magnetic self-pinching forces can never quite overcome the electric self-repulsion charges – although if the particle bunch is going very fast, they might come very close.

===Space charge===

===Self pinching===

==Energy loss in matter==

Sometimes, a particle beam needs to go through stuff to get it where you want it to go. Stuff like air, for example. Or the tissue in the way of getting the beam to a tumor. Also, sometimes, you want your particle beam to interact with stuff. Like the material you are trying to machine with an ion beam, or the bacteria you are trying to kill on medical equipment. All of this falls under the general realm of the interaction of the beam particles with matter. This is an enormously complex subject, but we can cover many of the basics here.

The primary effect on the beam of any interactions with matter are loss of energy of the beam particles (<b>stopping</b>), random changes to the particle directions that make the beam spread out (<b>straggling</b>), and knocking particles clean out of the beam altogether (<b>scattering</b>).

===Ionization===

The particles coming out of your particle accelerator are charged. When a fast moving charged particle goes through matter, the electric field from the charge pushes and pulls on the various charged particles that make up that matter – mainly electrons and the atomic nuclei around which the electrons orbit. At speeds above approximately 1% of the speed of light, it is primarily light and nimble electrons being affected while the ponderous nuclei just sit there barely perturbed by the passage of the particle. When the passing particle yanks on an electron hard enough, it can pop it clean off of the atom it was attached to, leaving a positively charged ion behind and giving you a free electron that can go and glom on to a previously neutral atom to make a negative ion. This process, unsurprisingly, is called <b>ionization</b>. For most particle beams, ionization is the dominant interaction with matter; although when you get very fast or very slow particles other effects can become more important.

if your particle was not initially charged when it goes into matter, it soon will be. The electric charges of the electrons and nuclei in the matter will do the same thing to the particle that a charged particle would do to them. This will quickly strip off any electrons around a high speed atom, leaving an ion and bare electrons to go through the matter. Only particles that are fundamentally neutral (like neutrons or photons) will not be immediately ionized.

It takes about 1 to 10 eV to ionize an atom. This energy must come from the kinetic energy of the beam particle. Particles from an accelerator have millions to billions of eV. So it takes a lot of ionizations to meaningfully affect the particles. However, one thing about matter is that it has lots of atoms that can be ionized so you do get those lots of ionizations. Because each ionization is so small compared to the total energy this is often approximated as a continuous slowing down of the beam particles. So many individual ionizations tend to average out, leading to all the beam particles being slowed down uniformly.

When the particle is yanking on nearby electrons to tear them out of atoms, those electrons are yanking back. This jitters the beam particle slightly from side to side. Again, these sideways yanks tend to average out. The statistical variations from the average slowing down and side-to-size jerking are what lead to straggling. Protons and ions are heavy, and will be deflected little from their original paths - like bowling balls smashing through matchsticks. These particles tend to go in mostly straight lines. Electrons, on the other hand, have the same mass as the electrons they are knocking about. So electron tracks through matter tend to zig-zag around a lot more.

As the particle whizzes past, it leaves a trail of atomic destruction in its wake - a trail of ions where the atoms are literally torn apart (not the nuclei, just the atoms). These particle tracks can sometimes be seen in detectors like cloud chambers. The faster the particle is going, the less time it has to yank on nearby electrons and so the less ionization it leaves behind. As a result, fast particles are not slowed down much but as they start to slow down they continue to be slowed down faster and faster. A consequence of this is that the particle dumps most of its energy into the material at the end of its track, just before it comes to a stop. This sudden peak in energy deposition is called the <b>Bragg peak</b>.

The electrons that a particle knocks out can end up going pretty fast. So these secondary electrons can produce further ionization in what is called an <b>electron cascade</b>. if an inner core electron is knocked out of an atom, other electrons falling in to fill that empty spot in the atom can release x-rays (<b>x-ray fluorescence</b>) or cause shake-ups in the outer electrons that knock other electrons off the atom (<b>Auger effect</b>).

===Brehmsstrahlung===

A charged particle is surrounded by an electric field. If it is moving, it is also surrounded by a circulating magnetic field. These fields move along with the particle.

If the particle knocks into something, it will slow down and probably change direction. The information about this bump only goes through the electromagnetic field at the speed of light. So a lot of that field that was moving along with the particle might not get the memo in time and keep right on going. Stray electromagnetic fields like this get this self-inductance effect going on where they start making themselves over and over again, allowing them to propagate as freely moving waves of electromagnetic radiation. This EM radiation from a slowing down particle is given the delightful name of <b>bremsstrahlung</b>.

Particles from an accelerator can be moving so fast that they make bremsstrahlung in the x-ray or gamma ray part of the spectrum. If you get very high energy bremsstrahlung gamma rays, the gamma rays can interact with matter by boiling an electron and its antimatter counterpart, a positron, out of the vacuum (using the nearby matter as the necessary momentum sink for the process). These positrons and electrons can themselves be very high energy and can in turn create more bremsstrahlung, and the process can repeat until the particles are no longer high enough energy to continue making each other. This is called an <b>electron-gamma shower</b>.

The x-ray machines used by dentists and radiologists use both bremsstrahlung and x-ray fluorescence that you get when shining an electron beam onto tungsten to make their x-rays.

At very high speeds, bremsstrahlung can be more a important source of energy loss than ionization.

===Electron capture and screening===

When an ion has slowed down to about 1% of the speed of light or less, it is moving slower than the electrons zipping around their atoms in the matter the ion is going through. This allows the electrons to stick on to the ion, now becoming the electrons of the atom that is moving through the material rather than of the material itself. The ion begins to lose its charge, and the rate of ionization goes down. Soon, ionization becomes negligible compared to the beam atom bumping in to other atoms.

===Phonons and displacement===

When an atom or ion moving through a material bumps into another atom, it will cause that other atom to move. This can have one of two primary effects.

First, if the atoms whack into each other hard enough, the atom in the material can be knocked clean out of place. Like a pool ball being thwacked by the cue ball, the struck atom will go zinging through the material. Atoms getting knocked out of their place like this is called <b>displacement</b>. Displacement creates chemical changes to molecules as atoms are removed, and creates defects called <b>vacancies</b> in crystals (which, as you might imagine, is just a missing atom in the crystal lattice). When the atom comes to rest, it can chemically attack other molecules, again changing the chemistry, or create an extra vacancy called an <b>interstitial</b> jammed into the crystal lattice where it is not supposed to be. If the displaced atom is going fast enough, it can whack other atoms and displace them, leading to a <b>displacement cascade</b> like tenpins knocking each other down. As a result, at the end of its track, an ion can leave a region of disordered material and messed up chemistry where the atoms have all crashed into each other like a freeway pileup.

If an atom is displaced near the surface of a material, the displaced atom can shoot out of the surface and escape the material altogether. This is called <b>sputtering</b>.

Second, the beam atom might only deliver a glancing blow. Or the beam particle might be an electron, which is so light that it is usually incapable of displacing atoms. In this case, the struck atom gets kicked to the side a bit and runs into other atoms; maybe it is also pulled back by the chemical bonds of other atoms that it is attached to. This transfers the motion of the atom to other atoms around it in the form of pressure waves (or, in a solid, shear waves). This adds heat to the material without causing direct damage – although as you might imagine, enough heat can also cause localized melting, evaporation, or decomposition. The individual quanta of vibration – fundamental particles of sound, if you want to think of them like that – are called phonons in analogy with the photons that are the fundamental particles of light. Much like ionization, a particle that excites pressure waves slows down gradually and continuously, unlike the sudden changes in speed and direction you get from knock-out displacement.

===Nuclear collisions===

So far, we have only discussed the things that happen when beam particles interact with matter particles by electromagnetic forces. Even atoms knocking into each other to produce phonons or displacement happens through the mutual repulsion of the electrons around the atoms (although this is dressed by other weird effects like how two electrons can't be in the same place at the same time, or how the fields of one electron can affect other nearby electrons that in turn affect the original electrons, leading to odd correlation effects and charge density waves and other fun stuff that keeps condensed matter physicists and quantum chemists busy).

But electromagnetism isn't the only thing you can get going on. below the level of structure of the atom, you have the atomic nucleus. And a direct hit on the nucleus can end up giving very different effects.

The simplest thing that can happen is an <b>elastic collision</b>, where no energy goes in to any internal nuclear state or into breaking up the nucleus. An elastic collision just ends up giving you displacement (or possibly phonons, but if you hit the nucleus you usually hit it hard enough to send it flying).

But beyond that you can cause any number of nuclear reactions. You can leave the nucleus in an excited nuclear state, from which it usually relaxes back down to its original state by emitting gamma rays. You can knock out nuclear particles, such as neutrons, protons, deuterons (a proton stuck to a neutron), tritons (a proton stuck to two neutrons), helions (a neutron stuck to two protons), alpha particles (two protons stuck to two neutrons), or potentially even larger nuclear fragments. If you hit a heavy nucleus like uranium, you can cause it to split apart by fission. And if you hit the nucleus really hard you might just shatter it into many tiny fragments of those protons, neutrons, deuterons, tritons, helions, alpha particles, and heavier nuclear fragments mentioned earlier. To get any of these <b>inelastic collisions</b>, you need to hit the nucleus hard enough. If your beam particle doesn't have enough energy to excite the nucleus or knock particles, these things simply can't happen. As it gets enough energy to excite a particular interaction channel (as they are called), that process becomes possible and as the energy increases the process becomes more and more likely.

A nucleus is only about a femtometer across (<math>1 \times 10^{-15}</math> m). Which makes it only about <math>1 \times 10^{-30}</math> m² in cross sectional area. In the meantime, chemical bonds between atoms are usually about 1/10th of a nanometer (<math>1 \times 10^{-10}</math> m) long. In condensed matter (liquids and solids), this means that you have somewhere about <math>1 \times 10^{30}</math> atoms per cubic meter. A meter thickness of atoms thus presents just about about enough atoms that their nuclei can cover the projected area – in other words, you can expect a particle to go through (very roughly) about a meter of condensed matter before it whacks into a nucleus.

This assumes that a particle that goes through a nucleus interacts with it. This is usually a good assumption for particles that interact by the strong nuclear force, such as protons, neutrons, mesons, and the atomic nuclei of ions. But many particles cannot interact by the strong nuclear force, such as electrons or muons. These are much less likely to interact with an atomic nucleus even if they pass through it, so they will end up going considerably farther between hitting nuclei than the above assumption. But nuclei can interact electromagnetically (they are made up of charged protons, after all, and at a smaller level of charged quarks), so electrons and muons can produce some nuclear interactions; just far less than actual nuclear particles like protons and ions.

A nuclear strike is a discrete event that will remove the impacting particle from the beam and will (except for elastic collisions) probably produce a spray of radiation.

===High energy physics===

As you get to higher and higher energies, you unlock all sorts of new and weird and interesting processes that can happen. Boiling exotic particles out of the vacuum, unifying forces that had been divorced since the big bang, and producing bizarre unstable resonances in the quantum fields. This is what physics build accelerators to measure. From the point of view of getting a beam to its target, or what happens to the target, however, the main effect is mostly that the interacting particle is removed from the beam and a bunch of additional radiation is produced.

= Operational Considerations =
== Logistics ==
The logistics of particle beam can be a mixed bag. By itself, any particle accelerator is a complicated piece of machinery full of delicate parts that have to deliver high performance. This may require cryogenic temperatures with associated challenges on the materials and mountings used (insulation, and distortions under temperature changes). When working, particle accelerators invariably generate some radiation across the whole assembly, which irradiates and activates the structure. Some parts may be exposed to significantly higher radiation flux due to their working principles. Often the particle beam is a dynamic system that must be managed with active, closed control loops, involving sensors, modeling and control software, actuators, and the computing and communication hardware that brings it all together. This poses engineering challenges across all disciplines.

A practical particle beam is inherently a counterintuitively sturdy device. If it was too delicate, it would break too much to be useful, though of course this is a sliding scale. Still, while sturdy, this doesn’t eliminate complexity, it manages complexity. The result is a system that is the product of a high-tech civilization. A superior machine shop and well-trained engineers are a must to maintain a particle beam system. Some parts of a particle accelerator may be mundane - others may be custom-machined parts that require their own dedicated industries to produce. That said, the manifold utility of particle accelerators is on your side here. Since there are so many practical applications, it is economic to produce parts for industrial demand. (that said, these may not be the kind of components you want in specific high-performance applications.)

The main thing a particle accelerator requires the most is electricity. Any mature power production facility can supply the power.

However, all particle beams require at the least a source of neutralizing mass. The projection of a charged beam will leave the projecting object oppositely charged. In an atmosphere the natural plasma formed by the beam will ground everything out, but in space the diffuse vacuum will not do the job.

Furthermore particle beams using ions also require a feedstock for their ions to produce from, and in an appropriate format. This can be a gas, a fluid or an easily ionized solid. It can be the feedstock species in its pure form or in an easily split molecule, though every complexity adds to the amount of preparation work that has to be done before the feedstock enters the particle beam.

Don’t worry about the mass of your source or neutralizing ions though. Large particle accelerators, like the LHC in Switzerland, only consume tiny gas bottles worth of feed gas every couple years. The mass of a single 5.56 NATO bullet worth of hydrogen could feed a 10 GW, 10 GeV ion beam for nearly 4 days of firing! You will not be running out of beam feedstock easily.

== Cooling ==
Cooling a particle accelerator can be a serious challenge. Various bits of a particle accelerator love to be supercooled for efficiency or basic working principle - any bit of superconductor used in the accelerator, or magnets, or EM generators, will require cooling to its generally supercool operating temperature.

Even if Standard Temperature and Pressure Superconductors (273 Kelvin, 100 Kilopascal) are available, present superconductors have demonstrated superior performance by being supercooled. That means to achieve even stronger fields, even STP superconductors may be cooled to cryogenic conditions.

Cooling cryogenic systems sucks. Thermodynamics shows itself from its worst side. You can forget running cryogenic radiators - the Stefan-Boltzman Law penalizes the direct radiative cooling of anything cryogenic severely as much as it empowers very high-temperature radiators. This only leaves stages of heat pumps to provide cryogenic fluids that can in turn, cool cryogenic systems. Heat pumps draw power, and for such a giant temperature gradient as we can have here, they will draw a lot of power.

However, we have other options. Cryogenic systems also tend to be highly efficient, so while they do heat up, they do so very slowly and there is little overall energy absorbed. This provides the option of running off of bunkered cryogenic fluid in dewars for the duration of operating the equipment and not running any heat pumps all the time.

Other bits of a particle accelerator system aren’t as required, but really nothing is a fan of high temperatures. Going higher than room temperature can be treated as an oddity. High-performance lasers may not require cryogenic conditions, but they certainly like it cool. Radiofrequency producers of any kind generally fare worse with rising temperatures. That means cooling overall remains a challenge - 300 Kelvin is still not that great of a hot loop temperature for radiators.

== Safety <ref>Thomas Otto, "Safety for particle accelerators", Springer 2021, Available online at: https://library.oapen.org/handle/20.500.12657/46120</ref>==
=== Ionizing Radiation ===
There are surprisingly many sources of ionizing radiation in a particle accelerator, besides the obvious threat of the output beam itself. Not all particles are accelerated correctly or in the correct direction. When such particles interact with the structure of the accelerator, they release a shower of secondary ionizing radiation. In electron accelerators and sources, stray electrons may be accelerated and release x-rays from bremsstrahlung as they collide with surrounding matter. The resulting doses can be significant and dangerous. Also, such collisions leave behind residual activation, making the accelerator a radiation hazard for some time even after it has been shut down. Generally, people should not work near the particle accelerator while it is functioning, and radiation barriers isolate the core accelerator from adjacent facilities. (In real-life facilities, you will often see accelerators inside larger buildings being cocooned inside doped concrete plates to shield the rest of the room.)

=== Radiofrequency ===
RF energy is used in many kinds of particle accelerators. This energy may leak from the structures it is generated, contained and used in due to damages or incorrect assembly. As microwave radiation and above is used, the radiation can be readily absorbed by bodies, hydrogen-rich materials or other objects. The end result of this can be rapid heating and thermal effects. Especially in accelerators designed to throughput a lot of energy (say a GW or more), this may become a serious issue if a waveguide is damaged. It’s not recommendable to stand in front of a multi-MW radar either!

=== High-powered electrical and electronic systems ===
The operation of the magnetic systems and accelerators often involves high-powered electrical and electronic systems. High voltages and/or amperes are involved. This poses risks like shock hazards, electrical arcs, and electrical fires. Some electrical systems such as capacitors or inductors may have combustive or explosive failure modes.

=== Flammables ===
Parts of an accelerator or its infrastructure may be manufactured from flammable substances such as hydrocarbons. In the case of an ignition source such as a high-powered beam cutting into a physical substance or an electrical fault, these may catch fire.

=== Lasers ===
Some accelerator systems use high-powered lasers. Laser plasma accelerators use very high-powered pulse lasers to provide pumping power. The safety concerns of lasers for blinding, physical damage, and possible exposure to ionizing radiation as a secondary effect of the beam impact, apply. Some high-powered lasers can also drive electrons which generate strong EMPs, which may damage or destroy other systems in turn. Parts of a laser assembly or laser beam guide may shatter if exposed to overloading conditions.

=== Magnetic systems ===
Parts of a particle accelerator may use strong magnets, up to superconductive magnets with multi-Tesla fields. While the magnets are integrated into chokes that shape the magnetic fields, fringe fields invariably extend beyond the core magnet installation.

Multi-Tesla transient electromagnetic field can affect the human nervous system temporarily, inducing nausea, vertigo, metallic taste and optical phenomena, if the person moves in relation to the magnetic field. Otherwise, the risk of exposure is low. That said, electronic systems that are not shielded properly, such as say, implants, could be adversely affected by the magnetic fields.

The biggest physical risk are reactions of other magnetic objects with the magnets. You can look up cases where metallic objects interacted with the multi-Tesla fields of MRI scanners. Such parts may fly across the room towards the magnet at increasing speeds, and damage anything or injure anyone in their path, possibly ending with the object or person squashed (or impaled!) to the magnet.

Also, large superconductive magnets store not insignificant amounts of energy. If the superconductive property breaks down, the magnet quenches and becomes a resistor. The resistance site heats up as current tries to flow through it, expanding the quench and possibly being heated until it sublimates into gas. This could possibly happen explosively. In such a case, up to many GJ of energy might be unleashed in a conflagration or explosion (subsonic or supersonic), damaging surrounding systems and ejecting shrapnell.

=== Cryogenic cooling systems ===
Particle accelerators often involve cryogenically cooled systems. This means cryogenic fluids like nitrogen and helium are present in the accelerator infrastructure. Such gasses have both a very low temperature and a high ratio of gaseous volume at high temperatures to their liquid volume.

Exposure to the leaked fluid can lead to cryogenic cold damage, injury, and death. Furthermore such gasses are efficient at displacing ambient breathing gasses such as oxygen, so they pose an invisible, and rapidly spreading, suffocation hazard.

Also, if insulation is damaged, ambient heat may leak into the cryogenic circuits, and cryogenic fluid may begin evaporating. This can lead to dramatic rises in internal pressure. Relief valves should trigger to safely direct such pressure spikes away from important places and keep the system from failing catastrophically, but otherwise, pipes and cryogenic storage devices may fail, up to explosively.

=== Beam excursions ===
If the magnets responsible for manipulating the beam break down, the beam may excurse from its designed paths. The beam may become defocused, or maintain its focus. The beam begins cutting into the accelerator structure, evaporating material and depositing energy. This will definitely foul the ultra-high vacuum inside the accelerator, leading to secondary radiation production as the beam collides with gas inside the accelerator. Some sensitive facilities requiring ultra-high vacuum such as some RF cavities may “crash” and become ruined. Further systems may become damaged or destroyed from either physical stress or the radiation pulse. If the beam bunches contain a lot of energy and maybe are heavy particles, the deposition may lead to an energetic local explosion. Alternatively the beam may escape and cut into structure beyond the particle accelerator itself, damaging and destroying equipment and seriously hurting or killing personnel.

In the worst cases, a beam excursion on a very powerful beam may lead to an explosion that destroys the accelerator utterly. Even in a less bad case, the radiation pulse might damage or destroy critical equipment around the accelerator and take the entire facility offline.

=== Beam dumping ===
If any other systems fail, the accelerator probably has to be shut down. This poses a problem - what do you do with the beam energy that is still in the accelerator system?

Sometimes, the beam is designed to leave the accelerator immediately, and shutting down the beam is easy. But in the case of cyclotrons and storage rings, the beam may circulate for a while. What do you do then? You have to dump the beam from the accelerator. Special beam stops are used on research and industrial accelerators, large blocks of dense material able to absorb the beam. During this process the blocks both heat up and become radioactive. For particle beams storing very high energies, this may become an increasing challenge. How do you safely dumb a beam containing many GJ or more? In some cases you may have no other choice but to dump the beam out into the environment. Where and when can you do this safely without threatening third parties?

= Application areas =
Particle accelerators have many, many, many areas of application.

The public examples of particle accelerators are the big research accelerators used to create conditions with which one can interrogate fundamental physics. These accelerators are after specific particles under specific, as finely controlled as possible, conditions.

However, particle accelerators are also used in imaging. The electron microscope is a classical particle accelerator-based imaging system, using the fact that electron are very tiny particles to bypass classical imaging limitations of photons. Thanks to them, the biological and material science world can be interrogated in great detail, down to single atoms, and advances in the sensors and processing software keep pushing the boundaries.

On a bigger scale, particle accelerators can be used to produce high-energy x-rays with very small wavelengths. The x-rays are projected at prepared crystaline targets. The x-rays interact with the electrons in the crystals, giving information about bond structures. Such crystallography has great importance in biochemistry, but also material science.

Similiarly electrons, ions and neutrons can be used to probe at the structure of materials of interest. Their different behaviors allow for surface or in-depth scanning with particular properties.

In manufacturing, particle beams can be used both to shape materials and to image them for quality control. Particle beams can deliver thermal energy to cut apart materials or weld them. In additive manufacture, particle beams can provide fine precision melting or even meld together volumetric composites with penetrating beams. At fine scales, precision beams can break down gasous feedswtock to deposit materials in micrometer-sized structures. Directed ion guns implant traces of other elements into surfaces for manufacture.

Industries have a certain demand for specific isotopes. Accelerator-driven manufacture of isotopes covers these needs.

And in medicine, particle accelerators of ever-greater sophistication find increasing use in cancer treatment and beyond, destroying life-threatening tissues with ever-reduced doses to surrounding healthy tissue.

For military purposes, particle beams have been probed for a long time, especially for use as ground- or space-based weapons to shoot down incoming intercontinental balistic missiles. Particle beams can be made to propagate well enough both in air and space with various methods. They can deliver ionizing radiation that damages electronics, cause fission and breakdown in nuclear materials, or deliver sufficient thermal energy in a tight spot to destroy hard materials. With modern and future technologies, particle beams can offer equal and superior performance characteristics for effective range as lasers, and with various target interactions paradigmas of interest. They can also play a role as part of sensors, producing high-energy radiation from their target interactions that sensors can pick up and analyze.

= Further material =
* [[Particle Beamed Propulsion]]
* [[Plasma Guns]]
* [[Particle Beam Weapons]]

== External reading ==
The US Particle Accelerator School posts material of its courses publically at times: https://uspas.fnal.gov/materials/materials-table.shtml

= Sources =

== Authors ==
Sevoris Doe, Gerrit Bruhaug, Luke Campbell.

== References ==

Particle Accelerators

2022-08-18T12:50:33Z

Sevoris: /* Application areas */

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Particle accelerators are an incredibly useful piece of technology and a diverse set of artifacts produced for many purposes. While particle accelerators are often invoked as weapons of war in science fiction, and have captured the public attention as giant research systems the size of cities, they have an incredible widespread range of applications. There are also many, many ways to build a particle accelerator, depending on what effects you need to achieve and what circumstances you work under.

This article aims to be a general primer on particle accelerators, their physics and a selection of technologies and construction styles. In detail, all particle accelerators are high-tech engineering customized to their circumstances in many, many aspects.

Further pages covering application fields are in the work.

= Accelerator physics =
The simple explanation is that we use electric fields to accelerate charged particles. But from this basic principle of leveraging the electromagnetic forces springs an entire set of accelerator technologies with different performances, trade-offs and characteristics. The most basic particle accelerators are just a big high-voltage source and can still be found in old CRT TV’s and dentist x-ray machines around the world. Things rapidly get more complex from there, but first we need to lay out some terms so we can all have a common conversation.

The first thing to keep in mind with particle accelerators is that we are talking about charged beams here, even if we neutralize them later. So no neutrons, photons, or other innately neutral particles. Your choices range from the simple electron up to anti-uranium and more, but they all must be charged. The particle of choice determines just about everything else about the accelerator and any potential uses. Really heavy particles (like large ions) don’t accelerate easily and take a lot of energy to get relativistic enough for time dilation tricks to take into place. Really light particles (like electrons) accelerate super easily, but can have other radiation problems to worry about.

The second most important thing to know about a given accelerator is the particle energy. This is a measurement of the kinetic energy of any individual electron, ion, whatever in the beam and is measured in a unit called the electron-volt or eV for short. The lower bound for any worthwhile particle accelerator is in the MeV (or million eV) range and we typically want a lot more! Inside of the accelerator we tend to talk about the electric field doing the accelerating (measured in Volts/meter). For singly charged particles (like electrons) this gets really to figure out the energy because of how the eV is defined. We just take the electric field (also called the accelerating gradient) and multiply by the length of the machine! Boom, now you have particle energy! For more complex particles you have to do some scaling based on mass and charge state.

Also a quick trick to help you with deciding if something is relativistic or not is to divide the particle energy <math>E</math> by the mass energy <math>M</math> of the accelerated particle (0.511 MeV for electrons/positrons and about 1 GeV for a proton) and add one. The resulting number is called the Lorentz gamma value and is used all over in accelerator engineering.
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>\gamma = 1+\frac{E}{M}</math></div>
Once you get a value much larger than 1, you have relativistic particles. Once that number is in the hundreds you get an ultra-relativistic beam. Figure 1 helps show the relationship between Lorentz gamma and speed of the particle.

[[File:LorentzGvsSped.svg|900px|thumb|center|Lorentz gamma versus fraction of the speed of light]]

The third most important feature of a particle accelerator is the emittance which is just a fancy term for divergence of the beam and can be thought of similar to the wavelength of lasers. No beam is made perfect and they will naturally want to expand outwards. The smaller the emittance, the longer the beam will stay in a tight spot. Emittance can be altered with a wide variety of beam engineering techniques that take entire books to discuss, but one other cheating thing you can do is drive the beam energy higher. This naturally lowers the emittance by a factor of Lorentz gamma due to the cheating effect of special relativity! Thank god for Einstein!

== Acceleration forces ==
Every accelerator tries to build an electric gradient appropriate for the charge of the particles it is accelerating. The greater the electric gradient, the bigger the amount of work that can be done, the more energy is imparted on a particle as it travels through the gradient, and thus the higher the acceleration. The simplest way to build up a gradient is simply two charged electrodes with a strong potential between them. Many other accelerator methods use electromagnetic fields, in the form of waves confined in electrically conductive structures. In this case we exploit that electromagnetic fields have both an electric and a magnetic component. The magnetic component cannot be used to accelerate particles (only bend their trajectory - which can be exploited for effects like deflection or focusing) but as the electromagnetic waves oscillate, so does an electric field. This naturally creates a charge difference across half of the phase of an electromagnetic wave - this electrical gradient can accelerate particles. Both of these technologies encounter problems with the physical materials asked to contain the electric or electromagnetic energy. Eventually the fields get so intense that solid materials get broken down and an arc is formed. As such, various advanced concepts use clouds of charged particles and plasmas, which are not subject to the material limitations of solid matter.

The strong and weak nuclear force as well as gravity (which are all of the forces we know of at the time of writing) are not suitable for accelerating particles. This is related to the power and range of these forces. The strong and weak nuclear force have extremely short ranges - even if we had a means to artificially shape them like with electromagnets, their range would be too short. Gravity has the range but is incredibly weak. To generate a measly 1 G of acceleration, 9.81 m/s², requires the mass-energy of the entire earth, some 5.97e24 kg of mass, to do the job! The acceleration involved in particle accelerators is much, much, much larger (on the order of 10^13 times larger or more).

{{MascotSays|Interestingly, one source of very high-energy particles are giant gravity wells. Black holes generate conditions around them that can eject particles with enormous energies. The jets of particles that shoot from the north and south poles of many such stellar entities are particle accelerators.}}

If you want to use a force other than electromagnetism to accelerate particles, it’ll have to be a fictional one. The forcefields of various sci-fi properties could be considered, since they can often apply forces to particles. If you propagate a wave of such forcefields down a tube, you could have something quite a lot like many particle accelerator systems!

== The particle bunch ==
Essentially all high energy particle beams are not constant streams of particles, but a time sequence of discrete groups of particles, commonly referred to as “bunches”. These bunches can be described both in their distribution in lateral space to the traveling direction (x and y axis) and longitude (z axis) along the traveling direction, anchored on an origin point (0,0,0) that moves in time.

[[File:ParticleBunch.png|center|900px]]

Bunches can have various profiles. Generally they are somewhat distributed in the lateral direction just because the charged particles have slightly different starting positions and lateral velocities (also called transverse emittance). In longitudinal space, differing acceleration experienced by individual particles as well as dynamics within the bunch scatter them out, with “noses” and “tails” that contain fewer particles. The end results is that most beam profiles follow a gaussian distribution, though there are exceptions.

This bunching behavior allows us to analyze other properties of particle beams very similar to lasers. There is a “pulse frequency” given in Hertz. Each bunch contains a certain energy, which can be large enough that we give it in Joules and not electronvolts. There is an average output energy over a given timespan, given in Watts. We have a beam peak power, given by how quickly a bunch arrives at a target and deposits its energy, which we give in Watts. Related to this we also have a related peak intensity depending on how much area/volume the beam is deposited into, with units of either W/m² or W/m³. If we have charged particles being accelerated, we get a beam voltage, the same way as in a wire. Beams can also have a temperature, the same way as any other group of particles with random motion from their energy, given in Kelvin or eV.

The systems that generate particle beams of course have efficiencies, with the wall plug efficiency (power inserted into the entire system versus power deposited in the particle beam) being the most interesting for first-order engineering. They also have an operating temperature.

== Particles for acceleration ==
With the constraint that our particles must have an electric charge in order to be accelerated with anything but fantastic technology, we can discuss quite a large family of particles, from the conventional to the exotic. They differ in the applications they can be interesting for. Some can be similar, but no two particles behave entirely alike.

=== Electrons ===
The ubiquitous electron is the most common particle to be accelerated and the first one ever pushed up near the speed of light by humans. Due to its excellent charge to mass ratio (the best possible due to the extremely low mass and charge of 1e) electrons are very easy to get up to relativistic speeds. Electron accelerators for radiography and sterilization are often portable by a single person and not much larger than a laptop in size (although they can be much bigger for power output reasons). The ease in which electrons reach relativistic speeds means that electron RF accelerators can be designed with no speed changes assumed (since the electrons are already moving at effectively light speed) and thus tend to be highly efficient and have the highest acceleration gradients achievable. For similar reasons electrons are also able to be accelerated in plasma based accelerators, allowing for even higher acceleration gradients and smaller machines for a given energy. The major downside of electrons is that due to their large charge to mass ratio they will radiate energy like crazy when bent by a magnetic field. Thus ring shaped accelerators are of limited use for electrons since you have to make the ring extremely large to not just lose all of your energy as beams of x-rays coming out of the machine! This very trick is used to make high quality x-ray sources for industrial and research applications.

When electrons hit matter they tend to release their energy as photons in a process called bremsstrahlung, or braking radiation. These extremely high energy photons (roughly ~⅓ the peak energy of the electron on average) then proceed to make electron/positron pairs that then make photons that then make pairs that then make photons… This whole thing is a massive cascade of radiation that ionizes, heats and can even damage on a nuclear level! At high enough energies you will generate every particle known and might even make some new ones!

=== Proton ===
Protons are the next most common particle to be accelerated and are currently used by the absolute largest particle accelerator ever made (the Large Hadron Collider or LHC) as the primary particle of choice. Since they have ~2000X lower charge to mass ratio than electrons, protons are about ~2000X harder to get up to speed. Unlike electrons they are not promptly at light speed so we have to send protons through a series of various accelerators depending on the energy of choice. For example the protons in the LHC the protons go through 5 different particle accelerators plus the extremely low energy stage that first makes the beam of ions! This difficulty in reaching highly relativistic speeds has two other bad side effects. Firstly, if we want to transmit our protons through empty space we will probably need to neutralize them with a beam of electrons.

{{Note|''When shooting extremely high current (kA), relatively low energy (not highly relativistic or in some cases relativistic at all) beams in the atmosphere the higher mass and positive charge of protons gives them many beam propagation advantages over electrons if the beams can be made. In this case no neutralization is required as the atmosphere itself becomes a plasma and neutralizes the beam.''}}
Luckily this is not very hard and will add almost nothing to your comparatively large proton accelerator, but may increase your beam spread a little bit via some neutralization physics. This does mean your beam is technically plasma, which is cool! Second, we effectively cannot use those neat plasma based accelerators for protons since we would need nearly TeV class proton beams to inject into the plasma accelerator and by the time you reach those energies you have basically already done all of the work. Maybe in the future there will be ways around this, but as of right now we are limited! There is one major benefit to this poor charge to mass ratio though, and that is the lack of synchrotron radiation! Ring shaped accelerators are ideal for protons, and basically all ions.

When protons hit matter they can interact with all of the four fundamental forces, although only two of them (Strong Nuclear and Electromagnetism) tend to play big roles. Protons barrel on in causing huge ionization cascades and can also release bremsstrahlung photons at high enough energies. They are also ideal at knocking neutrons, protons, deuterons and more out of the target nuclei which just adds to the fun! Protons also come with a neat trick known as the Bragg peak, where they deposit a large portion of their energy right at the end of their trajectory through a target material. This is a very useful trick for medicine and means you can “tune” the beam to almost ignore large portions of material and primarily heat, ionize, and destroy the area of interest. You can imagine how useful this is for cancer therapy or more nefarious things!

=== Light ions ===
Light ions are a rather fuzzy term but here we will use the definition of any ion heavier than a proton but lighter cobalt. The fuzzy reasoning here is that you get on the other end of the nuclear binding curve (i.e. can’t get net energy from fusion) at iron, so cobalt will then no longer be “light” and has to be made in a different process. Feel free to pick your own cut off! Historically beams up to neon have been experimented with for nuclear fission, nuclear fusion and medical therapy applications. Light ions have even worse charge to mass ratios than protons (Z/A times worse) and typically you will find light ions have about ½ the charge to mass ratio of a proton on average, although the isotope 3He is a notable exception. This means they are even slower to accelerate, although often the exact same machines as protons can be used with minor tweaks. Light ions have similar requirements for neutralization as protons, although their heavier mass can help fight beam bloom for a similar beam speed (not energy since the increased mass means the particles are higher energy for a given speed). The decreased charge to mass ratio also means that synchrotron radiation is even lower so beams can be driven to higher speeds before radiating. Light ions have similar target interaction physics as protons, although the ion itself will break up at very low speeds and add to the radiation environment.

=== Heavy ions ===
Heavy ions are being defined here as cobalt up to whatever unstable element you can jam into your accelerator! Typically heavy ion beams are made of elements like lead (used in some runs on the LHC), gold (used in the Relativistic Heavy Ion Collider) and even californium (used in some runs on the Argonne Tandem Linac Accelerator System). These beams tend to be for pure nuclear physics research, but there has been a long standing interest in heavy ion beams for inertial confinement fusion research and as potential methods of propelling spacecraft. These extremely heavy ions will be even harder to accelerate to ultra high speeds than light ions and much slower for a given energy, but they do provide a lot of momentum to make up for that! They have the same neutralization requirements and same potential upsides for beam bloom as light ions, but will most likely be orders of magnitude slower reaching our targets than the lighter beams. Heavy ions can also be used in the same machines as protons with small modifications and generate almost no synchrotron radiation as a result of their much smaller charge to mass ratio. They also have similar interaction physics to protons and light ions, but will stop in much thinner layers of materials. While electron, proton and light ion beams tend to both mechanically destroy and irradiate targets through-out, heavy ion beams will act much more like lasers and burn in from the outside of the target on in. There will still be a spray of ionizing radiation, but far less of one than an equivalent energy electron, proton, or light ion beam. The much higher momentum per particle, lower radiation concerns and ease of deflection makes heavy ion beams very interesting for beamed propulsion purposes.

=== Macrons ===
Macrons are typically defined as macroscopic particles, such as extremely large molecules or even grains of sand, that are charged up and accelerated via similar technologies as traditional particle accelerators. There are many difficulties with getting macrons up to appreciable speeds due to the extremely low charge to mass ratio, engineering difficulties with getting consistent and good charging, phase matching with accelerating cavities and lack of effective focusing optics. There are so called “dust accelerators” in existence that utilize the first stage of ion accelerators to generate their “beams” and proposals have been made to create better versions using updated technology. Macrons will interact just like a very small and very fast bullet unless they are designed to carry a nuclear payload to boost this output. Antimatter is an assured option for getting energy gain from a macron impacting a target, although there are potentially options for fusion, fission and fission/fusion given the correct sizes (mm scale or bigger), clever target design and the right impact speeds (most likely >>100 km/sec). The various energy gains versus speeds can be seen here:

[[File:IvKmacron.svg|900px|thumb|center]]

It can be seen that antimatter proves to be worth it up near the speed of light, but both fission and fusion stop providing much gain well below the relativistic regime. This also helps explain why heavy ion uranium beams are not going to provide some incredible advantage over any other ion.

=== Antimatter ===

Antimatter is the evil cousin of normal matter and whenever the two meet a release of energy equivalent to their combined mass energy follows. Now this does sound like an amazing option for beams but there are two downsides. First, antimatter is energetically expensive to make! Positrons (antimatter electrons) are the “cheapest” at 1.02 MeV of input energy at a theoretical best, but they only give you that much energy back when hitting a target. Since a typical positron beam can be expected to be equivalent to an electron beam, there is little gain in adding 1.02 MeV per particle to a multiple GeV (or even TeV) beam. You can see this explicitly:

[[File:IvKhBarPostron.svg|900px|thumb|center]]

For anti-protons the math becomes more favorable, but creating these particles is much more difficult and energy consuming. At best anti-protons require ~2 GeV of input energy to create, but in practice >400 GeV per anti-proton has been needed. Luckily significant energy gain is had up to several GeV of beam energy, so if anti-protons can be made at a central facility of some kind and easily stored (neither being easy things) then there may be some advantage to such a beam. Heavier ions are significantly harder to make since fusion with antimatter must be performed to generate those ions. Barring a naturally occurring source being found, antimatter heavier than a proton is most likely not worth using.

{{Note|'' Positrons can potentially fill an interesting niche either as a co-propagating neutralizing beam for electrons or perhaps as a higher performance atmospheric beam than electrons due to the positive charge. In this case the annihilation properties are no what is of interest, rather the positive charge and relative ease of acceleration compared to ions. Since positrons share all properties with electrons except for charge, they can be accelerated just as easily and even co-accelerated with electrons in the same machine.}}

=== Muons ===
Muons are an interesting cousin of electrons and positrons. They are unstable and heavy cousins of electrons, massing about 200X more than electrons and living about 2 microseconds in the lab frame before decaying into an electron (or positron) and a neutrino. If we accelerate them up to relativistic speeds (about 10X easier than protons and the muons are typically born relativistically thanks to how we make them) then the muons can be made to “live” far longer. Muons have long been of interest to accelerator physicists due to their position as an almost ideal intermediary between electrons and protons. The requirement to make them on site has always been an impediment though and muons are currently only used for certain niche experiments. During the planning stages for a potential muon collider it was also noted that a muon collider would also have a serious issue with neutrino radiation! This is due to how neutrinos are more likely to interact with matter at high energies and a muon collider would be generating a LOT of really high energy neutrinos in a ring around the machine. This has been proposed as a sort of super weapon even!

Muons should interact with matter in a similar fashion to electron beams and the beams will probably be treated in a similar manner. They can propagate through the atmosphere quite well in theory and should outperform electron and proton beams for a given beam energy, although at quite a cost… Muons are typically generated with proton beams impacting targets and will create an equal number of positive and negative muons at high energies that must be carefully captured. This would allow for the easy creation of a neutral muon beam though, which may make for a great giant doom beam! Do note that negative muons also can help catalyze nuclear fusion so a very high power beam hitting a fusion fuel tank could produce some rather… Interesting results!

=== Other exotic options ===
There are a wide variety of odd particles that could also be chosen for use in beams given a specific need for them, although generation and utilization of these particles will be quite difficult! Extremely short lived particles like pions, kaons and taus can be generated and accelerated up to extreme speeds to help propagate them far enough to matter, although that will need some very extreme speeds. At the upper end of extreme there are even proposals for accelerating tiny charged blackholes!

= Accelerator technologies =
There are a wide variety of accelerator technologies and here we try to summarize them as best as possible. Different accelerator types may be sequenced one after another. Some particle accelerator technologies depend on the particles fulfilling certain requirements of speed (thus energy) and hence different systems may be employed in sequence. Accelerator systems are described in terms of their efficiency of turning input power into particle energy, their acceleration gradient measured in electric field which translates to volts per meter, and the mass per length meter. An interesting secondary parameter is their operating temperature. Some accelerators can operate well at high temperatures. Others require cold or cryogenic conditions to work well.

== Core accelerator assembly ==
We define the core accelerator assembly as the structure where energy is imparted onto the particles.

=== DC Accelerators ===
For very low beam energies we can utilize large DC electric fields between plates or grids to accelerate particles up to speed. This is the same technique as is used in ion drives and is the most common industrial accelerator used. Every dentist's office and old CRT television uses this acceleration technique. Ion accelerators often also use a DC accelerator to generate the initial beam of ions, while electron beams utilize other techniques. These machines are limited by material breakdown to about 0.5 MV/m accelerating gradient and typically no more than a couple of MeV total particle energy for extremely large machines. The highest energy DC accelerators can reach 40 MeV for proton beams, but this is for machines massing in the 100s of tons and at low total beam powers. Very large DC accelerators are currently the only tested method for accelerating macrons up to 10’s of km/sec.These accelerators are limited by the risk of arcs jumping between the electrodes at too high voltage differentials.

=== Radiofrequency Quadrupole ===
A more advanced first stage accelerator for ion beams is the radiofrequency quadrupole or RFQ. These machines utilize low frequency radiowaves and carefully designed conductive bar structures to efficiently generate high current ion beams up to energies of ~7 MeV in a couple of meters. RFQs were a revolution in ion accelerator physics and caused quite a stir when the USSR first showed them off as they provided a much higher power and higher quality ion source in a smaller package than anything the USA had built! RFQs have also been proposed as methods of efficiently accelerating macrons up to >100 km/sec. These machines can be normal or superconducting and can operate with efficiencies as high as RF LINACs in theory.

=== Cyclotrons and Betatrons ===
One of the earliest ways to accelerate particles up to relativistic speeds is with circular accelerators known as cyclotrons (for ions) and betatrons (for electrons). In Cyclotrons the accelerator is a round chamber which houses two half-circle electrodes roughly shaped like a D (and thus called “dees”) separated by a gap, sandwiched between the poles of a large electromagnet. The magnet creates a static magnetic field which bends the path of charged particles into a circular motion. An oscillating alternative voltage is applied to the electrodes (what we often call RF or radiofrequency) to accelerate the ions. This radiofrequency field’s oscillation is matched so that one cycle of the voltage matches one circuit of the particles. As a consequence when crossing the gap from one electrode to the other, the charged ions experience a voltage gradient and are accelerated. As the particles gain energy and speed, their radius in the magnetic field gets larger. The result is an overall spiraling trajectory from an injection point in the center to an exit point at the rim where the charged particles are bent out of the cyclotron. Cyclotrons use the same voltage gradient many times over, defeating the electrostatic breakdown that limits the single-time electrostatic accelerators. As a consequence, proton energies of up to 70 MeV may be reached. Cyclotrons energies can also be increased by switching from normal magnets and RF resonators to superconductive components. This eliminates various losses from electric resistance and absorption and also allows for much stronger bending magnets, which allows the cyclotron to contain particles up to a higher energy if desired. Superconducting cyclotrons have achieved up to 520 MeV. Still, cyclotrons suffer from poor beam current, pulse rate, and beam quality issues that make them a less than ideal choice. The large magnet required (measured in meters) also does not lend itself to a lightweight weapon and as such, cyclotrons are regulated to niche medical and physics uses.

A close cousin is the betatron, which is used to accelerate electrons. In this case the magnetic field strength is increased as the electrons speed up, which prevents any spiraling trajectories of the particles. Betatrons have been operated up to electrons energies of 300 MeV, but are more typically in the ~10 MeV range. In principle a betatron could also be made superconducting, although magnet ramping may prove to be difficult. Other types of ring shaped accelerators may prove to be more useful, although electrons are heavily limited by synchrotron radiation and it is often better to use linear accelerators for them. For now betatrons remain limited to small (sometimes portable) accelerators for nuclear physics, industry and security work.

=== Radiofrequency Linear Accelerator ===
The real work-horse of the particle accelerator world is the RF linear accelerator. These babies use the same alternating field trick that cyclotrons use, but in a straight line rather than needing to bend the beam around in a circle. This provides a couple of nice advantages over the more traditional cyclotron. First and foremost you get to leave that large, bulky magnet behind which saves you a ton of mass, cooling and power consumption! You also get to now have much higher RF frequencies (1-10 GHz right now compared to the MHz used in cyclotrons), which means higher accelerating gradients. To generate these powerful, high frequency RF waves (also known as microwaves) that drive the accelerator we typically use military radar power supplies or (in more advanced cases) even lower energy particle accelerators! State of the art RF LINAC technology is well into the 100 MV/m range and have no upper limit on beam energy! The Stanford Linear Accelerator reaches 50 GeV and was built back in the 1960’s! There have been traditional RF LINAC designs for machines reaching up to 1 TeV, with the only limit being how much space you have to put these monsters. They do operate best (i.e. with high accelerating gradients and efficiencies) with particles already moving at relativistic speeds, so any ions fed in need to use a series of lower gradients LINACs or other pre-accelerators. Electrons and positrons can naturally be used in LINACs since they accelerate up to relativistic speeds so quickly.

LINACs also have been shown to operate at high average powers (MW scale is now in use) and high pulse energy (10’s of kJ pulses fired at high rep rate) while also maintaining a high quality beam that stays well focused. The two newest advancements have been to push the frequency ever higher for higher acceleration gradients (300+ MV/m has been shown and there are plans for even more) and to switch to superconducting RF accelerating cavities (the name of the actual piece of equipment that does the accelerating) which then dramatically increases the electrical to beam efficiency (80-90% has been shown and even copper waveguides have been operated at 70%). All in all, LINACs beat the tar out of cyclotrons which is why we don’t build cyclotrons anymore. To quote an old Fermilab engineer “cyclotrons are an expensive and bulky way to get a really crappy beam or a cheap way to get a giant chunk of lightly radioactive steel that can never leave your lab”.

LINACs work in detail by shaping the radiofrequency energy in a confining waveguide. The oscillations of the RF energy may be held in space, or allowed to travel down the waveguide over time. Radiofrequency waves have connected magnetic and electric components, which oscillate.

[[File:2022 08 16 GalLib ElectromagneticWave.svg|center|800px]]

Inside the RF accelerator, the electromagnetic radiation is oriented 90 degrees to the particle travel direction, which points the associated electrical field in the travel direction of the particles.The radiofrequency field naturally oscilates along its wave, and the connected electrical and magnetic fields oscilate as well: building up, reaching a peak, and diminishing, inversing, building up, reaching a peak, and diminishing, and the cycle repeats. The detailed behavior depends on the specific cavity.

[[File:CavityWithEM.png|600px|thumb|A resonant RF acceleration cavity and the electric field inside of it, created by resonating RF fields, at one moment in time.]]

Particles of the right charge in the right space at the right time experience an acceleration from the electrical field, being drawn to the opposite electric charge. If they land in the other region, they are instead decelerated. Inbetween, particles experience less to no acceleration. This behavior naturally separates a stream of particles into discrete '''bunches'''.

[[File:CavityWithEMAndParticles.svg|600px|thumb|Bunches being accelerated inside the electric field. ]]

It also creates an implicit limit to what can be effectively accelerated using a specific radiofrequency accelerator design. The particles driven through it have to keep pace with the oscillations of the electric field, reaching the next correct region inside the cavity at that time when the electric field is oriented correctly. If they are too slow to keep inside the accelerating regions, the accelerating action will be ineffective. This requires RF cavities to be matched to the particle's velocity gain under energy imparture and their velocity when entering, being inside, and exiting the accelerator. For electrons and positrons, this isn’t that complicated. Since these lightweight particles accelerate readily, they have few problems “staying in the pocket”. Particles that are slower to gain speed, such as ions and even larger particles, must however be accomodated. This in turn impacts how much acceleration and energy per meter can be fitted into the accelerator in the first place. The acceleration of heavy particles requires specifically designed and sequenced RF accelerators for specific regions of particle speed. This in turn complicates the design of the larger accelerator infrastructure, with multiple discrete stages getting the particles to the energy required so that the next stage in sequence can take over.

RF cavity accelerators are limited in their eV/m capability by the critical current at the containing waveguide. To increase the acceleration provided, more intense electrical fields must be built up. Eventually the electrical charge of the RF waves becomes so strong it can rip electrons out of the cavity walls, causing an arc to form and the cavity material to break down physically.

=== Advanced Accelerator technologies ===
Present accelerator technology caps out at around 250 MeV/m for the most overdriven designs. This poses issues for a variety of applications, where higher energies in shorter distances are desired. In science, "desktop" and "chipset" accelerators are of interest. When militarizing particle beams, some effects only become available at very high energies - but you want something compact enough to turret.

urrent and even near future accelerator isn’t really ready to supply that sort of scale quite yet, so we have to move on to the cutting edge of accelerator physics. There are three big options with sub technologies within these broad fields. We have extreme RF cavities using much higher frequencies, plasma accelerators using lasers for power and plasma accelerators using particle beams for power. Each technology provides a unique series of advantages and disadvantages, but fundamentally they are all a way to make linear accelerators that are much smaller.

==== Dielectric Wakefield Accelerator ====
LINACs using much higher frequencies than traditional LINACs have two fundamental problems to fight. The first is electrical breakdown and the second is generation of those mm-wave (10-100 GHz) or even THz (100-1000+ GHz) sources at sufficient power and efficiency to power a particle accelerator. As the frequency of the driving power goes up the electric field climbs up, which means the accelerating gradient gets larger (which is good) but also means that we run a higher risk of a huge arc forming in the accelerator and breaking it (which is bad). This is often fought by changing the accelerator cavity material from a metal (or superconductor) to a dielectric material like ceramic, or by using a coating of dielectric, which can sometimes earn them the moniker of Dielectric Wake Field Accelerator (DWFA). The issue of creating powerful and efficient enough sources of this higher frequency drive power is being actively pursued for a variety of fields (5G cell signals, beamed power, etc) but is still quite immature compared to the microwave RF sources of traditional LINACs. There are options using essentially "over-clocked" microwave sources, low energy particle beams, and even laser-plasmas. There is still a lot to be done here, but these sorts of problems are being actively worked on there are working accelerators using this technology. There is potential for GV/m or greater acceleration gradiatens and all of the same advantages of traditional RF LINACs (i.e. high powers, high fire rates, high energy pulses, quality beams) should stay the same. The wall plug efficiency is liable to be a bit lower, but that will be highly dependent on the efficiency of the source of these high frequency waves.

==== Laser-Plasma Accelerator ====
Plasma accelerators using lasers for power are a whole world in and of themselves. There are a wide variety of ways to accelerate particles with lasers and plasmas, but we will focus on the most studied technique known as Laser Plasma Accelerators (LPA). LPA accelerators utilize ultrashort (<1psec) and ultraintense (>10^18 W/cm^2) lasers to drive a plasma wake (thus sometimes earning them the name laser wakefield accelerator) and accelerate particles through the resulting charge differential. They have only shown the ability to accelerate electrons and positrons, but any ions that could be injected at the right time and at extremely relativistic speeds could conceivably be accelerated this way. In the operation of a typical LPA the ultraintense laser is fired into a jet of gas (turning it into plasma) or a channel of lightly pre-ionized plasma. The laser then blasts through this plasma like a boat through water and leaves a void behind it (the wake). This void ends up having a HUGE accelerating gradient (100 GV/m) and suffers no risk of electrical breakdown since the plasma is already broken down! A pre-accelerated bunch of particles can be injected into that void at the right time (called the witness bunch) for a BIG acceleration or you can rely on the wake itself spewing electrons out into the void and make a beam out of the plasma itself. The gas or plasma then quickly is reformed and the laser fires as soon as its cooling and charging allows. Keep in mind that even though the beams can hit incredible energies in short distances, the laser itself is not a pocket sized device. These lasers are room sized at minimum and not easy to shrink. The pulse lengths required are also much shorter than most of our hypothetical weapons-grade pulse lasers that may operate in the nanoseconds regime - dual-use may prove difficult. The intensities reached also pose significant issues for the laser hardware. If you don’t spread the energies out, it becomes quite possible you will break parts of your laser from the produced intensities.

Current LPA systems suffer from a couple of problems that are being actively worked on. First and foremost they are incredibly efficiency limited by the laser tech being used. There are plans to move on to much more efficient fiber lasers and eventually make LPA machines that operate in the 25% or greater wall plug efficiency, but that is not currently doable. These lasers also need to be fired at higher rates than the current ~1 Hz, but that is much more of a limit of laser engineering than accelerating engineering. LPA’s also have yet to link more than one accelerator together and the lasers are limited in just how much plasma they can accelerate through right now (~20 cm max) so the beam energy runs into a wall quite quickly. Aside from R&D on linking LPA’s together in a way similar to RF LINACs there are two other options. The acceleration gradient is higher the denser the plasma, so a move to denser plasmas and shorter wavelength lasers (to penetrate the plasma) could provide higher beam energies in similar distances. The other option is to use clever optical techniques like the so-called “flying focus” to allow for the laser to go through much longer plasmas. This could in theory allow for more efficient LWFA’s and single stage machines that provide ~1 TeV electron beams in meters! The final issue is beam quality. Even the best LPAs will not make beams that have as tight of a beam spot, energy spread, or emittance as a normal RF LINAC. This means they will end up needing some beam “cleaning” if you want to get a lot of range out of them.

LPAs are of engineering interest for very short wavelength X-ray free-electron lasers in a compact format. Currently these facilities require linear electron accelerators many kilometers in length. With an LPA as the electron accelerator, an XFEL able to image molecular structures and chemical reactions as they happen might be shrunken down to a much more compact format. (Sometimes this is discussed under the ultimate promise of the “benchtop” or even “chip-format”(sic!) XFEL)

==== Beam-Plasma Accelerator ====
Plasma accelerators using particle beams for power are very similar to LPAs in principle but use a particle beam rather than a laser. We typically just call these things Beam Plasma Accelerators (BPA) for short. The concept is very similar to the LPA in that a high intensity beam comes in and blasts a void into a plasma, which particles can then be accelerated in. BPA provides some interesting advantages over LPA in that you can use highly efficient superconducting RF LINACs (or synchrotrons if you want) to provide the particles at very high efficiency. They also have little in the way of plasma density limits and the accelerated beam takes on the characteristics of the beam driving the BPA. Thus little if any beam cleaning will be needed after acceleration. The big downside is the need for a relativistic particle beam to drive the effect, and typically one in the 100’s of MeV to low GeV range, depending on the particle used. Thus your final accelerator may be short, but you still won’t be making a handheld particle beam device.

== Accelerator Shapes ==
The accelerator technologies discussed above are the means by which we create situations for particles to gain energy. The accelerators become embedded in a larger structure that produces, directs, accelerates, even stores the particles until they are directed towards their final use.

There are two principle shapes for accelerator assemblies. Ones where the beampath is not closed onto itself, and designs where the beam path is closed on itself in part of the assembly, forming a loop.

=== Linear accelerator ===
Everything from electron generation to acceleration to conditioning is built in a straight line, one after another. This is the Linear Accelerator or LINAC. The main catch here is that there is only one single passage for the particles through the entire accelerator. Re-using accelerator hardware to drive the same particle to higher energies by repeated passage is out. The particles have to reach their desired energy in a single pass. Practically this puts limitations on the energy achievable with heavy particles in a LINAC. The length required gets impractically large. Even for electrons, lengths of hundreds of meters to kilometers may be required, unless advanced accelerators with high gradients are used. This said, every accelerator contains linear sections. Since so much high-performance accelerator technology depends on particles having a minimum velocity, ions must be “pre-conditioned” in a set of linear accelerators before the re-use of an accelerator through looped beam paths can be considered.

LINACs of low energy may be used in many applications. Medicinal they find use in radiation therapy and radiosurgery.

=== Synchrotrons ===
The other shape which has some interesting properties are accelerators where beampaths are looped onto themselves. These are called synchrotrons. Since beam paths are closed onto themselves, particles can pass through the same accelerator unit repeatedly, gaining more energy on every passage. This is advantageous for heavy particles which do not gain much speed on any single passage. They can pass through the same accelerator thousands or even millions of times instead. This is obviously advantageous for facility bulk and mass and allows us to reach energies that would otherwise be impractical. The accelerated beam bunches can also be held in storage, looping like the rotating mass of a flywheel, holding energy. The LHC can store some 700 Megajoules of energy in the recirculating beam after full speed has been reached.

That said, the devil is in the details. When charged particles are bent in circles they emit electromagnetic radiation which scales very rapidly (to the fourth power!) with relativistic gamma factor and with bending radius (inversely to the second power). This phenomenon is called synchrotron radiation. Due to the scaling, high energy particles emit more synchrotron radiation than lower energy ones and lighter particles (like electrons) at those high energies emit much more than heavier particles. In addition, the bigger you can make the bending radius of the synchrotron (often called “the ring”) the less synchrotron radiation you get.

[[File:SynchrVsE.svg|800px|center|thumb|Synchrotron radiation per particle versus particle energy for electrons and protons in a 50 meter in radius ring.]]

[[File:SynchroVsRing.svg|800px|center|thumb|Synchrotron radiation emission per particle for different particles at different energies, over varying bending radii. ]]

Although these numbers look small, keep in mind that this is PER PARTICLE. A typical pulse in an electron accelerator will have more than more than a billion electrons in it! Large synchrotrons like the LHC will have over 300 trillion ions circulating through it! Synchrotron radiation necessarily has to be managed to reduce radiation shielding, the radiation load (and possibility of hardware failure) on the accelerator, and to reach high energies in the first place - you might lose all the energy your accelerators are adding to synchrotron radiation!

Deflecting the beam also requires the application of magnetic fields, and higher degrees of deflection, in a smaller length, at higher particle energies, require stronger and stronger magnetic fields. Eventually multi-Tesla fields from superconductors will be required. And due to action and reaction, those magnets need to be braced against the pressure the beam exerts as it is deflected. Ultimately this leads to the same limitations as with flywheels - the mechanical strength of the braces and static connections that keep the magnets together as a larger structure is the accelerator’s limitation on J/kg held in storage. A further complication: The magnet strength has to be matched to particle energy. As a beam is accelerated, the deflectors have to ramp up their power to keep the degree of deflection constant.

Sometimes, synchrotron radiation from electrons is intentionally induced in a ring-shaped accelerator. Since synchrotron radiation of high-energy electrons tends to be hard x-rays emitted along narrow vectors, it’s a way to get x-rays used in various imaging techniques in material science and molecular physics.

Synchrotrons used for ions also need to respect the limitations of the involved accelerators. Synchrotrons often use radiofrequency accelerators. Since radiofrequency accelerators must match the speed and speed gain of their particles, an ion synchrotron cannot be made up of a single accelerator loop. Instead there are sequential loops connected in sequence, each designed to boost the particle through a specific velocity and energy region so that the next accelerator takes over. A designed example is the LHC. Particles headed for the main LHC actually circulate in two other synchrotrons beforehand, gaining sufficient energy so that the LHCs RF cavities can take over. (Also, before being injected into the first synchrotron ring, the ions must first go through multiple linear accelerators using different technologies to reach a base energy.)

An interesting design consideration: Synchrotrons don’t necessarily have to take the form of a (near-ideal) ring. Funamentally any dipole represents a single point of deflection of a straight path - they're polygons! Any geometric shape that contains a loop, and where any single bending curve keeps within the limits of the synchrotron radiation you are willing to tolerate, works. As such, complicated two- or three-dimensional “folded-up” shapes are also possible formats for a synchrotron.

=== Storage Rings ===
Since synchrotrons have to match their deflection magnet strength to the energy of the particles being accelerated, a single synchrotron track cannot hold more than one small section of beam bunches in storage at a time, and becomes unavailable to accelerate further bunches until it discharges the stored beam. This is disadvantageous for various applications where we may want to store a fully accelerated particle beam “bolt” for a longer time.

Storage rings are separate synchrotrons designed to hold multiple sets of bunches in circulation at a target energy at a time. The bunches are delivered from an accelerating synchrotron (sometimes called a booster ring) and held in circulation in the storage ring. Since there are synchrotron radiation losses, a few accelerator sections must compensate for these.

Storage rings can also be used to condition a stored beam. A common thing to do in storage rings is to “cool” a beam, draining unwanted movement energy out of the particles. Beam cooling may take significant time, which makes it ideal to conduct in a storage ring. The result is a beam with less tendency to drift apart, advantageous for many applications.

== Other parts ==
=== Particle sources ===
Particle sources are assemblies which supply particles to accelerators. Here we can group them into three categories: electron sources, ion sources, and exotic particle sources. The later involve the production of particles by means of gathering them from collision reactions (or the handwavium blackbox of your choice) and feeding them into another accelerator for further use. Positrons for example may be manufactured in beam collisions.

Electron sources require the production of a free stream of electrons. The oldest method here are thermionic sources. A hot piece of material ejects free electrons into a vacuum, where they can be drawn in using electric differentials and forwarded to the accelerator.

Field emission electron sources instead use a strong potential between a needle-like cathode and an anode plate with a hole. The electrons are drawn from the needle and through the anode due to the high charge differential. This method produces a “cooler” beam with less random divergence than thermionic sources.

A further improvement still are photoemission sources. Here the precise application of laser light onto a photosensitive material energizes electrons in the photocathode until they break free. This process is thermodynamically very precise and due to the nature of the laser light, also very controllable. The result is a low-emittance stream of electrons. These may be accelerated using electrostatic attraction with an anode at one end of the vacuum chamber, or directly using an RF field.

Ion beam sources are more complicated. The physics of ion formation are complex, which opens many design choices when creating an ion beam. Generally a plasma must be produced, from which the ions can be extracted.

The plasma can be formed in a variety of ways. Electrons from a local cathode-anode array, plasma frequency heating, the injection of electron beams, the application of laser light. The end results can be as diverse. Some methods only produce weakly charged ions and on many vectors, so with high emittance. Other methods offer greater ion charges or reduced emittance.

=== Magnets ===
While acceleration of charged particles is effected with electric fields, magnetic fields can be used to bend the trajectories of particle beams. As such, electromagnets can be used to bend the trajectories of charged particles in various ways, and in a controllable fashion.

==== Dipole Magnets ====
Dipole electromagnet are used to steer the entire particle beam in one axis. (To steer in two axes, we must either rotate one dipole about the beam axis, or couple two dipoles in sequence.) They can be used to direct a beam inside a particle accelerator, inject and extract beams out of looping constructs by switching on and off at the right moment, and point a beam at a target.

In a dipole magnet, a positive and negative coil oppose each other, embedded in a “choke” assembly often made of steel. Charged particles moving through the magnetic field between the coils experience a deflection in one direction specified by the polarization of the field, and a degree of deflection specific to the field strength.

==== Quadrupole Magnet ====
Dipoles can steer, but they influence all particles equally. As such, they cannot be used to manipulate the beam's shape. The first type of magnet able to influence a particle beam’s shape are quadrupole magnets.

They are assembled from four coils at 90 degree angles, with coils of the same polarity opposing each other. The passing beam is compressed on one axis, and expanded on the other. To focus the beam overall, at least two quadpoles must be arrayed in sequence.

Quadrupoles are the basic and essential “magnetic lens” of particle beams. Without them, charged beams would inevitably defocus and run havoc.

==== Hexapole Magnets ====
Hexapole magnets exist to address one problem of quadrupoles - they focus differently depending on the energy of the particle passing through the magnetic field. The effect is likened by particle accelerator physicists to the different diffraction of light in a prism - and hence the effect is called “chromatic aberration”. Hexapoles are used to correct for this, stronger deflecting higher-energy beams.

==== Integrated optics ====
As the field of particle accelerator science and engineering advances, more complex designs are proposed to handle more complex phenomena inside accelerators. An upcoming field is “integrated optics” - complex magnetic systems with nonlinear behaviors. These could provide more advanced manipulation of electron beams, building upon existing capabilities and adding new ones for shaping a beam.

=== Plasma lenses ===
Plasma lenses are an up-and-coming technology, where very high currents in plasma are used to create very strong magnetic fields. Similar to plasma-based accelerator technology, the fact that charges are already free in plasma is used to circumvent conventional charge breakdown and critical current limitations on solid-matter devices. The resulting magnetic fields can be much stronger and bend beams over shorter distances. Furthermore, by creating complicated profiles in the plasma, elaborate beam manipulations could be performed, more akin to sophisticated optical elements. As well, the charges in the plasma neutralize the particle beam in the region of the plasma lense - affecting its propagation behavior further and in a positive manner, as self-repulsion is neutralized.

The catch is that this is plasma - that means you are dealing with all the complications of plasma physics. Heating of the plasma has been an early challenge to successfully forming and maintaining plasma lenses.

Plasma lenses require an appropriate chamber, possibly active containment to keep the plasma from losing energy to the walls or leaking into other accelerator parts, and the necessary plasma sources, maintenance heating devices and then lense shapers and control sensors. In high-powered plasma lenses there will also be a need to cool the plasma as it heats up from waste heat. Potentially we may manage this by open-cycle cooling - venting the plasma controlled in one direction into open space, rather than have a closed-cycle solution for reconditioning the plasma. This could cause some very cool visuals!

Plasma lenses for particle beams should not be confused with optical plasma lenses for lasers! Those exist as well, but are their own topic.

=== Chicanes ===
Chicanes are assemblies of multiple dipole magnets used to manage the longitudinal dispersion of a particle beam due to different particle energies. Since particles with higher energy move faster, they drift ahead in the beam distribution over time; and particles with less energy increasingly lag behind.

Chicanes deflect high-energy particles onto an outward curving trajectory. Since traversing the curve takes longer, the high-energy particles are effectively delayed, allowing lower-energy particles to catch up. By the end of the chicane assembly, the bunch has been compressed in the longitudinal direction.

Chicanes however, do not remove energy from too-fast particles. That is the responsibility of other structures.

=== Undulators and wigglers ===
Undulators and wigglers are magnetic assemblies used to transform particle energy into electromagnetic emissions, with different properties from the specific implementation of a shared basic principle.

Assembled from stacks of generally dipole magnets (other configurations exist), both undulators and wigglers force charged particles onto “wiggling” trajectories by bending their path left and right. Wigglers force very strong arcs; undulators less so. In both cases, the rapid deflection of the charged particles causes them to shed parts of their energy as electromagnetic emissions. Due to the energy of the particles, these photons are generally highly energetic x-rays.

Undulators may be used to build free-electron laser systems.

They can also be used as a means of cooling the beam. The removal of energy from particles can also suppress unwanted movement modes, allowing accelerator structures to re-add energy only to desired movement modes. Of course, this means the particle beam actively loses energy, and produces x-rays that have to be absorbed (and cooled away.)

=== Beam coolers ===
Beam coolers are a collection of systems and strategies that are used to remove unwanted movements from a particle beam. Undulators and wigglers are one way to do this.

One method set of beam cooling relies on finding particles who are very “hot” with random motion, and either removing the particles from the beam using a kick device, or applying a precise counter-force that acts against the unwanted motion.

If one is dealing with ions, one clever trick is to inject a beam of electrons. Due to the thermodynamics within the particle population, the lighter electrons absorb heat from the heavier ions, taking this unwanted motion energy with them when the electron beam is bent out.

As well, lasers can be used (what can’t lasers be used for!) to apply counter-force on ions and remove unwanted movement in this manner.

Beam coolers are generally not instantaneous devices. They work over time to make a beam better. Often they involve the removal of energy or particles from the stream, requiring accelerators and sources to make up the loss and creating waste heat.

=== Beam recovery ===
Some accelerator systems can actually be run in reverse - they can turn beam energy back into some other energy format and then electricity. This can be used to recover some to most of the energy from a particle beam before disposing of it. This is advantageous both for your energy bills and making a very high-energy beam liable to spray radiation everywhere, easier to absorb in a beam dump.

=== Beam dump ===
A beam dump is a construction where a particle beam can be routed in case it has to be disposed. Beam dumps are designed to safely absorb the particle's energy without leakage of dangeorus radiation or activated materials to the outside enviroment. Common materials are carbon and metals, embedded in an actively cooled assembly.

= Propagation physics =
==Magnetic fields==

Magnetic fields are things of motion. They are created by moving electric charges. And they only affect electric charges that are moving. An electric charge at rest in the presence of only a strong magnetic field will just sit there. But when it starts moving, the magnetic field interacts with it, starts deflecting it.

But not along the field. Oh no. The magnetic field does not point in the direction it will move the charge. No, the charge experiences a force that is perpendicular to the direction of the field. And also that is perpendicular to its own velocity. A charged particle moving straight down a magnetic field line will not experience any force.

On the other hand, if the particle is moving directly perpendicular to the field, the magnetic force will always push it to the side, even as it keeps turning. This results in the particle moving in a circle. The radius of this circle is called the <b>gyroradius</b>, and the rate at which the particle makes circles is called the <b>gyrofrequency</b> (or sometimes <b>cyclotron frequency</b>).
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Circular_motion.png|frameless]]
<td width=255>[[File:magnetic_forces.png|frameless]]
<td width=255>[[File:spiral_around_field_lines.png|frameless]]
<tr>
<td>To get circular motion, you need a force (blue) that is always the same amount but always pushing perpendicular to the direction of motion, toward the center of the circle. A charged particle in a constant magnetic field moving perpendicular to the magnetic field meets these conditions, and moves in a circular path.
<td>A diagram of how a magnetic field (magenta) exerts forces (blue) on a moving particle (with velocity shown in green).
<td>If a particle is moving partially along the magnetic field (magenta), it will drift down the field lines while spiraling around it (black path). Where the magnetic field is stronger (indicated by having the field lines closer together) it will execute tighter, faster circles in its spiral and will drift more slowly than when the field is weaker.
</table>

It can be convenient to split up a particle's velocity into a part parallel to the magnetic field (the <b>longitudinal velocity</b>) and a part perpendicular to the field (the <b>transverse velocity</b>). The longitudinal velocity describes how fast the particle drifts along the magnetic field line. The transverse velocity determines how tightly and how quickly the particle circles around the field lines.

If the gyroradius is much smaller than the extent of the magnetic field, you get a case where the particle approximately cannot cross magnetic field lines. It just spirals around them. On the other hand, if the magnetic field ends before the particle can make a significant portion of a full revolution, then the particle ends up being somewhat deflected from its original trajectory. If the gydroradius is enogh to deflect the particle into a region where the magnetic field has a different strength or direction, then things get complicated!

When a particle is spiraling around the magnetic field while drifting down the field lines into regions where the magnetic field is stronger, it will circle around the field lines in a faster, smaller circle. This increases the kinetic energy in the spiraling motion; this kinetic energy must come from the kinetic energy of the longitudinal motion. If the field becomes too strong, the particle will slow down all the way and come to a stop. Then it will be reflected back, pushed the other way down the field lines toward the region where the field is not as strong. This phenomenon is called magnetic mirroring, and is responsible for the way radiation gets trapped in the Van Allen belts around Earth with the particles bouncing back and forth along the Earth's field lines and getting reflected back and forth at the poles where the field is strongest.

If you want to calculate for yourself what the gyroradius <math>r_c</math> and gyrofrequency <math>f_c</math> are
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
r_c = \frac{\gamma \, m \, v_\perp}{q \, B}
</math></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
f_c = \frac{q \, B}{\gamma \, m}
</math></div>
where <math>B</math> is the strength of the magnetic field, <math>q</math> is the amount of electric charge of the particle, <math>m</math> is the mass of the particle, <math>\gamma</math> is the Lorentz factor, and <math>v_\perp</math> is the part of the velocity that is perpendicular to the magnetic field. To get these formula to work, you need to use a consistent set of units. So for example if you measure the magnetic field in tesla you need to measure the charge in coulombs (not fundamental charges) and the mass in kilograms (not eV).

One consequence of this is that you can bend the beam coming out of your accelerator by placing a uniform magnetic field across the end. So you can steer your beam to where it needs to go. You can also bend the beam inside of your accelerator – to make it go around in a circle, for example.

Another consequence is that if your beam has to pass through regions with magnetic fields in it, it will get deflected. If you want the beam to end up at the right spot, you will need to account for this. If the exact strength or direction of the field are not known, you will get unpredictable deflections to your beam.

If your beam has been neutralized, the magnetic fields will work to bend the positively charged particles one way and the negatively charged particles the other way. If you have enough beam current, these separated charges will attract each other and counteract the magnetic force. If you don't, the beam will split and you'll get positive charges zipping off in one direction and negative charges going another direction. If you needed to neutralize your beam so that it doesn't electrostatically blow itself apart, well now it will be doing just that.

Even perfectly neutral atoms can be torn apart if the atoms are going fast enough and if the magnetic field is strong enough. Atoms are pretty resistant to this, but you can get conditions where it happens.

Magnetic fields affect the path of beams. But beams are made of charged particles and thus can have currents in them. Currents create magnetic fields themselves, which can influence the surrounding fields. Even a neutralized beam forming a plasma, when it encounters a magnetic field, will get currents induced in the plasma that helps to screen its interior from the field. A beam with enough punch can deform the fields around itself rather than having the fields bending it. As a rough rule of thumb, if the energy density of the beam over a regions larger than a gyroradius is more than the energy density of the field over that region, then the beam gets to have its way with the field, and vice versa. Highly focused, high current beams can thus be expected to better deal with ambient magnetic fields than those with less focus or lower current.

==Focusing==

===Focusing equipment===
In the previous section, we discussed how a particle beam can be steered by directing it through a uniform magnetic field. We can make this kind of field with a magnetic north pole directly opposite a magnetic south pole (with additional shimming magnets to keep things uniform). Because it has two poles, one north and one south, this kind of magnet is called a <b>dipole magnet</b>.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:dipole_beam_steering.png|frameless]]
</table>

But we want to do more than just steer the beam. we want to be able to expand it, contract it, and focus it to a point. To do this, we need a different kind of configuration. We will put two north poles opposite each other and, and 90 degrees, two south poles opposite each other. With four poles, this is called a <b>quadrupole magnet</b>, or sometimes just a "quad".
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_magnet.png|frameless]]
</table>
At the center of the quad arrangement, the magnetic fields from each side cancel out. there is no field there. A particle going through the center will travel straight through without deflection. However, the farther particles get from the center, the higher the field they encounter and the more they get deflected. A particle slightly off to one side may experience a force that curves it toward the center-line. If the quads are properly shimmed, a particle twice as far away will experience twice the force and twice the deflection from its trajectory back toward the center. This curves all the particles so they meet in one spot.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_focusing.png|frameless]]
</table>
However, there is a catch. Those particles that are offset from the center-line at 90 degrees from these focused trajectories are bent away from the center-line rather than towards it! A quad focuses in one direction and de-focuses in the other.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_defocusing.png|frameless]]
</table>

Is there anything e can do to get an arrangement of magnets that purely focuses the beam? Well yes. We can put another quad arrangement downstream of the first, rotated by 90 degrees. Now consider a set of particles in an arrangement that is de-focused by the first quad. When they get to the second quad, they are further from the center than they were when they went through the first quad. Because the second quad is aligned to focus anything the first quad de-focuses, these particles will be focused; and because they are farther from the center, the focusing deflection will be larger than their previous de-focusing deflection. So the net result is that the particles are focused.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Dual_quad_focusing.png|frameless]]
</table>
For particles with the opposite orientation that are first focused, these will be de-focused by the second quad but because they will have converged they will be going through a region of lower field and will be defocused with a smaller deflection than they were initially focused. This dual quad arrangement will focus any particle that goes through it, regardless of orientation.

===Emittance===

For small deflection angles, a dual quad set-up acts like an ideal lens for particles. An ideal lens can focus all the particles diverging from a point onto another point, or into perfectly parallel lines. It can focus particles on perfectly parallel lines onto a point. So now we can use the particle beam to deliver focused particles on our target.

Or, at least we could if all the particles came from a point, or if they traveled on perfectly parallel lines. Unfortunately, they don't. When particles are made, they come from a source with a finite spot size and a finite temperature. The random motion of the particles moving around from their initial thermal motion means that they are never moving perfectly parallel to each other. And when you try to focus them, instead of all converging onto a single point they make a spot of finite size. And if your focusing and beam bending elements are not perfect, you add additional imperfections to the beam that further hider your ability to focus it.

Your ability to focus can be quantified by a measure called <b>emittance</b>. Emittance is measured in units of a length scale times an angle - nominally meters-radians (m<math>\cdot</math>rad) but the technical literature more commonly gives units of millimeters-milliradians (or mm<math>\cdot</math>mrad) or even 2 <math>\pi</math> mm<math>\cdot</math>mrad. The divergence angle of a beam that you try to make parallel will the the emittance divided by the width of the beam. If you focus a beam down so it converges at an angle, the minimum spot size you can get at the focal point is the emittance divided by the beam angle. If you try to focus a beam with emittance <math>\epsilon</math> on a spot a distance <math>R</math> away with an initial beam diameter of <math>D</math>, the spot size <math>S</math> on the target you can achieve is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
S = R \frac{\epsilon}{D}
</math></div>
It is interesting to note that this is very nearly the same spot size you get from a laser due to [[Diffraction#Spot_Size|diffraction]] with a wavelength equal to the particle beam's enissivity. And, like lasers, the more you can expand the beam to large diameters at the focusing equipment, the tighter the spot you can achieve on the target.

So to get good focusing, we want low emittance. Is there a way we can lower the emittance beyond what we get from our source? Engineers are pretty bright folks, and they have found some ways to do this. You can, for example, pass a beam through a thin foil. This causes some stopping, slowing the beam down. You can then accelerate the beam back up to its original speed. However, the stopping is along the direction of the beam particle's trajectory, even if that trajectory is at an angle. Meanwhile, all the acceleration is in the direction of the beam axis. So the net result is to reduce the random motion perpendicular to the beam, and thus reducing its emittance. Similar tricks can be played with magnet set-ups that force particles going through them to emit radiation.

Another method can be used with a relatively high emittance beam, say a beam of protons or ions or muons. Electron beams can be made with substantially lower emittances. So you can send a low emittance electron beam to co-propagate along with your primary particle beam. The beams will interact, with the particles scattering off each other. This exchanges heat between the two beams and so the temperatures, and thus emittances, of the two beams get closer to each other (and, if given enough time, would become equal). If you repeat this often enough, you can get the emittance down to the emittance of the electron beams. This is more practical in beams going around circular accelerators, because they will have time to get cooled over and over.

A final method that has been used is to install sensors to measure the beam fluctuations. For beams going in circles, the time to get a light-speed signal directly across the diameter to the opposite side wil always be less than the time it takes for the particles to go around the curve to that spot. So you can send signals to give electromagnetic kicks to the beam particles to knock them back into alignment when they get to that spot.

It is intriguing to note that these beam cooling techniques can take <i>hours</i> to work!<ref name="RHIC emittance">[https://www.bnl.gov/isd/documents/86783.pdf M. Minty, R. Connolly, C. Liu,T. Summers, and S. Tepikian, "Absolute beam emittance measurements at RHIC using ionization profile monitors", Brookhaven National Laboratory formal report BNL-105970-2014-IR]</ref> However, without beam cooling methods like this, the emittance of ions or protons in storage rings can slowly increase on a time scale of hours.<ref>[https://arxiv.org/abs/1204.6022 Vladimir Shiltsev and Alvin Tollestrup. "Emittance growth mechanisms in the Tevatron beams", arXiv:1204.6022 [physics.acc-ph]]</ref>

But perhaps the simplest method to reduce emittance is just to speed the particles up. For particles moving at a fraction of light speed <math>\beta</math> and Lorentz factor <math>\gamma</math>, the emittance of particles from the same source with the same temperature and same random motion will be proportional to <math>1/(\beta \, \gamma)</math>. To reflect this, it is convenient to define a normalized emittance <math>\epsilon_n</math> such that for the usual geometric emittance <math>\epsilon</math> that we have been talking about
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
\epsilon_n = \beta \, \gamma \, \epsilon.
</math></div>

Normalized emittance measures makes it convenient to get basic estimates of particle beam performance and compare emittance of different sources and beams. For example, various high-performing electron beams around the world have managed to push their normalized emittances down to about <math>2 \times 10^{-6}</math> m<math>\cdot</math>rad to <math>3 \times 10^{-6}</math> m<math>\cdot</math>rad. Facilities with proton or ion beams seem to have about an order of magnitude higher normalized emittance.
<table>
<tr><td width=200>Machine <td width=300>normalized emittance (m<math>\cdot</math>rad) <td width=100>particles<td>reference
<tr><td>European XFEL <td><math>2.3\times10^{-6}</math> <td> electrons <td><ref>[https://journals.aps.org/prab/pdf/10.1103/PhysRevAccelBeams.24.110702 T. Hara <i>et al.</i>, "Low-emittance beam injection for a synchrotron radiation source using an X-ray free-electron laser linear accelerator", Physical Review Accelerators and Beams <b>24</b>, 110702 (2021)]</ref>, using reported <math>\epsilon \approx 1.5\times 10^{-10}</math> m<math>\cdot</math>rad and electrons at 8 GeV for <math>\beta \to 1</math> and <math>\gamma = 15700</math>.
<tr><td>DESY PETRA III<td><math>0.93\times10^{-6}</math> <td> electrons <td><ref>[https://www.desy.de/news/news_search/index_eng.html?openDirectAnchor=444&two_columns=0 New emittance world record at PETRA III]</ref>, using reported <math>\epsilon = 1.6\times 10^{-10}</math> m<math>\cdot</math>rad and electrons at 3 GeV for <math>\beta \to 1</math> and <math>\gamma = 5870</math>.
<tr><td>SLAC <td><math>3\times 10^{-6}</math> <td> electrons <td><ref>[https://accelconf.web.cern.ch/pac97/papers/pdf/4W009.PDF M. Hernandez <i>et al.</i>, "Emittance Measurements for the SLAC Gun Test Facility", 1998 IEEE]</ref>
<tr><td>LHC <td><math>3.75\times 10^{-5}</math> <td> protons <td><ref>[https://www.lhc-closer.es/taking_a_closer_look_at_lhc/0.beta___emittance Beta & Emittance; Taking a closer look at LHC]</ref>
<tr><td>Fermilab Tevatron <td><math>\approx 5\times 10^{-5}</math> <td> protons <td><ref>[https://accelconf.web.cern.ch/p03/PAPERS/TPPB070.PDF J. Slaughter, J. Estrada, K. Genser, A. Jansson, P. Lebrun, and J. C. Yun, "Tevatron Run II Luminosity, Emittance and Collision Point Size", IEEE Proceedings of the 2003 Particle Accelerator Conference, pages 1763-1765]</ref>
<tr><td>RHIC (with protons) <td><math>\approx 5\times 10^{-5}</math> <td> protons <td><ref>[http://www.osti.gov/servlets/purl/910428-6sKfP0/ Zhang, S, and Ptitsyn, V., "Proton Beam Emittance Growth at RHIC", Brookhaven National Laboratory formal report BNL-77563-2007-CP]</ref>
<tr><td>RHIC (on startup) <td><math>\approx 1.5\times 10^{-5}</math> <td> Au-197 <td><ref name="RHIC emittance"></ref>, using reported <math>\epsilon \approx 12\times 10^{-6}</math> m<math>\cdot</math>rad and Au-197 ions at 100 GeV for <math>\beta = 0.76</math> and <math>\gamma = 1.545</math>.
<tr><td>RHIC (after several hours cooling) <td><math>\approx 5\times 10^{-6}</math> <td> Au-197 <td><ref name="RHIC emittance"></ref>, using reported <math>\epsilon \approx 4\times 10^{-6}</math> m<math>\cdot</math>rad and Au-197 ions at 100 GeV for <math>\beta = 0.76</math> and <math>\gamma = 1.545</math>.
</table>
So as a plausible first estimate for your sci-fi electron guns, you can take <math>\epsilon_n \approx 10^{-6}</math> m<math>\cdot</math>rad and for proton guns <math>\epsilon_n \approx 10^{-5}</math> m<math>\cdot</math>rad. Modify for any improved future-tech that can bring the emittance down. Then divide by <math>\beta \, \gamma</math> for your beam to find the geometrical emittance. Which in turn can let you figure out how well you can focus your beam.

However, we should add one caution here. Unlike a laser, which can get a beam diameter nearly as wide as the width of the optical element (be it a lens or mirror) focusing it, a particle beam needs to go through a small hole inside much larger focusing elements. Thus, a particle beam width (what goes into the spot size calculation) is likely to be much smaller than the physical size of the focusing equipment. You may be able to overcome this limitation by making the geometrical emittance very low, but it will be an issue to keep in mind.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Laser_focus.png|400 px|frameless]]
<td width=150>
<td width=556>[[File:P-beam_focus.png|556 px|frameless]]
<tr>
<td>Laser focused with a mirror
<td>
<td>Particle beam focused with quadrupole magnets
</table>

==Beam self-forces==

particle beams are composed of moving charged particles. Moving charged particles are electric currents. And currents and charges exert forces on each other.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Charges_and_forces_pn.png|frameless]]
<td width=255>Opposite electric charges exert an attractive force on each other
<tr>
<td>[[File:Charges_and_forces_pp.png|frameless]]
<td rowspan=2>Electric charges of the same type exert repulsive forces on each other
<tr>
<td width=349>[[File:Charges_and_forces_nn.png|frameless]]
</table>
<td>
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Currents_and_forces_opposite.png|frameless]]
<td>[[File:Currents_and_forces_same.png|frameless]]
<tr>
<td width=200>Electric currents in opposite directions repel each other
<td width=200>Electric currents in the same direction attract each other
</table>
</table>

In a particle beam, the collection of like-charged particles and their mutual repulsion is called <b>space charge</b>, and can have various effects ranging from making beams expand and trumpet out if not properly neutralized to even making particle beams stop, turn around, and come back the way they came! The currents created by all the particles, on the other hand, are moving in the same direction so they attract each other. This attraction can be called a <b>z-pinch</b>, or sometimes a <b>Bennett pinch</b> after one of the first guys to study it.

===Looking at the same thing two different ways===

Imagine that you are sitting calmly, watching a collection of positively charged particles (as you do). Because the particles all have the same electric charge, they repel each other. You observe the particles pushing away each other and flying apart.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Charges_at_rest.png|478 px|frameless]]
</table>

Ah, but what we didn't tell you was that these particles were just shot out of a particle accelerator at high speed. You just happen to be moving at the same velocity as all the particles, so to you they appear to be at rest. But your colleague running the accelerator sees the particles (and, presumably, you) flying past at high speed. She sees not just particles of the same charge pushing out against each other, but also all the currents of these charges. And because the currents are all in the same direction, there is an attractive force pulling the particles together.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Charges_in_motion.png|750 px|frameless]]
</table>

With the extra forces from the current squeezing the particles together, your colleague should see the particles expanding less fast than you do. Maybe she even sees them collapse in on themselves if the current-current forces are higher than the charge-charge forces. What gives? Who is right?

It turns out, you both are. Because, as it turns out, magnetic fields are just a consequence of relativity balancing things out when you look at moving things from a different coordinate system. You may be familiar with the idea that in the theory of relativity, moving things experience time at a slower rate than things at rest (as observed by someone who is "at rest" in that frame of reference). So your colleague sees the things happening in your frame of reference happening slower than you do. Maybe only just a tiny bit slower, if you are not moving very fast at all. Maybe much, much slower if you are nearly relativistic with respect to her. But some degree slower, in any event. And so she won't see the particles expanding as fast as you do. Each of the particles has the same electric charge for both of you, so the electric forces will be the same. So your colleague must observe some other force in addition to the electric forces that act to partially compensate for them. And these are the magnetic forces, which are present in her frame of reference but not yours. And now you can see that, for bare isolated charges, the magnetic self-pinching forces can never quite overcome the electric self-repulsion charges – although if the particle bunch is going very fast, they might come very close.

===Space charge===

===Self pinching===

==Energy loss in matter==

Sometimes, a particle beam needs to go through stuff to get it where you want it to go. Stuff like air, for example. Or the tissue in the way of getting the beam to a tumor. Also, sometimes, you want your particle beam to interact with stuff. Like the material you are trying to machine with an ion beam, or the bacteria you are trying to kill on medical equipment. All of this falls under the general realm of the interaction of the beam particles with matter. This is an enormously complex subject, but we can cover many of the basics here.

The primary effect on the beam of any interactions with matter are loss of energy of the beam particles (<b>stopping</b>), random changes to the particle directions that make the beam spread out (<b>straggling</b>), and knocking particles clean out of the beam altogether (<b>scattering</b>).

===Ionization===

The particles coming out of your particle accelerator are charged. When a fast moving charged particle goes through matter, the electric field from the charge pushes and pulls on the various charged particles that make up that matter – mainly electrons and the atomic nuclei around which the electrons orbit. At speeds above approximately 1% of the speed of light, it is primarily light and nimble electrons being affected while the ponderous nuclei just sit there barely perturbed by the passage of the particle. When the passing particle yanks on an electron hard enough, it can pop it clean off of the atom it was attached to, leaving a positively charged ion behind and giving you a free electron that can go and glom on to a previously neutral atom to make a negative ion. This process, unsurprisingly, is called <b>ionization</b>. For most particle beams, ionization is the dominant interaction with matter; although when you get very fast or very slow particles other effects can become more important.

if your particle was not initially charged when it goes into matter, it soon will be. The electric charges of the electrons and nuclei in the matter will do the same thing to the particle that a charged particle would do to them. This will quickly strip off any electrons around a high speed atom, leaving an ion and bare electrons to go through the matter. Only particles that are fundamentally neutral (like neutrons or photons) will not be immediately ionized.

It takes about 1 to 10 eV to ionize an atom. This energy must come from the kinetic energy of the beam particle. Particles from an accelerator have millions to billions of eV. So it takes a lot of ionizations to meaningfully affect the particles. However, one thing about matter is that it has lots of atoms that can be ionized so you do get those lots of ionizations. Because each ionization is so small compared to the total energy this is often approximated as a continuous slowing down of the beam particles. So many individual ionizations tend to average out, leading to all the beam particles being slowed down uniformly.

When the particle is yanking on nearby electrons to tear them out of atoms, those electrons are yanking back. This jitters the beam particle slightly from side to side. Again, these sideways yanks tend to average out. The statistical variations from the average slowing down and side-to-size jerking are what lead to straggling. Protons and ions are heavy, and will be deflected little from their original paths - like bowling balls smashing through matchsticks. These particles tend to go in mostly straight lines. Electrons, on the other hand, have the same mass as the electrons they are knocking about. So electron tracks through matter tend to zig-zag around a lot more.

As the particle whizzes past, it leaves a trail of atomic destruction in its wake - a trail of ions where the atoms are literally torn apart (not the nuclei, just the atoms). These particle tracks can sometimes be seen in detectors like cloud chambers. The faster the particle is going, the less time it has to yank on nearby electrons and so the less ionization it leaves behind. As a result, fast particles are not slowed down much but as they start to slow down they continue to be slowed down faster and faster. A consequence of this is that the particle dumps most of its energy into the material at the end of its track, just before it comes to a stop. This sudden peak in energy deposition is called the <b>Bragg peak</b>.

The electrons that a particle knocks out can end up going pretty fast. So these secondary electrons can produce further ionization in what is called an <b>electron cascade</b>. if an inner core electron is knocked out of an atom, other electrons falling in to fill that empty spot in the atom can release x-rays (<b>x-ray fluorescence</b>) or cause shake-ups in the outer electrons that knock other electrons off the atom (<b>Auger effect</b>).

===Brehmsstrahlung===

A charged particle is surrounded by an electric field. If it is moving, it is also surrounded by a circulating magnetic field. These fields move along with the particle.

If the particle knocks into something, it will slow down and probably change direction. The information about this bump only goes through the electromagnetic field at the speed of light. So a lot of that field that was moving along with the particle might not get the memo in time and keep right on going. Stray electromagnetic fields like this get this self-inductance effect going on where they start making themselves over and over again, allowing them to propagate as freely moving waves of electromagnetic radiation. This EM radiation from a slowing down particle is given the delightful name of <b>bremsstrahlung</b>.

Particles from an accelerator can be moving so fast that they make bremsstrahlung in the x-ray or gamma ray part of the spectrum. If you get very high energy bremsstrahlung gamma rays, the gamma rays can interact with matter by boiling an electron and its antimatter counterpart, a positron, out of the vacuum (using the nearby matter as the necessary momentum sink for the process). These positrons and electrons can themselves be very high energy and can in turn create more bremsstrahlung, and the process can repeat until the particles are no longer high enough energy to continue making each other. This is called an <b>electron-gamma shower</b>.

The x-ray machines used by dentists and radiologists use both bremsstrahlung and x-ray fluorescence that you get when shining an electron beam onto tungsten to make their x-rays.

At very high speeds, bremsstrahlung can be more a important source of energy loss than ionization.

===Electron capture and screening===

When an ion has slowed down to about 1% of the speed of light or less, it is moving slower than the electrons zipping around their atoms in the matter the ion is going through. This allows the electrons to stick on to the ion, now becoming the electrons of the atom that is moving through the material rather than of the material itself. The ion begins to lose its charge, and the rate of ionization goes down. Soon, ionization becomes negligible compared to the beam atom bumping in to other atoms.

===Phonons and displacement===

When an atom or ion moving through a material bumps into another atom, it will cause that other atom to move. This can have one of two primary effects.

First, if the atoms whack into each other hard enough, the atom in the material can be knocked clean out of place. Like a pool ball being thwacked by the cue ball, the struck atom will go zinging through the material. Atoms getting knocked out of their place like this is called <b>displacement</b>. Displacement creates chemical changes to molecules as atoms are removed, and creates defects called <b>vacancies</b> in crystals (which, as you might imagine, is just a missing atom in the crystal lattice). When the atom comes to rest, it can chemically attack other molecules, again changing the chemistry, or create an extra vacancy called an <b>interstitial</b> jammed into the crystal lattice where it is not supposed to be. If the displaced atom is going fast enough, it can whack other atoms and displace them, leading to a <b>displacement cascade</b> like tenpins knocking each other down. As a result, at the end of its track, an ion can leave a region of disordered material and messed up chemistry where the atoms have all crashed into each other like a freeway pileup.

If an atom is displaced near the surface of a material, the displaced atom can shoot out of the surface and escape the material altogether. This is called <b>sputtering</b>.

Second, the beam atom might only deliver a glancing blow. Or the beam particle might be an electron, which is so light that it is usually incapable of displacing atoms. In this case, the struck atom gets kicked to the side a bit and runs into other atoms; maybe it is also pulled back by the chemical bonds of other atoms that it is attached to. This transfers the motion of the atom to other atoms around it in the form of pressure waves (or, in a solid, shear waves). This adds heat to the material without causing direct damage – although as you might imagine, enough heat can also cause localized melting, evaporation, or decomposition. The individual quanta of vibration – fundamental particles of sound, if you want to think of them like that – are called phonons in analogy with the photons that are the fundamental particles of light. Much like ionization, a particle that excites pressure waves slows down gradually and continuously, unlike the sudden changes in speed and direction you get from knock-out displacement.

===Nuclear collisions===

So far, we have only discussed the things that happen when beam particles interact with matter particles by electromagnetic forces. Even atoms knocking into each other to produce phonons or displacement happens through the mutual repulsion of the electrons around the atoms (although this is dressed by other weird effects like how two electrons can't be in the same place at the same time, or how the fields of one electron can affect other nearby electrons that in turn affect the original electrons, leading to odd correlation effects and charge density waves and other fun stuff that keeps condensed matter physicists and quantum chemists busy).

But electromagnetism isn't the only thing you can get going on. below the level of structure of the atom, you have the atomic nucleus. And a direct hit on the nucleus can end up giving very different effects.

The simplest thing that can happen is an <b>elastic collision</b>, where no energy goes in to any internal nuclear state or into breaking up the nucleus. An elastic collision just ends up giving you displacement (or possibly phonons, but if you hit the nucleus you usually hit it hard enough to send it flying).

But beyond that you can cause any number of nuclear reactions. You can leave the nucleus in an excited nuclear state, from which it usually relaxes back down to its original state by emitting gamma rays. You can knock out nuclear particles, such as neutrons, protons, deuterons (a proton stuck to a neutron), tritons (a proton stuck to two neutrons), helions (a neutron stuck to two protons), alpha particles (two protons stuck to two neutrons), or potentially even larger nuclear fragments. If you hit a heavy nucleus like uranium, you can cause it to split apart by fission. And if you hit the nucleus really hard you might just shatter it into many tiny fragments of those protons, neutrons, deuterons, tritons, helions, alpha particles, and heavier nuclear fragments mentioned earlier. To get any of these <b>inelastic collisions</b>, you need to hit the nucleus hard enough. If your beam particle doesn't have enough energy to excite the nucleus or knock particles, these things simply can't happen. As it gets enough energy to excite a particular interaction channel (as they are called), that process becomes possible and as the energy increases the process becomes more and more likely.

A nucleus is only about a femtometer across (<math>1 \times 10^{-15}</math> m). Which makes it only about <math>1 \times 10^{-30}</math> m² in cross sectional area. In the meantime, chemical bonds between atoms are usually about 1/10th of a nanometer (<math>1 \times 10^{-10}</math> m) long. In condensed matter (liquids and solids), this means that you have somewhere about <math>1 \times 10^{30}</math> atoms per cubic meter. A meter thickness of atoms thus presents just about about enough atoms that their nuclei can cover the projected area – in other words, you can expect a particle to go through (very roughly) about a meter of condensed matter before it whacks into a nucleus.

This assumes that a particle that goes through a nucleus interacts with it. This is usually a good assumption for particles that interact by the strong nuclear force, such as protons, neutrons, mesons, and the atomic nuclei of ions. But many particles cannot interact by the strong nuclear force, such as electrons or muons. These are much less likely to interact with an atomic nucleus even if they pass through it, so they will end up going considerably farther between hitting nuclei than the above assumption. But nuclei can interact electromagnetically (they are made up of charged protons, after all, and at a smaller level of charged quarks), so electrons and muons can produce some nuclear interactions; just far less than actual nuclear particles like protons and ions.

A nuclear strike is a discrete event that will remove the impacting particle from the beam and will (except for elastic collisions) probably produce a spray of radiation.

===High energy physics===

As you get to higher and higher energies, you unlock all sorts of new and weird and interesting processes that can happen. Boiling exotic particles out of the vacuum, unifying forces that had been divorced since the big bang, and producing bizarre unstable resonances in the quantum fields. This is what physics build accelerators to measure. From the point of view of getting a beam to its target, or what happens to the target, however, the main effect is mostly that the interacting particle is removed from the beam and a bunch of additional radiation is produced.

= Operational Considerations =
== Logistics ==
The logistics of particle beam can be a mixed bag. By itself, any particle accelerator is a complicated piece of machinery full of delicate parts that have to deliver high performance. This may require cryogenic temperatures with associated challenges on the materials and mountings used (insulation, and distortions under temperature changes). When working, particle accelerators invariably generate some radiation across the whole assembly, which irradiates and activates the structure. Some parts may be exposed to significantly higher radiation flux due to their working principles. Often the particle beam is a dynamic system that must be managed with active, closed control loops, involving sensors, modeling and control software, actuators, and the computing and communication hardware that brings it all together. This poses engineering challenges across all disciplines.

A practical particle beam is inherently a counterintuitively sturdy device. If it was too delicate, it would break too much to be useful, though of course this is a sliding scale. Still, while sturdy, this doesn’t eliminate complexity, it manages complexity. The result is a system that is the product of a high-tech civilization. A superior machine shop and well-trained engineers are a must to maintain a particle beam system. Some parts of a particle accelerator may be mundane - others may be custom-machined parts that require their own dedicated industries to produce. That said, the manifold utility of particle accelerators is on your side here. Since there are so many practical applications, it is economic to produce parts for industrial demand. (that said, these may not be the kind of components you want in specific high-performance applications.)

The main thing a particle accelerator requires the most is electricity. Any mature power production facility can supply the power.

However, all particle beams require at the least a source of neutralizing mass. The projection of a charged beam will leave the projecting object oppositely charged. In an atmosphere the natural plasma formed by the beam will ground everything out, but in space the diffuse vacuum will not do the job.

Furthermore particle beams using ions also require a feedstock for their ions to produce from, and in an appropriate format. This can be a gas, a fluid or an easily ionized solid. It can be the feedstock species in its pure form or in an easily split molecule, though every complexity adds to the amount of preparation work that has to be done before the feedstock enters the particle beam.

Don’t worry about the mass of your source or neutralizing ions though. Large particle accelerators, like the LHC in Switzerland, only consume tiny gas bottles worth of feed gas every couple years. The mass of a single 5.56 NATO bullet worth of hydrogen could feed a 10 GW, 10 GeV ion beam for nearly 4 days of firing! You will not be running out of beam feedstock easily.

== Cooling ==
Cooling a particle accelerator can be a serious challenge. Various bits of a particle accelerator love to be supercooled for efficiency or basic working principle - any bit of superconductor used in the accelerator, or magnets, or EM generators, will require cooling to its generally supercool operating temperature.

Even if Standard Temperature and Pressure Superconductors (273 Kelvin, 100 Kilopascal) are available, present superconductors have demonstrated superior performance by being supercooled. That means to achieve even stronger fields, even STP superconductors may be cooled to cryogenic conditions.

Cooling cryogenic systems sucks. Thermodynamics shows itself from its worst side. You can forget running cryogenic radiators - the Stefan-Boltzman Law penalizes the direct radiative cooling of anything cryogenic severely as much as it empowers very high-temperature radiators. This only leaves stages of heat pumps to provide cryogenic fluids that can in turn, cool cryogenic systems. Heat pumps draw power, and for such a giant temperature gradient as we can have here, they will draw a lot of power.

However, we have other options. Cryogenic systems also tend to be highly efficient, so while they do heat up, they do so very slowly and there is little overall energy absorbed. This provides the option of running off of bunkered cryogenic fluid in dewars for the duration of operating the equipment and not running any heat pumps all the time.

Other bits of a particle accelerator system aren’t as required, but really nothing is a fan of high temperatures. Going higher than room temperature can be treated as an oddity. High-performance lasers may not require cryogenic conditions, but they certainly like it cool. Radiofrequency producers of any kind generally fare worse with rising temperatures. That means cooling overall remains a challenge - 300 Kelvin is still not that great of a hot loop temperature for radiators.

== Safety <ref>Thomas Otto, "Safety for particle accelerators", Springer 2021, Available online at: https://library.oapen.org/handle/20.500.12657/46120</ref>==
=== Ionizing Radiation ===
There are surprisingly many sources of ionizing radiation in a particle accelerator, besides the obvious threat of the output beam itself. Not all particles are accelerated correctly or in the correct direction. When such particles interact with the structure of the accelerator, they release a shower of secondary ionizing radiation. In electron accelerators and sources, stray electrons may be accelerated and release x-rays from bremsstrahlung as they collide with surrounding matter. The resulting doses can be significant and dangerous. Also, such collisions leave behind residual activation, making the accelerator a radiation hazard for some time even after it has been shut down. Generally, people should not work near the particle accelerator while it is functioning, and radiation barriers isolate the core accelerator from adjacent facilities. (In real-life facilities, you will often see accelerators inside larger buildings being cocooned inside doped concrete plates to shield the rest of the room.)

=== Radiofrequency ===
RF energy is used in many kinds of particle accelerators. This energy may leak from the structures it is generated, contained and used in due to damages or incorrect assembly. As microwave radiation and above is used, the radiation can be readily absorbed by bodies, hydrogen-rich materials or other objects. The end result of this can be rapid heating and thermal effects. Especially in accelerators designed to throughput a lot of energy (say a GW or more), this may become a serious issue if a waveguide is damaged. It’s not recommendable to stand in front of a multi-MW radar either!

=== High-powered electrical and electronic systems ===
The operation of the magnetic systems and accelerators often involves high-powered electrical and electronic systems. High voltages and/or amperes are involved. This poses risks like shock hazards, electrical arcs, and electrical fires. Some electrical systems such as capacitors or inductors may have combustive or explosive failure modes.

=== Flammables ===
Parts of an accelerator or its infrastructure may be manufactured from flammable substances such as hydrocarbons. In the case of an ignition source such as a high-powered beam cutting into a physical substance or an electrical fault, these may catch fire.

=== Lasers ===
Some accelerator systems use high-powered lasers. Laser plasma accelerators use very high-powered pulse lasers to provide pumping power. The safety concerns of lasers for blinding, physical damage, and possible exposure to ionizing radiation as a secondary effect of the beam impact, apply. Some high-powered lasers can also drive electrons which generate strong EMPs, which may damage or destroy other systems in turn. Parts of a laser assembly or laser beam guide may shatter if exposed to overloading conditions.

=== Magnetic systems ===
Parts of a particle accelerator may use strong magnets, up to superconductive magnets with multi-Tesla fields. While the magnets are integrated into chokes that shape the magnetic fields, fringe fields invariably extend beyond the core magnet installation.

Multi-Tesla transient electromagnetic field can affect the human nervous system temporarily, inducing nausea, vertigo, metallic taste and optical phenomena, if the person moves in relation to the magnetic field. Otherwise, the risk of exposure is low. That said, electronic systems that are not shielded properly, such as say, implants, could be adversely affected by the magnetic fields.

The biggest physical risk are reactions of other magnetic objects with the magnets. You can look up cases where metallic objects interacted with the multi-Tesla fields of MRI scanners. Such parts may fly across the room towards the magnet at increasing speeds, and damage anything or injure anyone in their path, possibly ending with the object or person squashed (or impaled!) to the magnet.

Also, large superconductive magnets store not insignificant amounts of energy. If the superconductive property breaks down, the magnet quenches and becomes a resistor. The resistance site heats up as current tries to flow through it, expanding the quench and possibly being heated until it sublimates into gas. This could possibly happen explosively. In such a case, up to many GJ of energy might be unleashed in a conflagration or explosion (subsonic or supersonic), damaging surrounding systems and ejecting shrapnell.

=== Cryogenic cooling systems ===
Particle accelerators often involve cryogenically cooled systems. This means cryogenic fluids like nitrogen and helium are present in the accelerator infrastructure. Such gasses have both a very low temperature and a high ratio of gaseous volume at high temperatures to their liquid volume.

Exposure to the leaked fluid can lead to cryogenic cold damage, injury, and death. Furthermore such gasses are efficient at displacing ambient breathing gasses such as oxygen, so they pose an invisible, and rapidly spreading, suffocation hazard.

Also, if insulation is damaged, ambient heat may leak into the cryogenic circuits, and cryogenic fluid may begin evaporating. This can lead to dramatic rises in internal pressure. Relief valves should trigger to safely direct such pressure spikes away from important places and keep the system from failing catastrophically, but otherwise, pipes and cryogenic storage devices may fail, up to explosively.

=== Beam excursions ===
If the magnets responsible for manipulating the beam break down, the beam may excurse from its designed paths. The beam may become defocused, or maintain its focus. The beam begins cutting into the accelerator structure, evaporating material and depositing energy. This will definitely foul the ultra-high vacuum inside the accelerator, leading to secondary radiation production as the beam collides with gas inside the accelerator. Some sensitive facilities requiring ultra-high vacuum such as some RF cavities may “crash” and become ruined. Further systems may become damaged or destroyed from either physical stress or the radiation pulse. If the beam bunches contain a lot of energy and maybe are heavy particles, the deposition may lead to an energetic local explosion. Alternatively the beam may escape and cut into structure beyond the particle accelerator itself, damaging and destroying equipment and seriously hurting or killing personnel.

In the worst cases, a beam excursion on a very powerful beam may lead to an explosion that destroys the accelerator utterly. Even in a less bad case, the radiation pulse might damage or destroy critical equipment around the accelerator and take the entire facility offline.

=== Beam dumping ===
If any other systems fail, the accelerator probably has to be shut down. This poses a problem - what do you do with the beam energy that is still in the accelerator system?

Sometimes, the beam is designed to leave the accelerator immediately, and shutting down the beam is easy. But in the case of cyclotrons and storage rings, the beam may circulate for a while. What do you do then? You have to dump the beam from the accelerator. Special beam stops are used on research and industrial accelerators, large blocks of dense material able to absorb the beam. During this process the blocks both heat up and become radioactive. For particle beams storing very high energies, this may become an increasing challenge. How do you safely dumb a beam containing many GJ or more? In some cases you may have no other choice but to dump the beam out into the environment. Where and when can you do this safely without threatening third parties?

= Application areas =
Particle accelerators have many, many, many areas of application.

The public examples of particle accelerators are the big research accelerators used to create conditions with which one can interrogate fundamental physics. These accelerators are after specific particles under specific, as finely controlled as possible, conditions.

However, particle accelerators are also used in imaging. The electron microscope is a classical particle accelerator-based imaging system, using the fact that electron are very tiny particles to bypass classical imaging limitations of photons. Thanks to them, the biological and material science world can be interrogated in great detail, down to single atoms, and advances in the sensors and processing software keep pushing the boundaries.

On a bigger scale, particle accelerators can be used to produce high-energy x-rays with very small wavelengths. The x-rays are projected at prepared crystaline targets. The x-rays interact with the electrons in the crystals, giving information about bond structures. Such crystallography has great importance in biochemistry, but also material science.

Similiarly electrons, ions and neutrons can be used to probe at the structure of materials of interest. Their different behaviors allow for surface or in-depth scanning with particular properties.

In manufacturing, particle beams can be used both to shape materials and to image them for quality control. Particle beams can deliver thermal energy to cut apart materials or weld them. In additive manufacture, particle beams can provide fine precision melting or even meld together volumetric composites with penetrating beams. At fine scales, precision beams can break down gasous feedswtock to deposit materials in micrometer-sized structures. Directed ion guns implant traces of other elements into surfaces for manufacture.

Industries have a certain demand for specific isotopes. Accelerator-driven manufacture of isotopes covers these needs.

And in medicine, particle accelerators of ever-greater sophistication find increasing use in cancer treatment and beyond, destroying life-threatening tissues with ever-reduced doses to surrounding healthy tissue.

= Further material =
* [[Particle Beamed Propulsion]]
* [[Plasma Guns]]
* [[Particle Beam Weapons]]

== External reading ==
The US Particle Accelerator School posts material of its courses publically at times: https://uspas.fnal.gov/materials/materials-table.shtml

= Sources =

== Authors ==
Sevoris Doe, Gerrit Bruhaug, Luke Campbell.

== References ==

Particle Accelerators

2022-08-18T12:39:35Z

Sevoris: /* Further material */

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Particle accelerators are an incredibly useful piece of technology and a diverse set of artifacts produced for many purposes. While particle accelerators are often invoked as weapons of war in science fiction, and have captured the public attention as giant research systems the size of cities, they have an incredible widespread range of applications. There are also many, many ways to build a particle accelerator, depending on what effects you need to achieve and what circumstances you work under.

This article aims to be a general primer on particle accelerators, their physics and a selection of technologies and construction styles. In detail, all particle accelerators are high-tech engineering customized to their circumstances in many, many aspects.

Further pages covering application fields are in the work.

= Accelerator physics =
The simple explanation is that we use electric fields to accelerate charged particles. But from this basic principle of leveraging the electromagnetic forces springs an entire set of accelerator technologies with different performances, trade-offs and characteristics. The most basic particle accelerators are just a big high-voltage source and can still be found in old CRT TV’s and dentist x-ray machines around the world. Things rapidly get more complex from there, but first we need to lay out some terms so we can all have a common conversation.

The first thing to keep in mind with particle accelerators is that we are talking about charged beams here, even if we neutralize them later. So no neutrons, photons, or other innately neutral particles. Your choices range from the simple electron up to anti-uranium and more, but they all must be charged. The particle of choice determines just about everything else about the accelerator and any potential uses. Really heavy particles (like large ions) don’t accelerate easily and take a lot of energy to get relativistic enough for time dilation tricks to take into place. Really light particles (like electrons) accelerate super easily, but can have other radiation problems to worry about.

The second most important thing to know about a given accelerator is the particle energy. This is a measurement of the kinetic energy of any individual electron, ion, whatever in the beam and is measured in a unit called the electron-volt or eV for short. The lower bound for any worthwhile particle accelerator is in the MeV (or million eV) range and we typically want a lot more! Inside of the accelerator we tend to talk about the electric field doing the accelerating (measured in Volts/meter). For singly charged particles (like electrons) this gets really to figure out the energy because of how the eV is defined. We just take the electric field (also called the accelerating gradient) and multiply by the length of the machine! Boom, now you have particle energy! For more complex particles you have to do some scaling based on mass and charge state.

Also a quick trick to help you with deciding if something is relativistic or not is to divide the particle energy <math>E</math> by the mass energy <math>M</math> of the accelerated particle (0.511 MeV for electrons/positrons and about 1 GeV for a proton) and add one. The resulting number is called the Lorentz gamma value and is used all over in accelerator engineering.
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>\gamma = 1+\frac{E}{M}</math></div>
Once you get a value much larger than 1, you have relativistic particles. Once that number is in the hundreds you get an ultra-relativistic beam. Figure 1 helps show the relationship between Lorentz gamma and speed of the particle.

[[File:LorentzGvsSped.svg|900px|thumb|center|Lorentz gamma versus fraction of the speed of light]]

The third most important feature of a particle accelerator is the emittance which is just a fancy term for divergence of the beam and can be thought of similar to the wavelength of lasers. No beam is made perfect and they will naturally want to expand outwards. The smaller the emittance, the longer the beam will stay in a tight spot. Emittance can be altered with a wide variety of beam engineering techniques that take entire books to discuss, but one other cheating thing you can do is drive the beam energy higher. This naturally lowers the emittance by a factor of Lorentz gamma due to the cheating effect of special relativity! Thank god for Einstein!

== Acceleration forces ==
Every accelerator tries to build an electric gradient appropriate for the charge of the particles it is accelerating. The greater the electric gradient, the bigger the amount of work that can be done, the more energy is imparted on a particle as it travels through the gradient, and thus the higher the acceleration. The simplest way to build up a gradient is simply two charged electrodes with a strong potential between them. Many other accelerator methods use electromagnetic fields, in the form of waves confined in electrically conductive structures. In this case we exploit that electromagnetic fields have both an electric and a magnetic component. The magnetic component cannot be used to accelerate particles (only bend their trajectory - which can be exploited for effects like deflection or focusing) but as the electromagnetic waves oscillate, so does an electric field. This naturally creates a charge difference across half of the phase of an electromagnetic wave - this electrical gradient can accelerate particles. Both of these technologies encounter problems with the physical materials asked to contain the electric or electromagnetic energy. Eventually the fields get so intense that solid materials get broken down and an arc is formed. As such, various advanced concepts use clouds of charged particles and plasmas, which are not subject to the material limitations of solid matter.

The strong and weak nuclear force as well as gravity (which are all of the forces we know of at the time of writing) are not suitable for accelerating particles. This is related to the power and range of these forces. The strong and weak nuclear force have extremely short ranges - even if we had a means to artificially shape them like with electromagnets, their range would be too short. Gravity has the range but is incredibly weak. To generate a measly 1 G of acceleration, 9.81 m/s², requires the mass-energy of the entire earth, some 5.97e24 kg of mass, to do the job! The acceleration involved in particle accelerators is much, much, much larger (on the order of 10^13 times larger or more).

{{MascotSays|Interestingly, one source of very high-energy particles are giant gravity wells. Black holes generate conditions around them that can eject particles with enormous energies. The jets of particles that shoot from the north and south poles of many such stellar entities are particle accelerators.}}

If you want to use a force other than electromagnetism to accelerate particles, it’ll have to be a fictional one. The forcefields of various sci-fi properties could be considered, since they can often apply forces to particles. If you propagate a wave of such forcefields down a tube, you could have something quite a lot like many particle accelerator systems!

== The particle bunch ==
Essentially all high energy particle beams are not constant streams of particles, but a time sequence of discrete groups of particles, commonly referred to as “bunches”. These bunches can be described both in their distribution in lateral space to the traveling direction (x and y axis) and longitude (z axis) along the traveling direction, anchored on an origin point (0,0,0) that moves in time.

[[File:ParticleBunch.png|center|900px]]

Bunches can have various profiles. Generally they are somewhat distributed in the lateral direction just because the charged particles have slightly different starting positions and lateral velocities (also called transverse emittance). In longitudinal space, differing acceleration experienced by individual particles as well as dynamics within the bunch scatter them out, with “noses” and “tails” that contain fewer particles. The end results is that most beam profiles follow a gaussian distribution, though there are exceptions.

This bunching behavior allows us to analyze other properties of particle beams very similar to lasers. There is a “pulse frequency” given in Hertz. Each bunch contains a certain energy, which can be large enough that we give it in Joules and not electronvolts. There is an average output energy over a given timespan, given in Watts. We have a beam peak power, given by how quickly a bunch arrives at a target and deposits its energy, which we give in Watts. Related to this we also have a related peak intensity depending on how much area/volume the beam is deposited into, with units of either W/m² or W/m³. If we have charged particles being accelerated, we get a beam voltage, the same way as in a wire. Beams can also have a temperature, the same way as any other group of particles with random motion from their energy, given in Kelvin or eV.

The systems that generate particle beams of course have efficiencies, with the wall plug efficiency (power inserted into the entire system versus power deposited in the particle beam) being the most interesting for first-order engineering. They also have an operating temperature.

== Particles for acceleration ==
With the constraint that our particles must have an electric charge in order to be accelerated with anything but fantastic technology, we can discuss quite a large family of particles, from the conventional to the exotic. They differ in the applications they can be interesting for. Some can be similar, but no two particles behave entirely alike.

=== Electrons ===
The ubiquitous electron is the most common particle to be accelerated and the first one ever pushed up near the speed of light by humans. Due to its excellent charge to mass ratio (the best possible due to the extremely low mass and charge of 1e) electrons are very easy to get up to relativistic speeds. Electron accelerators for radiography and sterilization are often portable by a single person and not much larger than a laptop in size (although they can be much bigger for power output reasons). The ease in which electrons reach relativistic speeds means that electron RF accelerators can be designed with no speed changes assumed (since the electrons are already moving at effectively light speed) and thus tend to be highly efficient and have the highest acceleration gradients achievable. For similar reasons electrons are also able to be accelerated in plasma based accelerators, allowing for even higher acceleration gradients and smaller machines for a given energy. The major downside of electrons is that due to their large charge to mass ratio they will radiate energy like crazy when bent by a magnetic field. Thus ring shaped accelerators are of limited use for electrons since you have to make the ring extremely large to not just lose all of your energy as beams of x-rays coming out of the machine! This very trick is used to make high quality x-ray sources for industrial and research applications.

When electrons hit matter they tend to release their energy as photons in a process called bremsstrahlung, or braking radiation. These extremely high energy photons (roughly ~⅓ the peak energy of the electron on average) then proceed to make electron/positron pairs that then make photons that then make pairs that then make photons… This whole thing is a massive cascade of radiation that ionizes, heats and can even damage on a nuclear level! At high enough energies you will generate every particle known and might even make some new ones!

=== Proton ===
Protons are the next most common particle to be accelerated and are currently used by the absolute largest particle accelerator ever made (the Large Hadron Collider or LHC) as the primary particle of choice. Since they have ~2000X lower charge to mass ratio than electrons, protons are about ~2000X harder to get up to speed. Unlike electrons they are not promptly at light speed so we have to send protons through a series of various accelerators depending on the energy of choice. For example the protons in the LHC the protons go through 5 different particle accelerators plus the extremely low energy stage that first makes the beam of ions! This difficulty in reaching highly relativistic speeds has two other bad side effects. Firstly, if we want to transmit our protons through empty space we will probably need to neutralize them with a beam of electrons.

{{Note|''When shooting extremely high current (kA), relatively low energy (not highly relativistic or in some cases relativistic at all) beams in the atmosphere the higher mass and positive charge of protons gives them many beam propagation advantages over electrons if the beams can be made. In this case no neutralization is required as the atmosphere itself becomes a plasma and neutralizes the beam.''}}
Luckily this is not very hard and will add almost nothing to your comparatively large proton accelerator, but may increase your beam spread a little bit via some neutralization physics. This does mean your beam is technically plasma, which is cool! Second, we effectively cannot use those neat plasma based accelerators for protons since we would need nearly TeV class proton beams to inject into the plasma accelerator and by the time you reach those energies you have basically already done all of the work. Maybe in the future there will be ways around this, but as of right now we are limited! There is one major benefit to this poor charge to mass ratio though, and that is the lack of synchrotron radiation! Ring shaped accelerators are ideal for protons, and basically all ions.

When protons hit matter they can interact with all of the four fundamental forces, although only two of them (Strong Nuclear and Electromagnetism) tend to play big roles. Protons barrel on in causing huge ionization cascades and can also release bremsstrahlung photons at high enough energies. They are also ideal at knocking neutrons, protons, deuterons and more out of the target nuclei which just adds to the fun! Protons also come with a neat trick known as the Bragg peak, where they deposit a large portion of their energy right at the end of their trajectory through a target material. This is a very useful trick for medicine and means you can “tune” the beam to almost ignore large portions of material and primarily heat, ionize, and destroy the area of interest. You can imagine how useful this is for cancer therapy or more nefarious things!

=== Light ions ===
Light ions are a rather fuzzy term but here we will use the definition of any ion heavier than a proton but lighter cobalt. The fuzzy reasoning here is that you get on the other end of the nuclear binding curve (i.e. can’t get net energy from fusion) at iron, so cobalt will then no longer be “light” and has to be made in a different process. Feel free to pick your own cut off! Historically beams up to neon have been experimented with for nuclear fission, nuclear fusion and medical therapy applications. Light ions have even worse charge to mass ratios than protons (Z/A times worse) and typically you will find light ions have about ½ the charge to mass ratio of a proton on average, although the isotope 3He is a notable exception. This means they are even slower to accelerate, although often the exact same machines as protons can be used with minor tweaks. Light ions have similar requirements for neutralization as protons, although their heavier mass can help fight beam bloom for a similar beam speed (not energy since the increased mass means the particles are higher energy for a given speed). The decreased charge to mass ratio also means that synchrotron radiation is even lower so beams can be driven to higher speeds before radiating. Light ions have similar target interaction physics as protons, although the ion itself will break up at very low speeds and add to the radiation environment.

=== Heavy ions ===
Heavy ions are being defined here as cobalt up to whatever unstable element you can jam into your accelerator! Typically heavy ion beams are made of elements like lead (used in some runs on the LHC), gold (used in the Relativistic Heavy Ion Collider) and even californium (used in some runs on the Argonne Tandem Linac Accelerator System). These beams tend to be for pure nuclear physics research, but there has been a long standing interest in heavy ion beams for inertial confinement fusion research and as potential methods of propelling spacecraft. These extremely heavy ions will be even harder to accelerate to ultra high speeds than light ions and much slower for a given energy, but they do provide a lot of momentum to make up for that! They have the same neutralization requirements and same potential upsides for beam bloom as light ions, but will most likely be orders of magnitude slower reaching our targets than the lighter beams. Heavy ions can also be used in the same machines as protons with small modifications and generate almost no synchrotron radiation as a result of their much smaller charge to mass ratio. They also have similar interaction physics to protons and light ions, but will stop in much thinner layers of materials. While electron, proton and light ion beams tend to both mechanically destroy and irradiate targets through-out, heavy ion beams will act much more like lasers and burn in from the outside of the target on in. There will still be a spray of ionizing radiation, but far less of one than an equivalent energy electron, proton, or light ion beam. The much higher momentum per particle, lower radiation concerns and ease of deflection makes heavy ion beams very interesting for beamed propulsion purposes.

=== Macrons ===
Macrons are typically defined as macroscopic particles, such as extremely large molecules or even grains of sand, that are charged up and accelerated via similar technologies as traditional particle accelerators. There are many difficulties with getting macrons up to appreciable speeds due to the extremely low charge to mass ratio, engineering difficulties with getting consistent and good charging, phase matching with accelerating cavities and lack of effective focusing optics. There are so called “dust accelerators” in existence that utilize the first stage of ion accelerators to generate their “beams” and proposals have been made to create better versions using updated technology. Macrons will interact just like a very small and very fast bullet unless they are designed to carry a nuclear payload to boost this output. Antimatter is an assured option for getting energy gain from a macron impacting a target, although there are potentially options for fusion, fission and fission/fusion given the correct sizes (mm scale or bigger), clever target design and the right impact speeds (most likely >>100 km/sec). The various energy gains versus speeds can be seen here:

[[File:IvKmacron.svg|900px|thumb|center]]

It can be seen that antimatter proves to be worth it up near the speed of light, but both fission and fusion stop providing much gain well below the relativistic regime. This also helps explain why heavy ion uranium beams are not going to provide some incredible advantage over any other ion.

=== Antimatter ===

Antimatter is the evil cousin of normal matter and whenever the two meet a release of energy equivalent to their combined mass energy follows. Now this does sound like an amazing option for beams but there are two downsides. First, antimatter is energetically expensive to make! Positrons (antimatter electrons) are the “cheapest” at 1.02 MeV of input energy at a theoretical best, but they only give you that much energy back when hitting a target. Since a typical positron beam can be expected to be equivalent to an electron beam, there is little gain in adding 1.02 MeV per particle to a multiple GeV (or even TeV) beam. You can see this explicitly:

[[File:IvKhBarPostron.svg|900px|thumb|center]]

For anti-protons the math becomes more favorable, but creating these particles is much more difficult and energy consuming. At best anti-protons require ~2 GeV of input energy to create, but in practice >400 GeV per anti-proton has been needed. Luckily significant energy gain is had up to several GeV of beam energy, so if anti-protons can be made at a central facility of some kind and easily stored (neither being easy things) then there may be some advantage to such a beam. Heavier ions are significantly harder to make since fusion with antimatter must be performed to generate those ions. Barring a naturally occurring source being found, antimatter heavier than a proton is most likely not worth using.

{{Note|'' Positrons can potentially fill an interesting niche either as a co-propagating neutralizing beam for electrons or perhaps as a higher performance atmospheric beam than electrons due to the positive charge. In this case the annihilation properties are no what is of interest, rather the positive charge and relative ease of acceleration compared to ions. Since positrons share all properties with electrons except for charge, they can be accelerated just as easily and even co-accelerated with electrons in the same machine.}}

=== Muons ===
Muons are an interesting cousin of electrons and positrons. They are unstable and heavy cousins of electrons, massing about 200X more than electrons and living about 2 microseconds in the lab frame before decaying into an electron (or positron) and a neutrino. If we accelerate them up to relativistic speeds (about 10X easier than protons and the muons are typically born relativistically thanks to how we make them) then the muons can be made to “live” far longer. Muons have long been of interest to accelerator physicists due to their position as an almost ideal intermediary between electrons and protons. The requirement to make them on site has always been an impediment though and muons are currently only used for certain niche experiments. During the planning stages for a potential muon collider it was also noted that a muon collider would also have a serious issue with neutrino radiation! This is due to how neutrinos are more likely to interact with matter at high energies and a muon collider would be generating a LOT of really high energy neutrinos in a ring around the machine. This has been proposed as a sort of super weapon even!

Muons should interact with matter in a similar fashion to electron beams and the beams will probably be treated in a similar manner. They can propagate through the atmosphere quite well in theory and should outperform electron and proton beams for a given beam energy, although at quite a cost… Muons are typically generated with proton beams impacting targets and will create an equal number of positive and negative muons at high energies that must be carefully captured. This would allow for the easy creation of a neutral muon beam though, which may make for a great giant doom beam! Do note that negative muons also can help catalyze nuclear fusion so a very high power beam hitting a fusion fuel tank could produce some rather… Interesting results!

=== Other exotic options ===
There are a wide variety of odd particles that could also be chosen for use in beams given a specific need for them, although generation and utilization of these particles will be quite difficult! Extremely short lived particles like pions, kaons and taus can be generated and accelerated up to extreme speeds to help propagate them far enough to matter, although that will need some very extreme speeds. At the upper end of extreme there are even proposals for accelerating tiny charged blackholes!

= Accelerator technologies =
There are a wide variety of accelerator technologies and here we try to summarize them as best as possible. Different accelerator types may be sequenced one after another. Some particle accelerator technologies depend on the particles fulfilling certain requirements of speed (thus energy) and hence different systems may be employed in sequence. Accelerator systems are described in terms of their efficiency of turning input power into particle energy, their acceleration gradient measured in electric field which translates to volts per meter, and the mass per length meter. An interesting secondary parameter is their operating temperature. Some accelerators can operate well at high temperatures. Others require cold or cryogenic conditions to work well.

== Core accelerator assembly ==
We define the core accelerator assembly as the structure where energy is imparted onto the particles.

=== DC Accelerators ===
For very low beam energies we can utilize large DC electric fields between plates or grids to accelerate particles up to speed. This is the same technique as is used in ion drives and is the most common industrial accelerator used. Every dentist's office and old CRT television uses this acceleration technique. Ion accelerators often also use a DC accelerator to generate the initial beam of ions, while electron beams utilize other techniques. These machines are limited by material breakdown to about 0.5 MV/m accelerating gradient and typically no more than a couple of MeV total particle energy for extremely large machines. The highest energy DC accelerators can reach 40 MeV for proton beams, but this is for machines massing in the 100s of tons and at low total beam powers. Very large DC accelerators are currently the only tested method for accelerating macrons up to 10’s of km/sec.These accelerators are limited by the risk of arcs jumping between the electrodes at too high voltage differentials.

=== Radiofrequency Quadrupole ===
A more advanced first stage accelerator for ion beams is the radiofrequency quadrupole or RFQ. These machines utilize low frequency radiowaves and carefully designed conductive bar structures to efficiently generate high current ion beams up to energies of ~7 MeV in a couple of meters. RFQs were a revolution in ion accelerator physics and caused quite a stir when the USSR first showed them off as they provided a much higher power and higher quality ion source in a smaller package than anything the USA had built! RFQs have also been proposed as methods of efficiently accelerating macrons up to >100 km/sec. These machines can be normal or superconducting and can operate with efficiencies as high as RF LINACs in theory.

=== Cyclotrons and Betatrons ===
One of the earliest ways to accelerate particles up to relativistic speeds is with circular accelerators known as cyclotrons (for ions) and betatrons (for electrons). In Cyclotrons the accelerator is a round chamber which houses two half-circle electrodes roughly shaped like a D (and thus called “dees”) separated by a gap, sandwiched between the poles of a large electromagnet. The magnet creates a static magnetic field which bends the path of charged particles into a circular motion. An oscillating alternative voltage is applied to the electrodes (what we often call RF or radiofrequency) to accelerate the ions. This radiofrequency field’s oscillation is matched so that one cycle of the voltage matches one circuit of the particles. As a consequence when crossing the gap from one electrode to the other, the charged ions experience a voltage gradient and are accelerated. As the particles gain energy and speed, their radius in the magnetic field gets larger. The result is an overall spiraling trajectory from an injection point in the center to an exit point at the rim where the charged particles are bent out of the cyclotron. Cyclotrons use the same voltage gradient many times over, defeating the electrostatic breakdown that limits the single-time electrostatic accelerators. As a consequence, proton energies of up to 70 MeV may be reached. Cyclotrons energies can also be increased by switching from normal magnets and RF resonators to superconductive components. This eliminates various losses from electric resistance and absorption and also allows for much stronger bending magnets, which allows the cyclotron to contain particles up to a higher energy if desired. Superconducting cyclotrons have achieved up to 520 MeV. Still, cyclotrons suffer from poor beam current, pulse rate, and beam quality issues that make them a less than ideal choice. The large magnet required (measured in meters) also does not lend itself to a lightweight weapon and as such, cyclotrons are regulated to niche medical and physics uses.

A close cousin is the betatron, which is used to accelerate electrons. In this case the magnetic field strength is increased as the electrons speed up, which prevents any spiraling trajectories of the particles. Betatrons have been operated up to electrons energies of 300 MeV, but are more typically in the ~10 MeV range. In principle a betatron could also be made superconducting, although magnet ramping may prove to be difficult. Other types of ring shaped accelerators may prove to be more useful, although electrons are heavily limited by synchrotron radiation and it is often better to use linear accelerators for them. For now betatrons remain limited to small (sometimes portable) accelerators for nuclear physics, industry and security work.

=== Radiofrequency Linear Accelerator ===
The real work-horse of the particle accelerator world is the RF linear accelerator. These babies use the same alternating field trick that cyclotrons use, but in a straight line rather than needing to bend the beam around in a circle. This provides a couple of nice advantages over the more traditional cyclotron. First and foremost you get to leave that large, bulky magnet behind which saves you a ton of mass, cooling and power consumption! You also get to now have much higher RF frequencies (1-10 GHz right now compared to the MHz used in cyclotrons), which means higher accelerating gradients. To generate these powerful, high frequency RF waves (also known as microwaves) that drive the accelerator we typically use military radar power supplies or (in more advanced cases) even lower energy particle accelerators! State of the art RF LINAC technology is well into the 100 MV/m range and have no upper limit on beam energy! The Stanford Linear Accelerator reaches 50 GeV and was built back in the 1960’s! There have been traditional RF LINAC designs for machines reaching up to 1 TeV, with the only limit being how much space you have to put these monsters. They do operate best (i.e. with high accelerating gradients and efficiencies) with particles already moving at relativistic speeds, so any ions fed in need to use a series of lower gradients LINACs or other pre-accelerators. Electrons and positrons can naturally be used in LINACs since they accelerate up to relativistic speeds so quickly.

LINACs also have been shown to operate at high average powers (MW scale is now in use) and high pulse energy (10’s of kJ pulses fired at high rep rate) while also maintaining a high quality beam that stays well focused. The two newest advancements have been to push the frequency ever higher for higher acceleration gradients (300+ MV/m has been shown and there are plans for even more) and to switch to superconducting RF accelerating cavities (the name of the actual piece of equipment that does the accelerating) which then dramatically increases the electrical to beam efficiency (80-90% has been shown and even copper waveguides have been operated at 70%). All in all, LINACs beat the tar out of cyclotrons which is why we don’t build cyclotrons anymore. To quote an old Fermilab engineer “cyclotrons are an expensive and bulky way to get a really crappy beam or a cheap way to get a giant chunk of lightly radioactive steel that can never leave your lab”.

LINACs work in detail by shaping the radiofrequency energy in a confining waveguide. The oscillations of the RF energy may be held in space, or allowed to travel down the waveguide over time. Radiofrequency waves have connected magnetic and electric components, which oscillate.

[[File:2022 08 16 GalLib ElectromagneticWave.svg|center|800px]]

Inside the RF accelerator, the electromagnetic radiation is oriented 90 degrees to the particle travel direction, which points the associated electrical field in the travel direction of the particles.The radiofrequency field naturally oscilates along its wave, and the connected electrical and magnetic fields oscilate as well: building up, reaching a peak, and diminishing, inversing, building up, reaching a peak, and diminishing, and the cycle repeats. The detailed behavior depends on the specific cavity.

[[File:CavityWithEM.png|600px|thumb|A resonant RF acceleration cavity and the electric field inside of it, created by resonating RF fields, at one moment in time.]]

Particles of the right charge in the right space at the right time experience an acceleration from the electrical field, being drawn to the opposite electric charge. If they land in the other region, they are instead decelerated. Inbetween, particles experience less to no acceleration. This behavior naturally separates a stream of particles into discrete '''bunches'''.

[[File:CavityWithEMAndParticles.svg|600px|thumb|Bunches being accelerated inside the electric field. ]]

It also creates an implicit limit to what can be effectively accelerated using a specific radiofrequency accelerator design. The particles driven through it have to keep pace with the oscillations of the electric field, reaching the next correct region inside the cavity at that time when the electric field is oriented correctly. If they are too slow to keep inside the accelerating regions, the accelerating action will be ineffective. This requires RF cavities to be matched to the particle's velocity gain under energy imparture and their velocity when entering, being inside, and exiting the accelerator. For electrons and positrons, this isn’t that complicated. Since these lightweight particles accelerate readily, they have few problems “staying in the pocket”. Particles that are slower to gain speed, such as ions and even larger particles, must however be accomodated. This in turn impacts how much acceleration and energy per meter can be fitted into the accelerator in the first place. The acceleration of heavy particles requires specifically designed and sequenced RF accelerators for specific regions of particle speed. This in turn complicates the design of the larger accelerator infrastructure, with multiple discrete stages getting the particles to the energy required so that the next stage in sequence can take over.

RF cavity accelerators are limited in their eV/m capability by the critical current at the containing waveguide. To increase the acceleration provided, more intense electrical fields must be built up. Eventually the electrical charge of the RF waves becomes so strong it can rip electrons out of the cavity walls, causing an arc to form and the cavity material to break down physically.

=== Advanced Accelerator technologies ===
Present accelerator technology caps out at around 250 MeV/m for the most overdriven designs. This poses issues for a variety of applications, where higher energies in shorter distances are desired. In science, "desktop" and "chipset" accelerators are of interest. When militarizing particle beams, some effects only become available at very high energies - but you want something compact enough to turret.

urrent and even near future accelerator isn’t really ready to supply that sort of scale quite yet, so we have to move on to the cutting edge of accelerator physics. There are three big options with sub technologies within these broad fields. We have extreme RF cavities using much higher frequencies, plasma accelerators using lasers for power and plasma accelerators using particle beams for power. Each technology provides a unique series of advantages and disadvantages, but fundamentally they are all a way to make linear accelerators that are much smaller.

==== Dielectric Wakefield Accelerator ====
LINACs using much higher frequencies than traditional LINACs have two fundamental problems to fight. The first is electrical breakdown and the second is generation of those mm-wave (10-100 GHz) or even THz (100-1000+ GHz) sources at sufficient power and efficiency to power a particle accelerator. As the frequency of the driving power goes up the electric field climbs up, which means the accelerating gradient gets larger (which is good) but also means that we run a higher risk of a huge arc forming in the accelerator and breaking it (which is bad). This is often fought by changing the accelerator cavity material from a metal (or superconductor) to a dielectric material like ceramic, or by using a coating of dielectric, which can sometimes earn them the moniker of Dielectric Wake Field Accelerator (DWFA). The issue of creating powerful and efficient enough sources of this higher frequency drive power is being actively pursued for a variety of fields (5G cell signals, beamed power, etc) but is still quite immature compared to the microwave RF sources of traditional LINACs. There are options using essentially "over-clocked" microwave sources, low energy particle beams, and even laser-plasmas. There is still a lot to be done here, but these sorts of problems are being actively worked on there are working accelerators using this technology. There is potential for GV/m or greater acceleration gradiatens and all of the same advantages of traditional RF LINACs (i.e. high powers, high fire rates, high energy pulses, quality beams) should stay the same. The wall plug efficiency is liable to be a bit lower, but that will be highly dependent on the efficiency of the source of these high frequency waves.

==== Laser-Plasma Accelerator ====
Plasma accelerators using lasers for power are a whole world in and of themselves. There are a wide variety of ways to accelerate particles with lasers and plasmas, but we will focus on the most studied technique known as Laser Plasma Accelerators (LPA). LPA accelerators utilize ultrashort (<1psec) and ultraintense (>10^18 W/cm^2) lasers to drive a plasma wake (thus sometimes earning them the name laser wakefield accelerator) and accelerate particles through the resulting charge differential. They have only shown the ability to accelerate electrons and positrons, but any ions that could be injected at the right time and at extremely relativistic speeds could conceivably be accelerated this way. In the operation of a typical LPA the ultraintense laser is fired into a jet of gas (turning it into plasma) or a channel of lightly pre-ionized plasma. The laser then blasts through this plasma like a boat through water and leaves a void behind it (the wake). This void ends up having a HUGE accelerating gradient (100 GV/m) and suffers no risk of electrical breakdown since the plasma is already broken down! A pre-accelerated bunch of particles can be injected into that void at the right time (called the witness bunch) for a BIG acceleration or you can rely on the wake itself spewing electrons out into the void and make a beam out of the plasma itself. The gas or plasma then quickly is reformed and the laser fires as soon as its cooling and charging allows. Keep in mind that even though the beams can hit incredible energies in short distances, the laser itself is not a pocket sized device. These lasers are room sized at minimum and not easy to shrink. The pulse lengths required are also much shorter than most of our hypothetical weapons-grade pulse lasers that may operate in the nanoseconds regime - dual-use may prove difficult. The intensities reached also pose significant issues for the laser hardware. If you don’t spread the energies out, it becomes quite possible you will break parts of your laser from the produced intensities.

Current LPA systems suffer from a couple of problems that are being actively worked on. First and foremost they are incredibly efficiency limited by the laser tech being used. There are plans to move on to much more efficient fiber lasers and eventually make LPA machines that operate in the 25% or greater wall plug efficiency, but that is not currently doable. These lasers also need to be fired at higher rates than the current ~1 Hz, but that is much more of a limit of laser engineering than accelerating engineering. LPA’s also have yet to link more than one accelerator together and the lasers are limited in just how much plasma they can accelerate through right now (~20 cm max) so the beam energy runs into a wall quite quickly. Aside from R&D on linking LPA’s together in a way similar to RF LINACs there are two other options. The acceleration gradient is higher the denser the plasma, so a move to denser plasmas and shorter wavelength lasers (to penetrate the plasma) could provide higher beam energies in similar distances. The other option is to use clever optical techniques like the so-called “flying focus” to allow for the laser to go through much longer plasmas. This could in theory allow for more efficient LWFA’s and single stage machines that provide ~1 TeV electron beams in meters! The final issue is beam quality. Even the best LPAs will not make beams that have as tight of a beam spot, energy spread, or emittance as a normal RF LINAC. This means they will end up needing some beam “cleaning” if you want to get a lot of range out of them.

LPAs are of engineering interest for very short wavelength X-ray free-electron lasers in a compact format. Currently these facilities require linear electron accelerators many kilometers in length. With an LPA as the electron accelerator, an XFEL able to image molecular structures and chemical reactions as they happen might be shrunken down to a much more compact format. (Sometimes this is discussed under the ultimate promise of the “benchtop” or even “chip-format”(sic!) XFEL)

==== Beam-Plasma Accelerator ====
Plasma accelerators using particle beams for power are very similar to LPAs in principle but use a particle beam rather than a laser. We typically just call these things Beam Plasma Accelerators (BPA) for short. The concept is very similar to the LPA in that a high intensity beam comes in and blasts a void into a plasma, which particles can then be accelerated in. BPA provides some interesting advantages over LPA in that you can use highly efficient superconducting RF LINACs (or synchrotrons if you want) to provide the particles at very high efficiency. They also have little in the way of plasma density limits and the accelerated beam takes on the characteristics of the beam driving the BPA. Thus little if any beam cleaning will be needed after acceleration. The big downside is the need for a relativistic particle beam to drive the effect, and typically one in the 100’s of MeV to low GeV range, depending on the particle used. Thus your final accelerator may be short, but you still won’t be making a handheld particle beam device.

== Accelerator Shapes ==
The accelerator technologies discussed above are the means by which we create situations for particles to gain energy. The accelerators become embedded in a larger structure that produces, directs, accelerates, even stores the particles until they are directed towards their final use.

There are two principle shapes for accelerator assemblies. Ones where the beampath is not closed onto itself, and designs where the beam path is closed on itself in part of the assembly, forming a loop.

=== Linear accelerator ===
Everything from electron generation to acceleration to conditioning is built in a straight line, one after another. This is the Linear Accelerator or LINAC. The main catch here is that there is only one single passage for the particles through the entire accelerator. Re-using accelerator hardware to drive the same particle to higher energies by repeated passage is out. The particles have to reach their desired energy in a single pass. Practically this puts limitations on the energy achievable with heavy particles in a LINAC. The length required gets impractically large. Even for electrons, lengths of hundreds of meters to kilometers may be required, unless advanced accelerators with high gradients are used. This said, every accelerator contains linear sections. Since so much high-performance accelerator technology depends on particles having a minimum velocity, ions must be “pre-conditioned” in a set of linear accelerators before the re-use of an accelerator through looped beam paths can be considered.

LINACs of low energy may be used in many applications. Medicinal they find use in radiation therapy and radiosurgery.

=== Synchrotrons ===
The other shape which has some interesting properties are accelerators where beampaths are looped onto themselves. These are called synchrotrons. Since beam paths are closed onto themselves, particles can pass through the same accelerator unit repeatedly, gaining more energy on every passage. This is advantageous for heavy particles which do not gain much speed on any single passage. They can pass through the same accelerator thousands or even millions of times instead. This is obviously advantageous for facility bulk and mass and allows us to reach energies that would otherwise be impractical. The accelerated beam bunches can also be held in storage, looping like the rotating mass of a flywheel, holding energy. The LHC can store some 700 Megajoules of energy in the recirculating beam after full speed has been reached.

That said, the devil is in the details. When charged particles are bent in circles they emit electromagnetic radiation which scales very rapidly (to the fourth power!) with relativistic gamma factor and with bending radius (inversely to the second power). This phenomenon is called synchrotron radiation. Due to the scaling, high energy particles emit more synchrotron radiation than lower energy ones and lighter particles (like electrons) at those high energies emit much more than heavier particles. In addition, the bigger you can make the bending radius of the synchrotron (often called “the ring”) the less synchrotron radiation you get.

[[File:SynchrVsE.svg|800px|center|thumb|Synchrotron radiation per particle versus particle energy for electrons and protons in a 50 meter in radius ring.]]

[[File:SynchroVsRing.svg|800px|center|thumb|Synchrotron radiation emission per particle for different particles at different energies, over varying bending radii. ]]

Although these numbers look small, keep in mind that this is PER PARTICLE. A typical pulse in an electron accelerator will have more than more than a billion electrons in it! Large synchrotrons like the LHC will have over 300 trillion ions circulating through it! Synchrotron radiation necessarily has to be managed to reduce radiation shielding, the radiation load (and possibility of hardware failure) on the accelerator, and to reach high energies in the first place - you might lose all the energy your accelerators are adding to synchrotron radiation!

Deflecting the beam also requires the application of magnetic fields, and higher degrees of deflection, in a smaller length, at higher particle energies, require stronger and stronger magnetic fields. Eventually multi-Tesla fields from superconductors will be required. And due to action and reaction, those magnets need to be braced against the pressure the beam exerts as it is deflected. Ultimately this leads to the same limitations as with flywheels - the mechanical strength of the braces and static connections that keep the magnets together as a larger structure is the accelerator’s limitation on J/kg held in storage. A further complication: The magnet strength has to be matched to particle energy. As a beam is accelerated, the deflectors have to ramp up their power to keep the degree of deflection constant.

Sometimes, synchrotron radiation from electrons is intentionally induced in a ring-shaped accelerator. Since synchrotron radiation of high-energy electrons tends to be hard x-rays emitted along narrow vectors, it’s a way to get x-rays used in various imaging techniques in material science and molecular physics.

Synchrotrons used for ions also need to respect the limitations of the involved accelerators. Synchrotrons often use radiofrequency accelerators. Since radiofrequency accelerators must match the speed and speed gain of their particles, an ion synchrotron cannot be made up of a single accelerator loop. Instead there are sequential loops connected in sequence, each designed to boost the particle through a specific velocity and energy region so that the next accelerator takes over. A designed example is the LHC. Particles headed for the main LHC actually circulate in two other synchrotrons beforehand, gaining sufficient energy so that the LHCs RF cavities can take over. (Also, before being injected into the first synchrotron ring, the ions must first go through multiple linear accelerators using different technologies to reach a base energy.)

An interesting design consideration: Synchrotrons don’t necessarily have to take the form of a (near-ideal) ring. Funamentally any dipole represents a single point of deflection of a straight path - they're polygons! Any geometric shape that contains a loop, and where any single bending curve keeps within the limits of the synchrotron radiation you are willing to tolerate, works. As such, complicated two- or three-dimensional “folded-up” shapes are also possible formats for a synchrotron.

=== Storage Rings ===
Since synchrotrons have to match their deflection magnet strength to the energy of the particles being accelerated, a single synchrotron track cannot hold more than one small section of beam bunches in storage at a time, and becomes unavailable to accelerate further bunches until it discharges the stored beam. This is disadvantageous for various applications where we may want to store a fully accelerated particle beam “bolt” for a longer time.

Storage rings are separate synchrotrons designed to hold multiple sets of bunches in circulation at a target energy at a time. The bunches are delivered from an accelerating synchrotron (sometimes called a booster ring) and held in circulation in the storage ring. Since there are synchrotron radiation losses, a few accelerator sections must compensate for these.

Storage rings can also be used to condition a stored beam. A common thing to do in storage rings is to “cool” a beam, draining unwanted movement energy out of the particles. Beam cooling may take significant time, which makes it ideal to conduct in a storage ring. The result is a beam with less tendency to drift apart, advantageous for many applications.

== Other parts ==
=== Particle sources ===
Particle sources are assemblies which supply particles to accelerators. Here we can group them into three categories: electron sources, ion sources, and exotic particle sources. The later involve the production of particles by means of gathering them from collision reactions (or the handwavium blackbox of your choice) and feeding them into another accelerator for further use. Positrons for example may be manufactured in beam collisions.

Electron sources require the production of a free stream of electrons. The oldest method here are thermionic sources. A hot piece of material ejects free electrons into a vacuum, where they can be drawn in using electric differentials and forwarded to the accelerator.

Field emission electron sources instead use a strong potential between a needle-like cathode and an anode plate with a hole. The electrons are drawn from the needle and through the anode due to the high charge differential. This method produces a “cooler” beam with less random divergence than thermionic sources.

A further improvement still are photoemission sources. Here the precise application of laser light onto a photosensitive material energizes electrons in the photocathode until they break free. This process is thermodynamically very precise and due to the nature of the laser light, also very controllable. The result is a low-emittance stream of electrons. These may be accelerated using electrostatic attraction with an anode at one end of the vacuum chamber, or directly using an RF field.

Ion beam sources are more complicated. The physics of ion formation are complex, which opens many design choices when creating an ion beam. Generally a plasma must be produced, from which the ions can be extracted.

The plasma can be formed in a variety of ways. Electrons from a local cathode-anode array, plasma frequency heating, the injection of electron beams, the application of laser light. The end results can be as diverse. Some methods only produce weakly charged ions and on many vectors, so with high emittance. Other methods offer greater ion charges or reduced emittance.

=== Magnets ===
While acceleration of charged particles is effected with electric fields, magnetic fields can be used to bend the trajectories of particle beams. As such, electromagnets can be used to bend the trajectories of charged particles in various ways, and in a controllable fashion.

==== Dipole Magnets ====
Dipole electromagnet are used to steer the entire particle beam in one axis. (To steer in two axes, we must either rotate one dipole about the beam axis, or couple two dipoles in sequence.) They can be used to direct a beam inside a particle accelerator, inject and extract beams out of looping constructs by switching on and off at the right moment, and point a beam at a target.

In a dipole magnet, a positive and negative coil oppose each other, embedded in a “choke” assembly often made of steel. Charged particles moving through the magnetic field between the coils experience a deflection in one direction specified by the polarization of the field, and a degree of deflection specific to the field strength.

==== Quadrupole Magnet ====
Dipoles can steer, but they influence all particles equally. As such, they cannot be used to manipulate the beam's shape. The first type of magnet able to influence a particle beam’s shape are quadrupole magnets.

They are assembled from four coils at 90 degree angles, with coils of the same polarity opposing each other. The passing beam is compressed on one axis, and expanded on the other. To focus the beam overall, at least two quadpoles must be arrayed in sequence.

Quadrupoles are the basic and essential “magnetic lens” of particle beams. Without them, charged beams would inevitably defocus and run havoc.

==== Hexapole Magnets ====
Hexapole magnets exist to address one problem of quadrupoles - they focus differently depending on the energy of the particle passing through the magnetic field. The effect is likened by particle accelerator physicists to the different diffraction of light in a prism - and hence the effect is called “chromatic aberration”. Hexapoles are used to correct for this, stronger deflecting higher-energy beams.

==== Integrated optics ====
As the field of particle accelerator science and engineering advances, more complex designs are proposed to handle more complex phenomena inside accelerators. An upcoming field is “integrated optics” - complex magnetic systems with nonlinear behaviors. These could provide more advanced manipulation of electron beams, building upon existing capabilities and adding new ones for shaping a beam.

=== Plasma lenses ===
Plasma lenses are an up-and-coming technology, where very high currents in plasma are used to create very strong magnetic fields. Similar to plasma-based accelerator technology, the fact that charges are already free in plasma is used to circumvent conventional charge breakdown and critical current limitations on solid-matter devices. The resulting magnetic fields can be much stronger and bend beams over shorter distances. Furthermore, by creating complicated profiles in the plasma, elaborate beam manipulations could be performed, more akin to sophisticated optical elements. As well, the charges in the plasma neutralize the particle beam in the region of the plasma lense - affecting its propagation behavior further and in a positive manner, as self-repulsion is neutralized.

The catch is that this is plasma - that means you are dealing with all the complications of plasma physics. Heating of the plasma has been an early challenge to successfully forming and maintaining plasma lenses.

Plasma lenses require an appropriate chamber, possibly active containment to keep the plasma from losing energy to the walls or leaking into other accelerator parts, and the necessary plasma sources, maintenance heating devices and then lense shapers and control sensors. In high-powered plasma lenses there will also be a need to cool the plasma as it heats up from waste heat. Potentially we may manage this by open-cycle cooling - venting the plasma controlled in one direction into open space, rather than have a closed-cycle solution for reconditioning the plasma. This could cause some very cool visuals!

Plasma lenses for particle beams should not be confused with optical plasma lenses for lasers! Those exist as well, but are their own topic.

=== Chicanes ===
Chicanes are assemblies of multiple dipole magnets used to manage the longitudinal dispersion of a particle beam due to different particle energies. Since particles with higher energy move faster, they drift ahead in the beam distribution over time; and particles with less energy increasingly lag behind.

Chicanes deflect high-energy particles onto an outward curving trajectory. Since traversing the curve takes longer, the high-energy particles are effectively delayed, allowing lower-energy particles to catch up. By the end of the chicane assembly, the bunch has been compressed in the longitudinal direction.

Chicanes however, do not remove energy from too-fast particles. That is the responsibility of other structures.

=== Undulators and wigglers ===
Undulators and wigglers are magnetic assemblies used to transform particle energy into electromagnetic emissions, with different properties from the specific implementation of a shared basic principle.

Assembled from stacks of generally dipole magnets (other configurations exist), both undulators and wigglers force charged particles onto “wiggling” trajectories by bending their path left and right. Wigglers force very strong arcs; undulators less so. In both cases, the rapid deflection of the charged particles causes them to shed parts of their energy as electromagnetic emissions. Due to the energy of the particles, these photons are generally highly energetic x-rays.

Undulators may be used to build free-electron laser systems.

They can also be used as a means of cooling the beam. The removal of energy from particles can also suppress unwanted movement modes, allowing accelerator structures to re-add energy only to desired movement modes. Of course, this means the particle beam actively loses energy, and produces x-rays that have to be absorbed (and cooled away.)

=== Beam coolers ===
Beam coolers are a collection of systems and strategies that are used to remove unwanted movements from a particle beam. Undulators and wigglers are one way to do this.

One method set of beam cooling relies on finding particles who are very “hot” with random motion, and either removing the particles from the beam using a kick device, or applying a precise counter-force that acts against the unwanted motion.

If one is dealing with ions, one clever trick is to inject a beam of electrons. Due to the thermodynamics within the particle population, the lighter electrons absorb heat from the heavier ions, taking this unwanted motion energy with them when the electron beam is bent out.

As well, lasers can be used (what can’t lasers be used for!) to apply counter-force on ions and remove unwanted movement in this manner.

Beam coolers are generally not instantaneous devices. They work over time to make a beam better. Often they involve the removal of energy or particles from the stream, requiring accelerators and sources to make up the loss and creating waste heat.

=== Beam recovery ===
Some accelerator systems can actually be run in reverse - they can turn beam energy back into some other energy format and then electricity. This can be used to recover some to most of the energy from a particle beam before disposing of it. This is advantageous both for your energy bills and making a very high-energy beam liable to spray radiation everywhere, easier to absorb in a beam dump.

=== Beam dump ===
A beam dump is a construction where a particle beam can be routed in case it has to be disposed. Beam dumps are designed to safely absorb the particle's energy without leakage of dangeorus radiation or activated materials to the outside enviroment. Common materials are carbon and metals, embedded in an actively cooled assembly.

= Propagation physics =
==Magnetic fields==

Magnetic fields are things of motion. They are created by moving electric charges. And they only affect electric charges that are moving. An electric charge at rest in the presence of only a strong magnetic field will just sit there. But when it starts moving, the magnetic field interacts with it, starts deflecting it.

But not along the field. Oh no. The magnetic field does not point in the direction it will move the charge. No, the charge experiences a force that is perpendicular to the direction of the field. And also that is perpendicular to its own velocity. A charged particle moving straight down a magnetic field line will not experience any force.

On the other hand, if the particle is moving directly perpendicular to the field, the magnetic force will always push it to the side, even as it keeps turning. This results in the particle moving in a circle. The radius of this circle is called the <b>gyroradius</b>, and the rate at which the particle makes circles is called the <b>gyrofrequency</b> (or sometimes <b>cyclotron frequency</b>).
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Circular_motion.png|frameless]]
<td width=255>[[File:magnetic_forces.png|frameless]]
<td width=255>[[File:spiral_around_field_lines.png|frameless]]
<tr>
<td>To get circular motion, you need a force (blue) that is always the same amount but always pushing perpendicular to the direction of motion, toward the center of the circle. A charged particle in a constant magnetic field moving perpendicular to the magnetic field meets these conditions, and moves in a circular path.
<td>A diagram of how a magnetic field (magenta) exerts forces (blue) on a moving particle (with velocity shown in green).
<td>If a particle is moving partially along the magnetic field (magenta), it will drift down the field lines while spiraling around it (black path). Where the magnetic field is stronger (indicated by having the field lines closer together) it will execute tighter, faster circles in its spiral and will drift more slowly than when the field is weaker.
</table>

It can be convenient to split up a particle's velocity into a part parallel to the magnetic field (the <b>longitudinal velocity</b>) and a part perpendicular to the field (the <b>transverse velocity</b>). The longitudinal velocity describes how fast the particle drifts along the magnetic field line. The transverse velocity determines how tightly and how quickly the particle circles around the field lines.

If the gyroradius is much smaller than the extent of the magnetic field, you get a case where the particle approximately cannot cross magnetic field lines. It just spirals around them. On the other hand, if the magnetic field ends before the particle can make a significant portion of a full revolution, then the particle ends up being somewhat deflected from its original trajectory. If the gydroradius is enogh to deflect the particle into a region where the magnetic field has a different strength or direction, then things get complicated!

When a particle is spiraling around the magnetic field while drifting down the field lines into regions where the magnetic field is stronger, it will circle around the field lines in a faster, smaller circle. This increases the kinetic energy in the spiraling motion; this kinetic energy must come from the kinetic energy of the longitudinal motion. If the field becomes too strong, the particle will slow down all the way and come to a stop. Then it will be reflected back, pushed the other way down the field lines toward the region where the field is not as strong. This phenomenon is called magnetic mirroring, and is responsible for the way radiation gets trapped in the Van Allen belts around Earth with the particles bouncing back and forth along the Earth's field lines and getting reflected back and forth at the poles where the field is strongest.

If you want to calculate for yourself what the gyroradius <math>r_c</math> and gyrofrequency <math>f_c</math> are
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
r_c = \frac{\gamma \, m \, v_\perp}{q \, B}
</math></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
f_c = \frac{q \, B}{\gamma \, m}
</math></div>
where <math>B</math> is the strength of the magnetic field, <math>q</math> is the amount of electric charge of the particle, <math>m</math> is the mass of the particle, <math>\gamma</math> is the Lorentz factor, and <math>v_\perp</math> is the part of the velocity that is perpendicular to the magnetic field. To get these formula to work, you need to use a consistent set of units. So for example if you measure the magnetic field in tesla you need to measure the charge in coulombs (not fundamental charges) and the mass in kilograms (not eV).

One consequence of this is that you can bend the beam coming out of your accelerator by placing a uniform magnetic field across the end. So you can steer your beam to where it needs to go. You can also bend the beam inside of your accelerator – to make it go around in a circle, for example.

Another consequence is that if your beam has to pass through regions with magnetic fields in it, it will get deflected. If you want the beam to end up at the right spot, you will need to account for this. If the exact strength or direction of the field are not known, you will get unpredictable deflections to your beam.

If your beam has been neutralized, the magnetic fields will work to bend the positively charged particles one way and the negatively charged particles the other way. If you have enough beam current, these separated charges will attract each other and counteract the magnetic force. If you don't, the beam will split and you'll get positive charges zipping off in one direction and negative charges going another direction. If you needed to neutralize your beam so that it doesn't electrostatically blow itself apart, well now it will be doing just that.

Even perfectly neutral atoms can be torn apart if the atoms are going fast enough and if the magnetic field is strong enough. Atoms are pretty resistant to this, but you can get conditions where it happens.

Magnetic fields affect the path of beams. But beams are made of charged particles and thus can have currents in them. Currents create magnetic fields themselves, which can influence the surrounding fields. Even a neutralized beam forming a plasma, when it encounters a magnetic field, will get currents induced in the plasma that helps to screen its interior from the field. A beam with enough punch can deform the fields around itself rather than having the fields bending it. As a rough rule of thumb, if the energy density of the beam over a regions larger than a gyroradius is more than the energy density of the field over that region, then the beam gets to have its way with the field, and vice versa. Highly focused, high current beams can thus be expected to better deal with ambient magnetic fields than those with less focus or lower current.

==Focusing==

===Focusing equipment===
In the previous section, we discussed how a particle beam can be steered by directing it through a uniform magnetic field. We can make this kind of field with a magnetic north pole directly opposite a magnetic south pole (with additional shimming magnets to keep things uniform). Because it has two poles, one north and one south, this kind of magnet is called a <b>dipole magnet</b>.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:dipole_beam_steering.png|frameless]]
</table>

But we want to do more than just steer the beam. we want to be able to expand it, contract it, and focus it to a point. To do this, we need a different kind of configuration. We will put two north poles opposite each other and, and 90 degrees, two south poles opposite each other. With four poles, this is called a <b>quadrupole magnet</b>, or sometimes just a "quad".
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_magnet.png|frameless]]
</table>
At the center of the quad arrangement, the magnetic fields from each side cancel out. there is no field there. A particle going through the center will travel straight through without deflection. However, the farther particles get from the center, the higher the field they encounter and the more they get deflected. A particle slightly off to one side may experience a force that curves it toward the center-line. If the quads are properly shimmed, a particle twice as far away will experience twice the force and twice the deflection from its trajectory back toward the center. This curves all the particles so they meet in one spot.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_focusing.png|frameless]]
</table>
However, there is a catch. Those particles that are offset from the center-line at 90 degrees from these focused trajectories are bent away from the center-line rather than towards it! A quad focuses in one direction and de-focuses in the other.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_defocusing.png|frameless]]
</table>

Is there anything e can do to get an arrangement of magnets that purely focuses the beam? Well yes. We can put another quad arrangement downstream of the first, rotated by 90 degrees. Now consider a set of particles in an arrangement that is de-focused by the first quad. When they get to the second quad, they are further from the center than they were when they went through the first quad. Because the second quad is aligned to focus anything the first quad de-focuses, these particles will be focused; and because they are farther from the center, the focusing deflection will be larger than their previous de-focusing deflection. So the net result is that the particles are focused.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Dual_quad_focusing.png|frameless]]
</table>
For particles with the opposite orientation that are first focused, these will be de-focused by the second quad but because they will have converged they will be going through a region of lower field and will be defocused with a smaller deflection than they were initially focused. This dual quad arrangement will focus any particle that goes through it, regardless of orientation.

===Emittance===

For small deflection angles, a dual quad set-up acts like an ideal lens for particles. An ideal lens can focus all the particles diverging from a point onto another point, or into perfectly parallel lines. It can focus particles on perfectly parallel lines onto a point. So now we can use the particle beam to deliver focused particles on our target.

Or, at least we could if all the particles came from a point, or if they traveled on perfectly parallel lines. Unfortunately, they don't. When particles are made, they come from a source with a finite spot size and a finite temperature. The random motion of the particles moving around from their initial thermal motion means that they are never moving perfectly parallel to each other. And when you try to focus them, instead of all converging onto a single point they make a spot of finite size. And if your focusing and beam bending elements are not perfect, you add additional imperfections to the beam that further hider your ability to focus it.

Your ability to focus can be quantified by a measure called <b>emittance</b>. Emittance is measured in units of a length scale times an angle - nominally meters-radians (m<math>\cdot</math>rad) but the technical literature more commonly gives units of millimeters-milliradians (or mm<math>\cdot</math>mrad) or even 2 <math>\pi</math> mm<math>\cdot</math>mrad. The divergence angle of a beam that you try to make parallel will the the emittance divided by the width of the beam. If you focus a beam down so it converges at an angle, the minimum spot size you can get at the focal point is the emittance divided by the beam angle. If you try to focus a beam with emittance <math>\epsilon</math> on a spot a distance <math>R</math> away with an initial beam diameter of <math>D</math>, the spot size <math>S</math> on the target you can achieve is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
S = R \frac{\epsilon}{D}
</math></div>
It is interesting to note that this is very nearly the same spot size you get from a laser due to [[Diffraction#Spot_Size|diffraction]] with a wavelength equal to the particle beam's enissivity. And, like lasers, the more you can expand the beam to large diameters at the focusing equipment, the tighter the spot you can achieve on the target.

So to get good focusing, we want low emittance. Is there a way we can lower the emittance beyond what we get from our source? Engineers are pretty bright folks, and they have found some ways to do this. You can, for example, pass a beam through a thin foil. This causes some stopping, slowing the beam down. You can then accelerate the beam back up to its original speed. However, the stopping is along the direction of the beam particle's trajectory, even if that trajectory is at an angle. Meanwhile, all the acceleration is in the direction of the beam axis. So the net result is to reduce the random motion perpendicular to the beam, and thus reducing its emittance. Similar tricks can be played with magnet set-ups that force particles going through them to emit radiation.

Another method can be used with a relatively high emittance beam, say a beam of protons or ions or muons. Electron beams can be made with substantially lower emittances. So you can send a low emittance electron beam to co-propagate along with your primary particle beam. The beams will interact, with the particles scattering off each other. This exchanges heat between the two beams and so the temperatures, and thus emittances, of the two beams get closer to each other (and, if given enough time, would become equal). If you repeat this often enough, you can get the emittance down to the emittance of the electron beams. This is more practical in beams going around circular accelerators, because they will have time to get cooled over and over.

A final method that has been used is to install sensors to measure the beam fluctuations. For beams going in circles, the time to get a light-speed signal directly across the diameter to the opposite side wil always be less than the time it takes for the particles to go around the curve to that spot. So you can send signals to give electromagnetic kicks to the beam particles to knock them back into alignment when they get to that spot.

It is intriguing to note that these beam cooling techniques can take <i>hours</i> to work!<ref name="RHIC emittance">[https://www.bnl.gov/isd/documents/86783.pdf M. Minty, R. Connolly, C. Liu,T. Summers, and S. Tepikian, "Absolute beam emittance measurements at RHIC using ionization profile monitors", Brookhaven National Laboratory formal report BNL-105970-2014-IR]</ref> However, without beam cooling methods like this, the emittance of ions or protons in storage rings can slowly increase on a time scale of hours.<ref>[https://arxiv.org/abs/1204.6022 Vladimir Shiltsev and Alvin Tollestrup. "Emittance growth mechanisms in the Tevatron beams", arXiv:1204.6022 [physics.acc-ph]]</ref>

But perhaps the simplest method to reduce emittance is just to speed the particles up. For particles moving at a fraction of light speed <math>\beta</math> and Lorentz factor <math>\gamma</math>, the emittance of particles from the same source with the same temperature and same random motion will be proportional to <math>1/(\beta \, \gamma)</math>. To reflect this, it is convenient to define a normalized emittance <math>\epsilon_n</math> such that for the usual geometric emittance <math>\epsilon</math> that we have been talking about
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
\epsilon_n = \beta \, \gamma \, \epsilon.
</math></div>

Normalized emittance measures makes it convenient to get basic estimates of particle beam performance and compare emittance of different sources and beams. For example, various high-performing electron beams around the world have managed to push their normalized emittances down to about <math>2 \times 10^{-6}</math> m<math>\cdot</math>rad to <math>3 \times 10^{-6}</math> m<math>\cdot</math>rad. Facilities with proton or ion beams seem to have about an order of magnitude higher normalized emittance.
<table>
<tr><td width=200>Machine <td width=300>normalized emittance (m<math>\cdot</math>rad) <td width=100>particles<td>reference
<tr><td>European XFEL <td><math>2.3\times10^{-6}</math> <td> electrons <td><ref>[https://journals.aps.org/prab/pdf/10.1103/PhysRevAccelBeams.24.110702 T. Hara <i>et al.</i>, "Low-emittance beam injection for a synchrotron radiation source using an X-ray free-electron laser linear accelerator", Physical Review Accelerators and Beams <b>24</b>, 110702 (2021)]</ref>, using reported <math>\epsilon \approx 1.5\times 10^{-10}</math> m<math>\cdot</math>rad and electrons at 8 GeV for <math>\beta \to 1</math> and <math>\gamma = 15700</math>.
<tr><td>DESY PETRA III<td><math>0.93\times10^{-6}</math> <td> electrons <td><ref>[https://www.desy.de/news/news_search/index_eng.html?openDirectAnchor=444&two_columns=0 New emittance world record at PETRA III]</ref>, using reported <math>\epsilon = 1.6\times 10^{-10}</math> m<math>\cdot</math>rad and electrons at 3 GeV for <math>\beta \to 1</math> and <math>\gamma = 5870</math>.
<tr><td>SLAC <td><math>3\times 10^{-6}</math> <td> electrons <td><ref>[https://accelconf.web.cern.ch/pac97/papers/pdf/4W009.PDF M. Hernandez <i>et al.</i>, "Emittance Measurements for the SLAC Gun Test Facility", 1998 IEEE]</ref>
<tr><td>LHC <td><math>3.75\times 10^{-5}</math> <td> protons <td><ref>[https://www.lhc-closer.es/taking_a_closer_look_at_lhc/0.beta___emittance Beta & Emittance; Taking a closer look at LHC]</ref>
<tr><td>Fermilab Tevatron <td><math>\approx 5\times 10^{-5}</math> <td> protons <td><ref>[https://accelconf.web.cern.ch/p03/PAPERS/TPPB070.PDF J. Slaughter, J. Estrada, K. Genser, A. Jansson, P. Lebrun, and J. C. Yun, "Tevatron Run II Luminosity, Emittance and Collision Point Size", IEEE Proceedings of the 2003 Particle Accelerator Conference, pages 1763-1765]</ref>
<tr><td>RHIC (with protons) <td><math>\approx 5\times 10^{-5}</math> <td> protons <td><ref>[http://www.osti.gov/servlets/purl/910428-6sKfP0/ Zhang, S, and Ptitsyn, V., "Proton Beam Emittance Growth at RHIC", Brookhaven National Laboratory formal report BNL-77563-2007-CP]</ref>
<tr><td>RHIC (on startup) <td><math>\approx 1.5\times 10^{-5}</math> <td> Au-197 <td><ref name="RHIC emittance"></ref>, using reported <math>\epsilon \approx 12\times 10^{-6}</math> m<math>\cdot</math>rad and Au-197 ions at 100 GeV for <math>\beta = 0.76</math> and <math>\gamma = 1.545</math>.
<tr><td>RHIC (after several hours cooling) <td><math>\approx 5\times 10^{-6}</math> <td> Au-197 <td><ref name="RHIC emittance"></ref>, using reported <math>\epsilon \approx 4\times 10^{-6}</math> m<math>\cdot</math>rad and Au-197 ions at 100 GeV for <math>\beta = 0.76</math> and <math>\gamma = 1.545</math>.
</table>
So as a plausible first estimate for your sci-fi electron guns, you can take <math>\epsilon_n \approx 10^{-6}</math> m<math>\cdot</math>rad and for proton guns <math>\epsilon_n \approx 10^{-5}</math> m<math>\cdot</math>rad. Modify for any improved future-tech that can bring the emittance down. Then divide by <math>\beta \, \gamma</math> for your beam to find the geometrical emittance. Which in turn can let you figure out how well you can focus your beam.

However, we should add one caution here. Unlike a laser, which can get a beam diameter nearly as wide as the width of the optical element (be it a lens or mirror) focusing it, a particle beam needs to go through a small hole inside much larger focusing elements. Thus, a particle beam width (what goes into the spot size calculation) is likely to be much smaller than the physical size of the focusing equipment. You may be able to overcome this limitation by making the geometrical emittance very low, but it will be an issue to keep in mind.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Laser_focus.png|400 px|frameless]]
<td width=150>
<td width=556>[[File:P-beam_focus.png|556 px|frameless]]
<tr>
<td>Laser focused with a mirror
<td>
<td>Particle beam focused with quadrupole magnets
</table>

==Beam self-forces==

particle beams are composed of moving charged particles. Moving charged particles are electric currents. And currents and charges exert forces on each other.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Charges_and_forces_pn.png|frameless]]
<td width=255>Opposite electric charges exert an attractive force on each other
<tr>
<td>[[File:Charges_and_forces_pp.png|frameless]]
<td rowspan=2>Electric charges of the same type exert repulsive forces on each other
<tr>
<td width=349>[[File:Charges_and_forces_nn.png|frameless]]
</table>
<td>
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Currents_and_forces_opposite.png|frameless]]
<td>[[File:Currents_and_forces_same.png|frameless]]
<tr>
<td width=200>Electric currents in opposite directions repel each other
<td width=200>Electric currents in the same direction attract each other
</table>
</table>

In a particle beam, the collection of like-charged particles and their mutual repulsion is called <b>space charge</b>, and can have various effects ranging from making beams expand and trumpet out if not properly neutralized to even making particle beams stop, turn around, and come back the way they came! The currents created by all the particles, on the other hand, are moving in the same direction so they attract each other. This attraction can be called a <b>z-pinch</b>, or sometimes a <b>Bennett pinch</b> after one of the first guys to study it.

===Looking at the same thing two different ways===

Imagine that you are sitting calmly, watching a collection of positively charged particles (as you do). Because the particles all have the same electric charge, they repel each other. You observe the particles pushing away each other and flying apart.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Charges_at_rest.png|478 px|frameless]]
</table>

Ah, but what we didn't tell you was that these particles were just shot out of a particle accelerator at high speed. You just happen to be moving at the same velocity as all the particles, so to you they appear to be at rest. But your colleague running the accelerator sees the particles (and, presumably, you) flying past at high speed. She sees not just particles of the same charge pushing out against each other, but also all the currents of these charges. And because the currents are all in the same direction, there is an attractive force pulling the particles together.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Charges_in_motion.png|750 px|frameless]]
</table>

With the extra forces from the current squeezing the particles together, your colleague should see the particles expanding less fast than you do. Maybe she even sees them collapse in on themselves if the current-current forces are higher than the charge-charge forces. What gives? Who is right?

It turns out, you both are. Because, as it turns out, magnetic fields are just a consequence of relativity balancing things out when you look at moving things from a different coordinate system. You may be familiar with the idea that in the theory of relativity, moving things experience time at a slower rate than things at rest (as observed by someone who is "at rest" in that frame of reference). So your colleague sees the things happening in your frame of reference happening slower than you do. Maybe only just a tiny bit slower, if you are not moving very fast at all. Maybe much, much slower if you are nearly relativistic with respect to her. But some degree slower, in any event. And so she won't see the particles expanding as fast as you do. Each of the particles has the same electric charge for both of you, so the electric forces will be the same. So your colleague must observe some other force in addition to the electric forces that act to partially compensate for them. And these are the magnetic forces, which are present in her frame of reference but not yours. And now you can see that, for bare isolated charges, the magnetic self-pinching forces can never quite overcome the electric self-repulsion charges – although if the particle bunch is going very fast, they might come very close.

===Space charge===

===Self pinching===

==Energy loss in matter==

Sometimes, a particle beam needs to go through stuff to get it where you want it to go. Stuff like air, for example. Or the tissue in the way of getting the beam to a tumor. Also, sometimes, you want your particle beam to interact with stuff. Like the material you are trying to machine with an ion beam, or the bacteria you are trying to kill on medical equipment. All of this falls under the general realm of the interaction of the beam particles with matter. This is an enormously complex subject, but we can cover many of the basics here.

The primary effect on the beam of any interactions with matter are loss of energy of the beam particles (<b>stopping</b>), random changes to the particle directions that make the beam spread out (<b>straggling</b>), and knocking particles clean out of the beam altogether (<b>scattering</b>).

===Ionization===

The particles coming out of your particle accelerator are charged. When a fast moving charged particle goes through matter, the electric field from the charge pushes and pulls on the various charged particles that make up that matter – mainly electrons and the atomic nuclei around which the electrons orbit. At speeds above approximately 1% of the speed of light, it is primarily light and nimble electrons being affected while the ponderous nuclei just sit there barely perturbed by the passage of the particle. When the passing particle yanks on an electron hard enough, it can pop it clean off of the atom it was attached to, leaving a positively charged ion behind and giving you a free electron that can go and glom on to a previously neutral atom to make a negative ion. This process, unsurprisingly, is called <b>ionization</b>. For most particle beams, ionization is the dominant interaction with matter; although when you get very fast or very slow particles other effects can become more important.

if your particle was not initially charged when it goes into matter, it soon will be. The electric charges of the electrons and nuclei in the matter will do the same thing to the particle that a charged particle would do to them. This will quickly strip off any electrons around a high speed atom, leaving an ion and bare electrons to go through the matter. Only particles that are fundamentally neutral (like neutrons or photons) will not be immediately ionized.

It takes about 1 to 10 eV to ionize an atom. This energy must come from the kinetic energy of the beam particle. Particles from an accelerator have millions to billions of eV. So it takes a lot of ionizations to meaningfully affect the particles. However, one thing about matter is that it has lots of atoms that can be ionized so you do get those lots of ionizations. Because each ionization is so small compared to the total energy this is often approximated as a continuous slowing down of the beam particles. So many individual ionizations tend to average out, leading to all the beam particles being slowed down uniformly.

When the particle is yanking on nearby electrons to tear them out of atoms, those electrons are yanking back. This jitters the beam particle slightly from side to side. Again, these sideways yanks tend to average out. The statistical variations from the average slowing down and side-to-size jerking are what lead to straggling. Protons and ions are heavy, and will be deflected little from their original paths - like bowling balls smashing through matchsticks. These particles tend to go in mostly straight lines. Electrons, on the other hand, have the same mass as the electrons they are knocking about. So electron tracks through matter tend to zig-zag around a lot more.

As the particle whizzes past, it leaves a trail of atomic destruction in its wake - a trail of ions where the atoms are literally torn apart (not the nuclei, just the atoms). These particle tracks can sometimes be seen in detectors like cloud chambers. The faster the particle is going, the less time it has to yank on nearby electrons and so the less ionization it leaves behind. As a result, fast particles are not slowed down much but as they start to slow down they continue to be slowed down faster and faster. A consequence of this is that the particle dumps most of its energy into the material at the end of its track, just before it comes to a stop. This sudden peak in energy deposition is called the <b>Bragg peak</b>.

The electrons that a particle knocks out can end up going pretty fast. So these secondary electrons can produce further ionization in what is called an <b>electron cascade</b>. if an inner core electron is knocked out of an atom, other electrons falling in to fill that empty spot in the atom can release x-rays (<b>x-ray fluorescence</b>) or cause shake-ups in the outer electrons that knock other electrons off the atom (<b>Auger effect</b>).

===Brehmsstrahlung===

A charged particle is surrounded by an electric field. If it is moving, it is also surrounded by a circulating magnetic field. These fields move along with the particle.

If the particle knocks into something, it will slow down and probably change direction. The information about this bump only goes through the electromagnetic field at the speed of light. So a lot of that field that was moving along with the particle might not get the memo in time and keep right on going. Stray electromagnetic fields like this get this self-inductance effect going on where they start making themselves over and over again, allowing them to propagate as freely moving waves of electromagnetic radiation. This EM radiation from a slowing down particle is given the delightful name of <b>bremsstrahlung</b>.

Particles from an accelerator can be moving so fast that they make bremsstrahlung in the x-ray or gamma ray part of the spectrum. If you get very high energy bremsstrahlung gamma rays, the gamma rays can interact with matter by boiling an electron and its antimatter counterpart, a positron, out of the vacuum (using the nearby matter as the necessary momentum sink for the process). These positrons and electrons can themselves be very high energy and can in turn create more bremsstrahlung, and the process can repeat until the particles are no longer high enough energy to continue making each other. This is called an <b>electron-gamma shower</b>.

The x-ray machines used by dentists and radiologists use both bremsstrahlung and x-ray fluorescence that you get when shining an electron beam onto tungsten to make their x-rays.

At very high speeds, bremsstrahlung can be more a important source of energy loss than ionization.

===Electron capture and screening===

When an ion has slowed down to about 1% of the speed of light or less, it is moving slower than the electrons zipping around their atoms in the matter the ion is going through. This allows the electrons to stick on to the ion, now becoming the electrons of the atom that is moving through the material rather than of the material itself. The ion begins to lose its charge, and the rate of ionization goes down. Soon, ionization becomes negligible compared to the beam atom bumping in to other atoms.

===Phonons and displacement===

When an atom or ion moving through a material bumps into another atom, it will cause that other atom to move. This can have one of two primary effects.

First, if the atoms whack into each other hard enough, the atom in the material can be knocked clean out of place. Like a pool ball being thwacked by the cue ball, the struck atom will go zinging through the material. Atoms getting knocked out of their place like this is called <b>displacement</b>. Displacement creates chemical changes to molecules as atoms are removed, and creates defects called <b>vacancies</b> in crystals (which, as you might imagine, is just a missing atom in the crystal lattice). When the atom comes to rest, it can chemically attack other molecules, again changing the chemistry, or create an extra vacancy called an <b>interstitial</b> jammed into the crystal lattice where it is not supposed to be. If the displaced atom is going fast enough, it can whack other atoms and displace them, leading to a <b>displacement cascade</b> like tenpins knocking each other down. As a result, at the end of its track, an ion can leave a region of disordered material and messed up chemistry where the atoms have all crashed into each other like a freeway pileup.

If an atom is displaced near the surface of a material, the displaced atom can shoot out of the surface and escape the material altogether. This is called <b>sputtering</b>.

Second, the beam atom might only deliver a glancing blow. Or the beam particle might be an electron, which is so light that it is usually incapable of displacing atoms. In this case, the struck atom gets kicked to the side a bit and runs into other atoms; maybe it is also pulled back by the chemical bonds of other atoms that it is attached to. This transfers the motion of the atom to other atoms around it in the form of pressure waves (or, in a solid, shear waves). This adds heat to the material without causing direct damage – although as you might imagine, enough heat can also cause localized melting, evaporation, or decomposition. The individual quanta of vibration – fundamental particles of sound, if you want to think of them like that – are called phonons in analogy with the photons that are the fundamental particles of light. Much like ionization, a particle that excites pressure waves slows down gradually and continuously, unlike the sudden changes in speed and direction you get from knock-out displacement.

===Nuclear collisions===

So far, we have only discussed the things that happen when beam particles interact with matter particles by electromagnetic forces. Even atoms knocking into each other to produce phonons or displacement happens through the mutual repulsion of the electrons around the atoms (although this is dressed by other weird effects like how two electrons can't be in the same place at the same time, or how the fields of one electron can affect other nearby electrons that in turn affect the original electrons, leading to odd correlation effects and charge density waves and other fun stuff that keeps condensed matter physicists and quantum chemists busy).

But electromagnetism isn't the only thing you can get going on. below the level of structure of the atom, you have the atomic nucleus. And a direct hit on the nucleus can end up giving very different effects.

The simplest thing that can happen is an <b>elastic collision</b>, where no energy goes in to any internal nuclear state or into breaking up the nucleus. An elastic collision just ends up giving you displacement (or possibly phonons, but if you hit the nucleus you usually hit it hard enough to send it flying).

But beyond that you can cause any number of nuclear reactions. You can leave the nucleus in an excited nuclear state, from which it usually relaxes back down to its original state by emitting gamma rays. You can knock out nuclear particles, such as neutrons, protons, deuterons (a proton stuck to a neutron), tritons (a proton stuck to two neutrons), helions (a neutron stuck to two protons), alpha particles (two protons stuck to two neutrons), or potentially even larger nuclear fragments. If you hit a heavy nucleus like uranium, you can cause it to split apart by fission. And if you hit the nucleus really hard you might just shatter it into many tiny fragments of those protons, neutrons, deuterons, tritons, helions, alpha particles, and heavier nuclear fragments mentioned earlier. To get any of these <b>inelastic collisions</b>, you need to hit the nucleus hard enough. If your beam particle doesn't have enough energy to excite the nucleus or knock particles, these things simply can't happen. As it gets enough energy to excite a particular interaction channel (as they are called), that process becomes possible and as the energy increases the process becomes more and more likely.

A nucleus is only about a femtometer across (<math>1 \times 10^{-15}</math> m). Which makes it only about <math>1 \times 10^{-30}</math> m² in cross sectional area. In the meantime, chemical bonds between atoms are usually about 1/10th of a nanometer (<math>1 \times 10^{-10}</math> m) long. In condensed matter (liquids and solids), this means that you have somewhere about <math>1 \times 10^{30}</math> atoms per cubic meter. A meter thickness of atoms thus presents just about about enough atoms that their nuclei can cover the projected area – in other words, you can expect a particle to go through (very roughly) about a meter of condensed matter before it whacks into a nucleus.

This assumes that a particle that goes through a nucleus interacts with it. This is usually a good assumption for particles that interact by the strong nuclear force, such as protons, neutrons, mesons, and the atomic nuclei of ions. But many particles cannot interact by the strong nuclear force, such as electrons or muons. These are much less likely to interact with an atomic nucleus even if they pass through it, so they will end up going considerably farther between hitting nuclei than the above assumption. But nuclei can interact electromagnetically (they are made up of charged protons, after all, and at a smaller level of charged quarks), so electrons and muons can produce some nuclear interactions; just far less than actual nuclear particles like protons and ions.

A nuclear strike is a discrete event that will remove the impacting particle from the beam and will (except for elastic collisions) probably produce a spray of radiation.

===High energy physics===

As you get to higher and higher energies, you unlock all sorts of new and weird and interesting processes that can happen. Boiling exotic particles out of the vacuum, unifying forces that had been divorced since the big bang, and producing bizarre unstable resonances in the quantum fields. This is what physics build accelerators to measure. From the point of view of getting a beam to its target, or what happens to the target, however, the main effect is mostly that the interacting particle is removed from the beam and a bunch of additional radiation is produced.

= Operational Considerations =
== Logistics ==
The logistics of particle beam can be a mixed bag. By itself, any particle accelerator is a complicated piece of machinery full of delicate parts that have to deliver high performance. This may require cryogenic temperatures with associated challenges on the materials and mountings used (insulation, and distortions under temperature changes). When working, particle accelerators invariably generate some radiation across the whole assembly, which irradiates and activates the structure. Some parts may be exposed to significantly higher radiation flux due to their working principles. Often the particle beam is a dynamic system that must be managed with active, closed control loops, involving sensors, modeling and control software, actuators, and the computing and communication hardware that brings it all together. This poses engineering challenges across all disciplines.

A practical particle beam is inherently a counterintuitively sturdy device. If it was too delicate, it would break too much to be useful, though of course this is a sliding scale. Still, while sturdy, this doesn’t eliminate complexity, it manages complexity. The result is a system that is the product of a high-tech civilization. A superior machine shop and well-trained engineers are a must to maintain a particle beam system. Some parts of a particle accelerator may be mundane - others may be custom-machined parts that require their own dedicated industries to produce. That said, the manifold utility of particle accelerators is on your side here. Since there are so many practical applications, it is economic to produce parts for industrial demand. (that said, these may not be the kind of components you want in specific high-performance applications.)

The main thing a particle accelerator requires the most is electricity. Any mature power production facility can supply the power.

However, all particle beams require at the least a source of neutralizing mass. The projection of a charged beam will leave the projecting object oppositely charged. In an atmosphere the natural plasma formed by the beam will ground everything out, but in space the diffuse vacuum will not do the job.

Furthermore particle beams using ions also require a feedstock for their ions to produce from, and in an appropriate format. This can be a gas, a fluid or an easily ionized solid. It can be the feedstock species in its pure form or in an easily split molecule, though every complexity adds to the amount of preparation work that has to be done before the feedstock enters the particle beam.

Don’t worry about the mass of your source or neutralizing ions though. Large particle accelerators, like the LHC in Switzerland, only consume tiny gas bottles worth of feed gas every couple years. The mass of a single 5.56 NATO bullet worth of hydrogen could feed a 10 GW, 10 GeV ion beam for nearly 4 days of firing! You will not be running out of beam feedstock easily.

== Cooling ==
Cooling a particle accelerator can be a serious challenge. Various bits of a particle accelerator love to be supercooled for efficiency or basic working principle - any bit of superconductor used in the accelerator, or magnets, or EM generators, will require cooling to its generally supercool operating temperature.

Even if Standard Temperature and Pressure Superconductors (273 Kelvin, 100 Kilopascal) are available, present superconductors have demonstrated superior performance by being supercooled. That means to achieve even stronger fields, even STP superconductors may be cooled to cryogenic conditions.

Cooling cryogenic systems sucks. Thermodynamics shows itself from its worst side. You can forget running cryogenic radiators - the Stefan-Boltzman Law penalizes the direct radiative cooling of anything cryogenic severely as much as it empowers very high-temperature radiators. This only leaves stages of heat pumps to provide cryogenic fluids that can in turn, cool cryogenic systems. Heat pumps draw power, and for such a giant temperature gradient as we can have here, they will draw a lot of power.

However, we have other options. Cryogenic systems also tend to be highly efficient, so while they do heat up, they do so very slowly and there is little overall energy absorbed. This provides the option of running off of bunkered cryogenic fluid in dewars for the duration of operating the equipment and not running any heat pumps all the time.

Other bits of a particle accelerator system aren’t as required, but really nothing is a fan of high temperatures. Going higher than room temperature can be treated as an oddity. High-performance lasers may not require cryogenic conditions, but they certainly like it cool. Radiofrequency producers of any kind generally fare worse with rising temperatures. That means cooling overall remains a challenge - 300 Kelvin is still not that great of a hot loop temperature for radiators.

== Safety <ref>Thomas Otto, "Safety for particle accelerators", Springer 2021, Available online at: https://library.oapen.org/handle/20.500.12657/46120</ref>==
=== Ionizing Radiation ===
There are surprisingly many sources of ionizing radiation in a particle accelerator, besides the obvious threat of the output beam itself. Not all particles are accelerated correctly or in the correct direction. When such particles interact with the structure of the accelerator, they release a shower of secondary ionizing radiation. In electron accelerators and sources, stray electrons may be accelerated and release x-rays from bremsstrahlung as they collide with surrounding matter. The resulting doses can be significant and dangerous. Also, such collisions leave behind residual activation, making the accelerator a radiation hazard for some time even after it has been shut down. Generally, people should not work near the particle accelerator while it is functioning, and radiation barriers isolate the core accelerator from adjacent facilities. (In real-life facilities, you will often see accelerators inside larger buildings being cocooned inside doped concrete plates to shield the rest of the room.)

=== Radiofrequency ===
RF energy is used in many kinds of particle accelerators. This energy may leak from the structures it is generated, contained and used in due to damages or incorrect assembly. As microwave radiation and above is used, the radiation can be readily absorbed by bodies, hydrogen-rich materials or other objects. The end result of this can be rapid heating and thermal effects. Especially in accelerators designed to throughput a lot of energy (say a GW or more), this may become a serious issue if a waveguide is damaged. It’s not recommendable to stand in front of a multi-MW radar either!

=== High-powered electrical and electronic systems ===
The operation of the magnetic systems and accelerators often involves high-powered electrical and electronic systems. High voltages and/or amperes are involved. This poses risks like shock hazards, electrical arcs, and electrical fires. Some electrical systems such as capacitors or inductors may have combustive or explosive failure modes.

=== Flammables ===
Parts of an accelerator or its infrastructure may be manufactured from flammable substances such as hydrocarbons. In the case of an ignition source such as a high-powered beam cutting into a physical substance or an electrical fault, these may catch fire.

=== Lasers ===
Some accelerator systems use high-powered lasers. Laser plasma accelerators use very high-powered pulse lasers to provide pumping power. The safety concerns of lasers for blinding, physical damage, and possible exposure to ionizing radiation as a secondary effect of the beam impact, apply. Some high-powered lasers can also drive electrons which generate strong EMPs, which may damage or destroy other systems in turn. Parts of a laser assembly or laser beam guide may shatter if exposed to overloading conditions.

=== Magnetic systems ===
Parts of a particle accelerator may use strong magnets, up to superconductive magnets with multi-Tesla fields. While the magnets are integrated into chokes that shape the magnetic fields, fringe fields invariably extend beyond the core magnet installation.

Multi-Tesla transient electromagnetic field can affect the human nervous system temporarily, inducing nausea, vertigo, metallic taste and optical phenomena, if the person moves in relation to the magnetic field. Otherwise, the risk of exposure is low. That said, electronic systems that are not shielded properly, such as say, implants, could be adversely affected by the magnetic fields.

The biggest physical risk are reactions of other magnetic objects with the magnets. You can look up cases where metallic objects interacted with the multi-Tesla fields of MRI scanners. Such parts may fly across the room towards the magnet at increasing speeds, and damage anything or injure anyone in their path, possibly ending with the object or person squashed (or impaled!) to the magnet.

Also, large superconductive magnets store not insignificant amounts of energy. If the superconductive property breaks down, the magnet quenches and becomes a resistor. The resistance site heats up as current tries to flow through it, expanding the quench and possibly being heated until it sublimates into gas. This could possibly happen explosively. In such a case, up to many GJ of energy might be unleashed in a conflagration or explosion (subsonic or supersonic), damaging surrounding systems and ejecting shrapnell.

=== Cryogenic cooling systems ===
Particle accelerators often involve cryogenically cooled systems. This means cryogenic fluids like nitrogen and helium are present in the accelerator infrastructure. Such gasses have both a very low temperature and a high ratio of gaseous volume at high temperatures to their liquid volume.

Exposure to the leaked fluid can lead to cryogenic cold damage, injury, and death. Furthermore such gasses are efficient at displacing ambient breathing gasses such as oxygen, so they pose an invisible, and rapidly spreading, suffocation hazard.

Also, if insulation is damaged, ambient heat may leak into the cryogenic circuits, and cryogenic fluid may begin evaporating. This can lead to dramatic rises in internal pressure. Relief valves should trigger to safely direct such pressure spikes away from important places and keep the system from failing catastrophically, but otherwise, pipes and cryogenic storage devices may fail, up to explosively.

=== Beam excursions ===
If the magnets responsible for manipulating the beam break down, the beam may excurse from its designed paths. The beam may become defocused, or maintain its focus. The beam begins cutting into the accelerator structure, evaporating material and depositing energy. This will definitely foul the ultra-high vacuum inside the accelerator, leading to secondary radiation production as the beam collides with gas inside the accelerator. Some sensitive facilities requiring ultra-high vacuum such as some RF cavities may “crash” and become ruined. Further systems may become damaged or destroyed from either physical stress or the radiation pulse. If the beam bunches contain a lot of energy and maybe are heavy particles, the deposition may lead to an energetic local explosion. Alternatively the beam may escape and cut into structure beyond the particle accelerator itself, damaging and destroying equipment and seriously hurting or killing personnel.

In the worst cases, a beam excursion on a very powerful beam may lead to an explosion that destroys the accelerator utterly. Even in a less bad case, the radiation pulse might damage or destroy critical equipment around the accelerator and take the entire facility offline.

=== Beam dumping ===
If any other systems fail, the accelerator probably has to be shut down. This poses a problem - what do you do with the beam energy that is still in the accelerator system?

Sometimes, the beam is designed to leave the accelerator immediately, and shutting down the beam is easy. But in the case of cyclotrons and storage rings, the beam may circulate for a while. What do you do then? You have to dump the beam from the accelerator. Special beam stops are used on research and industrial accelerators, large blocks of dense material able to absorb the beam. During this process the blocks both heat up and become radioactive. For particle beams storing very high energies, this may become an increasing challenge. How do you safely dumb a beam containing many GJ or more? In some cases you may have no other choice but to dump the beam out into the environment. Where and when can you do this safely without threatening third parties?

= Application areas =
(This is a skeletal overview summary)
Particle accelerators have many, many, many areas of application.

In science, the big particle accelerators like the LHC are common knowledge. But electron microscopes also use components of particle accelerators. Many particle accelerators worldwide are used to produce high-energy x-rays in synchrotrons or linear accelerators, imaging samples for material science, chemistry, biochemistry and other areas.

In manufacture, particle beams can be used to cut material, but also weld material. Precision beams can break down gaseous feedstocks to deposit metal in micrometer-sized structures. Directed ion guns are used to implant traces of elements into surfaces for the manufacture of electronic devices.

And in medicine, particle accelerators of ever-greater sophistication find increasing use in cancer treatment and beyond, destroying life-threatening tissues with ever-reduced doses to surrounding healthy tissue.

= Further material =
* [[Particle Beamed Propulsion]]
* [[Plasma Guns]]
* [[Particle Beam Weapons]]

== External reading ==
The US Particle Accelerator School posts material of its courses publically at times: https://uspas.fnal.gov/materials/materials-table.shtml

= Sources =

== Authors ==
Sevoris Doe, Gerrit Bruhaug, Luke Campbell.

== References ==

Particle Accelerators

2022-08-18T12:35:51Z

Sevoris: /* Further material */

{{PageConstructionNotice}}

Particle accelerators are an incredibly useful piece of technology and a diverse set of artifacts produced for many purposes. While particle accelerators are often invoked as weapons of war in science fiction, and have captured the public attention as giant research systems the size of cities, they have an incredible widespread range of applications. There are also many, many ways to build a particle accelerator, depending on what effects you need to achieve and what circumstances you work under.

This article aims to be a general primer on particle accelerators, their physics and a selection of technologies and construction styles. In detail, all particle accelerators are high-tech engineering customized to their circumstances in many, many aspects.

Further pages covering application fields are in the work.

= Accelerator physics =
The simple explanation is that we use electric fields to accelerate charged particles. But from this basic principle of leveraging the electromagnetic forces springs an entire set of accelerator technologies with different performances, trade-offs and characteristics. The most basic particle accelerators are just a big high-voltage source and can still be found in old CRT TV’s and dentist x-ray machines around the world. Things rapidly get more complex from there, but first we need to lay out some terms so we can all have a common conversation.

The first thing to keep in mind with particle accelerators is that we are talking about charged beams here, even if we neutralize them later. So no neutrons, photons, or other innately neutral particles. Your choices range from the simple electron up to anti-uranium and more, but they all must be charged. The particle of choice determines just about everything else about the accelerator and any potential uses. Really heavy particles (like large ions) don’t accelerate easily and take a lot of energy to get relativistic enough for time dilation tricks to take into place. Really light particles (like electrons) accelerate super easily, but can have other radiation problems to worry about.

The second most important thing to know about a given accelerator is the particle energy. This is a measurement of the kinetic energy of any individual electron, ion, whatever in the beam and is measured in a unit called the electron-volt or eV for short. The lower bound for any worthwhile particle accelerator is in the MeV (or million eV) range and we typically want a lot more! Inside of the accelerator we tend to talk about the electric field doing the accelerating (measured in Volts/meter). For singly charged particles (like electrons) this gets really to figure out the energy because of how the eV is defined. We just take the electric field (also called the accelerating gradient) and multiply by the length of the machine! Boom, now you have particle energy! For more complex particles you have to do some scaling based on mass and charge state.

Also a quick trick to help you with deciding if something is relativistic or not is to divide the particle energy <math>E</math> by the mass energy <math>M</math> of the accelerated particle (0.511 MeV for electrons/positrons and about 1 GeV for a proton) and add one. The resulting number is called the Lorentz gamma value and is used all over in accelerator engineering.
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>\gamma = 1+\frac{E}{M}</math></div>
Once you get a value much larger than 1, you have relativistic particles. Once that number is in the hundreds you get an ultra-relativistic beam. Figure 1 helps show the relationship between Lorentz gamma and speed of the particle.

[[File:LorentzGvsSped.svg|900px|thumb|center|Lorentz gamma versus fraction of the speed of light]]

The third most important feature of a particle accelerator is the emittance which is just a fancy term for divergence of the beam and can be thought of similar to the wavelength of lasers. No beam is made perfect and they will naturally want to expand outwards. The smaller the emittance, the longer the beam will stay in a tight spot. Emittance can be altered with a wide variety of beam engineering techniques that take entire books to discuss, but one other cheating thing you can do is drive the beam energy higher. This naturally lowers the emittance by a factor of Lorentz gamma due to the cheating effect of special relativity! Thank god for Einstein!

== Acceleration forces ==
Every accelerator tries to build an electric gradient appropriate for the charge of the particles it is accelerating. The greater the electric gradient, the bigger the amount of work that can be done, the more energy is imparted on a particle as it travels through the gradient, and thus the higher the acceleration. The simplest way to build up a gradient is simply two charged electrodes with a strong potential between them. Many other accelerator methods use electromagnetic fields, in the form of waves confined in electrically conductive structures. In this case we exploit that electromagnetic fields have both an electric and a magnetic component. The magnetic component cannot be used to accelerate particles (only bend their trajectory - which can be exploited for effects like deflection or focusing) but as the electromagnetic waves oscillate, so does an electric field. This naturally creates a charge difference across half of the phase of an electromagnetic wave - this electrical gradient can accelerate particles. Both of these technologies encounter problems with the physical materials asked to contain the electric or electromagnetic energy. Eventually the fields get so intense that solid materials get broken down and an arc is formed. As such, various advanced concepts use clouds of charged particles and plasmas, which are not subject to the material limitations of solid matter.

The strong and weak nuclear force as well as gravity (which are all of the forces we know of at the time of writing) are not suitable for accelerating particles. This is related to the power and range of these forces. The strong and weak nuclear force have extremely short ranges - even if we had a means to artificially shape them like with electromagnets, their range would be too short. Gravity has the range but is incredibly weak. To generate a measly 1 G of acceleration, 9.81 m/s², requires the mass-energy of the entire earth, some 5.97e24 kg of mass, to do the job! The acceleration involved in particle accelerators is much, much, much larger (on the order of 10^13 times larger or more).

{{MascotSays|Interestingly, one source of very high-energy particles are giant gravity wells. Black holes generate conditions around them that can eject particles with enormous energies. The jets of particles that shoot from the north and south poles of many such stellar entities are particle accelerators.}}

If you want to use a force other than electromagnetism to accelerate particles, it’ll have to be a fictional one. The forcefields of various sci-fi properties could be considered, since they can often apply forces to particles. If you propagate a wave of such forcefields down a tube, you could have something quite a lot like many particle accelerator systems!

== The particle bunch ==
Essentially all high energy particle beams are not constant streams of particles, but a time sequence of discrete groups of particles, commonly referred to as “bunches”. These bunches can be described both in their distribution in lateral space to the traveling direction (x and y axis) and longitude (z axis) along the traveling direction, anchored on an origin point (0,0,0) that moves in time.

[[File:ParticleBunch.png|center|900px]]

Bunches can have various profiles. Generally they are somewhat distributed in the lateral direction just because the charged particles have slightly different starting positions and lateral velocities (also called transverse emittance). In longitudinal space, differing acceleration experienced by individual particles as well as dynamics within the bunch scatter them out, with “noses” and “tails” that contain fewer particles. The end results is that most beam profiles follow a gaussian distribution, though there are exceptions.

This bunching behavior allows us to analyze other properties of particle beams very similar to lasers. There is a “pulse frequency” given in Hertz. Each bunch contains a certain energy, which can be large enough that we give it in Joules and not electronvolts. There is an average output energy over a given timespan, given in Watts. We have a beam peak power, given by how quickly a bunch arrives at a target and deposits its energy, which we give in Watts. Related to this we also have a related peak intensity depending on how much area/volume the beam is deposited into, with units of either W/m² or W/m³. If we have charged particles being accelerated, we get a beam voltage, the same way as in a wire. Beams can also have a temperature, the same way as any other group of particles with random motion from their energy, given in Kelvin or eV.

The systems that generate particle beams of course have efficiencies, with the wall plug efficiency (power inserted into the entire system versus power deposited in the particle beam) being the most interesting for first-order engineering. They also have an operating temperature.

== Particles for acceleration ==
With the constraint that our particles must have an electric charge in order to be accelerated with anything but fantastic technology, we can discuss quite a large family of particles, from the conventional to the exotic. They differ in the applications they can be interesting for. Some can be similar, but no two particles behave entirely alike.

=== Electrons ===
The ubiquitous electron is the most common particle to be accelerated and the first one ever pushed up near the speed of light by humans. Due to its excellent charge to mass ratio (the best possible due to the extremely low mass and charge of 1e) electrons are very easy to get up to relativistic speeds. Electron accelerators for radiography and sterilization are often portable by a single person and not much larger than a laptop in size (although they can be much bigger for power output reasons). The ease in which electrons reach relativistic speeds means that electron RF accelerators can be designed with no speed changes assumed (since the electrons are already moving at effectively light speed) and thus tend to be highly efficient and have the highest acceleration gradients achievable. For similar reasons electrons are also able to be accelerated in plasma based accelerators, allowing for even higher acceleration gradients and smaller machines for a given energy. The major downside of electrons is that due to their large charge to mass ratio they will radiate energy like crazy when bent by a magnetic field. Thus ring shaped accelerators are of limited use for electrons since you have to make the ring extremely large to not just lose all of your energy as beams of x-rays coming out of the machine! This very trick is used to make high quality x-ray sources for industrial and research applications.

When electrons hit matter they tend to release their energy as photons in a process called bremsstrahlung, or braking radiation. These extremely high energy photons (roughly ~⅓ the peak energy of the electron on average) then proceed to make electron/positron pairs that then make photons that then make pairs that then make photons… This whole thing is a massive cascade of radiation that ionizes, heats and can even damage on a nuclear level! At high enough energies you will generate every particle known and might even make some new ones!

=== Proton ===
Protons are the next most common particle to be accelerated and are currently used by the absolute largest particle accelerator ever made (the Large Hadron Collider or LHC) as the primary particle of choice. Since they have ~2000X lower charge to mass ratio than electrons, protons are about ~2000X harder to get up to speed. Unlike electrons they are not promptly at light speed so we have to send protons through a series of various accelerators depending on the energy of choice. For example the protons in the LHC the protons go through 5 different particle accelerators plus the extremely low energy stage that first makes the beam of ions! This difficulty in reaching highly relativistic speeds has two other bad side effects. Firstly, if we want to transmit our protons through empty space we will probably need to neutralize them with a beam of electrons.

{{Note|''When shooting extremely high current (kA), relatively low energy (not highly relativistic or in some cases relativistic at all) beams in the atmosphere the higher mass and positive charge of protons gives them many beam propagation advantages over electrons if the beams can be made. In this case no neutralization is required as the atmosphere itself becomes a plasma and neutralizes the beam.''}}
Luckily this is not very hard and will add almost nothing to your comparatively large proton accelerator, but may increase your beam spread a little bit via some neutralization physics. This does mean your beam is technically plasma, which is cool! Second, we effectively cannot use those neat plasma based accelerators for protons since we would need nearly TeV class proton beams to inject into the plasma accelerator and by the time you reach those energies you have basically already done all of the work. Maybe in the future there will be ways around this, but as of right now we are limited! There is one major benefit to this poor charge to mass ratio though, and that is the lack of synchrotron radiation! Ring shaped accelerators are ideal for protons, and basically all ions.

When protons hit matter they can interact with all of the four fundamental forces, although only two of them (Strong Nuclear and Electromagnetism) tend to play big roles. Protons barrel on in causing huge ionization cascades and can also release bremsstrahlung photons at high enough energies. They are also ideal at knocking neutrons, protons, deuterons and more out of the target nuclei which just adds to the fun! Protons also come with a neat trick known as the Bragg peak, where they deposit a large portion of their energy right at the end of their trajectory through a target material. This is a very useful trick for medicine and means you can “tune” the beam to almost ignore large portions of material and primarily heat, ionize, and destroy the area of interest. You can imagine how useful this is for cancer therapy or more nefarious things!

=== Light ions ===
Light ions are a rather fuzzy term but here we will use the definition of any ion heavier than a proton but lighter cobalt. The fuzzy reasoning here is that you get on the other end of the nuclear binding curve (i.e. can’t get net energy from fusion) at iron, so cobalt will then no longer be “light” and has to be made in a different process. Feel free to pick your own cut off! Historically beams up to neon have been experimented with for nuclear fission, nuclear fusion and medical therapy applications. Light ions have even worse charge to mass ratios than protons (Z/A times worse) and typically you will find light ions have about ½ the charge to mass ratio of a proton on average, although the isotope 3He is a notable exception. This means they are even slower to accelerate, although often the exact same machines as protons can be used with minor tweaks. Light ions have similar requirements for neutralization as protons, although their heavier mass can help fight beam bloom for a similar beam speed (not energy since the increased mass means the particles are higher energy for a given speed). The decreased charge to mass ratio also means that synchrotron radiation is even lower so beams can be driven to higher speeds before radiating. Light ions have similar target interaction physics as protons, although the ion itself will break up at very low speeds and add to the radiation environment.

=== Heavy ions ===
Heavy ions are being defined here as cobalt up to whatever unstable element you can jam into your accelerator! Typically heavy ion beams are made of elements like lead (used in some runs on the LHC), gold (used in the Relativistic Heavy Ion Collider) and even californium (used in some runs on the Argonne Tandem Linac Accelerator System). These beams tend to be for pure nuclear physics research, but there has been a long standing interest in heavy ion beams for inertial confinement fusion research and as potential methods of propelling spacecraft. These extremely heavy ions will be even harder to accelerate to ultra high speeds than light ions and much slower for a given energy, but they do provide a lot of momentum to make up for that! They have the same neutralization requirements and same potential upsides for beam bloom as light ions, but will most likely be orders of magnitude slower reaching our targets than the lighter beams. Heavy ions can also be used in the same machines as protons with small modifications and generate almost no synchrotron radiation as a result of their much smaller charge to mass ratio. They also have similar interaction physics to protons and light ions, but will stop in much thinner layers of materials. While electron, proton and light ion beams tend to both mechanically destroy and irradiate targets through-out, heavy ion beams will act much more like lasers and burn in from the outside of the target on in. There will still be a spray of ionizing radiation, but far less of one than an equivalent energy electron, proton, or light ion beam. The much higher momentum per particle, lower radiation concerns and ease of deflection makes heavy ion beams very interesting for beamed propulsion purposes.

=== Macrons ===
Macrons are typically defined as macroscopic particles, such as extremely large molecules or even grains of sand, that are charged up and accelerated via similar technologies as traditional particle accelerators. There are many difficulties with getting macrons up to appreciable speeds due to the extremely low charge to mass ratio, engineering difficulties with getting consistent and good charging, phase matching with accelerating cavities and lack of effective focusing optics. There are so called “dust accelerators” in existence that utilize the first stage of ion accelerators to generate their “beams” and proposals have been made to create better versions using updated technology. Macrons will interact just like a very small and very fast bullet unless they are designed to carry a nuclear payload to boost this output. Antimatter is an assured option for getting energy gain from a macron impacting a target, although there are potentially options for fusion, fission and fission/fusion given the correct sizes (mm scale or bigger), clever target design and the right impact speeds (most likely >>100 km/sec). The various energy gains versus speeds can be seen here:

[[File:IvKmacron.svg|900px|thumb|center]]

It can be seen that antimatter proves to be worth it up near the speed of light, but both fission and fusion stop providing much gain well below the relativistic regime. This also helps explain why heavy ion uranium beams are not going to provide some incredible advantage over any other ion.

=== Antimatter ===

Antimatter is the evil cousin of normal matter and whenever the two meet a release of energy equivalent to their combined mass energy follows. Now this does sound like an amazing option for beams but there are two downsides. First, antimatter is energetically expensive to make! Positrons (antimatter electrons) are the “cheapest” at 1.02 MeV of input energy at a theoretical best, but they only give you that much energy back when hitting a target. Since a typical positron beam can be expected to be equivalent to an electron beam, there is little gain in adding 1.02 MeV per particle to a multiple GeV (or even TeV) beam. You can see this explicitly:

[[File:IvKhBarPostron.svg|900px|thumb|center]]

For anti-protons the math becomes more favorable, but creating these particles is much more difficult and energy consuming. At best anti-protons require ~2 GeV of input energy to create, but in practice >400 GeV per anti-proton has been needed. Luckily significant energy gain is had up to several GeV of beam energy, so if anti-protons can be made at a central facility of some kind and easily stored (neither being easy things) then there may be some advantage to such a beam. Heavier ions are significantly harder to make since fusion with antimatter must be performed to generate those ions. Barring a naturally occurring source being found, antimatter heavier than a proton is most likely not worth using.

{{Note|'' Positrons can potentially fill an interesting niche either as a co-propagating neutralizing beam for electrons or perhaps as a higher performance atmospheric beam than electrons due to the positive charge. In this case the annihilation properties are no what is of interest, rather the positive charge and relative ease of acceleration compared to ions. Since positrons share all properties with electrons except for charge, they can be accelerated just as easily and even co-accelerated with electrons in the same machine.}}

=== Muons ===
Muons are an interesting cousin of electrons and positrons. They are unstable and heavy cousins of electrons, massing about 200X more than electrons and living about 2 microseconds in the lab frame before decaying into an electron (or positron) and a neutrino. If we accelerate them up to relativistic speeds (about 10X easier than protons and the muons are typically born relativistically thanks to how we make them) then the muons can be made to “live” far longer. Muons have long been of interest to accelerator physicists due to their position as an almost ideal intermediary between electrons and protons. The requirement to make them on site has always been an impediment though and muons are currently only used for certain niche experiments. During the planning stages for a potential muon collider it was also noted that a muon collider would also have a serious issue with neutrino radiation! This is due to how neutrinos are more likely to interact with matter at high energies and a muon collider would be generating a LOT of really high energy neutrinos in a ring around the machine. This has been proposed as a sort of super weapon even!

Muons should interact with matter in a similar fashion to electron beams and the beams will probably be treated in a similar manner. They can propagate through the atmosphere quite well in theory and should outperform electron and proton beams for a given beam energy, although at quite a cost… Muons are typically generated with proton beams impacting targets and will create an equal number of positive and negative muons at high energies that must be carefully captured. This would allow for the easy creation of a neutral muon beam though, which may make for a great giant doom beam! Do note that negative muons also can help catalyze nuclear fusion so a very high power beam hitting a fusion fuel tank could produce some rather… Interesting results!

=== Other exotic options ===
There are a wide variety of odd particles that could also be chosen for use in beams given a specific need for them, although generation and utilization of these particles will be quite difficult! Extremely short lived particles like pions, kaons and taus can be generated and accelerated up to extreme speeds to help propagate them far enough to matter, although that will need some very extreme speeds. At the upper end of extreme there are even proposals for accelerating tiny charged blackholes!

= Accelerator technologies =
There are a wide variety of accelerator technologies and here we try to summarize them as best as possible. Different accelerator types may be sequenced one after another. Some particle accelerator technologies depend on the particles fulfilling certain requirements of speed (thus energy) and hence different systems may be employed in sequence. Accelerator systems are described in terms of their efficiency of turning input power into particle energy, their acceleration gradient measured in electric field which translates to volts per meter, and the mass per length meter. An interesting secondary parameter is their operating temperature. Some accelerators can operate well at high temperatures. Others require cold or cryogenic conditions to work well.

== Core accelerator assembly ==
We define the core accelerator assembly as the structure where energy is imparted onto the particles.

=== DC Accelerators ===
For very low beam energies we can utilize large DC electric fields between plates or grids to accelerate particles up to speed. This is the same technique as is used in ion drives and is the most common industrial accelerator used. Every dentist's office and old CRT television uses this acceleration technique. Ion accelerators often also use a DC accelerator to generate the initial beam of ions, while electron beams utilize other techniques. These machines are limited by material breakdown to about 0.5 MV/m accelerating gradient and typically no more than a couple of MeV total particle energy for extremely large machines. The highest energy DC accelerators can reach 40 MeV for proton beams, but this is for machines massing in the 100s of tons and at low total beam powers. Very large DC accelerators are currently the only tested method for accelerating macrons up to 10’s of km/sec.These accelerators are limited by the risk of arcs jumping between the electrodes at too high voltage differentials.

=== Radiofrequency Quadrupole ===
A more advanced first stage accelerator for ion beams is the radiofrequency quadrupole or RFQ. These machines utilize low frequency radiowaves and carefully designed conductive bar structures to efficiently generate high current ion beams up to energies of ~7 MeV in a couple of meters. RFQs were a revolution in ion accelerator physics and caused quite a stir when the USSR first showed them off as they provided a much higher power and higher quality ion source in a smaller package than anything the USA had built! RFQs have also been proposed as methods of efficiently accelerating macrons up to >100 km/sec. These machines can be normal or superconducting and can operate with efficiencies as high as RF LINACs in theory.

=== Cyclotrons and Betatrons ===
One of the earliest ways to accelerate particles up to relativistic speeds is with circular accelerators known as cyclotrons (for ions) and betatrons (for electrons). In Cyclotrons the accelerator is a round chamber which houses two half-circle electrodes roughly shaped like a D (and thus called “dees”) separated by a gap, sandwiched between the poles of a large electromagnet. The magnet creates a static magnetic field which bends the path of charged particles into a circular motion. An oscillating alternative voltage is applied to the electrodes (what we often call RF or radiofrequency) to accelerate the ions. This radiofrequency field’s oscillation is matched so that one cycle of the voltage matches one circuit of the particles. As a consequence when crossing the gap from one electrode to the other, the charged ions experience a voltage gradient and are accelerated. As the particles gain energy and speed, their radius in the magnetic field gets larger. The result is an overall spiraling trajectory from an injection point in the center to an exit point at the rim where the charged particles are bent out of the cyclotron. Cyclotrons use the same voltage gradient many times over, defeating the electrostatic breakdown that limits the single-time electrostatic accelerators. As a consequence, proton energies of up to 70 MeV may be reached. Cyclotrons energies can also be increased by switching from normal magnets and RF resonators to superconductive components. This eliminates various losses from electric resistance and absorption and also allows for much stronger bending magnets, which allows the cyclotron to contain particles up to a higher energy if desired. Superconducting cyclotrons have achieved up to 520 MeV. Still, cyclotrons suffer from poor beam current, pulse rate, and beam quality issues that make them a less than ideal choice. The large magnet required (measured in meters) also does not lend itself to a lightweight weapon and as such, cyclotrons are regulated to niche medical and physics uses.

A close cousin is the betatron, which is used to accelerate electrons. In this case the magnetic field strength is increased as the electrons speed up, which prevents any spiraling trajectories of the particles. Betatrons have been operated up to electrons energies of 300 MeV, but are more typically in the ~10 MeV range. In principle a betatron could also be made superconducting, although magnet ramping may prove to be difficult. Other types of ring shaped accelerators may prove to be more useful, although electrons are heavily limited by synchrotron radiation and it is often better to use linear accelerators for them. For now betatrons remain limited to small (sometimes portable) accelerators for nuclear physics, industry and security work.

=== Radiofrequency Linear Accelerator ===
The real work-horse of the particle accelerator world is the RF linear accelerator. These babies use the same alternating field trick that cyclotrons use, but in a straight line rather than needing to bend the beam around in a circle. This provides a couple of nice advantages over the more traditional cyclotron. First and foremost you get to leave that large, bulky magnet behind which saves you a ton of mass, cooling and power consumption! You also get to now have much higher RF frequencies (1-10 GHz right now compared to the MHz used in cyclotrons), which means higher accelerating gradients. To generate these powerful, high frequency RF waves (also known as microwaves) that drive the accelerator we typically use military radar power supplies or (in more advanced cases) even lower energy particle accelerators! State of the art RF LINAC technology is well into the 100 MV/m range and have no upper limit on beam energy! The Stanford Linear Accelerator reaches 50 GeV and was built back in the 1960’s! There have been traditional RF LINAC designs for machines reaching up to 1 TeV, with the only limit being how much space you have to put these monsters. They do operate best (i.e. with high accelerating gradients and efficiencies) with particles already moving at relativistic speeds, so any ions fed in need to use a series of lower gradients LINACs or other pre-accelerators. Electrons and positrons can naturally be used in LINACs since they accelerate up to relativistic speeds so quickly.

LINACs also have been shown to operate at high average powers (MW scale is now in use) and high pulse energy (10’s of kJ pulses fired at high rep rate) while also maintaining a high quality beam that stays well focused. The two newest advancements have been to push the frequency ever higher for higher acceleration gradients (300+ MV/m has been shown and there are plans for even more) and to switch to superconducting RF accelerating cavities (the name of the actual piece of equipment that does the accelerating) which then dramatically increases the electrical to beam efficiency (80-90% has been shown and even copper waveguides have been operated at 70%). All in all, LINACs beat the tar out of cyclotrons which is why we don’t build cyclotrons anymore. To quote an old Fermilab engineer “cyclotrons are an expensive and bulky way to get a really crappy beam or a cheap way to get a giant chunk of lightly radioactive steel that can never leave your lab”.

LINACs work in detail by shaping the radiofrequency energy in a confining waveguide. The oscillations of the RF energy may be held in space, or allowed to travel down the waveguide over time. Radiofrequency waves have connected magnetic and electric components, which oscillate.

[[File:2022 08 16 GalLib ElectromagneticWave.svg|center|800px]]

Inside the RF accelerator, the electromagnetic radiation is oriented 90 degrees to the particle travel direction, which points the associated electrical field in the travel direction of the particles.The radiofrequency field naturally oscilates along its wave, and the connected electrical and magnetic fields oscilate as well: building up, reaching a peak, and diminishing, inversing, building up, reaching a peak, and diminishing, and the cycle repeats. The detailed behavior depends on the specific cavity.

[[File:CavityWithEM.png|600px|thumb|A resonant RF acceleration cavity and the electric field inside of it, created by resonating RF fields, at one moment in time.]]

Particles of the right charge in the right space at the right time experience an acceleration from the electrical field, being drawn to the opposite electric charge. If they land in the other region, they are instead decelerated. Inbetween, particles experience less to no acceleration. This behavior naturally separates a stream of particles into discrete '''bunches'''.

[[File:CavityWithEMAndParticles.svg|600px|thumb|Bunches being accelerated inside the electric field. ]]

It also creates an implicit limit to what can be effectively accelerated using a specific radiofrequency accelerator design. The particles driven through it have to keep pace with the oscillations of the electric field, reaching the next correct region inside the cavity at that time when the electric field is oriented correctly. If they are too slow to keep inside the accelerating regions, the accelerating action will be ineffective. This requires RF cavities to be matched to the particle's velocity gain under energy imparture and their velocity when entering, being inside, and exiting the accelerator. For electrons and positrons, this isn’t that complicated. Since these lightweight particles accelerate readily, they have few problems “staying in the pocket”. Particles that are slower to gain speed, such as ions and even larger particles, must however be accomodated. This in turn impacts how much acceleration and energy per meter can be fitted into the accelerator in the first place. The acceleration of heavy particles requires specifically designed and sequenced RF accelerators for specific regions of particle speed. This in turn complicates the design of the larger accelerator infrastructure, with multiple discrete stages getting the particles to the energy required so that the next stage in sequence can take over.

RF cavity accelerators are limited in their eV/m capability by the critical current at the containing waveguide. To increase the acceleration provided, more intense electrical fields must be built up. Eventually the electrical charge of the RF waves becomes so strong it can rip electrons out of the cavity walls, causing an arc to form and the cavity material to break down physically.

=== Advanced Accelerator technologies ===
Present accelerator technology caps out at around 250 MeV/m for the most overdriven designs. This poses issues for a variety of applications, where higher energies in shorter distances are desired. In science, "desktop" and "chipset" accelerators are of interest. When militarizing particle beams, some effects only become available at very high energies - but you want something compact enough to turret.

urrent and even near future accelerator isn’t really ready to supply that sort of scale quite yet, so we have to move on to the cutting edge of accelerator physics. There are three big options with sub technologies within these broad fields. We have extreme RF cavities using much higher frequencies, plasma accelerators using lasers for power and plasma accelerators using particle beams for power. Each technology provides a unique series of advantages and disadvantages, but fundamentally they are all a way to make linear accelerators that are much smaller.

==== Dielectric Wakefield Accelerator ====
LINACs using much higher frequencies than traditional LINACs have two fundamental problems to fight. The first is electrical breakdown and the second is generation of those mm-wave (10-100 GHz) or even THz (100-1000+ GHz) sources at sufficient power and efficiency to power a particle accelerator. As the frequency of the driving power goes up the electric field climbs up, which means the accelerating gradient gets larger (which is good) but also means that we run a higher risk of a huge arc forming in the accelerator and breaking it (which is bad). This is often fought by changing the accelerator cavity material from a metal (or superconductor) to a dielectric material like ceramic, or by using a coating of dielectric, which can sometimes earn them the moniker of Dielectric Wake Field Accelerator (DWFA). The issue of creating powerful and efficient enough sources of this higher frequency drive power is being actively pursued for a variety of fields (5G cell signals, beamed power, etc) but is still quite immature compared to the microwave RF sources of traditional LINACs. There are options using essentially "over-clocked" microwave sources, low energy particle beams, and even laser-plasmas. There is still a lot to be done here, but these sorts of problems are being actively worked on there are working accelerators using this technology. There is potential for GV/m or greater acceleration gradiatens and all of the same advantages of traditional RF LINACs (i.e. high powers, high fire rates, high energy pulses, quality beams) should stay the same. The wall plug efficiency is liable to be a bit lower, but that will be highly dependent on the efficiency of the source of these high frequency waves.

==== Laser-Plasma Accelerator ====
Plasma accelerators using lasers for power are a whole world in and of themselves. There are a wide variety of ways to accelerate particles with lasers and plasmas, but we will focus on the most studied technique known as Laser Plasma Accelerators (LPA). LPA accelerators utilize ultrashort (<1psec) and ultraintense (>10^18 W/cm^2) lasers to drive a plasma wake (thus sometimes earning them the name laser wakefield accelerator) and accelerate particles through the resulting charge differential. They have only shown the ability to accelerate electrons and positrons, but any ions that could be injected at the right time and at extremely relativistic speeds could conceivably be accelerated this way. In the operation of a typical LPA the ultraintense laser is fired into a jet of gas (turning it into plasma) or a channel of lightly pre-ionized plasma. The laser then blasts through this plasma like a boat through water and leaves a void behind it (the wake). This void ends up having a HUGE accelerating gradient (100 GV/m) and suffers no risk of electrical breakdown since the plasma is already broken down! A pre-accelerated bunch of particles can be injected into that void at the right time (called the witness bunch) for a BIG acceleration or you can rely on the wake itself spewing electrons out into the void and make a beam out of the plasma itself. The gas or plasma then quickly is reformed and the laser fires as soon as its cooling and charging allows. Keep in mind that even though the beams can hit incredible energies in short distances, the laser itself is not a pocket sized device. These lasers are room sized at minimum and not easy to shrink. The pulse lengths required are also much shorter than most of our hypothetical weapons-grade pulse lasers that may operate in the nanoseconds regime - dual-use may prove difficult. The intensities reached also pose significant issues for the laser hardware. If you don’t spread the energies out, it becomes quite possible you will break parts of your laser from the produced intensities.

Current LPA systems suffer from a couple of problems that are being actively worked on. First and foremost they are incredibly efficiency limited by the laser tech being used. There are plans to move on to much more efficient fiber lasers and eventually make LPA machines that operate in the 25% or greater wall plug efficiency, but that is not currently doable. These lasers also need to be fired at higher rates than the current ~1 Hz, but that is much more of a limit of laser engineering than accelerating engineering. LPA’s also have yet to link more than one accelerator together and the lasers are limited in just how much plasma they can accelerate through right now (~20 cm max) so the beam energy runs into a wall quite quickly. Aside from R&D on linking LPA’s together in a way similar to RF LINACs there are two other options. The acceleration gradient is higher the denser the plasma, so a move to denser plasmas and shorter wavelength lasers (to penetrate the plasma) could provide higher beam energies in similar distances. The other option is to use clever optical techniques like the so-called “flying focus” to allow for the laser to go through much longer plasmas. This could in theory allow for more efficient LWFA’s and single stage machines that provide ~1 TeV electron beams in meters! The final issue is beam quality. Even the best LPAs will not make beams that have as tight of a beam spot, energy spread, or emittance as a normal RF LINAC. This means they will end up needing some beam “cleaning” if you want to get a lot of range out of them.

LPAs are of engineering interest for very short wavelength X-ray free-electron lasers in a compact format. Currently these facilities require linear electron accelerators many kilometers in length. With an LPA as the electron accelerator, an XFEL able to image molecular structures and chemical reactions as they happen might be shrunken down to a much more compact format. (Sometimes this is discussed under the ultimate promise of the “benchtop” or even “chip-format”(sic!) XFEL)

==== Beam-Plasma Accelerator ====
Plasma accelerators using particle beams for power are very similar to LPAs in principle but use a particle beam rather than a laser. We typically just call these things Beam Plasma Accelerators (BPA) for short. The concept is very similar to the LPA in that a high intensity beam comes in and blasts a void into a plasma, which particles can then be accelerated in. BPA provides some interesting advantages over LPA in that you can use highly efficient superconducting RF LINACs (or synchrotrons if you want) to provide the particles at very high efficiency. They also have little in the way of plasma density limits and the accelerated beam takes on the characteristics of the beam driving the BPA. Thus little if any beam cleaning will be needed after acceleration. The big downside is the need for a relativistic particle beam to drive the effect, and typically one in the 100’s of MeV to low GeV range, depending on the particle used. Thus your final accelerator may be short, but you still won’t be making a handheld particle beam device.

== Accelerator Shapes ==
The accelerator technologies discussed above are the means by which we create situations for particles to gain energy. The accelerators become embedded in a larger structure that produces, directs, accelerates, even stores the particles until they are directed towards their final use.

There are two principle shapes for accelerator assemblies. Ones where the beampath is not closed onto itself, and designs where the beam path is closed on itself in part of the assembly, forming a loop.

=== Linear accelerator ===
Everything from electron generation to acceleration to conditioning is built in a straight line, one after another. This is the Linear Accelerator or LINAC. The main catch here is that there is only one single passage for the particles through the entire accelerator. Re-using accelerator hardware to drive the same particle to higher energies by repeated passage is out. The particles have to reach their desired energy in a single pass. Practically this puts limitations on the energy achievable with heavy particles in a LINAC. The length required gets impractically large. Even for electrons, lengths of hundreds of meters to kilometers may be required, unless advanced accelerators with high gradients are used. This said, every accelerator contains linear sections. Since so much high-performance accelerator technology depends on particles having a minimum velocity, ions must be “pre-conditioned” in a set of linear accelerators before the re-use of an accelerator through looped beam paths can be considered.

LINACs of low energy may be used in many applications. Medicinal they find use in radiation therapy and radiosurgery.

=== Synchrotrons ===
The other shape which has some interesting properties are accelerators where beampaths are looped onto themselves. These are called synchrotrons. Since beam paths are closed onto themselves, particles can pass through the same accelerator unit repeatedly, gaining more energy on every passage. This is advantageous for heavy particles which do not gain much speed on any single passage. They can pass through the same accelerator thousands or even millions of times instead. This is obviously advantageous for facility bulk and mass and allows us to reach energies that would otherwise be impractical. The accelerated beam bunches can also be held in storage, looping like the rotating mass of a flywheel, holding energy. The LHC can store some 700 Megajoules of energy in the recirculating beam after full speed has been reached.

That said, the devil is in the details. When charged particles are bent in circles they emit electromagnetic radiation which scales very rapidly (to the fourth power!) with relativistic gamma factor and with bending radius (inversely to the second power). This phenomenon is called synchrotron radiation. Due to the scaling, high energy particles emit more synchrotron radiation than lower energy ones and lighter particles (like electrons) at those high energies emit much more than heavier particles. In addition, the bigger you can make the bending radius of the synchrotron (often called “the ring”) the less synchrotron radiation you get.

[[File:SynchrVsE.svg|800px|center|thumb|Synchrotron radiation per particle versus particle energy for electrons and protons in a 50 meter in radius ring.]]

[[File:SynchroVsRing.svg|800px|center|thumb|Synchrotron radiation emission per particle for different particles at different energies, over varying bending radii. ]]

Although these numbers look small, keep in mind that this is PER PARTICLE. A typical pulse in an electron accelerator will have more than more than a billion electrons in it! Large synchrotrons like the LHC will have over 300 trillion ions circulating through it! Synchrotron radiation necessarily has to be managed to reduce radiation shielding, the radiation load (and possibility of hardware failure) on the accelerator, and to reach high energies in the first place - you might lose all the energy your accelerators are adding to synchrotron radiation!

Deflecting the beam also requires the application of magnetic fields, and higher degrees of deflection, in a smaller length, at higher particle energies, require stronger and stronger magnetic fields. Eventually multi-Tesla fields from superconductors will be required. And due to action and reaction, those magnets need to be braced against the pressure the beam exerts as it is deflected. Ultimately this leads to the same limitations as with flywheels - the mechanical strength of the braces and static connections that keep the magnets together as a larger structure is the accelerator’s limitation on J/kg held in storage. A further complication: The magnet strength has to be matched to particle energy. As a beam is accelerated, the deflectors have to ramp up their power to keep the degree of deflection constant.

Sometimes, synchrotron radiation from electrons is intentionally induced in a ring-shaped accelerator. Since synchrotron radiation of high-energy electrons tends to be hard x-rays emitted along narrow vectors, it’s a way to get x-rays used in various imaging techniques in material science and molecular physics.

Synchrotrons used for ions also need to respect the limitations of the involved accelerators. Synchrotrons often use radiofrequency accelerators. Since radiofrequency accelerators must match the speed and speed gain of their particles, an ion synchrotron cannot be made up of a single accelerator loop. Instead there are sequential loops connected in sequence, each designed to boost the particle through a specific velocity and energy region so that the next accelerator takes over. A designed example is the LHC. Particles headed for the main LHC actually circulate in two other synchrotrons beforehand, gaining sufficient energy so that the LHCs RF cavities can take over. (Also, before being injected into the first synchrotron ring, the ions must first go through multiple linear accelerators using different technologies to reach a base energy.)

An interesting design consideration: Synchrotrons don’t necessarily have to take the form of a (near-ideal) ring. Funamentally any dipole represents a single point of deflection of a straight path - they're polygons! Any geometric shape that contains a loop, and where any single bending curve keeps within the limits of the synchrotron radiation you are willing to tolerate, works. As such, complicated two- or three-dimensional “folded-up” shapes are also possible formats for a synchrotron.

=== Storage Rings ===
Since synchrotrons have to match their deflection magnet strength to the energy of the particles being accelerated, a single synchrotron track cannot hold more than one small section of beam bunches in storage at a time, and becomes unavailable to accelerate further bunches until it discharges the stored beam. This is disadvantageous for various applications where we may want to store a fully accelerated particle beam “bolt” for a longer time.

Storage rings are separate synchrotrons designed to hold multiple sets of bunches in circulation at a target energy at a time. The bunches are delivered from an accelerating synchrotron (sometimes called a booster ring) and held in circulation in the storage ring. Since there are synchrotron radiation losses, a few accelerator sections must compensate for these.

Storage rings can also be used to condition a stored beam. A common thing to do in storage rings is to “cool” a beam, draining unwanted movement energy out of the particles. Beam cooling may take significant time, which makes it ideal to conduct in a storage ring. The result is a beam with less tendency to drift apart, advantageous for many applications.

== Other parts ==
=== Particle sources ===
Particle sources are assemblies which supply particles to accelerators. Here we can group them into three categories: electron sources, ion sources, and exotic particle sources. The later involve the production of particles by means of gathering them from collision reactions (or the handwavium blackbox of your choice) and feeding them into another accelerator for further use. Positrons for example may be manufactured in beam collisions.

Electron sources require the production of a free stream of electrons. The oldest method here are thermionic sources. A hot piece of material ejects free electrons into a vacuum, where they can be drawn in using electric differentials and forwarded to the accelerator.

Field emission electron sources instead use a strong potential between a needle-like cathode and an anode plate with a hole. The electrons are drawn from the needle and through the anode due to the high charge differential. This method produces a “cooler” beam with less random divergence than thermionic sources.

A further improvement still are photoemission sources. Here the precise application of laser light onto a photosensitive material energizes electrons in the photocathode until they break free. This process is thermodynamically very precise and due to the nature of the laser light, also very controllable. The result is a low-emittance stream of electrons. These may be accelerated using electrostatic attraction with an anode at one end of the vacuum chamber, or directly using an RF field.

Ion beam sources are more complicated. The physics of ion formation are complex, which opens many design choices when creating an ion beam. Generally a plasma must be produced, from which the ions can be extracted.

The plasma can be formed in a variety of ways. Electrons from a local cathode-anode array, plasma frequency heating, the injection of electron beams, the application of laser light. The end results can be as diverse. Some methods only produce weakly charged ions and on many vectors, so with high emittance. Other methods offer greater ion charges or reduced emittance.

=== Magnets ===
While acceleration of charged particles is effected with electric fields, magnetic fields can be used to bend the trajectories of particle beams. As such, electromagnets can be used to bend the trajectories of charged particles in various ways, and in a controllable fashion.

==== Dipole Magnets ====
Dipole electromagnet are used to steer the entire particle beam in one axis. (To steer in two axes, we must either rotate one dipole about the beam axis, or couple two dipoles in sequence.) They can be used to direct a beam inside a particle accelerator, inject and extract beams out of looping constructs by switching on and off at the right moment, and point a beam at a target.

In a dipole magnet, a positive and negative coil oppose each other, embedded in a “choke” assembly often made of steel. Charged particles moving through the magnetic field between the coils experience a deflection in one direction specified by the polarization of the field, and a degree of deflection specific to the field strength.

==== Quadrupole Magnet ====
Dipoles can steer, but they influence all particles equally. As such, they cannot be used to manipulate the beam's shape. The first type of magnet able to influence a particle beam’s shape are quadrupole magnets.

They are assembled from four coils at 90 degree angles, with coils of the same polarity opposing each other. The passing beam is compressed on one axis, and expanded on the other. To focus the beam overall, at least two quadpoles must be arrayed in sequence.

Quadrupoles are the basic and essential “magnetic lens” of particle beams. Without them, charged beams would inevitably defocus and run havoc.

==== Hexapole Magnets ====
Hexapole magnets exist to address one problem of quadrupoles - they focus differently depending on the energy of the particle passing through the magnetic field. The effect is likened by particle accelerator physicists to the different diffraction of light in a prism - and hence the effect is called “chromatic aberration”. Hexapoles are used to correct for this, stronger deflecting higher-energy beams.

==== Integrated optics ====
As the field of particle accelerator science and engineering advances, more complex designs are proposed to handle more complex phenomena inside accelerators. An upcoming field is “integrated optics” - complex magnetic systems with nonlinear behaviors. These could provide more advanced manipulation of electron beams, building upon existing capabilities and adding new ones for shaping a beam.

=== Plasma lenses ===
Plasma lenses are an up-and-coming technology, where very high currents in plasma are used to create very strong magnetic fields. Similar to plasma-based accelerator technology, the fact that charges are already free in plasma is used to circumvent conventional charge breakdown and critical current limitations on solid-matter devices. The resulting magnetic fields can be much stronger and bend beams over shorter distances. Furthermore, by creating complicated profiles in the plasma, elaborate beam manipulations could be performed, more akin to sophisticated optical elements. As well, the charges in the plasma neutralize the particle beam in the region of the plasma lense - affecting its propagation behavior further and in a positive manner, as self-repulsion is neutralized.

The catch is that this is plasma - that means you are dealing with all the complications of plasma physics. Heating of the plasma has been an early challenge to successfully forming and maintaining plasma lenses.

Plasma lenses require an appropriate chamber, possibly active containment to keep the plasma from losing energy to the walls or leaking into other accelerator parts, and the necessary plasma sources, maintenance heating devices and then lense shapers and control sensors. In high-powered plasma lenses there will also be a need to cool the plasma as it heats up from waste heat. Potentially we may manage this by open-cycle cooling - venting the plasma controlled in one direction into open space, rather than have a closed-cycle solution for reconditioning the plasma. This could cause some very cool visuals!

Plasma lenses for particle beams should not be confused with optical plasma lenses for lasers! Those exist as well, but are their own topic.

=== Chicanes ===
Chicanes are assemblies of multiple dipole magnets used to manage the longitudinal dispersion of a particle beam due to different particle energies. Since particles with higher energy move faster, they drift ahead in the beam distribution over time; and particles with less energy increasingly lag behind.

Chicanes deflect high-energy particles onto an outward curving trajectory. Since traversing the curve takes longer, the high-energy particles are effectively delayed, allowing lower-energy particles to catch up. By the end of the chicane assembly, the bunch has been compressed in the longitudinal direction.

Chicanes however, do not remove energy from too-fast particles. That is the responsibility of other structures.

=== Undulators and wigglers ===
Undulators and wigglers are magnetic assemblies used to transform particle energy into electromagnetic emissions, with different properties from the specific implementation of a shared basic principle.

Assembled from stacks of generally dipole magnets (other configurations exist), both undulators and wigglers force charged particles onto “wiggling” trajectories by bending their path left and right. Wigglers force very strong arcs; undulators less so. In both cases, the rapid deflection of the charged particles causes them to shed parts of their energy as electromagnetic emissions. Due to the energy of the particles, these photons are generally highly energetic x-rays.

Undulators may be used to build free-electron laser systems.

They can also be used as a means of cooling the beam. The removal of energy from particles can also suppress unwanted movement modes, allowing accelerator structures to re-add energy only to desired movement modes. Of course, this means the particle beam actively loses energy, and produces x-rays that have to be absorbed (and cooled away.)

=== Beam coolers ===
Beam coolers are a collection of systems and strategies that are used to remove unwanted movements from a particle beam. Undulators and wigglers are one way to do this.

One method set of beam cooling relies on finding particles who are very “hot” with random motion, and either removing the particles from the beam using a kick device, or applying a precise counter-force that acts against the unwanted motion.

If one is dealing with ions, one clever trick is to inject a beam of electrons. Due to the thermodynamics within the particle population, the lighter electrons absorb heat from the heavier ions, taking this unwanted motion energy with them when the electron beam is bent out.

As well, lasers can be used (what can’t lasers be used for!) to apply counter-force on ions and remove unwanted movement in this manner.

Beam coolers are generally not instantaneous devices. They work over time to make a beam better. Often they involve the removal of energy or particles from the stream, requiring accelerators and sources to make up the loss and creating waste heat.

=== Beam recovery ===
Some accelerator systems can actually be run in reverse - they can turn beam energy back into some other energy format and then electricity. This can be used to recover some to most of the energy from a particle beam before disposing of it. This is advantageous both for your energy bills and making a very high-energy beam liable to spray radiation everywhere, easier to absorb in a beam dump.

=== Beam dump ===
A beam dump is a construction where a particle beam can be routed in case it has to be disposed. Beam dumps are designed to safely absorb the particle's energy without leakage of dangeorus radiation or activated materials to the outside enviroment. Common materials are carbon and metals, embedded in an actively cooled assembly.

= Propagation physics =
==Magnetic fields==

Magnetic fields are things of motion. They are created by moving electric charges. And they only affect electric charges that are moving. An electric charge at rest in the presence of only a strong magnetic field will just sit there. But when it starts moving, the magnetic field interacts with it, starts deflecting it.

But not along the field. Oh no. The magnetic field does not point in the direction it will move the charge. No, the charge experiences a force that is perpendicular to the direction of the field. And also that is perpendicular to its own velocity. A charged particle moving straight down a magnetic field line will not experience any force.

On the other hand, if the particle is moving directly perpendicular to the field, the magnetic force will always push it to the side, even as it keeps turning. This results in the particle moving in a circle. The radius of this circle is called the <b>gyroradius</b>, and the rate at which the particle makes circles is called the <b>gyrofrequency</b> (or sometimes <b>cyclotron frequency</b>).
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Circular_motion.png|frameless]]
<td width=255>[[File:magnetic_forces.png|frameless]]
<td width=255>[[File:spiral_around_field_lines.png|frameless]]
<tr>
<td>To get circular motion, you need a force (blue) that is always the same amount but always pushing perpendicular to the direction of motion, toward the center of the circle. A charged particle in a constant magnetic field moving perpendicular to the magnetic field meets these conditions, and moves in a circular path.
<td>A diagram of how a magnetic field (magenta) exerts forces (blue) on a moving particle (with velocity shown in green).
<td>If a particle is moving partially along the magnetic field (magenta), it will drift down the field lines while spiraling around it (black path). Where the magnetic field is stronger (indicated by having the field lines closer together) it will execute tighter, faster circles in its spiral and will drift more slowly than when the field is weaker.
</table>

It can be convenient to split up a particle's velocity into a part parallel to the magnetic field (the <b>longitudinal velocity</b>) and a part perpendicular to the field (the <b>transverse velocity</b>). The longitudinal velocity describes how fast the particle drifts along the magnetic field line. The transverse velocity determines how tightly and how quickly the particle circles around the field lines.

If the gyroradius is much smaller than the extent of the magnetic field, you get a case where the particle approximately cannot cross magnetic field lines. It just spirals around them. On the other hand, if the magnetic field ends before the particle can make a significant portion of a full revolution, then the particle ends up being somewhat deflected from its original trajectory. If the gydroradius is enogh to deflect the particle into a region where the magnetic field has a different strength or direction, then things get complicated!

When a particle is spiraling around the magnetic field while drifting down the field lines into regions where the magnetic field is stronger, it will circle around the field lines in a faster, smaller circle. This increases the kinetic energy in the spiraling motion; this kinetic energy must come from the kinetic energy of the longitudinal motion. If the field becomes too strong, the particle will slow down all the way and come to a stop. Then it will be reflected back, pushed the other way down the field lines toward the region where the field is not as strong. This phenomenon is called magnetic mirroring, and is responsible for the way radiation gets trapped in the Van Allen belts around Earth with the particles bouncing back and forth along the Earth's field lines and getting reflected back and forth at the poles where the field is strongest.

If you want to calculate for yourself what the gyroradius <math>r_c</math> and gyrofrequency <math>f_c</math> are
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
r_c = \frac{\gamma \, m \, v_\perp}{q \, B}
</math></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
f_c = \frac{q \, B}{\gamma \, m}
</math></div>
where <math>B</math> is the strength of the magnetic field, <math>q</math> is the amount of electric charge of the particle, <math>m</math> is the mass of the particle, <math>\gamma</math> is the Lorentz factor, and <math>v_\perp</math> is the part of the velocity that is perpendicular to the magnetic field. To get these formula to work, you need to use a consistent set of units. So for example if you measure the magnetic field in tesla you need to measure the charge in coulombs (not fundamental charges) and the mass in kilograms (not eV).

One consequence of this is that you can bend the beam coming out of your accelerator by placing a uniform magnetic field across the end. So you can steer your beam to where it needs to go. You can also bend the beam inside of your accelerator – to make it go around in a circle, for example.

Another consequence is that if your beam has to pass through regions with magnetic fields in it, it will get deflected. If you want the beam to end up at the right spot, you will need to account for this. If the exact strength or direction of the field are not known, you will get unpredictable deflections to your beam.

If your beam has been neutralized, the magnetic fields will work to bend the positively charged particles one way and the negatively charged particles the other way. If you have enough beam current, these separated charges will attract each other and counteract the magnetic force. If you don't, the beam will split and you'll get positive charges zipping off in one direction and negative charges going another direction. If you needed to neutralize your beam so that it doesn't electrostatically blow itself apart, well now it will be doing just that.

Even perfectly neutral atoms can be torn apart if the atoms are going fast enough and if the magnetic field is strong enough. Atoms are pretty resistant to this, but you can get conditions where it happens.

Magnetic fields affect the path of beams. But beams are made of charged particles and thus can have currents in them. Currents create magnetic fields themselves, which can influence the surrounding fields. Even a neutralized beam forming a plasma, when it encounters a magnetic field, will get currents induced in the plasma that helps to screen its interior from the field. A beam with enough punch can deform the fields around itself rather than having the fields bending it. As a rough rule of thumb, if the energy density of the beam over a regions larger than a gyroradius is more than the energy density of the field over that region, then the beam gets to have its way with the field, and vice versa. Highly focused, high current beams can thus be expected to better deal with ambient magnetic fields than those with less focus or lower current.

==Focusing==

===Focusing equipment===
In the previous section, we discussed how a particle beam can be steered by directing it through a uniform magnetic field. We can make this kind of field with a magnetic north pole directly opposite a magnetic south pole (with additional shimming magnets to keep things uniform). Because it has two poles, one north and one south, this kind of magnet is called a <b>dipole magnet</b>.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:dipole_beam_steering.png|frameless]]
</table>

But we want to do more than just steer the beam. we want to be able to expand it, contract it, and focus it to a point. To do this, we need a different kind of configuration. We will put two north poles opposite each other and, and 90 degrees, two south poles opposite each other. With four poles, this is called a <b>quadrupole magnet</b>, or sometimes just a "quad".
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_magnet.png|frameless]]
</table>
At the center of the quad arrangement, the magnetic fields from each side cancel out. there is no field there. A particle going through the center will travel straight through without deflection. However, the farther particles get from the center, the higher the field they encounter and the more they get deflected. A particle slightly off to one side may experience a force that curves it toward the center-line. If the quads are properly shimmed, a particle twice as far away will experience twice the force and twice the deflection from its trajectory back toward the center. This curves all the particles so they meet in one spot.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_focusing.png|frameless]]
</table>
However, there is a catch. Those particles that are offset from the center-line at 90 degrees from these focused trajectories are bent away from the center-line rather than towards it! A quad focuses in one direction and de-focuses in the other.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_defocusing.png|frameless]]
</table>

Is there anything e can do to get an arrangement of magnets that purely focuses the beam? Well yes. We can put another quad arrangement downstream of the first, rotated by 90 degrees. Now consider a set of particles in an arrangement that is de-focused by the first quad. When they get to the second quad, they are further from the center than they were when they went through the first quad. Because the second quad is aligned to focus anything the first quad de-focuses, these particles will be focused; and because they are farther from the center, the focusing deflection will be larger than their previous de-focusing deflection. So the net result is that the particles are focused.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Dual_quad_focusing.png|frameless]]
</table>
For particles with the opposite orientation that are first focused, these will be de-focused by the second quad but because they will have converged they will be going through a region of lower field and will be defocused with a smaller deflection than they were initially focused. This dual quad arrangement will focus any particle that goes through it, regardless of orientation.

===Emittance===

For small deflection angles, a dual quad set-up acts like an ideal lens for particles. An ideal lens can focus all the particles diverging from a point onto another point, or into perfectly parallel lines. It can focus particles on perfectly parallel lines onto a point. So now we can use the particle beam to deliver focused particles on our target.

Or, at least we could if all the particles came from a point, or if they traveled on perfectly parallel lines. Unfortunately, they don't. When particles are made, they come from a source with a finite spot size and a finite temperature. The random motion of the particles moving around from their initial thermal motion means that they are never moving perfectly parallel to each other. And when you try to focus them, instead of all converging onto a single point they make a spot of finite size. And if your focusing and beam bending elements are not perfect, you add additional imperfections to the beam that further hider your ability to focus it.

Your ability to focus can be quantified by a measure called <b>emittance</b>. Emittance is measured in units of a length scale times an angle - nominally meters-radians (m<math>\cdot</math>rad) but the technical literature more commonly gives units of millimeters-milliradians (or mm<math>\cdot</math>mrad) or even 2 <math>\pi</math> mm<math>\cdot</math>mrad. The divergence angle of a beam that you try to make parallel will the the emittance divided by the width of the beam. If you focus a beam down so it converges at an angle, the minimum spot size you can get at the focal point is the emittance divided by the beam angle. If you try to focus a beam with emittance <math>\epsilon</math> on a spot a distance <math>R</math> away with an initial beam diameter of <math>D</math>, the spot size <math>S</math> on the target you can achieve is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
S = R \frac{\epsilon}{D}
</math></div>
It is interesting to note that this is very nearly the same spot size you get from a laser due to [[Diffraction#Spot_Size|diffraction]] with a wavelength equal to the particle beam's enissivity. And, like lasers, the more you can expand the beam to large diameters at the focusing equipment, the tighter the spot you can achieve on the target.

So to get good focusing, we want low emittance. Is there a way we can lower the emittance beyond what we get from our source? Engineers are pretty bright folks, and they have found some ways to do this. You can, for example, pass a beam through a thin foil. This causes some stopping, slowing the beam down. You can then accelerate the beam back up to its original speed. However, the stopping is along the direction of the beam particle's trajectory, even if that trajectory is at an angle. Meanwhile, all the acceleration is in the direction of the beam axis. So the net result is to reduce the random motion perpendicular to the beam, and thus reducing its emittance. Similar tricks can be played with magnet set-ups that force particles going through them to emit radiation.

Another method can be used with a relatively high emittance beam, say a beam of protons or ions or muons. Electron beams can be made with substantially lower emittances. So you can send a low emittance electron beam to co-propagate along with your primary particle beam. The beams will interact, with the particles scattering off each other. This exchanges heat between the two beams and so the temperatures, and thus emittances, of the two beams get closer to each other (and, if given enough time, would become equal). If you repeat this often enough, you can get the emittance down to the emittance of the electron beams. This is more practical in beams going around circular accelerators, because they will have time to get cooled over and over.

A final method that has been used is to install sensors to measure the beam fluctuations. For beams going in circles, the time to get a light-speed signal directly across the diameter to the opposite side wil always be less than the time it takes for the particles to go around the curve to that spot. So you can send signals to give electromagnetic kicks to the beam particles to knock them back into alignment when they get to that spot.

It is intriguing to note that these beam cooling techniques can take <i>hours</i> to work!<ref name="RHIC emittance">[https://www.bnl.gov/isd/documents/86783.pdf M. Minty, R. Connolly, C. Liu,T. Summers, and S. Tepikian, "Absolute beam emittance measurements at RHIC using ionization profile monitors", Brookhaven National Laboratory formal report BNL-105970-2014-IR]</ref> However, without beam cooling methods like this, the emittance of ions or protons in storage rings can slowly increase on a time scale of hours.<ref>[https://arxiv.org/abs/1204.6022 Vladimir Shiltsev and Alvin Tollestrup. "Emittance growth mechanisms in the Tevatron beams", arXiv:1204.6022 [physics.acc-ph]]</ref>

But perhaps the simplest method to reduce emittance is just to speed the particles up. For particles moving at a fraction of light speed <math>\beta</math> and Lorentz factor <math>\gamma</math>, the emittance of particles from the same source with the same temperature and same random motion will be proportional to <math>1/(\beta \, \gamma)</math>. To reflect this, it is convenient to define a normalized emittance <math>\epsilon_n</math> such that for the usual geometric emittance <math>\epsilon</math> that we have been talking about
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
\epsilon_n = \beta \, \gamma \, \epsilon.
</math></div>

Normalized emittance measures makes it convenient to get basic estimates of particle beam performance and compare emittance of different sources and beams. For example, various high-performing electron beams around the world have managed to push their normalized emittances down to about <math>2 \times 10^{-6}</math> m<math>\cdot</math>rad to <math>3 \times 10^{-6}</math> m<math>\cdot</math>rad. Facilities with proton or ion beams seem to have about an order of magnitude higher normalized emittance.
<table>
<tr><td width=200>Machine <td width=300>normalized emittance (m<math>\cdot</math>rad) <td width=100>particles<td>reference
<tr><td>European XFEL <td><math>2.3\times10^{-6}</math> <td> electrons <td><ref>[https://journals.aps.org/prab/pdf/10.1103/PhysRevAccelBeams.24.110702 T. Hara <i>et al.</i>, "Low-emittance beam injection for a synchrotron radiation source using an X-ray free-electron laser linear accelerator", Physical Review Accelerators and Beams <b>24</b>, 110702 (2021)]</ref>, using reported <math>\epsilon \approx 1.5\times 10^{-10}</math> m<math>\cdot</math>rad and electrons at 8 GeV for <math>\beta \to 1</math> and <math>\gamma = 15700</math>.
<tr><td>DESY PETRA III<td><math>0.93\times10^{-6}</math> <td> electrons <td><ref>[https://www.desy.de/news/news_search/index_eng.html?openDirectAnchor=444&two_columns=0 New emittance world record at PETRA III]</ref>, using reported <math>\epsilon = 1.6\times 10^{-10}</math> m<math>\cdot</math>rad and electrons at 3 GeV for <math>\beta \to 1</math> and <math>\gamma = 5870</math>.
<tr><td>SLAC <td><math>3\times 10^{-6}</math> <td> electrons <td><ref>[https://accelconf.web.cern.ch/pac97/papers/pdf/4W009.PDF M. Hernandez <i>et al.</i>, "Emittance Measurements for the SLAC Gun Test Facility", 1998 IEEE]</ref>
<tr><td>LHC <td><math>3.75\times 10^{-5}</math> <td> protons <td><ref>[https://www.lhc-closer.es/taking_a_closer_look_at_lhc/0.beta___emittance Beta & Emittance; Taking a closer look at LHC]</ref>
<tr><td>Fermilab Tevatron <td><math>\approx 5\times 10^{-5}</math> <td> protons <td><ref>[https://accelconf.web.cern.ch/p03/PAPERS/TPPB070.PDF J. Slaughter, J. Estrada, K. Genser, A. Jansson, P. Lebrun, and J. C. Yun, "Tevatron Run II Luminosity, Emittance and Collision Point Size", IEEE Proceedings of the 2003 Particle Accelerator Conference, pages 1763-1765]</ref>
<tr><td>RHIC (with protons) <td><math>\approx 5\times 10^{-5}</math> <td> protons <td><ref>[http://www.osti.gov/servlets/purl/910428-6sKfP0/ Zhang, S, and Ptitsyn, V., "Proton Beam Emittance Growth at RHIC", Brookhaven National Laboratory formal report BNL-77563-2007-CP]</ref>
<tr><td>RHIC (on startup) <td><math>\approx 1.5\times 10^{-5}</math> <td> Au-197 <td><ref name="RHIC emittance"></ref>, using reported <math>\epsilon \approx 12\times 10^{-6}</math> m<math>\cdot</math>rad and Au-197 ions at 100 GeV for <math>\beta = 0.76</math> and <math>\gamma = 1.545</math>.
<tr><td>RHIC (after several hours cooling) <td><math>\approx 5\times 10^{-6}</math> <td> Au-197 <td><ref name="RHIC emittance"></ref>, using reported <math>\epsilon \approx 4\times 10^{-6}</math> m<math>\cdot</math>rad and Au-197 ions at 100 GeV for <math>\beta = 0.76</math> and <math>\gamma = 1.545</math>.
</table>
So as a plausible first estimate for your sci-fi electron guns, you can take <math>\epsilon_n \approx 10^{-6}</math> m<math>\cdot</math>rad and for proton guns <math>\epsilon_n \approx 10^{-5}</math> m<math>\cdot</math>rad. Modify for any improved future-tech that can bring the emittance down. Then divide by <math>\beta \, \gamma</math> for your beam to find the geometrical emittance. Which in turn can let you figure out how well you can focus your beam.

However, we should add one caution here. Unlike a laser, which can get a beam diameter nearly as wide as the width of the optical element (be it a lens or mirror) focusing it, a particle beam needs to go through a small hole inside much larger focusing elements. Thus, a particle beam width (what goes into the spot size calculation) is likely to be much smaller than the physical size of the focusing equipment. You may be able to overcome this limitation by making the geometrical emittance very low, but it will be an issue to keep in mind.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=400>[[File:Laser_focus.png|400 px|frameless]]
<td width=150>
<td width=556>[[File:P-beam_focus.png|556 px|frameless]]
<tr>
<td>Laser focused with a mirror
<td>
<td>Particle beam focused with quadrupole magnets
</table>

==Beam self-forces==

particle beams are composed of moving charged particles. Moving charged particles are electric currents. And currents and charges exert forces on each other.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Charges_and_forces_pn.png|frameless]]
<td width=255>Opposite electric charges exert an attractive force on each other
<tr>
<td>[[File:Charges_and_forces_pp.png|frameless]]
<td rowspan=2>Electric charges of the same type exert repulsive forces on each other
<tr>
<td width=349>[[File:Charges_and_forces_nn.png|frameless]]
</table>
<td>
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Currents_and_forces_opposite.png|frameless]]
<td>[[File:Currents_and_forces_same.png|frameless]]
<tr>
<td width=200>Electric currents in opposite directions repel each other
<td width=200>Electric currents in the same direction attract each other
</table>
</table>

In a particle beam, the collection of like-charged particles and their mutual repulsion is called <b>space charge</b>, and can have various effects ranging from making beams expand and trumpet out if not properly neutralized to even making particle beams stop, turn around, and come back the way they came! The currents created by all the particles, on the other hand, are moving in the same direction so they attract each other. This attraction can be called a <b>z-pinch</b>, or sometimes a <b>Bennett pinch</b> after one of the first guys to study it.

===Looking at the same thing two different ways===

Imagine that you are sitting calmly, watching a collection of positively charged particles (as you do). Because the particles all have the same electric charge, they repel each other. You observe the particles pushing away each other and flying apart.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Charges_at_rest.png|478 px|frameless]]
</table>

Ah, but what we didn't tell you was that these particles were just shot out of a particle accelerator at high speed. You just happen to be moving at the same velocity as all the particles, so to you they appear to be at rest. But your colleague running the accelerator sees the particles (and, presumably, you) flying past at high speed. She sees not just particles of the same charge pushing out against each other, but also all the currents of these charges. And because the currents are all in the same direction, there is an attractive force pulling the particles together.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Charges_in_motion.png|750 px|frameless]]
</table>

With the extra forces from the current squeezing the particles together, your colleague should see the particles expanding less fast than you do. Maybe she even sees them collapse in on themselves if the current-current forces are higher than the charge-charge forces. What gives? Who is right?

It turns out, you both are. Because, as it turns out, magnetic fields are just a consequence of relativity balancing things out when you look at moving things from a different coordinate system. You may be familiar with the idea that in the theory of relativity, moving things experience time at a slower rate than things at rest (as observed by someone who is "at rest" in that frame of reference). So your colleague sees the things happening in your frame of reference happening slower than you do. Maybe only just a tiny bit slower, if you are not moving very fast at all. Maybe much, much slower if you are nearly relativistic with respect to her. But some degree slower, in any event. And so she won't see the particles expanding as fast as you do. Each of the particles has the same electric charge for both of you, so the electric forces will be the same. So your colleague must observe some other force in addition to the electric forces that act to partially compensate for them. And these are the magnetic forces, which are present in her frame of reference but not yours. And now you can see that, for bare isolated charges, the magnetic self-pinching forces can never quite overcome the electric self-repulsion charges – although if the particle bunch is going very fast, they might come very close.

===Space charge===

===Self pinching===

==Energy loss in matter==

Sometimes, a particle beam needs to go through stuff to get it where you want it to go. Stuff like air, for example. Or the tissue in the way of getting the beam to a tumor. Also, sometimes, you want your particle beam to interact with stuff. Like the material you are trying to machine with an ion beam, or the bacteria you are trying to kill on medical equipment. All of this falls under the general realm of the interaction of the beam particles with matter. This is an enormously complex subject, but we can cover many of the basics here.

The primary effect on the beam of any interactions with matter are loss of energy of the beam particles (<b>stopping</b>), random changes to the particle directions that make the beam spread out (<b>straggling</b>), and knocking particles clean out of the beam altogether (<b>scattering</b>).

===Ionization===

The particles coming out of your particle accelerator are charged. When a fast moving charged particle goes through matter, the electric field from the charge pushes and pulls on the various charged particles that make up that matter – mainly electrons and the atomic nuclei around which the electrons orbit. At speeds above approximately 1% of the speed of light, it is primarily light and nimble electrons being affected while the ponderous nuclei just sit there barely perturbed by the passage of the particle. When the passing particle yanks on an electron hard enough, it can pop it clean off of the atom it was attached to, leaving a positively charged ion behind and giving you a free electron that can go and glom on to a previously neutral atom to make a negative ion. This process, unsurprisingly, is called <b>ionization</b>. For most particle beams, ionization is the dominant interaction with matter; although when you get very fast or very slow particles other effects can become more important.

if your particle was not initially charged when it goes into matter, it soon will be. The electric charges of the electrons and nuclei in the matter will do the same thing to the particle that a charged particle would do to them. This will quickly strip off any electrons around a high speed atom, leaving an ion and bare electrons to go through the matter. Only particles that are fundamentally neutral (like neutrons or photons) will not be immediately ionized.

It takes about 1 to 10 eV to ionize an atom. This energy must come from the kinetic energy of the beam particle. Particles from an accelerator have millions to billions of eV. So it takes a lot of ionizations to meaningfully affect the particles. However, one thing about matter is that it has lots of atoms that can be ionized so you do get those lots of ionizations. Because each ionization is so small compared to the total energy this is often approximated as a continuous slowing down of the beam particles. So many individual ionizations tend to average out, leading to all the beam particles being slowed down uniformly.

When the particle is yanking on nearby electrons to tear them out of atoms, those electrons are yanking back. This jitters the beam particle slightly from side to side. Again, these sideways yanks tend to average out. The statistical variations from the average slowing down and side-to-size jerking are what lead to straggling. Protons and ions are heavy, and will be deflected little from their original paths - like bowling balls smashing through matchsticks. These particles tend to go in mostly straight lines. Electrons, on the other hand, have the same mass as the electrons they are knocking about. So electron tracks through matter tend to zig-zag around a lot more.

As the particle whizzes past, it leaves a trail of atomic destruction in its wake - a trail of ions where the atoms are literally torn apart (not the nuclei, just the atoms). These particle tracks can sometimes be seen in detectors like cloud chambers. The faster the particle is going, the less time it has to yank on nearby electrons and so the less ionization it leaves behind. As a result, fast particles are not slowed down much but as they start to slow down they continue to be slowed down faster and faster. A consequence of this is that the particle dumps most of its energy into the material at the end of its track, just before it comes to a stop. This sudden peak in energy deposition is called the <b>Bragg peak</b>.

The electrons that a particle knocks out can end up going pretty fast. So these secondary electrons can produce further ionization in what is called an <b>electron cascade</b>. if an inner core electron is knocked out of an atom, other electrons falling in to fill that empty spot in the atom can release x-rays (<b>x-ray fluorescence</b>) or cause shake-ups in the outer electrons that knock other electrons off the atom (<b>Auger effect</b>).

===Brehmsstrahlung===

A charged particle is surrounded by an electric field. If it is moving, it is also surrounded by a circulating magnetic field. These fields move along with the particle.

If the particle knocks into something, it will slow down and probably change direction. The information about this bump only goes through the electromagnetic field at the speed of light. So a lot of that field that was moving along with the particle might not get the memo in time and keep right on going. Stray electromagnetic fields like this get this self-inductance effect going on where they start making themselves over and over again, allowing them to propagate as freely moving waves of electromagnetic radiation. This EM radiation from a slowing down particle is given the delightful name of <b>bremsstrahlung</b>.

Particles from an accelerator can be moving so fast that they make bremsstrahlung in the x-ray or gamma ray part of the spectrum. If you get very high energy bremsstrahlung gamma rays, the gamma rays can interact with matter by boiling an electron and its antimatter counterpart, a positron, out of the vacuum (using the nearby matter as the necessary momentum sink for the process). These positrons and electrons can themselves be very high energy and can in turn create more bremsstrahlung, and the process can repeat until the particles are no longer high enough energy to continue making each other. This is called an <b>electron-gamma shower</b>.

The x-ray machines used by dentists and radiologists use both bremsstrahlung and x-ray fluorescence that you get when shining an electron beam onto tungsten to make their x-rays.

At very high speeds, bremsstrahlung can be more a important source of energy loss than ionization.

===Electron capture and screening===

When an ion has slowed down to about 1% of the speed of light or less, it is moving slower than the electrons zipping around their atoms in the matter the ion is going through. This allows the electrons to stick on to the ion, now becoming the electrons of the atom that is moving through the material rather than of the material itself. The ion begins to lose its charge, and the rate of ionization goes down. Soon, ionization becomes negligible compared to the beam atom bumping in to other atoms.

===Phonons and displacement===

When an atom or ion moving through a material bumps into another atom, it will cause that other atom to move. This can have one of two primary effects.

First, if the atoms whack into each other hard enough, the atom in the material can be knocked clean out of place. Like a pool ball being thwacked by the cue ball, the struck atom will go zinging through the material. Atoms getting knocked out of their place like this is called <b>displacement</b>. Displacement creates chemical changes to molecules as atoms are removed, and creates defects called <b>vacancies</b> in crystals (which, as you might imagine, is just a missing atom in the crystal lattice). When the atom comes to rest, it can chemically attack other molecules, again changing the chemistry, or create an extra vacancy called an <b>interstitial</b> jammed into the crystal lattice where it is not supposed to be. If the displaced atom is going fast enough, it can whack other atoms and displace them, leading to a <b>displacement cascade</b> like tenpins knocking each other down. As a result, at the end of its track, an ion can leave a region of disordered material and messed up chemistry where the atoms have all crashed into each other like a freeway pileup.

If an atom is displaced near the surface of a material, the displaced atom can shoot out of the surface and escape the material altogether. This is called <b>sputtering</b>.

Second, the beam atom might only deliver a glancing blow. Or the beam particle might be an electron, which is so light that it is usually incapable of displacing atoms. In this case, the struck atom gets kicked to the side a bit and runs into other atoms; maybe it is also pulled back by the chemical bonds of other atoms that it is attached to. This transfers the motion of the atom to other atoms around it in the form of pressure waves (or, in a solid, shear waves). This adds heat to the material without causing direct damage – although as you might imagine, enough heat can also cause localized melting, evaporation, or decomposition. The individual quanta of vibration – fundamental particles of sound, if you want to think of them like that – are called phonons in analogy with the photons that are the fundamental particles of light. Much like ionization, a particle that excites pressure waves slows down gradually and continuously, unlike the sudden changes in speed and direction you get from knock-out displacement.

===Nuclear collisions===

So far, we have only discussed the things that happen when beam particles interact with matter particles by electromagnetic forces. Even atoms knocking into each other to produce phonons or displacement happens through the mutual repulsion of the electrons around the atoms (although this is dressed by other weird effects like how two electrons can't be in the same place at the same time, or how the fields of one electron can affect other nearby electrons that in turn affect the original electrons, leading to odd correlation effects and charge density waves and other fun stuff that keeps condensed matter physicists and quantum chemists busy).

But electromagnetism isn't the only thing you can get going on. below the level of structure of the atom, you have the atomic nucleus. And a direct hit on the nucleus can end up giving very different effects.

The simplest thing that can happen is an <b>elastic collision</b>, where no energy goes in to any internal nuclear state or into breaking up the nucleus. An elastic collision just ends up giving you displacement (or possibly phonons, but if you hit the nucleus you usually hit it hard enough to send it flying).

But beyond that you can cause any number of nuclear reactions. You can leave the nucleus in an excited nuclear state, from which it usually relaxes back down to its original state by emitting gamma rays. You can knock out nuclear particles, such as neutrons, protons, deuterons (a proton stuck to a neutron), tritons (a proton stuck to two neutrons), helions (a neutron stuck to two protons), alpha particles (two protons stuck to two neutrons), or potentially even larger nuclear fragments. If you hit a heavy nucleus like uranium, you can cause it to split apart by fission. And if you hit the nucleus really hard you might just shatter it into many tiny fragments of those protons, neutrons, deuterons, tritons, helions, alpha particles, and heavier nuclear fragments mentioned earlier. To get any of these <b>inelastic collisions</b>, you need to hit the nucleus hard enough. If your beam particle doesn't have enough energy to excite the nucleus or knock particles, these things simply can't happen. As it gets enough energy to excite a particular interaction channel (as they are called), that process becomes possible and as the energy increases the process becomes more and more likely.

A nucleus is only about a femtometer across (<math>1 \times 10^{-15}</math> m). Which makes it only about <math>1 \times 10^{-30}</math> m² in cross sectional area. In the meantime, chemical bonds between atoms are usually about 1/10th of a nanometer (<math>1 \times 10^{-10}</math> m) long. In condensed matter (liquids and solids), this means that you have somewhere about <math>1 \times 10^{30}</math> atoms per cubic meter. A meter thickness of atoms thus presents just about about enough atoms that their nuclei can cover the projected area – in other words, you can expect a particle to go through (very roughly) about a meter of condensed matter before it whacks into a nucleus.

This assumes that a particle that goes through a nucleus interacts with it. This is usually a good assumption for particles that interact by the strong nuclear force, such as protons, neutrons, mesons, and the atomic nuclei of ions. But many particles cannot interact by the strong nuclear force, such as electrons or muons. These are much less likely to interact with an atomic nucleus even if they pass through it, so they will end up going considerably farther between hitting nuclei than the above assumption. But nuclei can interact electromagnetically (they are made up of charged protons, after all, and at a smaller level of charged quarks), so electrons and muons can produce some nuclear interactions; just far less than actual nuclear particles like protons and ions.

A nuclear strike is a discrete event that will remove the impacting particle from the beam and will (except for elastic collisions) probably produce a spray of radiation.

===High energy physics===

As you get to higher and higher energies, you unlock all sorts of new and weird and interesting processes that can happen. Boiling exotic particles out of the vacuum, unifying forces that had been divorced since the big bang, and producing bizarre unstable resonances in the quantum fields. This is what physics build accelerators to measure. From the point of view of getting a beam to its target, or what happens to the target, however, the main effect is mostly that the interacting particle is removed from the beam and a bunch of additional radiation is produced.

= Operational Considerations =
== Logistics ==
The logistics of particle beam can be a mixed bag. By itself, any particle accelerator is a complicated piece of machinery full of delicate parts that have to deliver high performance. This may require cryogenic temperatures with associated challenges on the materials and mountings used (insulation, and distortions under temperature changes). When working, particle accelerators invariably generate some radiation across the whole assembly, which irradiates and activates the structure. Some parts may be exposed to significantly higher radiation flux due to their working principles. Often the particle beam is a dynamic system that must be managed with active, closed control loops, involving sensors, modeling and control software, actuators, and the computing and communication hardware that brings it all together. This poses engineering challenges across all disciplines.

A practical particle beam is inherently a counterintuitively sturdy device. If it was too delicate, it would break too much to be useful, though of course this is a sliding scale. Still, while sturdy, this doesn’t eliminate complexity, it manages complexity. The result is a system that is the product of a high-tech civilization. A superior machine shop and well-trained engineers are a must to maintain a particle beam system. Some parts of a particle accelerator may be mundane - others may be custom-machined parts that require their own dedicated industries to produce. That said, the manifold utility of particle accelerators is on your side here. Since there are so many practical applications, it is economic to produce parts for industrial demand. (that said, these may not be the kind of components you want in specific high-performance applications.)

The main thing a particle accelerator requires the most is electricity. Any mature power production facility can supply the power.

However, all particle beams require at the least a source of neutralizing mass. The projection of a charged beam will leave the projecting object oppositely charged. In an atmosphere the natural plasma formed by the beam will ground everything out, but in space the diffuse vacuum will not do the job.

Furthermore particle beams using ions also require a feedstock for their ions to produce from, and in an appropriate format. This can be a gas, a fluid or an easily ionized solid. It can be the feedstock species in its pure form or in an easily split molecule, though every complexity adds to the amount of preparation work that has to be done before the feedstock enters the particle beam.

Don’t worry about the mass of your source or neutralizing ions though. Large particle accelerators, like the LHC in Switzerland, only consume tiny gas bottles worth of feed gas every couple years. The mass of a single 5.56 NATO bullet worth of hydrogen could feed a 10 GW, 10 GeV ion beam for nearly 4 days of firing! You will not be running out of beam feedstock easily.

== Cooling ==
Cooling a particle accelerator can be a serious challenge. Various bits of a particle accelerator love to be supercooled for efficiency or basic working principle - any bit of superconductor used in the accelerator, or magnets, or EM generators, will require cooling to its generally supercool operating temperature.

Even if Standard Temperature and Pressure Superconductors (273 Kelvin, 100 Kilopascal) are available, present superconductors have demonstrated superior performance by being supercooled. That means to achieve even stronger fields, even STP superconductors may be cooled to cryogenic conditions.

Cooling cryogenic systems sucks. Thermodynamics shows itself from its worst side. You can forget running cryogenic radiators - the Stefan-Boltzman Law penalizes the direct radiative cooling of anything cryogenic severely as much as it empowers very high-temperature radiators. This only leaves stages of heat pumps to provide cryogenic fluids that can in turn, cool cryogenic systems. Heat pumps draw power, and for such a giant temperature gradient as we can have here, they will draw a lot of power.

However, we have other options. Cryogenic systems also tend to be highly efficient, so while they do heat up, they do so very slowly and there is little overall energy absorbed. This provides the option of running off of bunkered cryogenic fluid in dewars for the duration of operating the equipment and not running any heat pumps all the time.

Other bits of a particle accelerator system aren’t as required, but really nothing is a fan of high temperatures. Going higher than room temperature can be treated as an oddity. High-performance lasers may not require cryogenic conditions, but they certainly like it cool. Radiofrequency producers of any kind generally fare worse with rising temperatures. That means cooling overall remains a challenge - 300 Kelvin is still not that great of a hot loop temperature for radiators.

== Safety <ref>Thomas Otto, "Safety for particle accelerators", Springer 2021, Available online at: https://library.oapen.org/handle/20.500.12657/46120</ref>==
=== Ionizing Radiation ===
There are surprisingly many sources of ionizing radiation in a particle accelerator, besides the obvious threat of the output beam itself. Not all particles are accelerated correctly or in the correct direction. When such particles interact with the structure of the accelerator, they release a shower of secondary ionizing radiation. In electron accelerators and sources, stray electrons may be accelerated and release x-rays from bremsstrahlung as they collide with surrounding matter. The resulting doses can be significant and dangerous. Also, such collisions leave behind residual activation, making the accelerator a radiation hazard for some time even after it has been shut down. Generally, people should not work near the particle accelerator while it is functioning, and radiation barriers isolate the core accelerator from adjacent facilities. (In real-life facilities, you will often see accelerators inside larger buildings being cocooned inside doped concrete plates to shield the rest of the room.)

=== Radiofrequency ===
RF energy is used in many kinds of particle accelerators. This energy may leak from the structures it is generated, contained and used in due to damages or incorrect assembly. As microwave radiation and above is used, the radiation can be readily absorbed by bodies, hydrogen-rich materials or other objects. The end result of this can be rapid heating and thermal effects. Especially in accelerators designed to throughput a lot of energy (say a GW or more), this may become a serious issue if a waveguide is damaged. It’s not recommendable to stand in front of a multi-MW radar either!

=== High-powered electrical and electronic systems ===
The operation of the magnetic systems and accelerators often involves high-powered electrical and electronic systems. High voltages and/or amperes are involved. This poses risks like shock hazards, electrical arcs, and electrical fires. Some electrical systems such as capacitors or inductors may have combustive or explosive failure modes.

=== Flammables ===
Parts of an accelerator or its infrastructure may be manufactured from flammable substances such as hydrocarbons. In the case of an ignition source such as a high-powered beam cutting into a physical substance or an electrical fault, these may catch fire.

=== Lasers ===
Some accelerator systems use high-powered lasers. Laser plasma accelerators use very high-powered pulse lasers to provide pumping power. The safety concerns of lasers for blinding, physical damage, and possible exposure to ionizing radiation as a secondary effect of the beam impact, apply. Some high-powered lasers can also drive electrons which generate strong EMPs, which may damage or destroy other systems in turn. Parts of a laser assembly or laser beam guide may shatter if exposed to overloading conditions.

=== Magnetic systems ===
Parts of a particle accelerator may use strong magnets, up to superconductive magnets with multi-Tesla fields. While the magnets are integrated into chokes that shape the magnetic fields, fringe fields invariably extend beyond the core magnet installation.

Multi-Tesla transient electromagnetic field can affect the human nervous system temporarily, inducing nausea, vertigo, metallic taste and optical phenomena, if the person moves in relation to the magnetic field. Otherwise, the risk of exposure is low. That said, electronic systems that are not shielded properly, such as say, implants, could be adversely affected by the magnetic fields.

The biggest physical risk are reactions of other magnetic objects with the magnets. You can look up cases where metallic objects interacted with the multi-Tesla fields of MRI scanners. Such parts may fly across the room towards the magnet at increasing speeds, and damage anything or injure anyone in their path, possibly ending with the object or person squashed (or impaled!) to the magnet.

Also, large superconductive magnets store not insignificant amounts of energy. If the superconductive property breaks down, the magnet quenches and becomes a resistor. The resistance site heats up as current tries to flow through it, expanding the quench and possibly being heated until it sublimates into gas. This could possibly happen explosively. In such a case, up to many GJ of energy might be unleashed in a conflagration or explosion (subsonic or supersonic), damaging surrounding systems and ejecting shrapnell.

=== Cryogenic cooling systems ===
Particle accelerators often involve cryogenically cooled systems. This means cryogenic fluids like nitrogen and helium are present in the accelerator infrastructure. Such gasses have both a very low temperature and a high ratio of gaseous volume at high temperatures to their liquid volume.

Exposure to the leaked fluid can lead to cryogenic cold damage, injury, and death. Furthermore such gasses are efficient at displacing ambient breathing gasses such as oxygen, so they pose an invisible, and rapidly spreading, suffocation hazard.

Also, if insulation is damaged, ambient heat may leak into the cryogenic circuits, and cryogenic fluid may begin evaporating. This can lead to dramatic rises in internal pressure. Relief valves should trigger to safely direct such pressure spikes away from important places and keep the system from failing catastrophically, but otherwise, pipes and cryogenic storage devices may fail, up to explosively.

=== Beam excursions ===
If the magnets responsible for manipulating the beam break down, the beam may excurse from its designed paths. The beam may become defocused, or maintain its focus. The beam begins cutting into the accelerator structure, evaporating material and depositing energy. This will definitely foul the ultra-high vacuum inside the accelerator, leading to secondary radiation production as the beam collides with gas inside the accelerator. Some sensitive facilities requiring ultra-high vacuum such as some RF cavities may “crash” and become ruined. Further systems may become damaged or destroyed from either physical stress or the radiation pulse. If the beam bunches contain a lot of energy and maybe are heavy particles, the deposition may lead to an energetic local explosion. Alternatively the beam may escape and cut into structure beyond the particle accelerator itself, damaging and destroying equipment and seriously hurting or killing personnel.

In the worst cases, a beam excursion on a very powerful beam may lead to an explosion that destroys the accelerator utterly. Even in a less bad case, the radiation pulse might damage or destroy critical equipment around the accelerator and take the entire facility offline.

=== Beam dumping ===
If any other systems fail, the accelerator probably has to be shut down. This poses a problem - what do you do with the beam energy that is still in the accelerator system?

Sometimes, the beam is designed to leave the accelerator immediately, and shutting down the beam is easy. But in the case of cyclotrons and storage rings, the beam may circulate for a while. What do you do then? You have to dump the beam from the accelerator. Special beam stops are used on research and industrial accelerators, large blocks of dense material able to absorb the beam. During this process the blocks both heat up and become radioactive. For particle beams storing very high energies, this may become an increasing challenge. How do you safely dumb a beam containing many GJ or more? In some cases you may have no other choice but to dump the beam out into the environment. Where and when can you do this safely without threatening third parties?

= Application areas =
(This is a skeletal overview summary)
Particle accelerators have many, many, many areas of application.

In science, the big particle accelerators like the LHC are common knowledge. But electron microscopes also use components of particle accelerators. Many particle accelerators worldwide are used to produce high-energy x-rays in synchrotrons or linear accelerators, imaging samples for material science, chemistry, biochemistry and other areas.

In manufacture, particle beams can be used to cut material, but also weld material. Precision beams can break down gaseous feedstocks to deposit metal in micrometer-sized structures. Directed ion guns are used to implant traces of elements into surfaces for the manufacture of electronic devices.

And in medicine, particle accelerators of ever-greater sophistication find increasing use in cancer treatment and beyond, destroying life-threatening tissues with ever-reduced doses to surrounding healthy tissue.

= Further material =
* [[Particle Beamed Propulsion]]
* [[Plasma Guns]]
* [[Particle Beam Weapons]]

= Sources =

== Authors ==
Sevoris Doe, Gerrit Bruhaug, Luke Campbell.

== References ==

Particle Beam Weapons

2022-08-17T13:24:01Z

Sevoris: Started editing on the article again, now that we have the particle accelerators reference article.

Particle Accelerators

2022-08-17T12:49:09Z

Sevoris: /* Safety "Safety for particle accelerators", Thomas Otto, Springer 2021, Available online at: https://library.oapen.org/handle/20.500.12657/46120 */

{{PageConstructionNotice}}

Particle accelerators are an incredibly useful piece of technology and a diverse set of artifacts produced for many purposes. While particle accelerators are often invoked as weapons of war in science fiction, and have captured the public attention as giant research systems the size of cities, they have an incredible widespread range of applications. There are also many, many ways to build a particle accelerator, depending on what effects you need to achieve and what circumstances you work under.

This article aims to be a general primer on particle accelerators, their physics and a selection of technologies and construction styles. In detail, all particle accelerators are high-tech engineering customized to their circumstances in many, many aspects.

Further pages covering application fields are in the work.

= Accelerator physics =
The simple explanation is that we use electric fields to accelerate charged particles. But from this basic principle of leveraging the electromagnetic forces springs an entire set of accelerator technologies with different performances, trade-offs and characteristics. The most basic particle accelerators are just a big high-voltage source and can still be found in old CRT TV’s and dentist x-ray machines around the world. Things rapidly get more complex from there, but first we need to lay out some terms so we can all have a common conversation.

The first thing to keep in mind with particle accelerators is that we are talking about charged beams here, even if we neutralize them later. So no neutrons, photons, or other innately neutral particles. Your choices range from the simple electron up to anti-uranium and more, but they all must be charged. The particle of choice determines just about everything else about the accelerator and any potential uses. Really heavy particles (like large ions) don’t accelerate easily and take a lot of energy to get relativistic enough for time dilation tricks to take into place. Really light particles (like electrons) accelerate super easily, but can have other radiation problems to worry about.

The second most important thing to know about a given accelerator is the particle energy. This is a measurement of the kinetic energy of any individual electron, ion, whatever in the beam and is measured in a unit called the electron-volt or eV for short. The lower bound for any worthwhile particle accelerator is in the MeV (or million eV) range and we typically want a lot more! Inside of the accelerator we tend to talk about the electric field doing the accelerating (measured in Volts/meter). For singly charged particles (like electrons) this gets really to figure out the energy because of how the eV is defined. We just take the electric field (also called the accelerating gradient) and multiply by the length of the machine! Boom, now you have particle energy! For more complex particles you have to do some scaling based on mass and charge state.

Also a quick trick to help you with deciding if something is relativistic or not is to divide the particle energy <math>E</math> by the mass energy <math>M</math> of the accelerated particle (0.511 MeV for electrons/positrons and about 1 GeV for a proton) and add one. The resulting number is called the Lorentz gamma value and is used all over in accelerator engineering.
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>\gamma = 1+\frac{E}{M}</math></div>
Once you get a value much larger than 1, you have relativistic particles. Once that number is in the hundreds you get an ultra-relativistic beam. Figure 1 helps show the relationship between Lorentz gamma and speed of the particle.

[[File:LorentzGvsSped.svg|900px|thumb|center|Lorentz gamma versus fraction of the speed of light]]

The third most important feature of a particle accelerator is the emittance which is just a fancy term for divergence of the beam and can be thought of similar to the wavelength of lasers. No beam is made perfect and they will naturally want to expand outwards. The smaller the emittance, the longer the beam will stay in a tight spot. Emittance can be altered with a wide variety of beam engineering techniques that take entire books to discuss, but one other cheating thing you can do is drive the beam energy higher. This naturally lowers the emittance by a factor of Lorentz gamma due to the cheating effect of special relativity! Thank god for Einstein!

== Acceleration forces ==
Every accelerator tries to build an electric gradient appropriate for the charge of the particles it is accelerating. The greater the electric gradient, the bigger the amount of work that can be done, the more energy is imparted on a particle as it travels through the gradient, and thus the higher the acceleration. The simplest way to build up a gradient is simply two charged electrodes with a strong potential between them. Many other accelerator methods use electromagnetic fields, in the form of waves confined in electrically conductive structures. In this case we exploit that electromagnetic fields have both an electric and a magnetic component. The magnetic component cannot be used to accelerate particles (only bend their trajectory - which can be exploited for effects like deflection or focusing) but as the electromagnetic waves oscillate, so does an electric field. This naturally creates a charge difference across half of the phase of an electromagnetic wave - this electrical gradient can accelerate particles. Both of these technologies encounter problems with the physical materials asked to contain the electric or electromagnetic energy. Eventually the fields get so intense that solid materials get broken down and an arc is formed. As such, various advanced concepts use clouds of charged particles and plasmas, which are not subject to the material limitations of solid matter.

The strong and weak nuclear force as well as gravity (which are all of the forces we know of at the time of writing) are not suitable for accelerating particles. This is related to the power and range of these forces. The strong and weak nuclear force have extremely short ranges - even if we had a means to artificially shape them like with electromagnets, their range would be too short. Gravity has the range but is incredibly weak. To generate a measly 1 G of acceleration, 9.81 m/s², requires the mass-energy of the entire earth, some 5.97e24 kg of mass, to do the job! The acceleration involved in particle accelerators is much, much, much larger (on the order of 10^13 times larger or more).

{{MascotSays|Interestingly, one source of very high-energy particles are giant gravity wells. Black holes generate conditions around them that can eject particles with enormous energies. The jets of particles that shoot from the north and south poles of many such stellar entities are particle accelerators.}}

If you want to use a force other than electromagnetism to accelerate particles, it’ll have to be a fictional one. The forcefields of various sci-fi properties could be considered, since they can often apply forces to particles. If you propagate a wave of such forcefields down a tube, you could have something quite a lot like many particle accelerator systems!

== The particle bunch ==
Essentially all high energy particle beams are not constant streams of particles, but a time sequence of discrete groups of particles, commonly referred to as “bunches”. These bunches can be described both in their distribution in lateral space to the traveling direction (x and y axis) and longitude (z axis) along the traveling direction, anchored on an origin point (0,0,0) that moves in time.

[[File:ParticleBunch.png|center|900px]]

Bunches can have various profiles. Generally they are somewhat distributed in the lateral direction just because the charged particles have slightly different starting positions and lateral velocities (also called transverse emittance). In longitudinal space, differing acceleration experienced by individual particles as well as dynamics within the bunch scatter them out, with “noses” and “tails” that contain fewer particles. The end results is that most beam profiles follow a gaussian distribution, though there are exceptions.

This bunching behavior allows us to analyze other properties of particle beams very similar to lasers. There is a “pulse frequency” given in Hertz. Each bunch contains a certain energy, which can be large enough that we give it in Joules and not electronvolts. There is an average output energy over a given timespan, given in Watts. We have a beam peak power, given by how quickly a bunch arrives at a target and deposits its energy, which we give in Watts. Related to this we also have a related peak intensity depending on how much area/volume the beam is deposited into, with units of either W/m² or W/m³. If we have charged particles being accelerated, we get a beam voltage, the same way as in a wire. Beams can also have a temperature, the same way as any other group of particles with random motion from their energy, given in Kelvin or eV.

The systems that generate particle beams of course have efficiencies, with the wall plug efficiency (power inserted into the entire system versus power deposited in the particle beam) being the most interesting for first-order engineering. They also have an operating temperature.

== Particles for acceleration ==
With the constraint that our particles must have an electric charge in order to be accelerated with anything but fantastic technology, we can discuss quite a large family of particles, from the conventional to the exotic. They differ in the applications they can be interesting for. Some can be similar, but no two particles behave entirely alike.

=== Electrons ===
The ubiquitous electron is the most common particle to be accelerated and the first one ever pushed up near the speed of light by humans. Due to its excellent charge to mass ratio (the best possible due to the extremely low mass and charge of 1e) electrons are very easy to get up to relativistic speeds. Electron accelerators for radiography and sterilization are often portable by a single person and not much larger than a laptop in size (although they can be much bigger for power output reasons). The ease in which electrons reach relativistic speeds means that electron RF accelerators can be designed with no speed changes assumed (since the electrons are already moving at effectively light speed) and thus tend to be highly efficient and have the highest acceleration gradients achievable. For similar reasons electrons are also able to be accelerated in plasma based accelerators, allowing for even higher acceleration gradients and smaller machines for a given energy. The major downside of electrons is that due to their large charge to mass ratio they will radiate energy like crazy when bent by a magnetic field. Thus ring shaped accelerators are of limited use for electrons since you have to make the ring extremely large to not just lose all of your energy as beams of x-rays coming out of the machine! This very trick is used to make high quality x-ray sources for industrial and research applications.

When electrons hit matter they tend to release their energy as photons in a process called bremsstrahlung, or braking radiation. These extremely high energy photons (roughly ~⅓ the peak energy of the electron on average) then proceed to make electron/positron pairs that then make photons that then make pairs that then make photons… This whole thing is a massive cascade of radiation that ionizes, heats and can even damage on a nuclear level! At high enough energies you will generate every particle known and might even make some new ones!

=== Proton ===
Protons are the next most common particle to be accelerated and are currently used by the absolute largest particle accelerator ever made (the Large Hadron Collider or LHC) as the primary particle of choice. Since they have ~2000X lower charge to mass ratio than electrons, protons are about ~2000X harder to get up to speed. Unlike electrons they are not promptly at light speed so we have to send protons through a series of various accelerators depending on the energy of choice. For example the protons in the LHC the protons go through 5 different particle accelerators plus the extremely low energy stage that first makes the beam of ions! This difficulty in reaching highly relativistic speeds has two other bad side effects. Firstly, if we want to transmit our protons through empty space we will probably need to neutralize them with a beam of electrons.

{{Note|''When shooting extremely high current (kA), relatively low energy (not highly relativistic or in some cases relativistic at all) beams in the atmosphere the higher mass and positive charge of protons gives them many beam propagation advantages over electrons if the beams can be made. In this case no neutralization is required as the atmosphere itself becomes a plasma and neutralizes the beam.''}}
Luckily this is not very hard and will add almost nothing to your comparatively large proton accelerator, but may increase your beam spread a little bit via some neutralization physics. This does mean your beam is technically plasma, which is cool! Second, we effectively cannot use those neat plasma based accelerators for protons since we would need nearly TeV class proton beams to inject into the plasma accelerator and by the time you reach those energies you have basically already done all of the work. Maybe in the future there will be ways around this, but as of right now we are limited! There is one major benefit to this poor charge to mass ratio though, and that is the lack of synchrotron radiation! Ring shaped accelerators are ideal for protons, and basically all ions.

When protons hit matter they can interact with all of the four fundamental forces, although only two of them (Strong Nuclear and Electromagnetism) tend to play big roles. Protons barrel on in causing huge ionization cascades and can also release bremsstrahlung photons at high enough energies. They are also ideal at knocking neutrons, protons, deuterons and more out of the target nuclei which just adds to the fun! Protons also come with a neat trick known as the Bragg peak, where they deposit a large portion of their energy right at the end of their trajectory through a target material. This is a very useful trick for medicine and means you can “tune” the beam to almost ignore large portions of material and primarily heat, ionize, and destroy the area of interest. You can imagine how useful this is for cancer therapy or more nefarious things!

=== Light ions ===
Light ions are a rather fuzzy term but here we will use the definition of any ion heavier than a proton but lighter cobalt. The fuzzy reasoning here is that you get on the other end of the nuclear binding curve (i.e. can’t get net energy from fusion) at iron, so cobalt will then no longer be “light” and has to be made in a different process. Feel free to pick your own cut off! Historically beams up to neon have been experimented with for nuclear fission, nuclear fusion and medical therapy applications. Light ions have even worse charge to mass ratios than protons (Z/A times worse) and typically you will find light ions have about ½ the charge to mass ratio of a proton on average, although the isotope 3He is a notable exception. This means they are even slower to accelerate, although often the exact same machines as protons can be used with minor tweaks. Light ions have similar requirements for neutralization as protons, although their heavier mass can help fight beam bloom for a similar beam speed (not energy since the increased mass means the particles are higher energy for a given speed). The decreased charge to mass ratio also means that synchrotron radiation is even lower so beams can be driven to higher speeds before radiating. Light ions have similar target interaction physics as protons, although the ion itself will break up at very low speeds and add to the radiation environment.

=== Heavy ions ===
Heavy ions are being defined here as cobalt up to whatever unstable element you can jam into your accelerator! Typically heavy ion beams are made of elements like lead (used in some runs on the LHC), gold (used in the Relativistic Heavy Ion Collider) and even californium (used in some runs on the Argonne Tandem Linac Accelerator System). These beams tend to be for pure nuclear physics research, but there has been a long standing interest in heavy ion beams for inertial confinement fusion research and as potential methods of propelling spacecraft. These extremely heavy ions will be even harder to accelerate to ultra high speeds than light ions and much slower for a given energy, but they do provide a lot of momentum to make up for that! They have the same neutralization requirements and same potential upsides for beam bloom as light ions, but will most likely be orders of magnitude slower reaching our targets than the lighter beams. Heavy ions can also be used in the same machines as protons with small modifications and generate almost no synchrotron radiation as a result of their much smaller charge to mass ratio. They also have similar interaction physics to protons and light ions, but will stop in much thinner layers of materials. While electron, proton and light ion beams tend to both mechanically destroy and irradiate targets through-out, heavy ion beams will act much more like lasers and burn in from the outside of the target on in. There will still be a spray of ionizing radiation, but far less of one than an equivalent energy electron, proton, or light ion beam. The much higher momentum per particle, lower radiation concerns and ease of deflection makes heavy ion beams very interesting for beamed propulsion purposes.

=== Macrons ===
Macrons are typically defined as macroscopic particles, such as extremely large molecules or even grains of sand, that are charged up and accelerated via similar technologies as traditional particle accelerators. There are many difficulties with getting macrons up to appreciable speeds due to the extremely low charge to mass ratio, engineering difficulties with getting consistent and good charging, phase matching with accelerating cavities and lack of effective focusing optics. There are so called “dust accelerators” in existence that utilize the first stage of ion accelerators to generate their “beams” and proposals have been made to create better versions using updated technology. Macrons will interact just like a very small and very fast bullet unless they are designed to carry a nuclear payload to boost this output. Antimatter is an assured option for getting energy gain from a macron impacting a target, although there are potentially options for fusion, fission and fission/fusion given the correct sizes (mm scale or bigger), clever target design and the right impact speeds (most likely >>100 km/sec). The various energy gains versus speeds can be seen here:

[[File:IvKmacron.svg|900px|thumb|center]]

It can be seen that antimatter proves to be worth it up near the speed of light, but both fission and fusion stop providing much gain well below the relativistic regime. This also helps explain why heavy ion uranium beams are not going to provide some incredible advantage over any other ion.

=== Antimatter ===

Antimatter is the evil cousin of normal matter and whenever the two meet a release of energy equivalent to their combined mass energy follows. Now this does sound like an amazing option for beams but there are two downsides. First, antimatter is energetically expensive to make! Positrons (antimatter electrons) are the “cheapest” at 1.02 MeV of input energy at a theoretical best, but they only give you that much energy back when hitting a target. Since a typical positron beam can be expected to be equivalent to an electron beam, there is little gain in adding 1.02 MeV per particle to a multiple GeV (or even TeV) beam. You can see this explicitly:

[[File:IvKhBarPostron.svg|900px|thumb|center]]

For anti-protons the math becomes more favorable, but creating these particles is much more difficult and energy consuming. At best anti-protons require ~2 GeV of input energy to create, but in practice >400 GeV per anti-proton has been needed. Luckily significant energy gain is had up to several GeV of beam energy, so if anti-protons can be made at a central facility of some kind and easily stored (neither being easy things) then there may be some advantage to such a beam. Heavier ions are significantly harder to make since fusion with antimatter must be performed to generate those ions. Barring a naturally occurring source being found, antimatter heavier than a proton is most likely not worth using.

{{Note|'' Positrons can potentially fill an interesting niche either as a co-propagating neutralizing beam for electrons or perhaps as a higher performance atmospheric beam than electrons due to the positive charge. In this case the annihilation properties are no what is of interest, rather the positive charge and relative ease of acceleration compared to ions. Since positrons share all properties with electrons except for charge, they can be accelerated just as easily and even co-accelerated with electrons in the same machine.}}

=== Muons ===
Muons are an interesting cousin of electrons and positrons. They are unstable and heavy cousins of electrons, massing about 200X more than electrons and living about 2 microseconds in the lab frame before decaying into an electron (or positron) and a neutrino. If we accelerate them up to relativistic speeds (about 10X easier than protons and the muons are typically born relativistically thanks to how we make them) then the muons can be made to “live” far longer. Muons have long been of interest to accelerator physicists due to their position as an almost ideal intermediary between electrons and protons. The requirement to make them on site has always been an impediment though and muons are currently only used for certain niche experiments. During the planning stages for a potential muon collider it was also noted that a muon collider would also have a serious issue with neutrino radiation! This is due to how neutrinos are more likely to interact with matter at high energies and a muon collider would be generating a LOT of really high energy neutrinos in a ring around the machine. This has been proposed as a sort of super weapon even!

Muons should interact with matter in a similar fashion to electron beams and the beams will probably be treated in a similar manner. They can propagate through the atmosphere quite well in theory and should outperform electron and proton beams for a given beam energy, although at quite a cost… Muons are typically generated with proton beams impacting targets and will create an equal number of positive and negative muons at high energies that must be carefully captured. This would allow for the easy creation of a neutral muon beam though, which may make for a great giant doom beam! Do note that negative muons also can help catalyze nuclear fusion so a very high power beam hitting a fusion fuel tank could produce some rather… Interesting results!

=== Other exotic options ===
There are a wide variety of odd particles that could also be chosen for use in beams given a specific need for them, although generation and utilization of these particles will be quite difficult! Extremely short lived particles like pions, kaons and taus can be generated and accelerated up to extreme speeds to help propagate them far enough to matter, although that will need some very extreme speeds. At the upper end of extreme there are even proposals for accelerating tiny charged blackholes!

= Accelerator technologies =
There are a wide variety of accelerator technologies and here we try to summarize them as best as possible. Different accelerator types may be sequenced one after another. Some particle accelerator technologies depend on the particles fulfilling certain requirements of speed (thus energy) and hence different systems may be employed in sequence. Accelerator systems are described in terms of their efficiency of turning input power into particle energy, their acceleration gradient measured in electric field which translates to volts per meter, and the mass per length meter. An interesting secondary parameter is their operating temperature. Some accelerators can operate well at high temperatures. Others require cold or cryogenic conditions to work well.

== Core accelerator assembly ==
We define the core accelerator assembly as the structure where energy is imparted onto the particles.

=== DC Accelerators ===
For very low beam energies we can utilize large DC electric fields between plates or grids to accelerate particles up to speed. This is the same technique as is used in ion drives and is the most common industrial accelerator used. Every dentist's office and old CRT television uses this acceleration technique. Ion accelerators often also use a DC accelerator to generate the initial beam of ions, while electron beams utilize other techniques. These machines are limited by material breakdown to about 0.5 MV/m accelerating gradient and typically no more than a couple of MeV total particle energy for extremely large machines. The highest energy DC accelerators can reach 40 MeV for proton beams, but this is for machines massing in the 100s of tons and at low total beam powers. Very large DC accelerators are currently the only tested method for accelerating macrons up to 10’s of km/sec.These accelerators are limited by the risk of arcs jumping between the electrodes at too high voltage differentials.

=== Radiofrequency Quadrupole ===
A more advanced first stage accelerator for ion beams is the radiofrequency quadrupole or RFQ. These machines utilize low frequency radiowaves and carefully designed conductive bar structures to efficiently generate high current ion beams up to energies of ~7 MeV in a couple of meters. RFQs were a revolution in ion accelerator physics and caused quite a stir when the USSR first showed them off as they provided a much higher power and higher quality ion source in a smaller package than anything the USA had built! RFQs have also been proposed as methods of efficiently accelerating macrons up to >100 km/sec. These machines can be normal or superconducting and can operate with efficiencies as high as RF LINACs in theory.

=== Cyclotrons and Betatrons ===
One of the earliest ways to accelerate particles up to relativistic speeds is with circular accelerators known as cyclotrons (for ions) and betatrons (for electrons). In Cyclotrons the accelerator is a round chamber which houses two half-circle electrodes roughly shaped like a D (and thus called “dees”) separated by a gap, sandwiched between the poles of a large electromagnet. The magnet creates a static magnetic field which bends the path of charged particles into a circular motion. An oscillating alternative voltage is applied to the electrodes (what we often call RF or radiofrequency) to accelerate the ions. This radiofrequency field’s oscillation is matched so that one cycle of the voltage matches one circuit of the particles. As a consequence when crossing the gap from one electrode to the other, the charged ions experience a voltage gradient and are accelerated. As the particles gain energy and speed, their radius in the magnetic field gets larger. The result is an overall spiraling trajectory from an injection point in the center to an exit point at the rim where the charged particles are bent out of the cyclotron. Cyclotrons use the same voltage gradient many times over, defeating the electrostatic breakdown that limits the single-time electrostatic accelerators. As a consequence, proton energies of up to 70 MeV may be reached. Cyclotrons energies can also be increased by switching from normal magnets and RF resonators to superconductive components. This eliminates various losses from electric resistance and absorption and also allows for much stronger bending magnets, which allows the cyclotron to contain particles up to a higher energy if desired. Superconducting cyclotrons have achieved up to 520 MeV. Still, cyclotrons suffer from poor beam current, pulse rate, and beam quality issues that make them a less than ideal choice. The large magnet required (measured in meters) also does not lend itself to a lightweight weapon and as such, cyclotrons are regulated to niche medical and physics uses.

A close cousin is the betatron, which is used to accelerate electrons. In this case the magnetic field strength is increased as the electrons speed up, which prevents any spiraling trajectories of the particles. Betatrons have been operated up to electrons energies of 300 MeV, but are more typically in the ~10 MeV range. In principle a betatron could also be made superconducting, although magnet ramping may prove to be difficult. Other types of ring shaped accelerators may prove to be more useful, although electrons are heavily limited by synchrotron radiation and it is often better to use linear accelerators for them. For now betatrons remain limited to small (sometimes portable) accelerators for nuclear physics, industry and security work.

=== Radiofrequency Linear Accelerator ===
The real work-horse of the particle accelerator world is the RF linear accelerator. These babies use the same alternating field trick that cyclotrons use, but in a straight line rather than needing to bend the beam around in a circle. This provides a couple of nice advantages over the more traditional cyclotron. First and foremost you get to leave that large, bulky magnet behind which saves you a ton of mass, cooling and power consumption! You also get to now have much higher RF frequencies (1-10 GHz right now compared to the MHz used in cyclotrons), which means higher accelerating gradients. To generate these powerful, high frequency RF waves (also known as microwaves) that drive the accelerator we typically use military radar power supplies or (in more advanced cases) even lower energy particle accelerators! State of the art RF LINAC technology is well into the 100 MV/m range and have no upper limit on beam energy! The Stanford Linear Accelerator reaches 50 GeV and was built back in the 1960’s! There have been traditional RF LINAC designs for machines reaching up to 1 TeV, with the only limit being how much space you have to put these monsters. They do operate best (i.e. with high accelerating gradients and efficiencies) with particles already moving at relativistic speeds, so any ions fed in need to use a series of lower gradients LINACs or other pre-accelerators. Electrons and positrons can naturally be used in LINACs since they accelerate up to relativistic speeds so quickly.

LINACs also have been shown to operate at high average powers (MW scale is now in use) and high pulse energy (10’s of kJ pulses fired at high rep rate) while also maintaining a high quality beam that stays well focused. The two newest advancements have been to push the frequency ever higher for higher acceleration gradients (300+ MV/m has been shown and there are plans for even more) and to switch to superconducting RF accelerating cavities (the name of the actual piece of equipment that does the accelerating) which then dramatically increases the electrical to beam efficiency (80-90% has been shown and even copper waveguides have been operated at 70%). All in all, LINACs beat the tar out of cyclotrons which is why we don’t build cyclotrons anymore. To quote an old Fermilab engineer “cyclotrons are an expensive and bulky way to get a really crappy beam or a cheap way to get a giant chunk of lightly radioactive steel that can never leave your lab”.

LINACs work in detail by shaping the radiofrequency energy in a confining waveguide. The oscillations of the RF energy may be held in space, or allowed to travel down the waveguide over time. Radiofrequency waves have connected magnetic and electric components, which oscillate.

[[File:2022 08 16 GalLib ElectromagneticWave.svg|center|800px]]

Inside the RF accelerator, the electromagnetic radiation is oriented 90 degrees to the particle travel direction, which points the associated electrical field in the travel direction of the particles.The radiofrequency field naturally oscilates along its wave, and the connected electrical and magnetic fields oscilate as well: building up, reaching a peak, and diminishing, inversing, building up, reaching a peak, and diminishing, and the cycle repeats. The detailed behavior depends on the specific cavity.

[[File:CavityWithEM.png|600px|thumb|A resonant RF acceleration cavity and the electric field inside of it, created by resonating RF fields, at one moment in time.]]

Particles of the right charge in the right space at the right time experience an acceleration from the electrical field, being drawn to the opposite electric charge. If they land in the other region, they are instead decelerated. Inbetween, particles experience less to no acceleration. This behavior naturally separates a stream of particles into discrete '''bunches'''.

[[File:CavityWithEMAndParticles.svg|600px|thumb|Bunches being accelerated inside the electric field. ]]

It also creates an implicit limit to what can be effectively accelerated using a specific radiofrequency accelerator design. The particles driven through it have to keep pace with the oscillations of the electric field, reaching the next correct region inside the cavity at that time when the electric field is oriented correctly. If they are too slow to keep inside the accelerating regions, the accelerating action will be ineffective. This requires RF cavities to be matched to the particle's velocity gain under energy imparture and their velocity when entering, being inside, and exiting the accelerator. For electrons and positrons, this isn’t that complicated. Since these lightweight particles accelerate readily, they have few problems “staying in the pocket”. Particles that are slower to gain speed, such as ions and even larger particles, must however be accomodated. This in turn impacts how much acceleration and energy per meter can be fitted into the accelerator in the first place. The acceleration of heavy particles requires specifically designed and sequenced RF accelerators for specific regions of particle speed. This in turn complicates the design of the larger accelerator infrastructure, with multiple discrete stages getting the particles to the energy required so that the next stage in sequence can take over.

RF cavity accelerators are limited in their eV/m capability by the critical current at the containing waveguide. To increase the acceleration provided, more intense electrical fields must be built up. Eventually the electrical charge of the RF waves becomes so strong it can rip electrons out of the cavity walls, causing an arc to form and the cavity material to break down physically.

=== Advanced Accelerator technologies ===
Present accelerator technology caps out at around 250 MeV/m for the most overdriven designs. This poses issues for a variety of applications, where higher energies in shorter distances are desired. In science, "desktop" and "chipset" accelerators are of interest. When militarizing particle beams, some effects only become available at very high energies - but you want something compact enough to turret.

urrent and even near future accelerator isn’t really ready to supply that sort of scale quite yet, so we have to move on to the cutting edge of accelerator physics. There are three big options with sub technologies within these broad fields. We have extreme RF cavities using much higher frequencies, plasma accelerators using lasers for power and plasma accelerators using particle beams for power. Each technology provides a unique series of advantages and disadvantages, but fundamentally they are all a way to make linear accelerators that are much smaller.

==== Dielectric Wakefield Accelerator ====
LINACs using much higher frequencies than traditional LINACs have two fundamental problems to fight. The first is electrical breakdown and the second is generation of those mm-wave (10-100 GHz) or even THz (100-1000+ GHz) sources at sufficient power and efficiency to power a particle accelerator. As the frequency of the driving power goes up the electric field climbs up, which means the accelerating gradient gets larger (which is good) but also means that we run a higher risk of a huge arc forming in the accelerator and breaking it (which is bad). This is often fought by changing the accelerator cavity material from a metal (or superconductor) to a dielectric material like ceramic, or by using a coating of dielectric, which can sometimes earn them the moniker of Dielectric Wake Field Accelerator (DWFA). The issue of creating powerful and efficient enough sources of this higher frequency drive power is being actively pursued for a variety of fields (5G cell signals, beamed power, etc) but is still quite immature compared to the microwave RF sources of traditional LINACs. There are options using essentially "over-clocked" microwave sources, low energy particle beams, and even laser-plasmas. There is still a lot to be done here, but these sorts of problems are being actively worked on there are working accelerators using this technology. There is potential for GV/m or greater acceleration gradiatens and all of the same advantages of traditional RF LINACs (i.e. high powers, high fire rates, high energy pulses, quality beams) should stay the same. The wall plug efficiency is liable to be a bit lower, but that will be highly dependent on the efficiency of the source of these high frequency waves.

==== Laser-Plasma Accelerator ====
Plasma accelerators using lasers for power are a whole world in and of themselves. There are a wide variety of ways to accelerate particles with lasers and plasmas, but we will focus on the most studied technique known as Laser Plasma Accelerators (LPA). LPA accelerators utilize ultrashort (<1psec) and ultraintense (>10^18 W/cm^2) lasers to drive a plasma wake (thus sometimes earning them the name laser wakefield accelerator) and accelerate particles through the resulting charge differential. They have only shown the ability to accelerate electrons and positrons, but any ions that could be injected at the right time and at extremely relativistic speeds could conceivably be accelerated this way. In the operation of a typical LPA the ultraintense laser is fired into a jet of gas (turning it into plasma) or a channel of lightly pre-ionized plasma. The laser then blasts through this plasma like a boat through water and leaves a void behind it (the wake). This void ends up having a HUGE accelerating gradient (100 GV/m) and suffers no risk of electrical breakdown since the plasma is already broken down! A pre-accelerated bunch of particles can be injected into that void at the right time (called the witness bunch) for a BIG acceleration or you can rely on the wake itself spewing electrons out into the void and make a beam out of the plasma itself. The gas or plasma then quickly is reformed and the laser fires as soon as its cooling and charging allows. Keep in mind that even though the beams can hit incredible energies in short distances, the laser itself is not a pocket sized device. These lasers are room sized at minimum and not easy to shrink. The pulse lengths required are also much shorter than most of our hypothetical weapons-grade pulse lasers that may operate in the nanoseconds regime - dual-use may prove difficult. The intensities reached also pose significant issues for the laser hardware. If you don’t spread the energies out, it becomes quite possible you will break parts of your laser from the produced intensities.

Current LPA systems suffer from a couple of problems that are being actively worked on. First and foremost they are incredibly efficiency limited by the laser tech being used. There are plans to move on to much more efficient fiber lasers and eventually make LPA machines that operate in the 25% or greater wall plug efficiency, but that is not currently doable. These lasers also need to be fired at higher rates than the current ~1 Hz, but that is much more of a limit of laser engineering than accelerating engineering. LPA’s also have yet to link more than one accelerator together and the lasers are limited in just how much plasma they can accelerate through right now (~20 cm max) so the beam energy runs into a wall quite quickly. Aside from R&D on linking LPA’s together in a way similar to RF LINACs there are two other options. The acceleration gradient is higher the denser the plasma, so a move to denser plasmas and shorter wavelength lasers (to penetrate the plasma) could provide higher beam energies in similar distances. The other option is to use clever optical techniques like the so-called “flying focus” to allow for the laser to go through much longer plasmas. This could in theory allow for more efficient LWFA’s and single stage machines that provide ~1 TeV electron beams in meters! The final issue is beam quality. Even the best LPAs will not make beams that have as tight of a beam spot, energy spread, or emittance as a normal RF LINAC. This means they will end up needing some beam “cleaning” if you want to get a lot of range out of them.

LPAs are of engineering interest for very short wavelength X-ray free-electron lasers in a compact format. Currently these facilities require linear electron accelerators many kilometers in length. With an LPA as the electron accelerator, an XFEL able to image molecular structures and chemical reactions as they happen might be shrunken down to a much more compact format. (Sometimes this is discussed under the ultimate promise of the “benchtop” or even “chip-format”(sic!) XFEL)

==== Beam-Plasma Accelerator ====
Plasma accelerators using particle beams for power are very similar to LPAs in principle but use a particle beam rather than a laser. We typically just call these things Beam Plasma Accelerators (BPA) for short. The concept is very similar to the LPA in that a high intensity beam comes in and blasts a void into a plasma, which particles can then be accelerated in. BPA provides some interesting advantages over LPA in that you can use highly efficient superconducting RF LINACs (or synchrotrons if you want) to provide the particles at very high efficiency. They also have little in the way of plasma density limits and the accelerated beam takes on the characteristics of the beam driving the BPA. Thus little if any beam cleaning will be needed after acceleration. The big downside is the need for a relativistic particle beam to drive the effect, and typically one in the 100’s of MeV to low GeV range, depending on the particle used. Thus your final accelerator may be short, but you still won’t be making a handheld particle beam device.

== Accelerator Shapes ==
The accelerator technologies discussed above are the means by which we create situations for particles to gain energy. The accelerators become embedded in a larger structure that produces, directs, accelerates, even stores the particles until they are directed towards their final use.

There are two principle shapes for accelerator assemblies. Ones where the beampath is not closed onto itself, and designs where the beam path is closed on itself in part of the assembly, forming a loop.

=== Linear accelerator ===
Everything from electron generation to acceleration to conditioning is built in a straight line, one after another. This is the Linear Accelerator or LINAC. The main catch here is that there is only one single passage for the particles through the entire accelerator. Re-using accelerator hardware to drive the same particle to higher energies by repeated passage is out. The particles have to reach their desired energy in a single pass. Practically this puts limitations on the energy achievable with heavy particles in a LINAC. The length required gets impractically large. Even for electrons, lengths of hundreds of meters to kilometers may be required, unless advanced accelerators with high gradients are used. This said, every accelerator contains linear sections. Since so much high-performance accelerator technology depends on particles having a minimum velocity, ions must be “pre-conditioned” in a set of linear accelerators before the re-use of an accelerator through looped beam paths can be considered.

LINACs of low energy may be used in many applications. Medicinal they find use in radiation therapy and radiosurgery.

=== Synchrotrons ===
The other shape which has some interesting properties are accelerators where beampaths are looped onto themselves. These are called synchrotrons. Since beam paths are closed onto themselves, particles can pass through the same accelerator unit repeatedly, gaining more energy on every passage. This is advantageous for heavy particles which do not gain much speed on any single passage. They can pass through the same accelerator thousands or even millions of times instead. This is obviously advantageous for facility bulk and mass and allows us to reach energies that would otherwise be impractical. The accelerated beam bunches can also be held in storage, looping like the rotating mass of a flywheel, holding energy. The LHC can store some 700 Megajoules of energy in the recirculating beam after full speed has been reached.

That said, the devil is in the details. When charged particles are bent in circles they emit electromagnetic radiation which scales very rapidly (to the fourth power!) with relativistic gamma factor and with bending radius (inversely to the second power). This phenomenon is called synchrotron radiation. Due to the scaling, high energy particles emit more synchrotron radiation than lower energy ones and lighter particles (like electrons) at those high energies emit much more than heavier particles. In addition, the bigger you can make the bending radius of the synchrotron (often called “the ring”) the less synchrotron radiation you get.

[[File:SynchrVsE.svg|800px|center|thumb|Synchrotron radiation per particle versus particle energy for electrons and protons in a 50 meter in radius ring.]]

[[File:SynchroVsRing.svg|800px|center|thumb|Synchrotron radiation emission per particle for different particles at different energies, over varying bending radii. ]]

Although these numbers look small, keep in mind that this is PER PARTICLE. A typical pulse in an electron accelerator will have more than more than a billion electrons in it! Large synchrotrons like the LHC will have over 300 trillion ions circulating through it! Synchrotron radiation necessarily has to be managed to reduce radiation shielding, the radiation load (and possibility of hardware failure) on the accelerator, and to reach high energies in the first place - you might lose all the energy your accelerators are adding to synchrotron radiation!

Deflecting the beam also requires the application of magnetic fields, and higher degrees of deflection, in a smaller length, at higher particle energies, require stronger and stronger magnetic fields. Eventually multi-Tesla fields from superconductors will be required. And due to action and reaction, those magnets need to be braced against the pressure the beam exerts as it is deflected. Ultimately this leads to the same limitations as with flywheels - the mechanical strength of the braces and static connections that keep the magnets together as a larger structure is the accelerator’s limitation on J/kg held in storage. A further complication: The magnet strength has to be matched to particle energy. As a beam is accelerated, the deflectors have to ramp up their power to keep the degree of deflection constant.

Sometimes, synchrotron radiation from electrons is intentionally induced in a ring-shaped accelerator. Since synchrotron radiation of high-energy electrons tends to be hard x-rays emitted along narrow vectors, it’s a way to get x-rays used in various imaging techniques in material science and molecular physics.

Synchrotrons used for ions also need to respect the limitations of the involved accelerators. Synchrotrons often use radiofrequency accelerators. Since radiofrequency accelerators must match the speed and speed gain of their particles, an ion synchrotron cannot be made up of a single accelerator loop. Instead there are sequential loops connected in sequence, each designed to boost the particle through a specific velocity and energy region so that the next accelerator takes over. A designed example is the LHC. Particles headed for the main LHC actually circulate in two other synchrotrons beforehand, gaining sufficient energy so that the LHCs RF cavities can take over. (Also, before being injected into the first synchrotron ring, the ions must first go through multiple linear accelerators using different technologies to reach a base energy.)

An interesting design consideration: Synchrotrons don’t necessarily have to take the form of a (near-ideal) ring. Funamentally any dipole represents a single point of deflection of a straight path - they're polygons! Any geometric shape that contains a loop, and where any single bending curve keeps within the limits of the synchrotron radiation you are willing to tolerate, works. As such, complicated two- or three-dimensional “folded-up” shapes are also possible formats for a synchrotron.

=== Storage Rings ===
Since synchrotrons have to match their deflection magnet strength to the energy of the particles being accelerated, a single synchrotron track cannot hold more than one small section of beam bunches in storage at a time, and becomes unavailable to accelerate further bunches until it discharges the stored beam. This is disadvantageous for various applications where we may want to store a fully accelerated particle beam “bolt” for a longer time.

Storage rings are separate synchrotrons designed to hold multiple sets of bunches in circulation at a target energy at a time. The bunches are delivered from an accelerating synchrotron (sometimes called a booster ring) and held in circulation in the storage ring. Since there are synchrotron radiation losses, a few accelerator sections must compensate for these.

Storage rings can also be used to condition a stored beam. A common thing to do in storage rings is to “cool” a beam, draining unwanted movement energy out of the particles. Beam cooling may take significant time, which makes it ideal to conduct in a storage ring. The result is a beam with less tendency to drift apart, advantageous for many applications.

== Other parts ==
=== Particle sources ===
Particle sources are assemblies which supply particles to accelerators. Here we can group them into three categories: electron sources, ion sources, and exotic particle sources. The later involve the production of particles by means of gathering them from collision reactions (or the handwavium blackbox of your choice) and feeding them into another accelerator for further use. Positrons for example may be manufactured in beam collisions.

Electron sources require the production of a free stream of electrons. The oldest method here are thermionic sources. A hot piece of material ejects free electrons into a vacuum, where they can be drawn in using electric differentials and forwarded to the accelerator.

Field emission electron sources instead use a strong potential between a needle-like cathode and an anode plate with a hole. The electrons are drawn from the needle and through the anode due to the high charge differential. This method produces a “cooler” beam with less random divergence than thermionic sources.

A further improvement still are photoemission sources. Here the precise application of laser light onto a photosensitive material energizes electrons in the photocathode until they break free. This process is thermodynamically very precise and due to the nature of the laser light, also very controllable. The result is a low-emittance stream of electrons. These may be accelerated using electrostatic attraction with an anode at one end of the vacuum chamber, or directly using an RF field.

Ion beam sources are more complicated. The physics of ion formation are complex, which opens many design choices when creating an ion beam. Generally a plasma must be produced, from which the ions can be extracted.

The plasma can be formed in a variety of ways. Electrons from a local cathode-anode array, plasma frequency heating, the injection of electron beams, the application of laser light. The end results can be as diverse. Some methods only produce weakly charged ions and on many vectors, so with high emittance. Other methods offer greater ion charges or reduced emittance.

=== Magnets ===
While acceleration of charged particles is effected with electric fields, magnetic fields can be used to bend the trajectories of particle beams. As such, electromagnets can be used to bend the trajectories of charged particles in various ways, and in a controllable fashion.

==== Dipole Magnets ====
Dipole electromagnet are used to steer the entire particle beam in one axis. (To steer in two axes, we must either rotate one dipole about the beam axis, or couple two dipoles in sequence.) They can be used to direct a beam inside a particle accelerator, inject and extract beams out of looping constructs by switching on and off at the right moment, and point a beam at a target.

In a dipole magnet, a positive and negative coil oppose each other, embedded in a “choke” assembly often made of steel. Charged particles moving through the magnetic field between the coils experience a deflection in one direction specified by the polarization of the field, and a degree of deflection specific to the field strength.

==== Quadrupole Magnet ====
Dipoles can steer, but they influence all particles equally. As such, they cannot be used to manipulate the beam's shape. The first type of magnet able to influence a particle beam’s shape are quadrupole magnets.

They are assembled from four coils at 90 degree angles, with coils of the same polarity opposing each other. The passing beam is compressed on one axis, and expanded on the other. To focus the beam overall, at least two quadpoles must be arrayed in sequence.

Quadrupoles are the basic and essential “magnetic lens” of particle beams. Without them, charged beams would inevitably defocus and run havoc.

==== Hexapole Magnets ====
Hexapole magnets exist to address one problem of quadrupoles - they focus differently depending on the energy of the particle passing through the magnetic field. The effect is likened by particle accelerator physicists to the different diffraction of light in a prism - and hence the effect is called “chromatic aberration”. Hexapoles are used to correct for this, stronger deflecting higher-energy beams.

==== Integrated optics ====
As the field of particle accelerator science and engineering advances, more complex designs are proposed to handle more complex phenomena inside accelerators. An upcoming field is “integrated optics” - complex magnetic systems with nonlinear behaviors. These could provide more advanced manipulation of electron beams, building upon existing capabilities and adding new ones for shaping a beam.

=== Plasma lenses ===
Plasma lenses are an up-and-coming technology, where very high currents in plasma are used to create very strong magnetic fields. Similar to plasma-based accelerator technology, the fact that charges are already free in plasma is used to circumvent conventional charge breakdown and critical current limitations on solid-matter devices. The resulting magnetic fields can be much stronger and bend beams over shorter distances. Furthermore, by creating complicated profiles in the plasma, elaborate beam manipulations could be performed, more akin to sophisticated optical elements. As well, the charges in the plasma neutralize the particle beam in the region of the plasma lense - affecting its propagation behavior further and in a positive manner, as self-repulsion is neutralized.

The catch is that this is plasma - that means you are dealing with all the complications of plasma physics. Heating of the plasma has been an early challenge to successfully forming and maintaining plasma lenses.

Plasma lenses require an appropriate chamber, possibly active containment to keep the plasma from losing energy to the walls or leaking into other accelerator parts, and the necessary plasma sources, maintenance heating devices and then lense shapers and control sensors. In high-powered plasma lenses there will also be a need to cool the plasma as it heats up from waste heat. Potentially we may manage this by open-cycle cooling - venting the plasma controlled in one direction into open space, rather than have a closed-cycle solution for reconditioning the plasma. This could cause some very cool visuals!

Plasma lenses for particle beams should not be confused with optical plasma lenses for lasers! Those exist as well, but are their own topic.

=== Chicanes ===
Chicanes are assemblies of multiple dipole magnets used to manage the longitudinal dispersion of a particle beam due to different particle energies. Since particles with higher energy move faster, they drift ahead in the beam distribution over time; and particles with less energy increasingly lag behind.

Chicanes deflect high-energy particles onto an outward curving trajectory. Since traversing the curve takes longer, the high-energy particles are effectively delayed, allowing lower-energy particles to catch up. By the end of the chicane assembly, the bunch has been compressed in the longitudinal direction.

Chicanes however, do not remove energy from too-fast particles. That is the responsibility of other structures.

=== Undulators and wigglers ===
Undulators and wigglers are magnetic assemblies used to transform particle energy into electromagnetic emissions, with different properties from the specific implementation of a shared basic principle.

Assembled from stacks of generally dipole magnets (other configurations exist), both undulators and wigglers force charged particles onto “wiggling” trajectories by bending their path left and right. Wigglers force very strong arcs; undulators less so. In both cases, the rapid deflection of the charged particles causes them to shed parts of their energy as electromagnetic emissions. Due to the energy of the particles, these photons are generally highly energetic x-rays.

Undulators may be used to build free-electron laser systems.

They can also be used as a means of cooling the beam. The removal of energy from particles can also suppress unwanted movement modes, allowing accelerator structures to re-add energy only to desired movement modes. Of course, this means the particle beam actively loses energy, and produces x-rays that have to be absorbed (and cooled away.)

=== Beam coolers ===
Beam coolers are a collection of systems and strategies that are used to remove unwanted movements from a particle beam. Undulators and wigglers are one way to do this.

One method set of beam cooling relies on finding particles who are very “hot” with random motion, and either removing the particles from the beam using a kick device, or applying a precise counter-force that acts against the unwanted motion.

If one is dealing with ions, one clever trick is to inject a beam of electrons. Due to the thermodynamics within the particle population, the lighter electrons absorb heat from the heavier ions, taking this unwanted motion energy with them when the electron beam is bent out.

As well, lasers can be used (what can’t lasers be used for!) to apply counter-force on ions and remove unwanted movement in this manner.

Beam coolers are generally not instantaneous devices. They work over time to make a beam better. Often they involve the removal of energy or particles from the stream, requiring accelerators and sources to make up the loss and creating waste heat.

=== Beam recovery ===
Some accelerator systems can actually be run in reverse - they can turn beam energy back into some other energy format and then electricity. This can be used to recover some to most of the energy from a particle beam before disposing of it. This is advantageous both for your energy bills and making a very high-energy beam liable to spray radiation everywhere, easier to absorb in a beam dump.

=== Beam dump ===
A beam dump is a construction where a particle beam can be routed in case it has to be disposed. Beam dumps are designed to safely absorb the particle's energy without leakage of dangeorus radiation or activated materials to the outside enviroment. Common materials are carbon and metals, embedded in an actively cooled assembly.

= Propagation physics =
==Magnetic fields==

Magnetic fields are things of motion. They are created by moving electric charges. And they only affect electric charges that are moving. An electric charge at rest in the presence of only a strong magnetic field will just sit there. But when it starts moving, the magnetic field interacts with it, starts deflecting it.

But not along the field. Oh no. The magnetic field does not point in the direction it will move the charge. No, the charge experiences a force that is perpendicular to the direction of the field. And also that is perpendicular to its own velocity. A charged particle moving straight down a magnetic field line will not experience any force.

On the other hand, if the particle is moving directly perpendicular to the field, the magnetic force will always push it to the side, even as it keeps turning. This results in the particle moving in a circle. The radius of this circle is called the <b>gyroradius</b>, and the rate at which the particle makes circles is called the <b>gyrofrequency</b> (or sometimes <b>cyclotron frequency</b>).
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Circular_motion.png|frameless]]
<td width=255>[[File:magnetic_forces.png|frameless]]
<td width=255>[[File:spiral_around_field_lines.png|frameless]]
<tr>
<td>To get circular motion, you need a force (blue) that is always the same amount but always pushing perpendicular to the direction of motion, toward the center of the circle. A charged particle in a constant magnetic field moving perpendicular to the magnetic field meets these conditions, and moves in a circular path.
<td>A diagram of how a magnetic field (magenta) exerts forces (blue) on a moving particle (with velocity shown in green).
<td>If a particle is moving partially along the magnetic field (magenta), it will drift down the field lines while spiraling around it (black path). Where the magnetic field is stronger (indicated by having the field lines closer together) it will execute tighter, faster circles in its spiral and will drift more slowly than when the field is weaker.
</table>

It can be convenient to split up a particle's velocity into a part parallel to the magnetic field (the <b>longitudinal velocity</b>) and a part perpendicular to the field (the <b>transverse velocity</b>). The longitudinal velocity describes how fast the particle drifts along the magnetic field line. The transverse velocity determines how tightly and how quickly the particle circles around the field lines.

If the gyroradius is much smaller than the extent of the magnetic field, you get a case where the particle approximately cannot cross magnetic field lines. It just spirals around them. On the other hand, if the magnetic field ends before the particle can make a significant portion of a full revolution, then the particle ends up being somewhat deflected from its original trajectory. If the gydroradius is enogh to deflect the particle into a region where the magnetic field has a different strength or direction, then things get complicated!

When a particle is spiraling around the magnetic field while drifting down the field lines into regions where the magnetic field is stronger, it will circle around the field lines in a faster, smaller circle. This increases the kinetic energy in the spiraling motion; this kinetic energy must come from the kinetic energy of the longitudinal motion. If the field becomes too strong, the particle will slow down all the way and come to a stop. Then it will be reflected back, pushed the other way down the field lines toward the region where the field is not as strong. This phenomenon is called magnetic mirroring, and is responsible for the way radiation gets trapped in the Van Allen belts around Earth with the particles bouncing back and forth along the Earth's field lines and getting reflected back and forth at the poles where the field is strongest.

If you want to calculate for yourself what the gyroradius <math>r_c</math> and gyrofrequency <math>f_c</math> are
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
r_c = \frac{\gamma \, m \, v_\perp}{q \, B}
</math></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
f_c = \frac{q \, B}{\gamma \, m}
</math></div>
where <math>B</math> is the strength of the magnetic field, <math>q</math> is the amount of electric charge of the particle, <math>m</math> is the mass of the particle, <math>\gamma</math> is the Lorentz factor, and <math>v_\perp</math> is the part of the velocity that is perpendicular to the magnetic field. To get these formula to work, you need to use a consistent set of units. So for example if you measure the magnetic field in tesla you need to measure the charge in coulombs (not fundamental charges) and the mass in kilograms (not eV).

One consequence of this is that you can bend the beam coming out of your accelerator by placing a uniform magnetic field across the end. So you can steer your beam to where it needs to go. You can also bend the beam inside of your accelerator – to make it go around in a circle, for example.

Another consequence is that if your beam has to pass through regions with magnetic fields in it, it will get deflected. If you want the beam to end up at the right spot, you will need to account for this. If the exact strength or direction of the field are not known, you will get unpredictable deflections to your beam.

If your beam has been neutralized, the magnetic fields will work to bend the positively charged particles one way and the negatively charged particles the other way. If you have enough beam current, these separated charges will attract each other and counteract the magnetic force. If you don't, the beam will split and you'll get positive charges zipping off in one direction and negative charges going another direction. If you needed to neutralize your beam so that it doesn't electrostatically blow itself apart, well now it will be doing just that.

Even perfectly neutral atoms can be torn apart if the atoms are going fast enough and if the magnetic field is strong enough. Atoms are pretty resistant to this, but you can get conditions where it happens.

Magnetic fields affect the path of beams. But beams are made of charged particles and thus can have currents in them. Currents create magnetic fields themselves, which can influence the surrounding fields. Even a neutralized beam forming a plasma, when it encounters a magnetic field, will get currents induced in the plasma that helps to screen its interior from the field. A beam with enough punch can deform the fields around itself rather than having the fields bending it. As a rough rule of thumb, if the energy density of the beam over a regions larger than a gyroradius is more than the energy density of the field over that region, then the beam gets to have its way with the field, and vice versa. Highly focused, high current beams can thus be expected to better deal with ambient magnetic fields than those with less focus or lower current.

==Focusing==

In the previous section, we discussed how a particle beam can be steered by directing it through a uniform magnetic field. We can make this kind of field with a magnetic north pole directly opposite a magnetic south pole (with additional shimming magnets to keep things uniform). Because it has two poles, one north and one south, this kind of magnet is called a <b>dipole magnet</b>.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:dipole_beam_steering.png|frameless]]
</table>

But we want to do more than just steer the beam. we want to be able to expand it, contract it, and focus it to a point. To do this, we need a different kind of configuration. We will put two north poles opposite each other and, and 90 degrees, two south poles opposite each other. With four poles, this is called a <b>quadrupole magnet</b>, or sometimes just a "quad".
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_magnet.png|frameless]]
</table>
At the center of the quad arrangement, the magnetic fields from each side cancel out. there is no field there. A particle going through the center will travel straight through without deflection. However, the farther particles get from the center, the higher the field they encounter and the more they get deflected. A particle slightly off to one side may experience a force that curves it toward the center-line. If the quads are properly shimmed, a particle twice as far away will experience twice the force and twice the deflection from its trajectory back toward the center. This curves all the particles so they meet in one spot.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_focusing.png|frameless]]
</table>
However, there is a catch. Those particles that are offset from the center-line at 90 degrees from these focused trajectories are bent away from the center-line rather than towards it! A quad focuses in one direction and de-focuses in the other.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_defocusing.png|frameless]]
</table>

Is there anything e can do to get an arrangement of magnets that purely focuses the beam? Well yes. We can put another quad arrangement downstream of the first, rotated by 90 degrees. Now consider a set of particles in an arrangement that is de-focused by the first quad. When they get to the second quad, they are further from the center than they were when they went through the first quad. Because the second quad is aligned to focus anything the first quad de-focuses, these particles will be focused; and because they are farther from the center, the focusing deflection will be larger than their previous de-focusing deflection. So the net result is that the particles are focused.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Dual_quad_focusing.png|frameless]]
</table>
For particles with the opposite orientation that are first defocused, these will be defocused by the second quad but because they will have converged they will be going through a region of lower field and will be defocused with a smaller deflection than they were initially focused. This dual quad arrangement will focus any particle that goes through it, regardless of orientation.

For small deflection angles, a dual quad set-up acts like an ideal lens for particles. An ideal lens can focus all the particles diverging from a point onto another point, or into perfectly parallel lines. It can focus particles on perfectly parallel lines onto a point. So now we can use the particle beam to deliver focused particles on our target.

Or, at least we could if all the particles came from a point, or if they traveled on perfectly parallel lines. Unfortunately, they don't. When particles are made, they come from a source with a finite spot size and a finite temperature. The random motion of the particles moving around from their initial thermal motion means that they are never moving perfectly parallel to each other. And when you try to focus them, instead of all converging onto a single point they make a spot of finite size. And if your focusing and beam bending elements are not perfect, you add additional imperfections to the beam that further hider your ability to focus it.

Your ability to focus can be quantified by a measure called <b>emittance</b>. Emittance is measured in units of a length scale times an angle - nominally meters-radians (m<math>\cdot</math>rad) but the technical literature more commonly gives units of millimeters-milliradians (or mm<math>\cdot</math>mrad) or even 2 <math>\pi</math> mm<math>\cdot</math>mrad. The divergence angle of a beam that you try to make parallel will the the emittance divided by the width of the beam. If you focus a beam down so it converges at an angle, the minimum spot size you can get at the focal point is the emittance divided by the beam angle. If you try to focus a beam with emittance <math>\epsilon</math> on a spot a distance <math>R</math> away with an initial beam diameter of <math>D</math>, the spot size <math>S</math> on the target you can achieve is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
S = R \frac{\epsilon}{D}
</math></div>

So to get good focusing, we want low emittance. Is there a way we can lower the meittance beyond what we get from our source? Engineers are pretty bright folks, and they have found some ways to do this. You can, for example, pass a beam through a thin foil. This causes some stopping, slowing the beam down. You can then accelerate the beam back up to its original speed. However, the stopping is along the direction of the beam particle's tranjectory, even if that trajectory is at an angle. Meanwhile, all the acceleration is in the direction of the beam axis. So the net result is to reduce the random motion perpendicular to the beam, and thus reducing its emittance. Similar tricks can be played with magnet set-ups that force particles going through them to emit radiation.

Another method can be used with a relatively high emittance beam, say a beam of protons or ions or muons. Electron beams can be made with substantially lower emittances. So you can send a low emittance electron beam to co-propagate along with your primary particle beam. The beams will interact, with the particles scattering off each other. This exchanges heat between the two beams and so the temperatures, and thus emittances, of the two beams get closer to each other (and, if given enough time, would become equal). If you repeat this often enough, you can get the emittance down to the emittance of the electron beams. This is more practical in beams going around circular accelerators, because they will have time to get cooled over and over.

A final method that has been used is to install sensors to measure the beam fluctuations. For beams going in circles, the time to get a light-speed signal directly across the diameter to the opposite side wil always be less than the time it takes for the particles to go around the curve to that spot. So you can send signals to give electromagnetic kicks to the beam particles to knock them back into alignment when they get to that spot.

It is intriguing to note that these beam cooling techniques can take <i>hours</i> to work! <ref>[https://www.bnl.gov/isd/documents/86783.pdf M. Minty, R. Connolly, C. Liu,T. Summers, and S. Tepikian, "Absolute beam emittance measurements at RHIC using ionization profile monitors", Brookhaven National Laboratory formal report BNL-105970-2014-IR]</ref>

But perhaps the simplest method to reduce emittance is just to speed the particles up. For particles moving at a fraction of light speed <math>\beta</math> and Lorentz factor <math>\gamma</math>, the emittance of particles from the same source with the same temperature and same random motion will be proportional to <math>1/(\beta \, \gamma)</math>. To reflect this, it is convenient to define a normalized emittance <math>\epsilon_n</math> such that for the usual geometric emittance <math>\epsilon</math> that we have been talking about
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
\epsilon_n = \beta \, \gamma \, \epsilon.
</math></div>

Normalized emittance measures makes it convenient to get basic estimates of particle beam performance and compare emittance of different sources and beams. For example, various high-performing electron beams around the world have managed to push their normalized emittances down to about <math>2 \times 10^{-6}</math> m<math>\cdot</math>rad to <math>3 \times 10^{-6}</math> m<math>\cdot</math>rad. Facilities with proton or ion beams seem to have about an order of magnitude higher normalized emittance.
<table>
<tr><td width=200>Machine <td width=300>normalized emittance (m<math>\cdot</math>rad) <td>reference
<tr><td>European XFEL <td><math>2.3\times10^{-6}</math> <td><ref>[https://journals.aps.org/prab/pdf/10.1103/PhysRevAccelBeams.24.110702 T. Hara <i>et al.</i>, "Low-emittance beam injection for a synchrotron radiation source using an X-ray free-electron laser linear accelerator", Physical Review Accelerators and Beams <b>24</b>, 110702 (2021)]</ref>
<tr><td>SLAC <td><math>3\times 10^{-6}</math> <td><ref>[https://accelconf.web.cern.ch/pac97/papers/pdf/4W009.PDF M. Hernandez <i>et al.</i>, "Emittance Measurements for the SLAC Gun Test Facility", 1998 IEEE]</ref>
<tr><td>LHC <td><math>3.75\times 10^{-5}</math> <td><ref>[https://www.lhc-closer.es/taking_a_closer_look_at_lhc/0.beta___emittance Beta & Emittance; Taking a closer look at LHC]</ref>
<tr><td>RHIC (with protons) <td><math>\approx 5\times 10^{-5}</math> <td><ref>[http://www.osti.gov/servlets/purl/910428-6sKfP0/ Zhang, S, and Ptitsyn, V., "Proton Beam Emittance Growth at RHIC", Brookhaven National Laboratory formal report BNL-77563-2007-CP]</ref>
</table>
So as a first estimate for your sci-fi electron guns, you can take <math>\epsilon_n \approx 3\times 10^{-6}</math> m<math>\cdot</math>rad and for proton guns <math>\epsilon_n \approx 4\times 10^{-5}</math> m<math>\cdot</math>rad. Then divide by <math>\beta \, \gamma</math> for your beam to find the geometrical emittance. Which in turn can let you figure out how well you can focus your beam.

==Space charge==

==Energy loss in matter==

Sometimes, a particle beam needs to go through stuff to get it where you want it to go. Stuff like air, for example. Or the tissue in the way of getting the beam to a tumor. Also, sometimes, you want your particle beam to interact with stuff. Like the material you are trying to machine with an ion beam, or the bacteria you are trying to kill on medical equipment. All of this falls under the general realm of the interaction of the beam particles with matter. This is an enormously complex subject, but we can cover many of the basics here.

The primary effect on the beam of any interactions with matter are loss of energy of the beam particles (<b>stopping</b>), random changes to the particle directions that make the beam spread out (<b>straggling</b>), and knocking particles clean out of the beam altogether (<b>scattering</b>).

===Ionization===

The particles coming out of your particle accelerator are charged. When a fast moving charged particle goes through matter, the electric field from the charge pushes and pulls on the various charged particles that make up that matter – mainly electrons and the atomic nuclei around which the electrons orbit. At speeds above approximately 1% of the speed of light, it is primarily light and nimble electrons being affected while the ponderous nuclei just sit there barely perturbed by the passage of the particle. When the passing particle yanks on an electron hard enough, it can pop it clean off of the atom it was attached to, leaving a positively charged ion behind and giving you a free electron that can go and glom on to a previously neutral atom to make a negative ion. This process, unsurprisingly, is called <b>ionization</b>. For most particle beams, ionization is the dominant interaction with matter; although when you get very fast or very slow particles other effects can become more important.

if your particle was not initially charged when it goes into matter, it soon will be. The electric charges of the electrons and nuclei in the matter will do the same thing to the particle that a charged particle would do to them. This will quickly strip off any electrons around a high speed atom, leaving an ion and bare electrons to go through the matter. Only particles that are fundamentally neutral (like neutrons or photons) will not be immediately ionized.

It takes about 1 to 10 eV to ionize an atom. This energy must come from the kinetic energy of the beam particle. Particles from an accelerator have millions to billions of eV. So it takes a lot of ionizations to meaningfully affect the particles. However, one thing about matter is that it has lots of atoms that can be ionized so you do get those lots of ionizations. Because each ionization is so small compared to the total energy this is often approximated as a continuous slowing down of the beam particles. So many individual ionizations tend to average out, leading to all the beam particles being slowed down uniformly.

When the particle is yanking on nearby electrons to tear them out of atoms, those electrons are yanking back. This jitters the beam particle slightly from side to side. Again, these sideways yanks tend to average out. The statistical variations from the average slowing down and side-to-size jerking are what lead to straggling. Protons and ions are heavy, and will be deflected little from their original paths - like bowling balls smashing through matchsticks. These particles tend to go in mostly straight lines. Electrons, on the other hand, have the same mass as the electrons they are knocking about. So electron tracks through matter tend to zig-zag around a lot more.

As the particle whizzes past, it leaves a trail of atomic destruction in its wake - a trail of ions where the atoms are literally torn apart (not the nuclei, just the atoms). These particle tracks can sometimes be seen in detectors like cloud chambers. The faster the particle is going, the less time it has to yank on nearby electrons and so the less ionization it leaves behind. As a result, fast particles are not slowed down much but as they start to slow down they continue to be slowed down faster and faster. A consequence of this is that the particle dumps most of its energy into the material at the end of its track, just before it comes to a stop. This sudden peak in energy deposition is called the <b>Bragg peak</b>.

The electrons that a particle knocks out can end up going pretty fast. So these secondary electrons can produce further ionization in what is called an <b>electron cascade</b>. if an inner core electron is knocked out of an atom, other electrons falling in to fill that empty spot in the atom can release x-rays (<b>x-ray fluorescence</b>) or cause shake-ups in the outer electrons that knock other electrons off the atom (<b>Auger effect</b>).

===Brehmsstrahlung===

A charged particle is surrounded by an electric field. If it is moving, it is also surrounded by a circulating magnetic field. These fields move along with the particle.

If the particle knocks into something, it will slow down and probably change direction. The information about this bump only goes through the electromagnetic field at the speed of light. So a lot of that field that was moving along with the particle might not get the memo in time and keep right on going. Stray electromagnetic fields like this get this self-inductance effect going on where they start making themselves over and over again, allowing them to propagate as freely moving waves of electromagnetic radiation. This EM radiation from a slowing down particle is given the delightful name of <b>bremsstrahlung</b>.

Particles from an accelerator can be moving so fast that they make bremsstrahlung in the x-ray or gamma ray part of the spectrum. If you get very high energy bremsstrahlung gamma rays, the gamma rays can interact with matter by boiling an electron and its antimatter counterpart, a positron, out of the vacuum (using the nearby matter as the necessary momentum sink for the process). These positrons and electrons can themselves be very high energy and can in turn create more bremsstrahlung, and the process can repeat until the particles are no longer high enough energy to continue making each other. This is called an <b>electron-gamma shower</b>.

The x-ray machines used by dentists and radiologists use both bremsstrahlung and x-ray fluorescence that you get when shining an electron beam onto tungsten to make their x-rays.

At very high speeds, bremsstrahlung can be more a important source of energy loss than ionization.

===Electron capture and screening===

When an ion has slowed down to about 1% of the speed of light or less, it is moving slower than the electrons zipping around their atoms in the matter the ion is going through. This allows the electrons to stick on to the ion, now becoming the electrons of the atom that is moving through the material rather than of the material itself. The ion begins to lose its charge, and the rate of ionization goes down. Soon, ionization becomes negligible compared to the beam atom bumping in to other atoms.

===Phonons and displacement===

When an atom or ion moving through a material bumps into another atom, it will cause that other atom to move. This can have one of two primary effects.

First, if the atoms whack into each other hard enough, the atom in the material can be knocked clean out of place. Like a pool ball being thwacked by the cue ball, the struck atom will go zinging through the material. Atoms getting knocked out of their place like this is called <b>displacement</b>. Displacement creates chemical changes to molecules as atoms are removed, and creates defects called <b>vacancies</b> in crystals (which, as you might imagine, is just a missing atom in the crystal lattice). When the atom comes to rest, it can chemically attack other molecules, again changing the chemistry, or create an extra vacancy called an <b>interstitial</b> jammed into the crystal lattice where it is not supposed to be. If the displaced atom is going fast enough, it can whack other atoms and displace them, leading to a <b>displacement cascade</b> like tenpins knocking each other down. As a result, at the end of its track, an ion can leave a region of disordered material and messed up chemistry where the atoms have all crashed into each other like a freeway pileup.

If an atom is displaced near the surface of a material, the displaced atom can shoot out of the surface and escape the material altogether. This is called <b>sputtering</b>.

Second, the beam atom might only deliver a glancing blow. Or the beam particle might be an electron, which is so light that it is usually incapable of displacing atoms. In this case, the struck atom gets kicked to the side a bit and runs into other atoms; maybe it is also pulled back by the chemical bonds of other atoms that it is attached to. This transfers the motion of the atom to other atoms around it in the form of pressure waves (or, in a solid, shear waves). This adds heat to the material without causing direct damage – although as you might imagine, enough heat can also cause localized melting, evaporation, or decomposition. The individual quanta of vibration – fundamental particles of sound, if you want to think of them like that – are called phonons in analogy with the photons that are the fundamental particles of light. Much like ionization, a particle that excites pressure waves slows down gradually and continuously, unlike the sudden changes in speed and direction you get from knock-out displacement.

===Nuclear collisions===

So far, we have only discussed the things that happen when beam particles interact with matter particles by electromagnetic forces. Even atoms knocking into each other to produce phonons or displacement happens through the mutual repulsion of the electrons around the atoms (although this is dressed by other weird effects like how two electrons can't be in the same place at the same time, or how the fields of one electron can affect other nearby electrons that in turn affect the original electrons, leading to odd correlation effects and charge density waves and other fun stuff that keeps condensed matter physicists and quantum chemists busy).

But electromagnetism isn't the only thing you can get going on. below the level of structure of the atom, you have the atomic nucleus. And a direct hit on the nucleus can end up giving very different effects.

The simplest thing that can happen is an <b>elastic collision</b>, where no energy goes in to any internal nuclear state or into breaking up the nucleus. An elastic collision just ends up giving you displacement (or possibly phonons, but if you hit the nucleus you usually hit it hard enough to send it flying).

But beyond that you can cause any number of nuclear reactions. You can leave the nucleus in an excited nuclear state, from which it usually relaxes back down to its original state by emitting gamma rays. You can knock out nuclear particles, such as neutrons, protons, deuterons (a proton stuck to a neutron), tritons (a proton stuck to two neutrons), helions (a neutron stuck to two protons), alpha particles (two protons stuck to two neutrons), or potentially even larger nuclear fragments. If you hit a heavy nucleus like uranium, you can cause it to split apart by fission. And if you hit the nucleus really hard you might just shatter it into many tiny fragments of those protons, neutrons, deuterons, tritons, helions, alpha particles, and heavier nuclear fragments mentioned earlier. To get any of these <b>inelastic collisions</b>, you need to hit the nucleus hard enough. If your beam particle doesn't have enough energy to excite the nucleus or knock particles, these things simply can't happen. As it gets enough energy to excite a particular interaction channel (as they are called), that process becomes possible and as the energy increases the process becomes more and more likely.

A nucleus is only about a femtometer across (<math>1 \times 10^{-15}</math> m). Which makes it only about <math>1 \times 10^{-30}</math> m² in cross sectional area. In the meantime, chemical bonds between atoms are usually about 1/10th of a nanometer (<math>1 \times 10^{-10}</math> m) long. In condensed matter (liquids and solids), this means that you have somewhere about <math>1 \times 10^{30}</math> atoms per cubic meter. A meter thickness of atoms thus presents just about about enough atoms that their nuclei can cover the projected area – in other words, you can expect a particle to go through (very roughly) about a meter of condensed matter before it whacks into a nucleus.

This assumes that a particle that goes through a nucleus interacts with it. This is usually a good assumption for particles that interact by the strong nuclear force, such as protons, neutrons, mesons, and the atomic nuclei of ions. But many particles cannot interact by the strong nuclear force, such as electrons or muons. These are much less likely to interact with an atomic nucleus even if they pass through it, so they will end up going considerably farther between hitting nuclei than the above assumption. But nuclei can interact electromagnetically (they are made up of charged protons, after all, and at a smaller level of charged quarks), so electrons and muons can produce some nuclear interactions; just far less than actual nuclear particles like protons and ions.

A nuclear strike is a discrete event that will remove the impacting particle from the beam and will (except for elastic collisions) probably produce a spray of radiation.

===High energy physics===

As you get to higher and higher energies, you unlock all sorts of new and weird and interesting processes that can happen. Boiling exotic particles out of the vacuum, unifying forces that had been divorced since the big bang, and producing bizarre unstable resonances in the quantum fields. This is what physics build accelerators to measure. From the point of view of getting a beam to its target, or what happens to the target, however, the main effect is mostly that the interacting particle is removed from the beam and a bunch of additional radiation is produced.

==Self pinching==

= Operational Considerations =
== Logistics ==
The logistics of particle beam can be a mixed bag. By itself, any particle accelerator is a complicated piece of machinery full of delicate parts that have to deliver high performance. This may require cryogenic temperatures with associated challenges on the materials and mountings used (insulation, and distortions under temperature changes). When working, particle accelerators invariably generate some radiation across the whole assembly, which irradiates and activates the structure. Some parts may be exposed to significantly higher radiation flux due to their working principles. Often the particle beam is a dynamic system that must be managed with active, closed control loops, involving sensors, modeling and control software, actuators, and the computing and communication hardware that brings it all together. This poses engineering challenges across all disciplines.

A practical particle beam is inherently a counterintuitively sturdy device. If it was too delicate, it would break too much to be useful, though of course this is a sliding scale. Still, while sturdy, this doesn’t eliminate complexity, it manages complexity. The result is a system that is the product of a high-tech civilization. A superior machine shop and well-trained engineers are a must to maintain a particle beam system. Some parts of a particle accelerator may be mundane - others may be custom-machined parts that require their own dedicated industries to produce. That said, the manifold utility of particle accelerators is on your side here. Since there are so many practical applications, it is economic to produce parts for industrial demand. (that said, these may not be the kind of components you want in specific high-performance applications.)

The main thing a particle accelerator requires the most is electricity. Any mature power production facility can supply the power.

However, all particle beams require at the least a source of neutralizing mass. The projection of a charged beam will leave the projecting object oppositely charged. In an atmosphere the natural plasma formed by the beam will ground everything out, but in space the diffuse vacuum will not do the job.

Furthermore particle beams using ions also require a feedstock for their ions to produce from, and in an appropriate format. This can be a gas, a fluid or an easily ionized solid. It can be the feedstock species in its pure form or in an easily split molecule, though every complexity adds to the amount of preparation work that has to be done before the feedstock enters the particle beam.

Don’t worry about the mass of your source or neutralizing ions though. Large particle accelerators, like the LHC in Switzerland, only consume tiny gas bottles worth of feed gas every couple years. The mass of a single 5.56 NATO bullet worth of hydrogen could feed a 10 GW, 10 GeV ion beam for nearly 4 days of firing! You will not be running out of beam feedstock easily.

== Cooling ==
Cooling a particle accelerator can be a serious challenge. Various bits of a particle accelerator love to be supercooled for efficiency or basic working principle - any bit of superconductor used in the accelerator, or magnets, or EM generators, will require cooling to its generally supercool operating temperature.

Even if Standard Temperature and Pressure Superconductors (273 Kelvin, 100 Kilopascal) are available, present superconductors have demonstrated superior performance by being supercooled. That means to achieve even stronger fields, even STP superconductors may be cooled to cryogenic conditions.

Cooling cryogenic systems sucks. Thermodynamics shows itself from its worst side. You can forget running cryogenic radiators - the Stefan-Boltzman Law penalizes the direct radiative cooling of anything cryogenic severely as much as it empowers very high-temperature radiators. This only leaves stages of heat pumps to provide cryogenic fluids that can in turn, cool cryogenic systems. Heat pumps draw power, and for such a giant temperature gradient as we can have here, they will draw a lot of power.

However, we have other options. Cryogenic systems also tend to be highly efficient, so while they do heat up, they do so very slowly and there is little overall energy absorbed. This provides the option of running off of bunkered cryogenic fluid in dewars for the duration of operating the equipment and not running any heat pumps all the time.

Other bits of a particle accelerator system aren’t as required, but really nothing is a fan of high temperatures. Going higher than room temperature can be treated as an oddity. High-performance lasers may not require cryogenic conditions, but they certainly like it cool. Radiofrequency producers of any kind generally fare worse with rising temperatures. That means cooling overall remains a challenge - 300 Kelvin is still not that great of a hot loop temperature for radiators.

== Safety <ref>Thomas Otto, "Safety for particle accelerators", Springer 2021, Available online at: https://library.oapen.org/handle/20.500.12657/46120</ref>==
=== Ionizing Radiation ===
There are surprisingly many sources of ionizing radiation in a particle accelerator, besides the obvious threat of the output beam itself. Not all particles are accelerated correctly or in the correct direction. When such particles interact with the structure of the accelerator, they release a shower of secondary ionizing radiation. In electron accelerators and sources, stray electrons may be accelerated and release x-rays from bremsstrahlung as they collide with surrounding matter. The resulting doses can be significant and dangerous. Also, such collisions leave behind residual activation, making the accelerator a radiation hazard for some time even after it has been shut down. Generally, people should not work near the particle accelerator while it is functioning, and radiation barriers isolate the core accelerator from adjacent facilities. (In real-life facilities, you will often see accelerators inside larger buildings being cocooned inside doped concrete plates to shield the rest of the room.)

=== Radiofrequency ===
RF energy is used in many kinds of particle accelerators. This energy may leak from the structures it is generated, contained and used in due to damages or incorrect assembly. As microwave radiation and above is used, the radiation can be readily absorbed by bodies, hydrogen-rich materials or other objects. The end result of this can be rapid heating and thermal effects. Especially in accelerators designed to throughput a lot of energy (say a GW or more), this may become a serious issue if a waveguide is damaged. It’s not recommendable to stand in front of a multi-MW radar either!

=== High-powered electrical and electronic systems ===
The operation of the magnetic systems and accelerators often involves high-powered electrical and electronic systems. High voltages and/or amperes are involved. This poses risks like shock hazards, electrical arcs, and electrical fires. Some electrical systems such as capacitors or inductors may have combustive or explosive failure modes.

=== Flammables ===
Parts of an accelerator or its infrastructure may be manufactured from flammable substances such as hydrocarbons. In the case of an ignition source such as a high-powered beam cutting into a physical substance or an electrical fault, these may catch fire.

=== Lasers ===
Some accelerator systems use high-powered lasers. Laser plasma accelerators use very high-powered pulse lasers to provide pumping power. The safety concerns of lasers for blinding, physical damage, and possible exposure to ionizing radiation as a secondary effect of the beam impact, apply. Some high-powered lasers can also drive electrons which generate strong EMPs, which may damage or destroy other systems in turn. Parts of a laser assembly or laser beam guide may shatter if exposed to overloading conditions.

=== Magnetic systems ===
Parts of a particle accelerator may use strong magnets, up to superconductive magnets with multi-Tesla fields. While the magnets are integrated into chokes that shape the magnetic fields, fringe fields invariably extend beyond the core magnet installation.

Multi-Tesla transient electromagnetic field can affect the human nervous system temporarily, inducing nausea, vertigo, metallic taste and optical phenomena, if the person moves in relation to the magnetic field. Otherwise, the risk of exposure is low. That said, electronic systems that are not shielded properly, such as say, implants, could be adversely affected by the magnetic fields.

The biggest physical risk are reactions of other magnetic objects with the magnets. You can look up cases where metallic objects interacted with the multi-Tesla fields of MRI scanners. Such parts may fly across the room towards the magnet at increasing speeds, and damage anything or injure anyone in their path, possibly ending with the object or person squashed (or impaled!) to the magnet.

Also, large superconductive magnets store not insignificant amounts of energy. If the superconductive property breaks down, the magnet quenches and becomes a resistor. The resistance site heats up as current tries to flow through it, expanding the quench and possibly being heated until it sublimates into gas. This could possibly happen explosively. In such a case, up to many GJ of energy might be unleashed in a conflagration or explosion (subsonic or supersonic), damaging surrounding systems and ejecting shrapnell.

=== Cryogenic cooling systems ===
Particle accelerators often involve cryogenically cooled systems. This means cryogenic fluids like nitrogen and helium are present in the accelerator infrastructure. Such gasses have both a very low temperature and a high ratio of gaseous volume at high temperatures to their liquid volume.

Exposure to the leaked fluid can lead to cryogenic cold damage, injury, and death. Furthermore such gasses are efficient at displacing ambient breathing gasses such as oxygen, so they pose an invisible, and rapidly spreading, suffocation hazard.

Also, if insulation is damaged, ambient heat may leak into the cryogenic circuits, and cryogenic fluid may begin evaporating. This can lead to dramatic rises in internal pressure. Relief valves should trigger to safely direct such pressure spikes away from important places and keep the system from failing catastrophically, but otherwise, pipes and cryogenic storage devices may fail, up to explosively.

=== Beam excursions ===
If the magnets responsible for manipulating the beam break down, the beam may excurse from its designed paths. The beam may become defocused, or maintain its focus. The beam begins cutting into the accelerator structure, evaporating material and depositing energy. This will definitely foul the ultra-high vacuum inside the accelerator, leading to secondary radiation production as the beam collides with gas inside the accelerator. Some sensitive facilities requiring ultra-high vacuum such as some RF cavities may “crash” and become ruined. Further systems may become damaged or destroyed from either physical stress or the radiation pulse. If the beam bunches contain a lot of energy and maybe are heavy particles, the deposition may lead to an energetic local explosion. Alternatively the beam may escape and cut into structure beyond the particle accelerator itself, damaging and destroying equipment and seriously hurting or killing personnel.

In the worst cases, a beam excursion on a very powerful beam may lead to an explosion that destroys the accelerator utterly. Even in a less bad case, the radiation pulse might damage or destroy critical equipment around the accelerator and take the entire facility offline.

=== Beam dumping ===
If any other systems fail, the accelerator probably has to be shut down. This poses a problem - what do you do with the beam energy that is still in the accelerator system?

Sometimes, the beam is designed to leave the accelerator immediately, and shutting down the beam is easy. But in the case of cyclotrons and storage rings, the beam may circulate for a while. What do you do then? You have to dump the beam from the accelerator. Special beam stops are used on research and industrial accelerators, large blocks of dense material able to absorb the beam. During this process the blocks both heat up and become radioactive. For particle beams storing very high energies, this may become an increasing challenge. How do you safely dumb a beam containing many GJ or more? In some cases you may have no other choice but to dump the beam out into the environment. Where and when can you do this safely without threatening third parties?

= Application areas =
(This is a skeletal overview summary)
Particle accelerators have many, many, many areas of application.

In science, the big particle accelerators like the LHC are common knowledge. But electron microscopes also use components of particle accelerators. Many particle accelerators worldwide are used to produce high-energy x-rays in synchrotrons or linear accelerators, imaging samples for material science, chemistry, biochemistry and other areas.

In manufacture, particle beams can be used to cut material, but also weld material. Precision beams can break down gaseous feedstocks to deposit metal in micrometer-sized structures. Directed ion guns are used to implant traces of elements into surfaces for the manufacture of electronic devices.

And in medicine, particle accelerators of ever-greater sophistication find increasing use in cancer treatment and beyond, destroying life-threatening tissues with ever-reduced doses to surrounding healthy tissue.

= Further material =

= Sources =

== Authors ==
Sevoris Doe, Gerrit Bruhaug, Luke Campbell.

== References ==

Particle Accelerators

2022-08-17T12:48:37Z

Sevoris: /* Safety */

{{PageConstructionNotice}}

Particle accelerators are an incredibly useful piece of technology and a diverse set of artifacts produced for many purposes. While particle accelerators are often invoked as weapons of war in science fiction, and have captured the public attention as giant research systems the size of cities, they have an incredible widespread range of applications. There are also many, many ways to build a particle accelerator, depending on what effects you need to achieve and what circumstances you work under.

This article aims to be a general primer on particle accelerators, their physics and a selection of technologies and construction styles. In detail, all particle accelerators are high-tech engineering customized to their circumstances in many, many aspects.

Further pages covering application fields are in the work.

= Accelerator physics =
The simple explanation is that we use electric fields to accelerate charged particles. But from this basic principle of leveraging the electromagnetic forces springs an entire set of accelerator technologies with different performances, trade-offs and characteristics. The most basic particle accelerators are just a big high-voltage source and can still be found in old CRT TV’s and dentist x-ray machines around the world. Things rapidly get more complex from there, but first we need to lay out some terms so we can all have a common conversation.

The first thing to keep in mind with particle accelerators is that we are talking about charged beams here, even if we neutralize them later. So no neutrons, photons, or other innately neutral particles. Your choices range from the simple electron up to anti-uranium and more, but they all must be charged. The particle of choice determines just about everything else about the accelerator and any potential uses. Really heavy particles (like large ions) don’t accelerate easily and take a lot of energy to get relativistic enough for time dilation tricks to take into place. Really light particles (like electrons) accelerate super easily, but can have other radiation problems to worry about.

The second most important thing to know about a given accelerator is the particle energy. This is a measurement of the kinetic energy of any individual electron, ion, whatever in the beam and is measured in a unit called the electron-volt or eV for short. The lower bound for any worthwhile particle accelerator is in the MeV (or million eV) range and we typically want a lot more! Inside of the accelerator we tend to talk about the electric field doing the accelerating (measured in Volts/meter). For singly charged particles (like electrons) this gets really to figure out the energy because of how the eV is defined. We just take the electric field (also called the accelerating gradient) and multiply by the length of the machine! Boom, now you have particle energy! For more complex particles you have to do some scaling based on mass and charge state.

Also a quick trick to help you with deciding if something is relativistic or not is to divide the particle energy <math>E</math> by the mass energy <math>M</math> of the accelerated particle (0.511 MeV for electrons/positrons and about 1 GeV for a proton) and add one. The resulting number is called the Lorentz gamma value and is used all over in accelerator engineering.
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>\gamma = 1+\frac{E}{M}</math></div>
Once you get a value much larger than 1, you have relativistic particles. Once that number is in the hundreds you get an ultra-relativistic beam. Figure 1 helps show the relationship between Lorentz gamma and speed of the particle.

[[File:LorentzGvsSped.svg|900px|thumb|center|Lorentz gamma versus fraction of the speed of light]]

The third most important feature of a particle accelerator is the emittance which is just a fancy term for divergence of the beam and can be thought of similar to the wavelength of lasers. No beam is made perfect and they will naturally want to expand outwards. The smaller the emittance, the longer the beam will stay in a tight spot. Emittance can be altered with a wide variety of beam engineering techniques that take entire books to discuss, but one other cheating thing you can do is drive the beam energy higher. This naturally lowers the emittance by a factor of Lorentz gamma due to the cheating effect of special relativity! Thank god for Einstein!

== Acceleration forces ==
Every accelerator tries to build an electric gradient appropriate for the charge of the particles it is accelerating. The greater the electric gradient, the bigger the amount of work that can be done, the more energy is imparted on a particle as it travels through the gradient, and thus the higher the acceleration. The simplest way to build up a gradient is simply two charged electrodes with a strong potential between them. Many other accelerator methods use electromagnetic fields, in the form of waves confined in electrically conductive structures. In this case we exploit that electromagnetic fields have both an electric and a magnetic component. The magnetic component cannot be used to accelerate particles (only bend their trajectory - which can be exploited for effects like deflection or focusing) but as the electromagnetic waves oscillate, so does an electric field. This naturally creates a charge difference across half of the phase of an electromagnetic wave - this electrical gradient can accelerate particles. Both of these technologies encounter problems with the physical materials asked to contain the electric or electromagnetic energy. Eventually the fields get so intense that solid materials get broken down and an arc is formed. As such, various advanced concepts use clouds of charged particles and plasmas, which are not subject to the material limitations of solid matter.

The strong and weak nuclear force as well as gravity (which are all of the forces we know of at the time of writing) are not suitable for accelerating particles. This is related to the power and range of these forces. The strong and weak nuclear force have extremely short ranges - even if we had a means to artificially shape them like with electromagnets, their range would be too short. Gravity has the range but is incredibly weak. To generate a measly 1 G of acceleration, 9.81 m/s², requires the mass-energy of the entire earth, some 5.97e24 kg of mass, to do the job! The acceleration involved in particle accelerators is much, much, much larger (on the order of 10^13 times larger or more).

{{MascotSays|Interestingly, one source of very high-energy particles are giant gravity wells. Black holes generate conditions around them that can eject particles with enormous energies. The jets of particles that shoot from the north and south poles of many such stellar entities are particle accelerators.}}

If you want to use a force other than electromagnetism to accelerate particles, it’ll have to be a fictional one. The forcefields of various sci-fi properties could be considered, since they can often apply forces to particles. If you propagate a wave of such forcefields down a tube, you could have something quite a lot like many particle accelerator systems!

== The particle bunch ==
Essentially all high energy particle beams are not constant streams of particles, but a time sequence of discrete groups of particles, commonly referred to as “bunches”. These bunches can be described both in their distribution in lateral space to the traveling direction (x and y axis) and longitude (z axis) along the traveling direction, anchored on an origin point (0,0,0) that moves in time.

[[File:ParticleBunch.png|center|900px]]

Bunches can have various profiles. Generally they are somewhat distributed in the lateral direction just because the charged particles have slightly different starting positions and lateral velocities (also called transverse emittance). In longitudinal space, differing acceleration experienced by individual particles as well as dynamics within the bunch scatter them out, with “noses” and “tails” that contain fewer particles. The end results is that most beam profiles follow a gaussian distribution, though there are exceptions.

This bunching behavior allows us to analyze other properties of particle beams very similar to lasers. There is a “pulse frequency” given in Hertz. Each bunch contains a certain energy, which can be large enough that we give it in Joules and not electronvolts. There is an average output energy over a given timespan, given in Watts. We have a beam peak power, given by how quickly a bunch arrives at a target and deposits its energy, which we give in Watts. Related to this we also have a related peak intensity depending on how much area/volume the beam is deposited into, with units of either W/m² or W/m³. If we have charged particles being accelerated, we get a beam voltage, the same way as in a wire. Beams can also have a temperature, the same way as any other group of particles with random motion from their energy, given in Kelvin or eV.

The systems that generate particle beams of course have efficiencies, with the wall plug efficiency (power inserted into the entire system versus power deposited in the particle beam) being the most interesting for first-order engineering. They also have an operating temperature.

== Particles for acceleration ==
With the constraint that our particles must have an electric charge in order to be accelerated with anything but fantastic technology, we can discuss quite a large family of particles, from the conventional to the exotic. They differ in the applications they can be interesting for. Some can be similar, but no two particles behave entirely alike.

=== Electrons ===
The ubiquitous electron is the most common particle to be accelerated and the first one ever pushed up near the speed of light by humans. Due to its excellent charge to mass ratio (the best possible due to the extremely low mass and charge of 1e) electrons are very easy to get up to relativistic speeds. Electron accelerators for radiography and sterilization are often portable by a single person and not much larger than a laptop in size (although they can be much bigger for power output reasons). The ease in which electrons reach relativistic speeds means that electron RF accelerators can be designed with no speed changes assumed (since the electrons are already moving at effectively light speed) and thus tend to be highly efficient and have the highest acceleration gradients achievable. For similar reasons electrons are also able to be accelerated in plasma based accelerators, allowing for even higher acceleration gradients and smaller machines for a given energy. The major downside of electrons is that due to their large charge to mass ratio they will radiate energy like crazy when bent by a magnetic field. Thus ring shaped accelerators are of limited use for electrons since you have to make the ring extremely large to not just lose all of your energy as beams of x-rays coming out of the machine! This very trick is used to make high quality x-ray sources for industrial and research applications.

When electrons hit matter they tend to release their energy as photons in a process called bremsstrahlung, or braking radiation. These extremely high energy photons (roughly ~⅓ the peak energy of the electron on average) then proceed to make electron/positron pairs that then make photons that then make pairs that then make photons… This whole thing is a massive cascade of radiation that ionizes, heats and can even damage on a nuclear level! At high enough energies you will generate every particle known and might even make some new ones!

=== Proton ===
Protons are the next most common particle to be accelerated and are currently used by the absolute largest particle accelerator ever made (the Large Hadron Collider or LHC) as the primary particle of choice. Since they have ~2000X lower charge to mass ratio than electrons, protons are about ~2000X harder to get up to speed. Unlike electrons they are not promptly at light speed so we have to send protons through a series of various accelerators depending on the energy of choice. For example the protons in the LHC the protons go through 5 different particle accelerators plus the extremely low energy stage that first makes the beam of ions! This difficulty in reaching highly relativistic speeds has two other bad side effects. Firstly, if we want to transmit our protons through empty space we will probably need to neutralize them with a beam of electrons.

{{Note|''When shooting extremely high current (kA), relatively low energy (not highly relativistic or in some cases relativistic at all) beams in the atmosphere the higher mass and positive charge of protons gives them many beam propagation advantages over electrons if the beams can be made. In this case no neutralization is required as the atmosphere itself becomes a plasma and neutralizes the beam.''}}
Luckily this is not very hard and will add almost nothing to your comparatively large proton accelerator, but may increase your beam spread a little bit via some neutralization physics. This does mean your beam is technically plasma, which is cool! Second, we effectively cannot use those neat plasma based accelerators for protons since we would need nearly TeV class proton beams to inject into the plasma accelerator and by the time you reach those energies you have basically already done all of the work. Maybe in the future there will be ways around this, but as of right now we are limited! There is one major benefit to this poor charge to mass ratio though, and that is the lack of synchrotron radiation! Ring shaped accelerators are ideal for protons, and basically all ions.

When protons hit matter they can interact with all of the four fundamental forces, although only two of them (Strong Nuclear and Electromagnetism) tend to play big roles. Protons barrel on in causing huge ionization cascades and can also release bremsstrahlung photons at high enough energies. They are also ideal at knocking neutrons, protons, deuterons and more out of the target nuclei which just adds to the fun! Protons also come with a neat trick known as the Bragg peak, where they deposit a large portion of their energy right at the end of their trajectory through a target material. This is a very useful trick for medicine and means you can “tune” the beam to almost ignore large portions of material and primarily heat, ionize, and destroy the area of interest. You can imagine how useful this is for cancer therapy or more nefarious things!

=== Light ions ===
Light ions are a rather fuzzy term but here we will use the definition of any ion heavier than a proton but lighter cobalt. The fuzzy reasoning here is that you get on the other end of the nuclear binding curve (i.e. can’t get net energy from fusion) at iron, so cobalt will then no longer be “light” and has to be made in a different process. Feel free to pick your own cut off! Historically beams up to neon have been experimented with for nuclear fission, nuclear fusion and medical therapy applications. Light ions have even worse charge to mass ratios than protons (Z/A times worse) and typically you will find light ions have about ½ the charge to mass ratio of a proton on average, although the isotope 3He is a notable exception. This means they are even slower to accelerate, although often the exact same machines as protons can be used with minor tweaks. Light ions have similar requirements for neutralization as protons, although their heavier mass can help fight beam bloom for a similar beam speed (not energy since the increased mass means the particles are higher energy for a given speed). The decreased charge to mass ratio also means that synchrotron radiation is even lower so beams can be driven to higher speeds before radiating. Light ions have similar target interaction physics as protons, although the ion itself will break up at very low speeds and add to the radiation environment.

=== Heavy ions ===
Heavy ions are being defined here as cobalt up to whatever unstable element you can jam into your accelerator! Typically heavy ion beams are made of elements like lead (used in some runs on the LHC), gold (used in the Relativistic Heavy Ion Collider) and even californium (used in some runs on the Argonne Tandem Linac Accelerator System). These beams tend to be for pure nuclear physics research, but there has been a long standing interest in heavy ion beams for inertial confinement fusion research and as potential methods of propelling spacecraft. These extremely heavy ions will be even harder to accelerate to ultra high speeds than light ions and much slower for a given energy, but they do provide a lot of momentum to make up for that! They have the same neutralization requirements and same potential upsides for beam bloom as light ions, but will most likely be orders of magnitude slower reaching our targets than the lighter beams. Heavy ions can also be used in the same machines as protons with small modifications and generate almost no synchrotron radiation as a result of their much smaller charge to mass ratio. They also have similar interaction physics to protons and light ions, but will stop in much thinner layers of materials. While electron, proton and light ion beams tend to both mechanically destroy and irradiate targets through-out, heavy ion beams will act much more like lasers and burn in from the outside of the target on in. There will still be a spray of ionizing radiation, but far less of one than an equivalent energy electron, proton, or light ion beam. The much higher momentum per particle, lower radiation concerns and ease of deflection makes heavy ion beams very interesting for beamed propulsion purposes.

=== Macrons ===
Macrons are typically defined as macroscopic particles, such as extremely large molecules or even grains of sand, that are charged up and accelerated via similar technologies as traditional particle accelerators. There are many difficulties with getting macrons up to appreciable speeds due to the extremely low charge to mass ratio, engineering difficulties with getting consistent and good charging, phase matching with accelerating cavities and lack of effective focusing optics. There are so called “dust accelerators” in existence that utilize the first stage of ion accelerators to generate their “beams” and proposals have been made to create better versions using updated technology. Macrons will interact just like a very small and very fast bullet unless they are designed to carry a nuclear payload to boost this output. Antimatter is an assured option for getting energy gain from a macron impacting a target, although there are potentially options for fusion, fission and fission/fusion given the correct sizes (mm scale or bigger), clever target design and the right impact speeds (most likely >>100 km/sec). The various energy gains versus speeds can be seen here:

[[File:IvKmacron.svg|900px|thumb|center]]

It can be seen that antimatter proves to be worth it up near the speed of light, but both fission and fusion stop providing much gain well below the relativistic regime. This also helps explain why heavy ion uranium beams are not going to provide some incredible advantage over any other ion.

=== Antimatter ===

Antimatter is the evil cousin of normal matter and whenever the two meet a release of energy equivalent to their combined mass energy follows. Now this does sound like an amazing option for beams but there are two downsides. First, antimatter is energetically expensive to make! Positrons (antimatter electrons) are the “cheapest” at 1.02 MeV of input energy at a theoretical best, but they only give you that much energy back when hitting a target. Since a typical positron beam can be expected to be equivalent to an electron beam, there is little gain in adding 1.02 MeV per particle to a multiple GeV (or even TeV) beam. You can see this explicitly:

[[File:IvKhBarPostron.svg|900px|thumb|center]]

For anti-protons the math becomes more favorable, but creating these particles is much more difficult and energy consuming. At best anti-protons require ~2 GeV of input energy to create, but in practice >400 GeV per anti-proton has been needed. Luckily significant energy gain is had up to several GeV of beam energy, so if anti-protons can be made at a central facility of some kind and easily stored (neither being easy things) then there may be some advantage to such a beam. Heavier ions are significantly harder to make since fusion with antimatter must be performed to generate those ions. Barring a naturally occurring source being found, antimatter heavier than a proton is most likely not worth using.

{{Note|'' Positrons can potentially fill an interesting niche either as a co-propagating neutralizing beam for electrons or perhaps as a higher performance atmospheric beam than electrons due to the positive charge. In this case the annihilation properties are no what is of interest, rather the positive charge and relative ease of acceleration compared to ions. Since positrons share all properties with electrons except for charge, they can be accelerated just as easily and even co-accelerated with electrons in the same machine.}}

=== Muons ===
Muons are an interesting cousin of electrons and positrons. They are unstable and heavy cousins of electrons, massing about 200X more than electrons and living about 2 microseconds in the lab frame before decaying into an electron (or positron) and a neutrino. If we accelerate them up to relativistic speeds (about 10X easier than protons and the muons are typically born relativistically thanks to how we make them) then the muons can be made to “live” far longer. Muons have long been of interest to accelerator physicists due to their position as an almost ideal intermediary between electrons and protons. The requirement to make them on site has always been an impediment though and muons are currently only used for certain niche experiments. During the planning stages for a potential muon collider it was also noted that a muon collider would also have a serious issue with neutrino radiation! This is due to how neutrinos are more likely to interact with matter at high energies and a muon collider would be generating a LOT of really high energy neutrinos in a ring around the machine. This has been proposed as a sort of super weapon even!

Muons should interact with matter in a similar fashion to electron beams and the beams will probably be treated in a similar manner. They can propagate through the atmosphere quite well in theory and should outperform electron and proton beams for a given beam energy, although at quite a cost… Muons are typically generated with proton beams impacting targets and will create an equal number of positive and negative muons at high energies that must be carefully captured. This would allow for the easy creation of a neutral muon beam though, which may make for a great giant doom beam! Do note that negative muons also can help catalyze nuclear fusion so a very high power beam hitting a fusion fuel tank could produce some rather… Interesting results!

=== Other exotic options ===
There are a wide variety of odd particles that could also be chosen for use in beams given a specific need for them, although generation and utilization of these particles will be quite difficult! Extremely short lived particles like pions, kaons and taus can be generated and accelerated up to extreme speeds to help propagate them far enough to matter, although that will need some very extreme speeds. At the upper end of extreme there are even proposals for accelerating tiny charged blackholes!

= Accelerator technologies =
There are a wide variety of accelerator technologies and here we try to summarize them as best as possible. Different accelerator types may be sequenced one after another. Some particle accelerator technologies depend on the particles fulfilling certain requirements of speed (thus energy) and hence different systems may be employed in sequence. Accelerator systems are described in terms of their efficiency of turning input power into particle energy, their acceleration gradient measured in electric field which translates to volts per meter, and the mass per length meter. An interesting secondary parameter is their operating temperature. Some accelerators can operate well at high temperatures. Others require cold or cryogenic conditions to work well.

== Core accelerator assembly ==
We define the core accelerator assembly as the structure where energy is imparted onto the particles.

=== DC Accelerators ===
For very low beam energies we can utilize large DC electric fields between plates or grids to accelerate particles up to speed. This is the same technique as is used in ion drives and is the most common industrial accelerator used. Every dentist's office and old CRT television uses this acceleration technique. Ion accelerators often also use a DC accelerator to generate the initial beam of ions, while electron beams utilize other techniques. These machines are limited by material breakdown to about 0.5 MV/m accelerating gradient and typically no more than a couple of MeV total particle energy for extremely large machines. The highest energy DC accelerators can reach 40 MeV for proton beams, but this is for machines massing in the 100s of tons and at low total beam powers. Very large DC accelerators are currently the only tested method for accelerating macrons up to 10’s of km/sec.These accelerators are limited by the risk of arcs jumping between the electrodes at too high voltage differentials.

=== Radiofrequency Quadrupole ===
A more advanced first stage accelerator for ion beams is the radiofrequency quadrupole or RFQ. These machines utilize low frequency radiowaves and carefully designed conductive bar structures to efficiently generate high current ion beams up to energies of ~7 MeV in a couple of meters. RFQs were a revolution in ion accelerator physics and caused quite a stir when the USSR first showed them off as they provided a much higher power and higher quality ion source in a smaller package than anything the USA had built! RFQs have also been proposed as methods of efficiently accelerating macrons up to >100 km/sec. These machines can be normal or superconducting and can operate with efficiencies as high as RF LINACs in theory.

=== Cyclotrons and Betatrons ===
One of the earliest ways to accelerate particles up to relativistic speeds is with circular accelerators known as cyclotrons (for ions) and betatrons (for electrons). In Cyclotrons the accelerator is a round chamber which houses two half-circle electrodes roughly shaped like a D (and thus called “dees”) separated by a gap, sandwiched between the poles of a large electromagnet. The magnet creates a static magnetic field which bends the path of charged particles into a circular motion. An oscillating alternative voltage is applied to the electrodes (what we often call RF or radiofrequency) to accelerate the ions. This radiofrequency field’s oscillation is matched so that one cycle of the voltage matches one circuit of the particles. As a consequence when crossing the gap from one electrode to the other, the charged ions experience a voltage gradient and are accelerated. As the particles gain energy and speed, their radius in the magnetic field gets larger. The result is an overall spiraling trajectory from an injection point in the center to an exit point at the rim where the charged particles are bent out of the cyclotron. Cyclotrons use the same voltage gradient many times over, defeating the electrostatic breakdown that limits the single-time electrostatic accelerators. As a consequence, proton energies of up to 70 MeV may be reached. Cyclotrons energies can also be increased by switching from normal magnets and RF resonators to superconductive components. This eliminates various losses from electric resistance and absorption and also allows for much stronger bending magnets, which allows the cyclotron to contain particles up to a higher energy if desired. Superconducting cyclotrons have achieved up to 520 MeV. Still, cyclotrons suffer from poor beam current, pulse rate, and beam quality issues that make them a less than ideal choice. The large magnet required (measured in meters) also does not lend itself to a lightweight weapon and as such, cyclotrons are regulated to niche medical and physics uses.

A close cousin is the betatron, which is used to accelerate electrons. In this case the magnetic field strength is increased as the electrons speed up, which prevents any spiraling trajectories of the particles. Betatrons have been operated up to electrons energies of 300 MeV, but are more typically in the ~10 MeV range. In principle a betatron could also be made superconducting, although magnet ramping may prove to be difficult. Other types of ring shaped accelerators may prove to be more useful, although electrons are heavily limited by synchrotron radiation and it is often better to use linear accelerators for them. For now betatrons remain limited to small (sometimes portable) accelerators for nuclear physics, industry and security work.

=== Radiofrequency Linear Accelerator ===
The real work-horse of the particle accelerator world is the RF linear accelerator. These babies use the same alternating field trick that cyclotrons use, but in a straight line rather than needing to bend the beam around in a circle. This provides a couple of nice advantages over the more traditional cyclotron. First and foremost you get to leave that large, bulky magnet behind which saves you a ton of mass, cooling and power consumption! You also get to now have much higher RF frequencies (1-10 GHz right now compared to the MHz used in cyclotrons), which means higher accelerating gradients. To generate these powerful, high frequency RF waves (also known as microwaves) that drive the accelerator we typically use military radar power supplies or (in more advanced cases) even lower energy particle accelerators! State of the art RF LINAC technology is well into the 100 MV/m range and have no upper limit on beam energy! The Stanford Linear Accelerator reaches 50 GeV and was built back in the 1960’s! There have been traditional RF LINAC designs for machines reaching up to 1 TeV, with the only limit being how much space you have to put these monsters. They do operate best (i.e. with high accelerating gradients and efficiencies) with particles already moving at relativistic speeds, so any ions fed in need to use a series of lower gradients LINACs or other pre-accelerators. Electrons and positrons can naturally be used in LINACs since they accelerate up to relativistic speeds so quickly.

LINACs also have been shown to operate at high average powers (MW scale is now in use) and high pulse energy (10’s of kJ pulses fired at high rep rate) while also maintaining a high quality beam that stays well focused. The two newest advancements have been to push the frequency ever higher for higher acceleration gradients (300+ MV/m has been shown and there are plans for even more) and to switch to superconducting RF accelerating cavities (the name of the actual piece of equipment that does the accelerating) which then dramatically increases the electrical to beam efficiency (80-90% has been shown and even copper waveguides have been operated at 70%). All in all, LINACs beat the tar out of cyclotrons which is why we don’t build cyclotrons anymore. To quote an old Fermilab engineer “cyclotrons are an expensive and bulky way to get a really crappy beam or a cheap way to get a giant chunk of lightly radioactive steel that can never leave your lab”.

LINACs work in detail by shaping the radiofrequency energy in a confining waveguide. The oscillations of the RF energy may be held in space, or allowed to travel down the waveguide over time. Radiofrequency waves have connected magnetic and electric components, which oscillate.

[[File:2022 08 16 GalLib ElectromagneticWave.svg|center|800px]]

Inside the RF accelerator, the electromagnetic radiation is oriented 90 degrees to the particle travel direction, which points the associated electrical field in the travel direction of the particles.The radiofrequency field naturally oscilates along its wave, and the connected electrical and magnetic fields oscilate as well: building up, reaching a peak, and diminishing, inversing, building up, reaching a peak, and diminishing, and the cycle repeats. The detailed behavior depends on the specific cavity.

[[File:CavityWithEM.png|600px|thumb|A resonant RF acceleration cavity and the electric field inside of it, created by resonating RF fields, at one moment in time.]]

Particles of the right charge in the right space at the right time experience an acceleration from the electrical field, being drawn to the opposite electric charge. If they land in the other region, they are instead decelerated. Inbetween, particles experience less to no acceleration. This behavior naturally separates a stream of particles into discrete '''bunches'''.

[[File:CavityWithEMAndParticles.svg|600px|thumb|Bunches being accelerated inside the electric field. ]]

It also creates an implicit limit to what can be effectively accelerated using a specific radiofrequency accelerator design. The particles driven through it have to keep pace with the oscillations of the electric field, reaching the next correct region inside the cavity at that time when the electric field is oriented correctly. If they are too slow to keep inside the accelerating regions, the accelerating action will be ineffective. This requires RF cavities to be matched to the particle's velocity gain under energy imparture and their velocity when entering, being inside, and exiting the accelerator. For electrons and positrons, this isn’t that complicated. Since these lightweight particles accelerate readily, they have few problems “staying in the pocket”. Particles that are slower to gain speed, such as ions and even larger particles, must however be accomodated. This in turn impacts how much acceleration and energy per meter can be fitted into the accelerator in the first place. The acceleration of heavy particles requires specifically designed and sequenced RF accelerators for specific regions of particle speed. This in turn complicates the design of the larger accelerator infrastructure, with multiple discrete stages getting the particles to the energy required so that the next stage in sequence can take over.

RF cavity accelerators are limited in their eV/m capability by the critical current at the containing waveguide. To increase the acceleration provided, more intense electrical fields must be built up. Eventually the electrical charge of the RF waves becomes so strong it can rip electrons out of the cavity walls, causing an arc to form and the cavity material to break down physically.

=== Advanced Accelerator technologies ===
Present accelerator technology caps out at around 250 MeV/m for the most overdriven designs. This poses issues for a variety of applications, where higher energies in shorter distances are desired. In science, "desktop" and "chipset" accelerators are of interest. When militarizing particle beams, some effects only become available at very high energies - but you want something compact enough to turret.

urrent and even near future accelerator isn’t really ready to supply that sort of scale quite yet, so we have to move on to the cutting edge of accelerator physics. There are three big options with sub technologies within these broad fields. We have extreme RF cavities using much higher frequencies, plasma accelerators using lasers for power and plasma accelerators using particle beams for power. Each technology provides a unique series of advantages and disadvantages, but fundamentally they are all a way to make linear accelerators that are much smaller.

==== Dielectric Wakefield Accelerator ====
LINACs using much higher frequencies than traditional LINACs have two fundamental problems to fight. The first is electrical breakdown and the second is generation of those mm-wave (10-100 GHz) or even THz (100-1000+ GHz) sources at sufficient power and efficiency to power a particle accelerator. As the frequency of the driving power goes up the electric field climbs up, which means the accelerating gradient gets larger (which is good) but also means that we run a higher risk of a huge arc forming in the accelerator and breaking it (which is bad). This is often fought by changing the accelerator cavity material from a metal (or superconductor) to a dielectric material like ceramic, or by using a coating of dielectric, which can sometimes earn them the moniker of Dielectric Wake Field Accelerator (DWFA). The issue of creating powerful and efficient enough sources of this higher frequency drive power is being actively pursued for a variety of fields (5G cell signals, beamed power, etc) but is still quite immature compared to the microwave RF sources of traditional LINACs. There are options using essentially "over-clocked" microwave sources, low energy particle beams, and even laser-plasmas. There is still a lot to be done here, but these sorts of problems are being actively worked on there are working accelerators using this technology. There is potential for GV/m or greater acceleration gradiatens and all of the same advantages of traditional RF LINACs (i.e. high powers, high fire rates, high energy pulses, quality beams) should stay the same. The wall plug efficiency is liable to be a bit lower, but that will be highly dependent on the efficiency of the source of these high frequency waves.

==== Laser-Plasma Accelerator ====
Plasma accelerators using lasers for power are a whole world in and of themselves. There are a wide variety of ways to accelerate particles with lasers and plasmas, but we will focus on the most studied technique known as Laser Plasma Accelerators (LPA). LPA accelerators utilize ultrashort (<1psec) and ultraintense (>10^18 W/cm^2) lasers to drive a plasma wake (thus sometimes earning them the name laser wakefield accelerator) and accelerate particles through the resulting charge differential. They have only shown the ability to accelerate electrons and positrons, but any ions that could be injected at the right time and at extremely relativistic speeds could conceivably be accelerated this way. In the operation of a typical LPA the ultraintense laser is fired into a jet of gas (turning it into plasma) or a channel of lightly pre-ionized plasma. The laser then blasts through this plasma like a boat through water and leaves a void behind it (the wake). This void ends up having a HUGE accelerating gradient (100 GV/m) and suffers no risk of electrical breakdown since the plasma is already broken down! A pre-accelerated bunch of particles can be injected into that void at the right time (called the witness bunch) for a BIG acceleration or you can rely on the wake itself spewing electrons out into the void and make a beam out of the plasma itself. The gas or plasma then quickly is reformed and the laser fires as soon as its cooling and charging allows. Keep in mind that even though the beams can hit incredible energies in short distances, the laser itself is not a pocket sized device. These lasers are room sized at minimum and not easy to shrink. The pulse lengths required are also much shorter than most of our hypothetical weapons-grade pulse lasers that may operate in the nanoseconds regime - dual-use may prove difficult. The intensities reached also pose significant issues for the laser hardware. If you don’t spread the energies out, it becomes quite possible you will break parts of your laser from the produced intensities.

Current LPA systems suffer from a couple of problems that are being actively worked on. First and foremost they are incredibly efficiency limited by the laser tech being used. There are plans to move on to much more efficient fiber lasers and eventually make LPA machines that operate in the 25% or greater wall plug efficiency, but that is not currently doable. These lasers also need to be fired at higher rates than the current ~1 Hz, but that is much more of a limit of laser engineering than accelerating engineering. LPA’s also have yet to link more than one accelerator together and the lasers are limited in just how much plasma they can accelerate through right now (~20 cm max) so the beam energy runs into a wall quite quickly. Aside from R&D on linking LPA’s together in a way similar to RF LINACs there are two other options. The acceleration gradient is higher the denser the plasma, so a move to denser plasmas and shorter wavelength lasers (to penetrate the plasma) could provide higher beam energies in similar distances. The other option is to use clever optical techniques like the so-called “flying focus” to allow for the laser to go through much longer plasmas. This could in theory allow for more efficient LWFA’s and single stage machines that provide ~1 TeV electron beams in meters! The final issue is beam quality. Even the best LPAs will not make beams that have as tight of a beam spot, energy spread, or emittance as a normal RF LINAC. This means they will end up needing some beam “cleaning” if you want to get a lot of range out of them.

LPAs are of engineering interest for very short wavelength X-ray free-electron lasers in a compact format. Currently these facilities require linear electron accelerators many kilometers in length. With an LPA as the electron accelerator, an XFEL able to image molecular structures and chemical reactions as they happen might be shrunken down to a much more compact format. (Sometimes this is discussed under the ultimate promise of the “benchtop” or even “chip-format”(sic!) XFEL)

==== Beam-Plasma Accelerator ====
Plasma accelerators using particle beams for power are very similar to LPAs in principle but use a particle beam rather than a laser. We typically just call these things Beam Plasma Accelerators (BPA) for short. The concept is very similar to the LPA in that a high intensity beam comes in and blasts a void into a plasma, which particles can then be accelerated in. BPA provides some interesting advantages over LPA in that you can use highly efficient superconducting RF LINACs (or synchrotrons if you want) to provide the particles at very high efficiency. They also have little in the way of plasma density limits and the accelerated beam takes on the characteristics of the beam driving the BPA. Thus little if any beam cleaning will be needed after acceleration. The big downside is the need for a relativistic particle beam to drive the effect, and typically one in the 100’s of MeV to low GeV range, depending on the particle used. Thus your final accelerator may be short, but you still won’t be making a handheld particle beam device.

== Accelerator Shapes ==
The accelerator technologies discussed above are the means by which we create situations for particles to gain energy. The accelerators become embedded in a larger structure that produces, directs, accelerates, even stores the particles until they are directed towards their final use.

There are two principle shapes for accelerator assemblies. Ones where the beampath is not closed onto itself, and designs where the beam path is closed on itself in part of the assembly, forming a loop.

=== Linear accelerator ===
Everything from electron generation to acceleration to conditioning is built in a straight line, one after another. This is the Linear Accelerator or LINAC. The main catch here is that there is only one single passage for the particles through the entire accelerator. Re-using accelerator hardware to drive the same particle to higher energies by repeated passage is out. The particles have to reach their desired energy in a single pass. Practically this puts limitations on the energy achievable with heavy particles in a LINAC. The length required gets impractically large. Even for electrons, lengths of hundreds of meters to kilometers may be required, unless advanced accelerators with high gradients are used. This said, every accelerator contains linear sections. Since so much high-performance accelerator technology depends on particles having a minimum velocity, ions must be “pre-conditioned” in a set of linear accelerators before the re-use of an accelerator through looped beam paths can be considered.

LINACs of low energy may be used in many applications. Medicinal they find use in radiation therapy and radiosurgery.

=== Synchrotrons ===
The other shape which has some interesting properties are accelerators where beampaths are looped onto themselves. These are called synchrotrons. Since beam paths are closed onto themselves, particles can pass through the same accelerator unit repeatedly, gaining more energy on every passage. This is advantageous for heavy particles which do not gain much speed on any single passage. They can pass through the same accelerator thousands or even millions of times instead. This is obviously advantageous for facility bulk and mass and allows us to reach energies that would otherwise be impractical. The accelerated beam bunches can also be held in storage, looping like the rotating mass of a flywheel, holding energy. The LHC can store some 700 Megajoules of energy in the recirculating beam after full speed has been reached.

That said, the devil is in the details. When charged particles are bent in circles they emit electromagnetic radiation which scales very rapidly (to the fourth power!) with relativistic gamma factor and with bending radius (inversely to the second power). This phenomenon is called synchrotron radiation. Due to the scaling, high energy particles emit more synchrotron radiation than lower energy ones and lighter particles (like electrons) at those high energies emit much more than heavier particles. In addition, the bigger you can make the bending radius of the synchrotron (often called “the ring”) the less synchrotron radiation you get.

[[File:SynchrVsE.svg|800px|center|thumb|Synchrotron radiation per particle versus particle energy for electrons and protons in a 50 meter in radius ring.]]

[[File:SynchroVsRing.svg|800px|center|thumb|Synchrotron radiation emission per particle for different particles at different energies, over varying bending radii. ]]

Although these numbers look small, keep in mind that this is PER PARTICLE. A typical pulse in an electron accelerator will have more than more than a billion electrons in it! Large synchrotrons like the LHC will have over 300 trillion ions circulating through it! Synchrotron radiation necessarily has to be managed to reduce radiation shielding, the radiation load (and possibility of hardware failure) on the accelerator, and to reach high energies in the first place - you might lose all the energy your accelerators are adding to synchrotron radiation!

Deflecting the beam also requires the application of magnetic fields, and higher degrees of deflection, in a smaller length, at higher particle energies, require stronger and stronger magnetic fields. Eventually multi-Tesla fields from superconductors will be required. And due to action and reaction, those magnets need to be braced against the pressure the beam exerts as it is deflected. Ultimately this leads to the same limitations as with flywheels - the mechanical strength of the braces and static connections that keep the magnets together as a larger structure is the accelerator’s limitation on J/kg held in storage. A further complication: The magnet strength has to be matched to particle energy. As a beam is accelerated, the deflectors have to ramp up their power to keep the degree of deflection constant.

Sometimes, synchrotron radiation from electrons is intentionally induced in a ring-shaped accelerator. Since synchrotron radiation of high-energy electrons tends to be hard x-rays emitted along narrow vectors, it’s a way to get x-rays used in various imaging techniques in material science and molecular physics.

Synchrotrons used for ions also need to respect the limitations of the involved accelerators. Synchrotrons often use radiofrequency accelerators. Since radiofrequency accelerators must match the speed and speed gain of their particles, an ion synchrotron cannot be made up of a single accelerator loop. Instead there are sequential loops connected in sequence, each designed to boost the particle through a specific velocity and energy region so that the next accelerator takes over. A designed example is the LHC. Particles headed for the main LHC actually circulate in two other synchrotrons beforehand, gaining sufficient energy so that the LHCs RF cavities can take over. (Also, before being injected into the first synchrotron ring, the ions must first go through multiple linear accelerators using different technologies to reach a base energy.)

An interesting design consideration: Synchrotrons don’t necessarily have to take the form of a (near-ideal) ring. Funamentally any dipole represents a single point of deflection of a straight path - they're polygons! Any geometric shape that contains a loop, and where any single bending curve keeps within the limits of the synchrotron radiation you are willing to tolerate, works. As such, complicated two- or three-dimensional “folded-up” shapes are also possible formats for a synchrotron.

=== Storage Rings ===
Since synchrotrons have to match their deflection magnet strength to the energy of the particles being accelerated, a single synchrotron track cannot hold more than one small section of beam bunches in storage at a time, and becomes unavailable to accelerate further bunches until it discharges the stored beam. This is disadvantageous for various applications where we may want to store a fully accelerated particle beam “bolt” for a longer time.

Storage rings are separate synchrotrons designed to hold multiple sets of bunches in circulation at a target energy at a time. The bunches are delivered from an accelerating synchrotron (sometimes called a booster ring) and held in circulation in the storage ring. Since there are synchrotron radiation losses, a few accelerator sections must compensate for these.

Storage rings can also be used to condition a stored beam. A common thing to do in storage rings is to “cool” a beam, draining unwanted movement energy out of the particles. Beam cooling may take significant time, which makes it ideal to conduct in a storage ring. The result is a beam with less tendency to drift apart, advantageous for many applications.

== Other parts ==
=== Particle sources ===
Particle sources are assemblies which supply particles to accelerators. Here we can group them into three categories: electron sources, ion sources, and exotic particle sources. The later involve the production of particles by means of gathering them from collision reactions (or the handwavium blackbox of your choice) and feeding them into another accelerator for further use. Positrons for example may be manufactured in beam collisions.

Electron sources require the production of a free stream of electrons. The oldest method here are thermionic sources. A hot piece of material ejects free electrons into a vacuum, where they can be drawn in using electric differentials and forwarded to the accelerator.

Field emission electron sources instead use a strong potential between a needle-like cathode and an anode plate with a hole. The electrons are drawn from the needle and through the anode due to the high charge differential. This method produces a “cooler” beam with less random divergence than thermionic sources.

A further improvement still are photoemission sources. Here the precise application of laser light onto a photosensitive material energizes electrons in the photocathode until they break free. This process is thermodynamically very precise and due to the nature of the laser light, also very controllable. The result is a low-emittance stream of electrons. These may be accelerated using electrostatic attraction with an anode at one end of the vacuum chamber, or directly using an RF field.

Ion beam sources are more complicated. The physics of ion formation are complex, which opens many design choices when creating an ion beam. Generally a plasma must be produced, from which the ions can be extracted.

The plasma can be formed in a variety of ways. Electrons from a local cathode-anode array, plasma frequency heating, the injection of electron beams, the application of laser light. The end results can be as diverse. Some methods only produce weakly charged ions and on many vectors, so with high emittance. Other methods offer greater ion charges or reduced emittance.

=== Magnets ===
While acceleration of charged particles is effected with electric fields, magnetic fields can be used to bend the trajectories of particle beams. As such, electromagnets can be used to bend the trajectories of charged particles in various ways, and in a controllable fashion.

==== Dipole Magnets ====
Dipole electromagnet are used to steer the entire particle beam in one axis. (To steer in two axes, we must either rotate one dipole about the beam axis, or couple two dipoles in sequence.) They can be used to direct a beam inside a particle accelerator, inject and extract beams out of looping constructs by switching on and off at the right moment, and point a beam at a target.

In a dipole magnet, a positive and negative coil oppose each other, embedded in a “choke” assembly often made of steel. Charged particles moving through the magnetic field between the coils experience a deflection in one direction specified by the polarization of the field, and a degree of deflection specific to the field strength.

==== Quadrupole Magnet ====
Dipoles can steer, but they influence all particles equally. As such, they cannot be used to manipulate the beam's shape. The first type of magnet able to influence a particle beam’s shape are quadrupole magnets.

They are assembled from four coils at 90 degree angles, with coils of the same polarity opposing each other. The passing beam is compressed on one axis, and expanded on the other. To focus the beam overall, at least two quadpoles must be arrayed in sequence.

Quadrupoles are the basic and essential “magnetic lens” of particle beams. Without them, charged beams would inevitably defocus and run havoc.

==== Hexapole Magnets ====
Hexapole magnets exist to address one problem of quadrupoles - they focus differently depending on the energy of the particle passing through the magnetic field. The effect is likened by particle accelerator physicists to the different diffraction of light in a prism - and hence the effect is called “chromatic aberration”. Hexapoles are used to correct for this, stronger deflecting higher-energy beams.

==== Integrated optics ====
As the field of particle accelerator science and engineering advances, more complex designs are proposed to handle more complex phenomena inside accelerators. An upcoming field is “integrated optics” - complex magnetic systems with nonlinear behaviors. These could provide more advanced manipulation of electron beams, building upon existing capabilities and adding new ones for shaping a beam.

=== Plasma lenses ===
Plasma lenses are an up-and-coming technology, where very high currents in plasma are used to create very strong magnetic fields. Similar to plasma-based accelerator technology, the fact that charges are already free in plasma is used to circumvent conventional charge breakdown and critical current limitations on solid-matter devices. The resulting magnetic fields can be much stronger and bend beams over shorter distances. Furthermore, by creating complicated profiles in the plasma, elaborate beam manipulations could be performed, more akin to sophisticated optical elements. As well, the charges in the plasma neutralize the particle beam in the region of the plasma lense - affecting its propagation behavior further and in a positive manner, as self-repulsion is neutralized.

The catch is that this is plasma - that means you are dealing with all the complications of plasma physics. Heating of the plasma has been an early challenge to successfully forming and maintaining plasma lenses.

Plasma lenses require an appropriate chamber, possibly active containment to keep the plasma from losing energy to the walls or leaking into other accelerator parts, and the necessary plasma sources, maintenance heating devices and then lense shapers and control sensors. In high-powered plasma lenses there will also be a need to cool the plasma as it heats up from waste heat. Potentially we may manage this by open-cycle cooling - venting the plasma controlled in one direction into open space, rather than have a closed-cycle solution for reconditioning the plasma. This could cause some very cool visuals!

Plasma lenses for particle beams should not be confused with optical plasma lenses for lasers! Those exist as well, but are their own topic.

=== Chicanes ===
Chicanes are assemblies of multiple dipole magnets used to manage the longitudinal dispersion of a particle beam due to different particle energies. Since particles with higher energy move faster, they drift ahead in the beam distribution over time; and particles with less energy increasingly lag behind.

Chicanes deflect high-energy particles onto an outward curving trajectory. Since traversing the curve takes longer, the high-energy particles are effectively delayed, allowing lower-energy particles to catch up. By the end of the chicane assembly, the bunch has been compressed in the longitudinal direction.

Chicanes however, do not remove energy from too-fast particles. That is the responsibility of other structures.

=== Undulators and wigglers ===
Undulators and wigglers are magnetic assemblies used to transform particle energy into electromagnetic emissions, with different properties from the specific implementation of a shared basic principle.

Assembled from stacks of generally dipole magnets (other configurations exist), both undulators and wigglers force charged particles onto “wiggling” trajectories by bending their path left and right. Wigglers force very strong arcs; undulators less so. In both cases, the rapid deflection of the charged particles causes them to shed parts of their energy as electromagnetic emissions. Due to the energy of the particles, these photons are generally highly energetic x-rays.

Undulators may be used to build free-electron laser systems.

They can also be used as a means of cooling the beam. The removal of energy from particles can also suppress unwanted movement modes, allowing accelerator structures to re-add energy only to desired movement modes. Of course, this means the particle beam actively loses energy, and produces x-rays that have to be absorbed (and cooled away.)

=== Beam coolers ===
Beam coolers are a collection of systems and strategies that are used to remove unwanted movements from a particle beam. Undulators and wigglers are one way to do this.

One method set of beam cooling relies on finding particles who are very “hot” with random motion, and either removing the particles from the beam using a kick device, or applying a precise counter-force that acts against the unwanted motion.

If one is dealing with ions, one clever trick is to inject a beam of electrons. Due to the thermodynamics within the particle population, the lighter electrons absorb heat from the heavier ions, taking this unwanted motion energy with them when the electron beam is bent out.

As well, lasers can be used (what can’t lasers be used for!) to apply counter-force on ions and remove unwanted movement in this manner.

Beam coolers are generally not instantaneous devices. They work over time to make a beam better. Often they involve the removal of energy or particles from the stream, requiring accelerators and sources to make up the loss and creating waste heat.

=== Beam recovery ===
Some accelerator systems can actually be run in reverse - they can turn beam energy back into some other energy format and then electricity. This can be used to recover some to most of the energy from a particle beam before disposing of it. This is advantageous both for your energy bills and making a very high-energy beam liable to spray radiation everywhere, easier to absorb in a beam dump.

=== Beam dump ===
A beam dump is a construction where a particle beam can be routed in case it has to be disposed. Beam dumps are designed to safely absorb the particle's energy without leakage of dangeorus radiation or activated materials to the outside enviroment. Common materials are carbon and metals, embedded in an actively cooled assembly.

= Propagation physics =
==Magnetic fields==

Magnetic fields are things of motion. They are created by moving electric charges. And they only affect electric charges that are moving. An electric charge at rest in the presence of only a strong magnetic field will just sit there. But when it starts moving, the magnetic field interacts with it, starts deflecting it.

But not along the field. Oh no. The magnetic field does not point in the direction it will move the charge. No, the charge experiences a force that is perpendicular to the direction of the field. And also that is perpendicular to its own velocity. A charged particle moving straight down a magnetic field line will not experience any force.

On the other hand, if the particle is moving directly perpendicular to the field, the magnetic force will always push it to the side, even as it keeps turning. This results in the particle moving in a circle. The radius of this circle is called the <b>gyroradius</b>, and the rate at which the particle makes circles is called the <b>gyrofrequency</b> (or sometimes <b>cyclotron frequency</b>).
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Circular_motion.png|frameless]]
<td width=255>[[File:magnetic_forces.png|frameless]]
<td width=255>[[File:spiral_around_field_lines.png|frameless]]
<tr>
<td>To get circular motion, you need a force (blue) that is always the same amount but always pushing perpendicular to the direction of motion, toward the center of the circle. A charged particle in a constant magnetic field moving perpendicular to the magnetic field meets these conditions, and moves in a circular path.
<td>A diagram of how a magnetic field (magenta) exerts forces (blue) on a moving particle (with velocity shown in green).
<td>If a particle is moving partially along the magnetic field (magenta), it will drift down the field lines while spiraling around it (black path). Where the magnetic field is stronger (indicated by having the field lines closer together) it will execute tighter, faster circles in its spiral and will drift more slowly than when the field is weaker.
</table>

It can be convenient to split up a particle's velocity into a part parallel to the magnetic field (the <b>longitudinal velocity</b>) and a part perpendicular to the field (the <b>transverse velocity</b>). The longitudinal velocity describes how fast the particle drifts along the magnetic field line. The transverse velocity determines how tightly and how quickly the particle circles around the field lines.

If the gyroradius is much smaller than the extent of the magnetic field, you get a case where the particle approximately cannot cross magnetic field lines. It just spirals around them. On the other hand, if the magnetic field ends before the particle can make a significant portion of a full revolution, then the particle ends up being somewhat deflected from its original trajectory. If the gydroradius is enogh to deflect the particle into a region where the magnetic field has a different strength or direction, then things get complicated!

When a particle is spiraling around the magnetic field while drifting down the field lines into regions where the magnetic field is stronger, it will circle around the field lines in a faster, smaller circle. This increases the kinetic energy in the spiraling motion; this kinetic energy must come from the kinetic energy of the longitudinal motion. If the field becomes too strong, the particle will slow down all the way and come to a stop. Then it will be reflected back, pushed the other way down the field lines toward the region where the field is not as strong. This phenomenon is called magnetic mirroring, and is responsible for the way radiation gets trapped in the Van Allen belts around Earth with the particles bouncing back and forth along the Earth's field lines and getting reflected back and forth at the poles where the field is strongest.

If you want to calculate for yourself what the gyroradius <math>r_c</math> and gyrofrequency <math>f_c</math> are
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
r_c = \frac{\gamma \, m \, v_\perp}{q \, B}
</math></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
f_c = \frac{q \, B}{\gamma \, m}
</math></div>
where <math>B</math> is the strength of the magnetic field, <math>q</math> is the amount of electric charge of the particle, <math>m</math> is the mass of the particle, <math>\gamma</math> is the Lorentz factor, and <math>v_\perp</math> is the part of the velocity that is perpendicular to the magnetic field. To get these formula to work, you need to use a consistent set of units. So for example if you measure the magnetic field in tesla you need to measure the charge in coulombs (not fundamental charges) and the mass in kilograms (not eV).

One consequence of this is that you can bend the beam coming out of your accelerator by placing a uniform magnetic field across the end. So you can steer your beam to where it needs to go. You can also bend the beam inside of your accelerator – to make it go around in a circle, for example.

Another consequence is that if your beam has to pass through regions with magnetic fields in it, it will get deflected. If you want the beam to end up at the right spot, you will need to account for this. If the exact strength or direction of the field are not known, you will get unpredictable deflections to your beam.

If your beam has been neutralized, the magnetic fields will work to bend the positively charged particles one way and the negatively charged particles the other way. If you have enough beam current, these separated charges will attract each other and counteract the magnetic force. If you don't, the beam will split and you'll get positive charges zipping off in one direction and negative charges going another direction. If you needed to neutralize your beam so that it doesn't electrostatically blow itself apart, well now it will be doing just that.

Even perfectly neutral atoms can be torn apart if the atoms are going fast enough and if the magnetic field is strong enough. Atoms are pretty resistant to this, but you can get conditions where it happens.

Magnetic fields affect the path of beams. But beams are made of charged particles and thus can have currents in them. Currents create magnetic fields themselves, which can influence the surrounding fields. Even a neutralized beam forming a plasma, when it encounters a magnetic field, will get currents induced in the plasma that helps to screen its interior from the field. A beam with enough punch can deform the fields around itself rather than having the fields bending it. As a rough rule of thumb, if the energy density of the beam over a regions larger than a gyroradius is more than the energy density of the field over that region, then the beam gets to have its way with the field, and vice versa. Highly focused, high current beams can thus be expected to better deal with ambient magnetic fields than those with less focus or lower current.

==Focusing==

In the previous section, we discussed how a particle beam can be steered by directing it through a uniform magnetic field. We can make this kind of field with a magnetic north pole directly opposite a magnetic south pole (with additional shimming magnets to keep things uniform). Because it has two poles, one north and one south, this kind of magnet is called a <b>dipole magnet</b>.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:dipole_beam_steering.png|frameless]]
</table>

But we want to do more than just steer the beam. we want to be able to expand it, contract it, and focus it to a point. To do this, we need a different kind of configuration. We will put two north poles opposite each other and, and 90 degrees, two south poles opposite each other. With four poles, this is called a <b>quadrupole magnet</b>, or sometimes just a "quad".
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_magnet.png|frameless]]
</table>
At the center of the quad arrangement, the magnetic fields from each side cancel out. there is no field there. A particle going through the center will travel straight through without deflection. However, the farther particles get from the center, the higher the field they encounter and the more they get deflected. A particle slightly off to one side may experience a force that curves it toward the center-line. If the quads are properly shimmed, a particle twice as far away will experience twice the force and twice the deflection from its trajectory back toward the center. This curves all the particles so they meet in one spot.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_focusing.png|frameless]]
</table>
However, there is a catch. Those particles that are offset from the center-line at 90 degrees from these focused trajectories are bent away from the center-line rather than towards it! A quad focuses in one direction and de-focuses in the other.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_defocusing.png|frameless]]
</table>

Is there anything e can do to get an arrangement of magnets that purely focuses the beam? Well yes. We can put another quad arrangement downstream of the first, rotated by 90 degrees. Now consider a set of particles in an arrangement that is de-focused by the first quad. When they get to the second quad, they are further from the center than they were when they went through the first quad. Because the second quad is aligned to focus anything the first quad de-focuses, these particles will be focused; and because they are farther from the center, the focusing deflection will be larger than their previous de-focusing deflection. So the net result is that the particles are focused.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Dual_quad_focusing.png|frameless]]
</table>
For particles with the opposite orientation that are first defocused, these will be defocused by the second quad but because they will have converged they will be going through a region of lower field and will be defocused with a smaller deflection than they were initially focused. This dual quad arrangement will focus any particle that goes through it, regardless of orientation.

For small deflection angles, a dual quad set-up acts like an ideal lens for particles. An ideal lens can focus all the particles diverging from a point onto another point, or into perfectly parallel lines. It can focus particles on perfectly parallel lines onto a point. So now we can use the particle beam to deliver focused particles on our target.

Or, at least we could if all the particles came from a point, or if they traveled on perfectly parallel lines. Unfortunately, they don't. When particles are made, they come from a source with a finite spot size and a finite temperature. The random motion of the particles moving around from their initial thermal motion means that they are never moving perfectly parallel to each other. And when you try to focus them, instead of all converging onto a single point they make a spot of finite size. And if your focusing and beam bending elements are not perfect, you add additional imperfections to the beam that further hider your ability to focus it.

Your ability to focus can be quantified by a measure called <b>emittance</b>. Emittance is measured in units of a length scale times an angle - nominally meters-radians (m<math>\cdot</math>rad) but the technical literature more commonly gives units of millimeters-milliradians (or mm<math>\cdot</math>mrad) or even 2 <math>\pi</math> mm<math>\cdot</math>mrad. The divergence angle of a beam that you try to make parallel will the the emittance divided by the width of the beam. If you focus a beam down so it converges at an angle, the minimum spot size you can get at the focal point is the emittance divided by the beam angle. If you try to focus a beam with emittance <math>\epsilon</math> on a spot a distance <math>R</math> away with an initial beam diameter of <math>D</math>, the spot size <math>S</math> on the target you can achieve is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
S = R \frac{\epsilon}{D}
</math></div>

So to get good focusing, we want low emittance. Is there a way we can lower the meittance beyond what we get from our source? Engineers are pretty bright folks, and they have found some ways to do this. You can, for example, pass a beam through a thin foil. This causes some stopping, slowing the beam down. You can then accelerate the beam back up to its original speed. However, the stopping is along the direction of the beam particle's tranjectory, even if that trajectory is at an angle. Meanwhile, all the acceleration is in the direction of the beam axis. So the net result is to reduce the random motion perpendicular to the beam, and thus reducing its emittance. Similar tricks can be played with magnet set-ups that force particles going through them to emit radiation.

Another method can be used with a relatively high emittance beam, say a beam of protons or ions or muons. Electron beams can be made with substantially lower emittances. So you can send a low emittance electron beam to co-propagate along with your primary particle beam. The beams will interact, with the particles scattering off each other. This exchanges heat between the two beams and so the temperatures, and thus emittances, of the two beams get closer to each other (and, if given enough time, would become equal). If you repeat this often enough, you can get the emittance down to the emittance of the electron beams. This is more practical in beams going around circular accelerators, because they will have time to get cooled over and over.

A final method that has been used is to install sensors to measure the beam fluctuations. For beams going in circles, the time to get a light-speed signal directly across the diameter to the opposite side wil always be less than the time it takes for the particles to go around the curve to that spot. So you can send signals to give electromagnetic kicks to the beam particles to knock them back into alignment when they get to that spot.

It is intriguing to note that these beam cooling techniques can take <i>hours</i> to work! <ref>[https://www.bnl.gov/isd/documents/86783.pdf M. Minty, R. Connolly, C. Liu,T. Summers, and S. Tepikian, "Absolute beam emittance measurements at RHIC using ionization profile monitors", Brookhaven National Laboratory formal report BNL-105970-2014-IR]</ref>

But perhaps the simplest method to reduce emittance is just to speed the particles up. For particles moving at a fraction of light speed <math>\beta</math> and Lorentz factor <math>\gamma</math>, the emittance of particles from the same source with the same temperature and same random motion will be proportional to <math>1/(\beta \, \gamma)</math>. To reflect this, it is convenient to define a normalized emittance <math>\epsilon_n</math> such that for the usual geometric emittance <math>\epsilon</math> that we have been talking about
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
\epsilon_n = \beta \, \gamma \, \epsilon.
</math></div>

Normalized emittance measures makes it convenient to get basic estimates of particle beam performance and compare emittance of different sources and beams. For example, various high-performing electron beams around the world have managed to push their normalized emittances down to about <math>2 \times 10^{-6}</math> m<math>\cdot</math>rad to <math>3 \times 10^{-6}</math> m<math>\cdot</math>rad. Facilities with proton or ion beams seem to have about an order of magnitude higher normalized emittance.
<table>
<tr><td width=200>Machine <td width=300>normalized emittance (m<math>\cdot</math>rad) <td>reference
<tr><td>European XFEL <td><math>2.3\times10^{-6}</math> <td><ref>[https://journals.aps.org/prab/pdf/10.1103/PhysRevAccelBeams.24.110702 T. Hara <i>et al.</i>, "Low-emittance beam injection for a synchrotron radiation source using an X-ray free-electron laser linear accelerator", Physical Review Accelerators and Beams <b>24</b>, 110702 (2021)]</ref>
<tr><td>SLAC <td><math>3\times 10^{-6}</math> <td><ref>[https://accelconf.web.cern.ch/pac97/papers/pdf/4W009.PDF M. Hernandez <i>et al.</i>, "Emittance Measurements for the SLAC Gun Test Facility", 1998 IEEE]</ref>
<tr><td>LHC <td><math>3.75\times 10^{-5}</math> <td><ref>[https://www.lhc-closer.es/taking_a_closer_look_at_lhc/0.beta___emittance Beta & Emittance; Taking a closer look at LHC]</ref>
<tr><td>RHIC (with protons) <td><math>\approx 5\times 10^{-5}</math> <td><ref>[http://www.osti.gov/servlets/purl/910428-6sKfP0/ Zhang, S, and Ptitsyn, V., "Proton Beam Emittance Growth at RHIC", Brookhaven National Laboratory formal report BNL-77563-2007-CP]</ref>
</table>
So as a first estimate for your sci-fi electron guns, you can take <math>\epsilon_n \approx 3\times 10^{-6}</math> m<math>\cdot</math>rad and for proton guns <math>\epsilon_n \approx 4\times 10^{-5}</math> m<math>\cdot</math>rad. Then divide by <math>\beta \, \gamma</math> for your beam to find the geometrical emittance. Which in turn can let you figure out how well you can focus your beam.

==Space charge==

==Energy loss in matter==

Sometimes, a particle beam needs to go through stuff to get it where you want it to go. Stuff like air, for example. Or the tissue in the way of getting the beam to a tumor. Also, sometimes, you want your particle beam to interact with stuff. Like the material you are trying to machine with an ion beam, or the bacteria you are trying to kill on medical equipment. All of this falls under the general realm of the interaction of the beam particles with matter. This is an enormously complex subject, but we can cover many of the basics here.

The primary effect on the beam of any interactions with matter are loss of energy of the beam particles (<b>stopping</b>), random changes to the particle directions that make the beam spread out (<b>straggling</b>), and knocking particles clean out of the beam altogether (<b>scattering</b>).

===Ionization===

The particles coming out of your particle accelerator are charged. When a fast moving charged particle goes through matter, the electric field from the charge pushes and pulls on the various charged particles that make up that matter – mainly electrons and the atomic nuclei around which the electrons orbit. At speeds above approximately 1% of the speed of light, it is primarily light and nimble electrons being affected while the ponderous nuclei just sit there barely perturbed by the passage of the particle. When the passing particle yanks on an electron hard enough, it can pop it clean off of the atom it was attached to, leaving a positively charged ion behind and giving you a free electron that can go and glom on to a previously neutral atom to make a negative ion. This process, unsurprisingly, is called <b>ionization</b>. For most particle beams, ionization is the dominant interaction with matter; although when you get very fast or very slow particles other effects can become more important.

if your particle was not initially charged when it goes into matter, it soon will be. The electric charges of the electrons and nuclei in the matter will do the same thing to the particle that a charged particle would do to them. This will quickly strip off any electrons around a high speed atom, leaving an ion and bare electrons to go through the matter. Only particles that are fundamentally neutral (like neutrons or photons) will not be immediately ionized.

It takes about 1 to 10 eV to ionize an atom. This energy must come from the kinetic energy of the beam particle. Particles from an accelerator have millions to billions of eV. So it takes a lot of ionizations to meaningfully affect the particles. However, one thing about matter is that it has lots of atoms that can be ionized so you do get those lots of ionizations. Because each ionization is so small compared to the total energy this is often approximated as a continuous slowing down of the beam particles. So many individual ionizations tend to average out, leading to all the beam particles being slowed down uniformly.

When the particle is yanking on nearby electrons to tear them out of atoms, those electrons are yanking back. This jitters the beam particle slightly from side to side. Again, these sideways yanks tend to average out. The statistical variations from the average slowing down and side-to-size jerking are what lead to straggling. Protons and ions are heavy, and will be deflected little from their original paths - like bowling balls smashing through matchsticks. These particles tend to go in mostly straight lines. Electrons, on the other hand, have the same mass as the electrons they are knocking about. So electron tracks through matter tend to zig-zag around a lot more.

As the particle whizzes past, it leaves a trail of atomic destruction in its wake - a trail of ions where the atoms are literally torn apart (not the nuclei, just the atoms). These particle tracks can sometimes be seen in detectors like cloud chambers. The faster the particle is going, the less time it has to yank on nearby electrons and so the less ionization it leaves behind. As a result, fast particles are not slowed down much but as they start to slow down they continue to be slowed down faster and faster. A consequence of this is that the particle dumps most of its energy into the material at the end of its track, just before it comes to a stop. This sudden peak in energy deposition is called the <b>Bragg peak</b>.

The electrons that a particle knocks out can end up going pretty fast. So these secondary electrons can produce further ionization in what is called an <b>electron cascade</b>. if an inner core electron is knocked out of an atom, other electrons falling in to fill that empty spot in the atom can release x-rays (<b>x-ray fluorescence</b>) or cause shake-ups in the outer electrons that knock other electrons off the atom (<b>Auger effect</b>).

===Brehmsstrahlung===

A charged particle is surrounded by an electric field. If it is moving, it is also surrounded by a circulating magnetic field. These fields move along with the particle.

If the particle knocks into something, it will slow down and probably change direction. The information about this bump only goes through the electromagnetic field at the speed of light. So a lot of that field that was moving along with the particle might not get the memo in time and keep right on going. Stray electromagnetic fields like this get this self-inductance effect going on where they start making themselves over and over again, allowing them to propagate as freely moving waves of electromagnetic radiation. This EM radiation from a slowing down particle is given the delightful name of <b>bremsstrahlung</b>.

Particles from an accelerator can be moving so fast that they make bremsstrahlung in the x-ray or gamma ray part of the spectrum. If you get very high energy bremsstrahlung gamma rays, the gamma rays can interact with matter by boiling an electron and its antimatter counterpart, a positron, out of the vacuum (using the nearby matter as the necessary momentum sink for the process). These positrons and electrons can themselves be very high energy and can in turn create more bremsstrahlung, and the process can repeat until the particles are no longer high enough energy to continue making each other. This is called an <b>electron-gamma shower</b>.

The x-ray machines used by dentists and radiologists use both bremsstrahlung and x-ray fluorescence that you get when shining an electron beam onto tungsten to make their x-rays.

At very high speeds, bremsstrahlung can be more a important source of energy loss than ionization.

===Electron capture and screening===

When an ion has slowed down to about 1% of the speed of light or less, it is moving slower than the electrons zipping around their atoms in the matter the ion is going through. This allows the electrons to stick on to the ion, now becoming the electrons of the atom that is moving through the material rather than of the material itself. The ion begins to lose its charge, and the rate of ionization goes down. Soon, ionization becomes negligible compared to the beam atom bumping in to other atoms.

===Phonons and displacement===

When an atom or ion moving through a material bumps into another atom, it will cause that other atom to move. This can have one of two primary effects.

First, if the atoms whack into each other hard enough, the atom in the material can be knocked clean out of place. Like a pool ball being thwacked by the cue ball, the struck atom will go zinging through the material. Atoms getting knocked out of their place like this is called <b>displacement</b>. Displacement creates chemical changes to molecules as atoms are removed, and creates defects called <b>vacancies</b> in crystals (which, as you might imagine, is just a missing atom in the crystal lattice). When the atom comes to rest, it can chemically attack other molecules, again changing the chemistry, or create an extra vacancy called an <b>interstitial</b> jammed into the crystal lattice where it is not supposed to be. If the displaced atom is going fast enough, it can whack other atoms and displace them, leading to a <b>displacement cascade</b> like tenpins knocking each other down. As a result, at the end of its track, an ion can leave a region of disordered material and messed up chemistry where the atoms have all crashed into each other like a freeway pileup.

If an atom is displaced near the surface of a material, the displaced atom can shoot out of the surface and escape the material altogether. This is called <b>sputtering</b>.

Second, the beam atom might only deliver a glancing blow. Or the beam particle might be an electron, which is so light that it is usually incapable of displacing atoms. In this case, the struck atom gets kicked to the side a bit and runs into other atoms; maybe it is also pulled back by the chemical bonds of other atoms that it is attached to. This transfers the motion of the atom to other atoms around it in the form of pressure waves (or, in a solid, shear waves). This adds heat to the material without causing direct damage – although as you might imagine, enough heat can also cause localized melting, evaporation, or decomposition. The individual quanta of vibration – fundamental particles of sound, if you want to think of them like that – are called phonons in analogy with the photons that are the fundamental particles of light. Much like ionization, a particle that excites pressure waves slows down gradually and continuously, unlike the sudden changes in speed and direction you get from knock-out displacement.

===Nuclear collisions===

So far, we have only discussed the things that happen when beam particles interact with matter particles by electromagnetic forces. Even atoms knocking into each other to produce phonons or displacement happens through the mutual repulsion of the electrons around the atoms (although this is dressed by other weird effects like how two electrons can't be in the same place at the same time, or how the fields of one electron can affect other nearby electrons that in turn affect the original electrons, leading to odd correlation effects and charge density waves and other fun stuff that keeps condensed matter physicists and quantum chemists busy).

But electromagnetism isn't the only thing you can get going on. below the level of structure of the atom, you have the atomic nucleus. And a direct hit on the nucleus can end up giving very different effects.

The simplest thing that can happen is an <b>elastic collision</b>, where no energy goes in to any internal nuclear state or into breaking up the nucleus. An elastic collision just ends up giving you displacement (or possibly phonons, but if you hit the nucleus you usually hit it hard enough to send it flying).

But beyond that you can cause any number of nuclear reactions. You can leave the nucleus in an excited nuclear state, from which it usually relaxes back down to its original state by emitting gamma rays. You can knock out nuclear particles, such as neutrons, protons, deuterons (a proton stuck to a neutron), tritons (a proton stuck to two neutrons), helions (a neutron stuck to two protons), alpha particles (two protons stuck to two neutrons), or potentially even larger nuclear fragments. If you hit a heavy nucleus like uranium, you can cause it to split apart by fission. And if you hit the nucleus really hard you might just shatter it into many tiny fragments of those protons, neutrons, deuterons, tritons, helions, alpha particles, and heavier nuclear fragments mentioned earlier. To get any of these <b>inelastic collisions</b>, you need to hit the nucleus hard enough. If your beam particle doesn't have enough energy to excite the nucleus or knock particles, these things simply can't happen. As it gets enough energy to excite a particular interaction channel (as they are called), that process becomes possible and as the energy increases the process becomes more and more likely.

A nucleus is only about a femtometer across (<math>1 \times 10^{-15}</math> m). Which makes it only about <math>1 \times 10^{-30}</math> m² in cross sectional area. In the meantime, chemical bonds between atoms are usually about 1/10th of a nanometer (<math>1 \times 10^{-10}</math> m) long. In condensed matter (liquids and solids), this means that you have somewhere about <math>1 \times 10^{30}</math> atoms per cubic meter. A meter thickness of atoms thus presents just about about enough atoms that their nuclei can cover the projected area – in other words, you can expect a particle to go through (very roughly) about a meter of condensed matter before it whacks into a nucleus.

This assumes that a particle that goes through a nucleus interacts with it. This is usually a good assumption for particles that interact by the strong nuclear force, such as protons, neutrons, mesons, and the atomic nuclei of ions. But many particles cannot interact by the strong nuclear force, such as electrons or muons. These are much less likely to interact with an atomic nucleus even if they pass through it, so they will end up going considerably farther between hitting nuclei than the above assumption. But nuclei can interact electromagnetically (they are made up of charged protons, after all, and at a smaller level of charged quarks), so electrons and muons can produce some nuclear interactions; just far less than actual nuclear particles like protons and ions.

A nuclear strike is a discrete event that will remove the impacting particle from the beam and will (except for elastic collisions) probably produce a spray of radiation.

===High energy physics===

As you get to higher and higher energies, you unlock all sorts of new and weird and interesting processes that can happen. Boiling exotic particles out of the vacuum, unifying forces that had been divorced since the big bang, and producing bizarre unstable resonances in the quantum fields. This is what physics build accelerators to measure. From the point of view of getting a beam to its target, or what happens to the target, however, the main effect is mostly that the interacting particle is removed from the beam and a bunch of additional radiation is produced.

==Self pinching==

= Operational Considerations =
== Logistics ==
The logistics of particle beam can be a mixed bag. By itself, any particle accelerator is a complicated piece of machinery full of delicate parts that have to deliver high performance. This may require cryogenic temperatures with associated challenges on the materials and mountings used (insulation, and distortions under temperature changes). When working, particle accelerators invariably generate some radiation across the whole assembly, which irradiates and activates the structure. Some parts may be exposed to significantly higher radiation flux due to their working principles. Often the particle beam is a dynamic system that must be managed with active, closed control loops, involving sensors, modeling and control software, actuators, and the computing and communication hardware that brings it all together. This poses engineering challenges across all disciplines.

A practical particle beam is inherently a counterintuitively sturdy device. If it was too delicate, it would break too much to be useful, though of course this is a sliding scale. Still, while sturdy, this doesn’t eliminate complexity, it manages complexity. The result is a system that is the product of a high-tech civilization. A superior machine shop and well-trained engineers are a must to maintain a particle beam system. Some parts of a particle accelerator may be mundane - others may be custom-machined parts that require their own dedicated industries to produce. That said, the manifold utility of particle accelerators is on your side here. Since there are so many practical applications, it is economic to produce parts for industrial demand. (that said, these may not be the kind of components you want in specific high-performance applications.)

The main thing a particle accelerator requires the most is electricity. Any mature power production facility can supply the power.

However, all particle beams require at the least a source of neutralizing mass. The projection of a charged beam will leave the projecting object oppositely charged. In an atmosphere the natural plasma formed by the beam will ground everything out, but in space the diffuse vacuum will not do the job.

Furthermore particle beams using ions also require a feedstock for their ions to produce from, and in an appropriate format. This can be a gas, a fluid or an easily ionized solid. It can be the feedstock species in its pure form or in an easily split molecule, though every complexity adds to the amount of preparation work that has to be done before the feedstock enters the particle beam.

Don’t worry about the mass of your source or neutralizing ions though. Large particle accelerators, like the LHC in Switzerland, only consume tiny gas bottles worth of feed gas every couple years. The mass of a single 5.56 NATO bullet worth of hydrogen could feed a 10 GW, 10 GeV ion beam for nearly 4 days of firing! You will not be running out of beam feedstock easily.

== Cooling ==
Cooling a particle accelerator can be a serious challenge. Various bits of a particle accelerator love to be supercooled for efficiency or basic working principle - any bit of superconductor used in the accelerator, or magnets, or EM generators, will require cooling to its generally supercool operating temperature.

Even if Standard Temperature and Pressure Superconductors (273 Kelvin, 100 Kilopascal) are available, present superconductors have demonstrated superior performance by being supercooled. That means to achieve even stronger fields, even STP superconductors may be cooled to cryogenic conditions.

Cooling cryogenic systems sucks. Thermodynamics shows itself from its worst side. You can forget running cryogenic radiators - the Stefan-Boltzman Law penalizes the direct radiative cooling of anything cryogenic severely as much as it empowers very high-temperature radiators. This only leaves stages of heat pumps to provide cryogenic fluids that can in turn, cool cryogenic systems. Heat pumps draw power, and for such a giant temperature gradient as we can have here, they will draw a lot of power.

However, we have other options. Cryogenic systems also tend to be highly efficient, so while they do heat up, they do so very slowly and there is little overall energy absorbed. This provides the option of running off of bunkered cryogenic fluid in dewars for the duration of operating the equipment and not running any heat pumps all the time.

Other bits of a particle accelerator system aren’t as required, but really nothing is a fan of high temperatures. Going higher than room temperature can be treated as an oddity. High-performance lasers may not require cryogenic conditions, but they certainly like it cool. Radiofrequency producers of any kind generally fare worse with rising temperatures. That means cooling overall remains a challenge - 300 Kelvin is still not that great of a hot loop temperature for radiators.

== Safety <ref>"Safety for particle accelerators", Thomas Otto, Springer 2021, Available online at: https://library.oapen.org/handle/20.500.12657/46120</ref>==
=== Ionizing Radiation ===
There are surprisingly many sources of ionizing radiation in a particle accelerator, besides the obvious threat of the output beam itself. Not all particles are accelerated correctly or in the correct direction. When such particles interact with the structure of the accelerator, they release a shower of secondary ionizing radiation. In electron accelerators and sources, stray electrons may be accelerated and release x-rays from bremsstrahlung as they collide with surrounding matter. The resulting doses can be significant and dangerous. Also, such collisions leave behind residual activation, making the accelerator a radiation hazard for some time even after it has been shut down. Generally, people should not work near the particle accelerator while it is functioning, and radiation barriers isolate the core accelerator from adjacent facilities. (In real-life facilities, you will often see accelerators inside larger buildings being cocooned inside doped concrete plates to shield the rest of the room.)

=== Radiofrequency ===
RF energy is used in many kinds of particle accelerators. This energy may leak from the structures it is generated, contained and used in due to damages or incorrect assembly. As microwave radiation and above is used, the radiation can be readily absorbed by bodies, hydrogen-rich materials or other objects. The end result of this can be rapid heating and thermal effects. Especially in accelerators designed to throughput a lot of energy (say a GW or more), this may become a serious issue if a waveguide is damaged. It’s not recommendable to stand in front of a multi-MW radar either!

=== High-powered electrical and electronic systems ===
The operation of the magnetic systems and accelerators often involves high-powered electrical and electronic systems. High voltages and/or amperes are involved. This poses risks like shock hazards, electrical arcs, and electrical fires. Some electrical systems such as capacitors or inductors may have combustive or explosive failure modes.

=== Flammables ===
Parts of an accelerator or its infrastructure may be manufactured from flammable substances such as hydrocarbons. In the case of an ignition source such as a high-powered beam cutting into a physical substance or an electrical fault, these may catch fire.

=== Lasers ===
Some accelerator systems use high-powered lasers. Laser plasma accelerators use very high-powered pulse lasers to provide pumping power. The safety concerns of lasers for blinding, physical damage, and possible exposure to ionizing radiation as a secondary effect of the beam impact, apply. Some high-powered lasers can also drive electrons which generate strong EMPs, which may damage or destroy other systems in turn. Parts of a laser assembly or laser beam guide may shatter if exposed to overloading conditions.

=== Magnetic systems ===
Parts of a particle accelerator may use strong magnets, up to superconductive magnets with multi-Tesla fields. While the magnets are integrated into chokes that shape the magnetic fields, fringe fields invariably extend beyond the core magnet installation.

Multi-Tesla transient electromagnetic field can affect the human nervous system temporarily, inducing nausea, vertigo, metallic taste and optical phenomena, if the person moves in relation to the magnetic field. Otherwise, the risk of exposure is low. That said, electronic systems that are not shielded properly, such as say, implants, could be adversely affected by the magnetic fields.

The biggest physical risk are reactions of other magnetic objects with the magnets. You can look up cases where metallic objects interacted with the multi-Tesla fields of MRI scanners. Such parts may fly across the room towards the magnet at increasing speeds, and damage anything or injure anyone in their path, possibly ending with the object or person squashed (or impaled!) to the magnet.

Also, large superconductive magnets store not insignificant amounts of energy. If the superconductive property breaks down, the magnet quenches and becomes a resistor. The resistance site heats up as current tries to flow through it, expanding the quench and possibly being heated until it sublimates into gas. This could possibly happen explosively. In such a case, up to many GJ of energy might be unleashed in a conflagration or explosion (subsonic or supersonic), damaging surrounding systems and ejecting shrapnell.

=== Cryogenic cooling systems ===
Particle accelerators often involve cryogenically cooled systems. This means cryogenic fluids like nitrogen and helium are present in the accelerator infrastructure. Such gasses have both a very low temperature and a high ratio of gaseous volume at high temperatures to their liquid volume.

Exposure to the leaked fluid can lead to cryogenic cold damage, injury, and death. Furthermore such gasses are efficient at displacing ambient breathing gasses such as oxygen, so they pose an invisible, and rapidly spreading, suffocation hazard.

Also, if insulation is damaged, ambient heat may leak into the cryogenic circuits, and cryogenic fluid may begin evaporating. This can lead to dramatic rises in internal pressure. Relief valves should trigger to safely direct such pressure spikes away from important places and keep the system from failing catastrophically, but otherwise, pipes and cryogenic storage devices may fail, up to explosively.

=== Beam excursions ===
If the magnets responsible for manipulating the beam break down, the beam may excurse from its designed paths. The beam may become defocused, or maintain its focus. The beam begins cutting into the accelerator structure, evaporating material and depositing energy. This will definitely foul the ultra-high vacuum inside the accelerator, leading to secondary radiation production as the beam collides with gas inside the accelerator. Some sensitive facilities requiring ultra-high vacuum such as some RF cavities may “crash” and become ruined. Further systems may become damaged or destroyed from either physical stress or the radiation pulse. If the beam bunches contain a lot of energy and maybe are heavy particles, the deposition may lead to an energetic local explosion. Alternatively the beam may escape and cut into structure beyond the particle accelerator itself, damaging and destroying equipment and seriously hurting or killing personnel.

In the worst cases, a beam excursion on a very powerful beam may lead to an explosion that destroys the accelerator utterly. Even in a less bad case, the radiation pulse might damage or destroy critical equipment around the accelerator and take the entire facility offline.

=== Beam dumping ===
If any other systems fail, the accelerator probably has to be shut down. This poses a problem - what do you do with the beam energy that is still in the accelerator system?

Sometimes, the beam is designed to leave the accelerator immediately, and shutting down the beam is easy. But in the case of cyclotrons and storage rings, the beam may circulate for a while. What do you do then? You have to dump the beam from the accelerator. Special beam stops are used on research and industrial accelerators, large blocks of dense material able to absorb the beam. During this process the blocks both heat up and become radioactive. For particle beams storing very high energies, this may become an increasing challenge. How do you safely dumb a beam containing many GJ or more? In some cases you may have no other choice but to dump the beam out into the environment. Where and when can you do this safely without threatening third parties?

= Application areas =
(This is a skeletal overview summary)
Particle accelerators have many, many, many areas of application.

In science, the big particle accelerators like the LHC are common knowledge. But electron microscopes also use components of particle accelerators. Many particle accelerators worldwide are used to produce high-energy x-rays in synchrotrons or linear accelerators, imaging samples for material science, chemistry, biochemistry and other areas.

In manufacture, particle beams can be used to cut material, but also weld material. Precision beams can break down gaseous feedstocks to deposit metal in micrometer-sized structures. Directed ion guns are used to implant traces of elements into surfaces for the manufacture of electronic devices.

And in medicine, particle accelerators of ever-greater sophistication find increasing use in cancer treatment and beyond, destroying life-threatening tissues with ever-reduced doses to surrounding healthy tissue.

= Further material =

= Sources =

== Authors ==
Sevoris Doe, Gerrit Bruhaug, Luke Campbell.

== References ==

Particle Accelerators

2022-08-17T12:42:39Z

Sevoris: /* Sources */

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Particle accelerators are an incredibly useful piece of technology and a diverse set of artifacts produced for many purposes. While particle accelerators are often invoked as weapons of war in science fiction, and have captured the public attention as giant research systems the size of cities, they have an incredible widespread range of applications. There are also many, many ways to build a particle accelerator, depending on what effects you need to achieve and what circumstances you work under.

This article aims to be a general primer on particle accelerators, their physics and a selection of technologies and construction styles. In detail, all particle accelerators are high-tech engineering customized to their circumstances in many, many aspects.

Further pages covering application fields are in the work.

= Accelerator physics =
The simple explanation is that we use electric fields to accelerate charged particles. But from this basic principle of leveraging the electromagnetic forces springs an entire set of accelerator technologies with different performances, trade-offs and characteristics. The most basic particle accelerators are just a big high-voltage source and can still be found in old CRT TV’s and dentist x-ray machines around the world. Things rapidly get more complex from there, but first we need to lay out some terms so we can all have a common conversation.

The first thing to keep in mind with particle accelerators is that we are talking about charged beams here, even if we neutralize them later. So no neutrons, photons, or other innately neutral particles. Your choices range from the simple electron up to anti-uranium and more, but they all must be charged. The particle of choice determines just about everything else about the accelerator and any potential uses. Really heavy particles (like large ions) don’t accelerate easily and take a lot of energy to get relativistic enough for time dilation tricks to take into place. Really light particles (like electrons) accelerate super easily, but can have other radiation problems to worry about.

The second most important thing to know about a given accelerator is the particle energy. This is a measurement of the kinetic energy of any individual electron, ion, whatever in the beam and is measured in a unit called the electron-volt or eV for short. The lower bound for any worthwhile particle accelerator is in the MeV (or million eV) range and we typically want a lot more! Inside of the accelerator we tend to talk about the electric field doing the accelerating (measured in Volts/meter). For singly charged particles (like electrons) this gets really to figure out the energy because of how the eV is defined. We just take the electric field (also called the accelerating gradient) and multiply by the length of the machine! Boom, now you have particle energy! For more complex particles you have to do some scaling based on mass and charge state.

Also a quick trick to help you with deciding if something is relativistic or not is to divide the particle energy <math>E</math> by the mass energy <math>M</math> of the accelerated particle (0.511 MeV for electrons/positrons and about 1 GeV for a proton) and add one. The resulting number is called the Lorentz gamma value and is used all over in accelerator engineering.
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>\gamma = 1+\frac{E}{M}</math></div>
Once you get a value much larger than 1, you have relativistic particles. Once that number is in the hundreds you get an ultra-relativistic beam. Figure 1 helps show the relationship between Lorentz gamma and speed of the particle.

[[File:LorentzGvsSped.svg|900px|thumb|center|Lorentz gamma versus fraction of the speed of light]]

The third most important feature of a particle accelerator is the emittance which is just a fancy term for divergence of the beam and can be thought of similar to the wavelength of lasers. No beam is made perfect and they will naturally want to expand outwards. The smaller the emittance, the longer the beam will stay in a tight spot. Emittance can be altered with a wide variety of beam engineering techniques that take entire books to discuss, but one other cheating thing you can do is drive the beam energy higher. This naturally lowers the emittance by a factor of Lorentz gamma due to the cheating effect of special relativity! Thank god for Einstein!

== Acceleration forces ==
Every accelerator tries to build an electric gradient appropriate for the charge of the particles it is accelerating. The greater the electric gradient, the bigger the amount of work that can be done, the more energy is imparted on a particle as it travels through the gradient, and thus the higher the acceleration. The simplest way to build up a gradient is simply two charged electrodes with a strong potential between them. Many other accelerator methods use electromagnetic fields, in the form of waves confined in electrically conductive structures. In this case we exploit that electromagnetic fields have both an electric and a magnetic component. The magnetic component cannot be used to accelerate particles (only bend their trajectory - which can be exploited for effects like deflection or focusing) but as the electromagnetic waves oscillate, so does an electric field. This naturally creates a charge difference across half of the phase of an electromagnetic wave - this electrical gradient can accelerate particles. Both of these technologies encounter problems with the physical materials asked to contain the electric or electromagnetic energy. Eventually the fields get so intense that solid materials get broken down and an arc is formed. As such, various advanced concepts use clouds of charged particles and plasmas, which are not subject to the material limitations of solid matter.

The strong and weak nuclear force as well as gravity (which are all of the forces we know of at the time of writing) are not suitable for accelerating particles. This is related to the power and range of these forces. The strong and weak nuclear force have extremely short ranges - even if we had a means to artificially shape them like with electromagnets, their range would be too short. Gravity has the range but is incredibly weak. To generate a measly 1 G of acceleration, 9.81 m/s², requires the mass-energy of the entire earth, some 5.97e24 kg of mass, to do the job! The acceleration involved in particle accelerators is much, much, much larger (on the order of 10^13 times larger or more).

{{MascotSays|Interestingly, one source of very high-energy particles are giant gravity wells. Black holes generate conditions around them that can eject particles with enormous energies. The jets of particles that shoot from the north and south poles of many such stellar entities are particle accelerators.}}

If you want to use a force other than electromagnetism to accelerate particles, it’ll have to be a fictional one. The forcefields of various sci-fi properties could be considered, since they can often apply forces to particles. If you propagate a wave of such forcefields down a tube, you could have something quite a lot like many particle accelerator systems!

== The particle bunch ==
Essentially all high energy particle beams are not constant streams of particles, but a time sequence of discrete groups of particles, commonly referred to as “bunches”. These bunches can be described both in their distribution in lateral space to the traveling direction (x and y axis) and longitude (z axis) along the traveling direction, anchored on an origin point (0,0,0) that moves in time.

[[File:ParticleBunch.png|center|900px]]

Bunches can have various profiles. Generally they are somewhat distributed in the lateral direction just because the charged particles have slightly different starting positions and lateral velocities (also called transverse emittance). In longitudinal space, differing acceleration experienced by individual particles as well as dynamics within the bunch scatter them out, with “noses” and “tails” that contain fewer particles. The end results is that most beam profiles follow a gaussian distribution, though there are exceptions.

This bunching behavior allows us to analyze other properties of particle beams very similar to lasers. There is a “pulse frequency” given in Hertz. Each bunch contains a certain energy, which can be large enough that we give it in Joules and not electronvolts. There is an average output energy over a given timespan, given in Watts. We have a beam peak power, given by how quickly a bunch arrives at a target and deposits its energy, which we give in Watts. Related to this we also have a related peak intensity depending on how much area/volume the beam is deposited into, with units of either W/m² or W/m³. If we have charged particles being accelerated, we get a beam voltage, the same way as in a wire. Beams can also have a temperature, the same way as any other group of particles with random motion from their energy, given in Kelvin or eV.

The systems that generate particle beams of course have efficiencies, with the wall plug efficiency (power inserted into the entire system versus power deposited in the particle beam) being the most interesting for first-order engineering. They also have an operating temperature.

== Particles for acceleration ==
With the constraint that our particles must have an electric charge in order to be accelerated with anything but fantastic technology, we can discuss quite a large family of particles, from the conventional to the exotic. They differ in the applications they can be interesting for. Some can be similar, but no two particles behave entirely alike.

=== Electrons ===
The ubiquitous electron is the most common particle to be accelerated and the first one ever pushed up near the speed of light by humans. Due to its excellent charge to mass ratio (the best possible due to the extremely low mass and charge of 1e) electrons are very easy to get up to relativistic speeds. Electron accelerators for radiography and sterilization are often portable by a single person and not much larger than a laptop in size (although they can be much bigger for power output reasons). The ease in which electrons reach relativistic speeds means that electron RF accelerators can be designed with no speed changes assumed (since the electrons are already moving at effectively light speed) and thus tend to be highly efficient and have the highest acceleration gradients achievable. For similar reasons electrons are also able to be accelerated in plasma based accelerators, allowing for even higher acceleration gradients and smaller machines for a given energy. The major downside of electrons is that due to their large charge to mass ratio they will radiate energy like crazy when bent by a magnetic field. Thus ring shaped accelerators are of limited use for electrons since you have to make the ring extremely large to not just lose all of your energy as beams of x-rays coming out of the machine! This very trick is used to make high quality x-ray sources for industrial and research applications.

When electrons hit matter they tend to release their energy as photons in a process called bremsstrahlung, or braking radiation. These extremely high energy photons (roughly ~⅓ the peak energy of the electron on average) then proceed to make electron/positron pairs that then make photons that then make pairs that then make photons… This whole thing is a massive cascade of radiation that ionizes, heats and can even damage on a nuclear level! At high enough energies you will generate every particle known and might even make some new ones!

=== Proton ===
Protons are the next most common particle to be accelerated and are currently used by the absolute largest particle accelerator ever made (the Large Hadron Collider or LHC) as the primary particle of choice. Since they have ~2000X lower charge to mass ratio than electrons, protons are about ~2000X harder to get up to speed. Unlike electrons they are not promptly at light speed so we have to send protons through a series of various accelerators depending on the energy of choice. For example the protons in the LHC the protons go through 5 different particle accelerators plus the extremely low energy stage that first makes the beam of ions! This difficulty in reaching highly relativistic speeds has two other bad side effects. Firstly, if we want to transmit our protons through empty space we will probably need to neutralize them with a beam of electrons.

{{Note|''When shooting extremely high current (kA), relatively low energy (not highly relativistic or in some cases relativistic at all) beams in the atmosphere the higher mass and positive charge of protons gives them many beam propagation advantages over electrons if the beams can be made. In this case no neutralization is required as the atmosphere itself becomes a plasma and neutralizes the beam.''}}
Luckily this is not very hard and will add almost nothing to your comparatively large proton accelerator, but may increase your beam spread a little bit via some neutralization physics. This does mean your beam is technically plasma, which is cool! Second, we effectively cannot use those neat plasma based accelerators for protons since we would need nearly TeV class proton beams to inject into the plasma accelerator and by the time you reach those energies you have basically already done all of the work. Maybe in the future there will be ways around this, but as of right now we are limited! There is one major benefit to this poor charge to mass ratio though, and that is the lack of synchrotron radiation! Ring shaped accelerators are ideal for protons, and basically all ions.

When protons hit matter they can interact with all of the four fundamental forces, although only two of them (Strong Nuclear and Electromagnetism) tend to play big roles. Protons barrel on in causing huge ionization cascades and can also release bremsstrahlung photons at high enough energies. They are also ideal at knocking neutrons, protons, deuterons and more out of the target nuclei which just adds to the fun! Protons also come with a neat trick known as the Bragg peak, where they deposit a large portion of their energy right at the end of their trajectory through a target material. This is a very useful trick for medicine and means you can “tune” the beam to almost ignore large portions of material and primarily heat, ionize, and destroy the area of interest. You can imagine how useful this is for cancer therapy or more nefarious things!

=== Light ions ===
Light ions are a rather fuzzy term but here we will use the definition of any ion heavier than a proton but lighter cobalt. The fuzzy reasoning here is that you get on the other end of the nuclear binding curve (i.e. can’t get net energy from fusion) at iron, so cobalt will then no longer be “light” and has to be made in a different process. Feel free to pick your own cut off! Historically beams up to neon have been experimented with for nuclear fission, nuclear fusion and medical therapy applications. Light ions have even worse charge to mass ratios than protons (Z/A times worse) and typically you will find light ions have about ½ the charge to mass ratio of a proton on average, although the isotope 3He is a notable exception. This means they are even slower to accelerate, although often the exact same machines as protons can be used with minor tweaks. Light ions have similar requirements for neutralization as protons, although their heavier mass can help fight beam bloom for a similar beam speed (not energy since the increased mass means the particles are higher energy for a given speed). The decreased charge to mass ratio also means that synchrotron radiation is even lower so beams can be driven to higher speeds before radiating. Light ions have similar target interaction physics as protons, although the ion itself will break up at very low speeds and add to the radiation environment.

=== Heavy ions ===
Heavy ions are being defined here as cobalt up to whatever unstable element you can jam into your accelerator! Typically heavy ion beams are made of elements like lead (used in some runs on the LHC), gold (used in the Relativistic Heavy Ion Collider) and even californium (used in some runs on the Argonne Tandem Linac Accelerator System). These beams tend to be for pure nuclear physics research, but there has been a long standing interest in heavy ion beams for inertial confinement fusion research and as potential methods of propelling spacecraft. These extremely heavy ions will be even harder to accelerate to ultra high speeds than light ions and much slower for a given energy, but they do provide a lot of momentum to make up for that! They have the same neutralization requirements and same potential upsides for beam bloom as light ions, but will most likely be orders of magnitude slower reaching our targets than the lighter beams. Heavy ions can also be used in the same machines as protons with small modifications and generate almost no synchrotron radiation as a result of their much smaller charge to mass ratio. They also have similar interaction physics to protons and light ions, but will stop in much thinner layers of materials. While electron, proton and light ion beams tend to both mechanically destroy and irradiate targets through-out, heavy ion beams will act much more like lasers and burn in from the outside of the target on in. There will still be a spray of ionizing radiation, but far less of one than an equivalent energy electron, proton, or light ion beam. The much higher momentum per particle, lower radiation concerns and ease of deflection makes heavy ion beams very interesting for beamed propulsion purposes.

=== Macrons ===
Macrons are typically defined as macroscopic particles, such as extremely large molecules or even grains of sand, that are charged up and accelerated via similar technologies as traditional particle accelerators. There are many difficulties with getting macrons up to appreciable speeds due to the extremely low charge to mass ratio, engineering difficulties with getting consistent and good charging, phase matching with accelerating cavities and lack of effective focusing optics. There are so called “dust accelerators” in existence that utilize the first stage of ion accelerators to generate their “beams” and proposals have been made to create better versions using updated technology. Macrons will interact just like a very small and very fast bullet unless they are designed to carry a nuclear payload to boost this output. Antimatter is an assured option for getting energy gain from a macron impacting a target, although there are potentially options for fusion, fission and fission/fusion given the correct sizes (mm scale or bigger), clever target design and the right impact speeds (most likely >>100 km/sec). The various energy gains versus speeds can be seen here:

[[File:IvKmacron.svg|900px|thumb|center]]

It can be seen that antimatter proves to be worth it up near the speed of light, but both fission and fusion stop providing much gain well below the relativistic regime. This also helps explain why heavy ion uranium beams are not going to provide some incredible advantage over any other ion.

=== Antimatter ===

Antimatter is the evil cousin of normal matter and whenever the two meet a release of energy equivalent to their combined mass energy follows. Now this does sound like an amazing option for beams but there are two downsides. First, antimatter is energetically expensive to make! Positrons (antimatter electrons) are the “cheapest” at 1.02 MeV of input energy at a theoretical best, but they only give you that much energy back when hitting a target. Since a typical positron beam can be expected to be equivalent to an electron beam, there is little gain in adding 1.02 MeV per particle to a multiple GeV (or even TeV) beam. You can see this explicitly:

[[File:IvKhBarPostron.svg|900px|thumb|center]]

For anti-protons the math becomes more favorable, but creating these particles is much more difficult and energy consuming. At best anti-protons require ~2 GeV of input energy to create, but in practice >400 GeV per anti-proton has been needed. Luckily significant energy gain is had up to several GeV of beam energy, so if anti-protons can be made at a central facility of some kind and easily stored (neither being easy things) then there may be some advantage to such a beam. Heavier ions are significantly harder to make since fusion with antimatter must be performed to generate those ions. Barring a naturally occurring source being found, antimatter heavier than a proton is most likely not worth using.

{{Note|'' Positrons can potentially fill an interesting niche either as a co-propagating neutralizing beam for electrons or perhaps as a higher performance atmospheric beam than electrons due to the positive charge. In this case the annihilation properties are no what is of interest, rather the positive charge and relative ease of acceleration compared to ions. Since positrons share all properties with electrons except for charge, they can be accelerated just as easily and even co-accelerated with electrons in the same machine.}}

=== Muons ===
Muons are an interesting cousin of electrons and positrons. They are unstable and heavy cousins of electrons, massing about 200X more than electrons and living about 2 microseconds in the lab frame before decaying into an electron (or positron) and a neutrino. If we accelerate them up to relativistic speeds (about 10X easier than protons and the muons are typically born relativistically thanks to how we make them) then the muons can be made to “live” far longer. Muons have long been of interest to accelerator physicists due to their position as an almost ideal intermediary between electrons and protons. The requirement to make them on site has always been an impediment though and muons are currently only used for certain niche experiments. During the planning stages for a potential muon collider it was also noted that a muon collider would also have a serious issue with neutrino radiation! This is due to how neutrinos are more likely to interact with matter at high energies and a muon collider would be generating a LOT of really high energy neutrinos in a ring around the machine. This has been proposed as a sort of super weapon even!

Muons should interact with matter in a similar fashion to electron beams and the beams will probably be treated in a similar manner. They can propagate through the atmosphere quite well in theory and should outperform electron and proton beams for a given beam energy, although at quite a cost… Muons are typically generated with proton beams impacting targets and will create an equal number of positive and negative muons at high energies that must be carefully captured. This would allow for the easy creation of a neutral muon beam though, which may make for a great giant doom beam! Do note that negative muons also can help catalyze nuclear fusion so a very high power beam hitting a fusion fuel tank could produce some rather… Interesting results!

=== Other exotic options ===
There are a wide variety of odd particles that could also be chosen for use in beams given a specific need for them, although generation and utilization of these particles will be quite difficult! Extremely short lived particles like pions, kaons and taus can be generated and accelerated up to extreme speeds to help propagate them far enough to matter, although that will need some very extreme speeds. At the upper end of extreme there are even proposals for accelerating tiny charged blackholes!

= Accelerator technologies =
There are a wide variety of accelerator technologies and here we try to summarize them as best as possible. Different accelerator types may be sequenced one after another. Some particle accelerator technologies depend on the particles fulfilling certain requirements of speed (thus energy) and hence different systems may be employed in sequence. Accelerator systems are described in terms of their efficiency of turning input power into particle energy, their acceleration gradient measured in electric field which translates to volts per meter, and the mass per length meter. An interesting secondary parameter is their operating temperature. Some accelerators can operate well at high temperatures. Others require cold or cryogenic conditions to work well.

== Core accelerator assembly ==
We define the core accelerator assembly as the structure where energy is imparted onto the particles.

=== DC Accelerators ===
For very low beam energies we can utilize large DC electric fields between plates or grids to accelerate particles up to speed. This is the same technique as is used in ion drives and is the most common industrial accelerator used. Every dentist's office and old CRT television uses this acceleration technique. Ion accelerators often also use a DC accelerator to generate the initial beam of ions, while electron beams utilize other techniques. These machines are limited by material breakdown to about 0.5 MV/m accelerating gradient and typically no more than a couple of MeV total particle energy for extremely large machines. The highest energy DC accelerators can reach 40 MeV for proton beams, but this is for machines massing in the 100s of tons and at low total beam powers. Very large DC accelerators are currently the only tested method for accelerating macrons up to 10’s of km/sec.These accelerators are limited by the risk of arcs jumping between the electrodes at too high voltage differentials.

=== Radiofrequency Quadrupole ===
A more advanced first stage accelerator for ion beams is the radiofrequency quadrupole or RFQ. These machines utilize low frequency radiowaves and carefully designed conductive bar structures to efficiently generate high current ion beams up to energies of ~7 MeV in a couple of meters. RFQs were a revolution in ion accelerator physics and caused quite a stir when the USSR first showed them off as they provided a much higher power and higher quality ion source in a smaller package than anything the USA had built! RFQs have also been proposed as methods of efficiently accelerating macrons up to >100 km/sec. These machines can be normal or superconducting and can operate with efficiencies as high as RF LINACs in theory.

=== Cyclotrons and Betatrons ===
One of the earliest ways to accelerate particles up to relativistic speeds is with circular accelerators known as cyclotrons (for ions) and betatrons (for electrons). In Cyclotrons the accelerator is a round chamber which houses two half-circle electrodes roughly shaped like a D (and thus called “dees”) separated by a gap, sandwiched between the poles of a large electromagnet. The magnet creates a static magnetic field which bends the path of charged particles into a circular motion. An oscillating alternative voltage is applied to the electrodes (what we often call RF or radiofrequency) to accelerate the ions. This radiofrequency field’s oscillation is matched so that one cycle of the voltage matches one circuit of the particles. As a consequence when crossing the gap from one electrode to the other, the charged ions experience a voltage gradient and are accelerated. As the particles gain energy and speed, their radius in the magnetic field gets larger. The result is an overall spiraling trajectory from an injection point in the center to an exit point at the rim where the charged particles are bent out of the cyclotron. Cyclotrons use the same voltage gradient many times over, defeating the electrostatic breakdown that limits the single-time electrostatic accelerators. As a consequence, proton energies of up to 70 MeV may be reached. Cyclotrons energies can also be increased by switching from normal magnets and RF resonators to superconductive components. This eliminates various losses from electric resistance and absorption and also allows for much stronger bending magnets, which allows the cyclotron to contain particles up to a higher energy if desired. Superconducting cyclotrons have achieved up to 520 MeV. Still, cyclotrons suffer from poor beam current, pulse rate, and beam quality issues that make them a less than ideal choice. The large magnet required (measured in meters) also does not lend itself to a lightweight weapon and as such, cyclotrons are regulated to niche medical and physics uses.

A close cousin is the betatron, which is used to accelerate electrons. In this case the magnetic field strength is increased as the electrons speed up, which prevents any spiraling trajectories of the particles. Betatrons have been operated up to electrons energies of 300 MeV, but are more typically in the ~10 MeV range. In principle a betatron could also be made superconducting, although magnet ramping may prove to be difficult. Other types of ring shaped accelerators may prove to be more useful, although electrons are heavily limited by synchrotron radiation and it is often better to use linear accelerators for them. For now betatrons remain limited to small (sometimes portable) accelerators for nuclear physics, industry and security work.

=== Radiofrequency Linear Accelerator ===
The real work-horse of the particle accelerator world is the RF linear accelerator. These babies use the same alternating field trick that cyclotrons use, but in a straight line rather than needing to bend the beam around in a circle. This provides a couple of nice advantages over the more traditional cyclotron. First and foremost you get to leave that large, bulky magnet behind which saves you a ton of mass, cooling and power consumption! You also get to now have much higher RF frequencies (1-10 GHz right now compared to the MHz used in cyclotrons), which means higher accelerating gradients. To generate these powerful, high frequency RF waves (also known as microwaves) that drive the accelerator we typically use military radar power supplies or (in more advanced cases) even lower energy particle accelerators! State of the art RF LINAC technology is well into the 100 MV/m range and have no upper limit on beam energy! The Stanford Linear Accelerator reaches 50 GeV and was built back in the 1960’s! There have been traditional RF LINAC designs for machines reaching up to 1 TeV, with the only limit being how much space you have to put these monsters. They do operate best (i.e. with high accelerating gradients and efficiencies) with particles already moving at relativistic speeds, so any ions fed in need to use a series of lower gradients LINACs or other pre-accelerators. Electrons and positrons can naturally be used in LINACs since they accelerate up to relativistic speeds so quickly.

LINACs also have been shown to operate at high average powers (MW scale is now in use) and high pulse energy (10’s of kJ pulses fired at high rep rate) while also maintaining a high quality beam that stays well focused. The two newest advancements have been to push the frequency ever higher for higher acceleration gradients (300+ MV/m has been shown and there are plans for even more) and to switch to superconducting RF accelerating cavities (the name of the actual piece of equipment that does the accelerating) which then dramatically increases the electrical to beam efficiency (80-90% has been shown and even copper waveguides have been operated at 70%). All in all, LINACs beat the tar out of cyclotrons which is why we don’t build cyclotrons anymore. To quote an old Fermilab engineer “cyclotrons are an expensive and bulky way to get a really crappy beam or a cheap way to get a giant chunk of lightly radioactive steel that can never leave your lab”.

LINACs work in detail by shaping the radiofrequency energy in a confining waveguide. The oscillations of the RF energy may be held in space, or allowed to travel down the waveguide over time. Radiofrequency waves have connected magnetic and electric components, which oscillate.

[[File:2022 08 16 GalLib ElectromagneticWave.svg|center|800px]]

Inside the RF accelerator, the electromagnetic radiation is oriented 90 degrees to the particle travel direction, which points the associated electrical field in the travel direction of the particles.The radiofrequency field naturally oscilates along its wave, and the connected electrical and magnetic fields oscilate as well: building up, reaching a peak, and diminishing, inversing, building up, reaching a peak, and diminishing, and the cycle repeats. The detailed behavior depends on the specific cavity.

[[File:CavityWithEM.png|600px|thumb|A resonant RF acceleration cavity and the electric field inside of it, created by resonating RF fields, at one moment in time.]]

Particles of the right charge in the right space at the right time experience an acceleration from the electrical field, being drawn to the opposite electric charge. If they land in the other region, they are instead decelerated. Inbetween, particles experience less to no acceleration. This behavior naturally separates a stream of particles into discrete '''bunches'''.

[[File:CavityWithEMAndParticles.svg|600px|thumb|Bunches being accelerated inside the electric field. ]]

It also creates an implicit limit to what can be effectively accelerated using a specific radiofrequency accelerator design. The particles driven through it have to keep pace with the oscillations of the electric field, reaching the next correct region inside the cavity at that time when the electric field is oriented correctly. If they are too slow to keep inside the accelerating regions, the accelerating action will be ineffective. This requires RF cavities to be matched to the particle's velocity gain under energy imparture and their velocity when entering, being inside, and exiting the accelerator. For electrons and positrons, this isn’t that complicated. Since these lightweight particles accelerate readily, they have few problems “staying in the pocket”. Particles that are slower to gain speed, such as ions and even larger particles, must however be accomodated. This in turn impacts how much acceleration and energy per meter can be fitted into the accelerator in the first place. The acceleration of heavy particles requires specifically designed and sequenced RF accelerators for specific regions of particle speed. This in turn complicates the design of the larger accelerator infrastructure, with multiple discrete stages getting the particles to the energy required so that the next stage in sequence can take over.

RF cavity accelerators are limited in their eV/m capability by the critical current at the containing waveguide. To increase the acceleration provided, more intense electrical fields must be built up. Eventually the electrical charge of the RF waves becomes so strong it can rip electrons out of the cavity walls, causing an arc to form and the cavity material to break down physically.

=== Advanced Accelerator technologies ===
Present accelerator technology caps out at around 250 MeV/m for the most overdriven designs. This poses issues for a variety of applications, where higher energies in shorter distances are desired. In science, "desktop" and "chipset" accelerators are of interest. When militarizing particle beams, some effects only become available at very high energies - but you want something compact enough to turret.

urrent and even near future accelerator isn’t really ready to supply that sort of scale quite yet, so we have to move on to the cutting edge of accelerator physics. There are three big options with sub technologies within these broad fields. We have extreme RF cavities using much higher frequencies, plasma accelerators using lasers for power and plasma accelerators using particle beams for power. Each technology provides a unique series of advantages and disadvantages, but fundamentally they are all a way to make linear accelerators that are much smaller.

==== Dielectric Wakefield Accelerator ====
LINACs using much higher frequencies than traditional LINACs have two fundamental problems to fight. The first is electrical breakdown and the second is generation of those mm-wave (10-100 GHz) or even THz (100-1000+ GHz) sources at sufficient power and efficiency to power a particle accelerator. As the frequency of the driving power goes up the electric field climbs up, which means the accelerating gradient gets larger (which is good) but also means that we run a higher risk of a huge arc forming in the accelerator and breaking it (which is bad). This is often fought by changing the accelerator cavity material from a metal (or superconductor) to a dielectric material like ceramic, or by using a coating of dielectric, which can sometimes earn them the moniker of Dielectric Wake Field Accelerator (DWFA). The issue of creating powerful and efficient enough sources of this higher frequency drive power is being actively pursued for a variety of fields (5G cell signals, beamed power, etc) but is still quite immature compared to the microwave RF sources of traditional LINACs. There are options using essentially "over-clocked" microwave sources, low energy particle beams, and even laser-plasmas. There is still a lot to be done here, but these sorts of problems are being actively worked on there are working accelerators using this technology. There is potential for GV/m or greater acceleration gradiatens and all of the same advantages of traditional RF LINACs (i.e. high powers, high fire rates, high energy pulses, quality beams) should stay the same. The wall plug efficiency is liable to be a bit lower, but that will be highly dependent on the efficiency of the source of these high frequency waves.

==== Laser-Plasma Accelerator ====
Plasma accelerators using lasers for power are a whole world in and of themselves. There are a wide variety of ways to accelerate particles with lasers and plasmas, but we will focus on the most studied technique known as Laser Plasma Accelerators (LPA). LPA accelerators utilize ultrashort (<1psec) and ultraintense (>10^18 W/cm^2) lasers to drive a plasma wake (thus sometimes earning them the name laser wakefield accelerator) and accelerate particles through the resulting charge differential. They have only shown the ability to accelerate electrons and positrons, but any ions that could be injected at the right time and at extremely relativistic speeds could conceivably be accelerated this way. In the operation of a typical LPA the ultraintense laser is fired into a jet of gas (turning it into plasma) or a channel of lightly pre-ionized plasma. The laser then blasts through this plasma like a boat through water and leaves a void behind it (the wake). This void ends up having a HUGE accelerating gradient (100 GV/m) and suffers no risk of electrical breakdown since the plasma is already broken down! A pre-accelerated bunch of particles can be injected into that void at the right time (called the witness bunch) for a BIG acceleration or you can rely on the wake itself spewing electrons out into the void and make a beam out of the plasma itself. The gas or plasma then quickly is reformed and the laser fires as soon as its cooling and charging allows. Keep in mind that even though the beams can hit incredible energies in short distances, the laser itself is not a pocket sized device. These lasers are room sized at minimum and not easy to shrink. The pulse lengths required are also much shorter than most of our hypothetical weapons-grade pulse lasers that may operate in the nanoseconds regime - dual-use may prove difficult. The intensities reached also pose significant issues for the laser hardware. If you don’t spread the energies out, it becomes quite possible you will break parts of your laser from the produced intensities.

Current LPA systems suffer from a couple of problems that are being actively worked on. First and foremost they are incredibly efficiency limited by the laser tech being used. There are plans to move on to much more efficient fiber lasers and eventually make LPA machines that operate in the 25% or greater wall plug efficiency, but that is not currently doable. These lasers also need to be fired at higher rates than the current ~1 Hz, but that is much more of a limit of laser engineering than accelerating engineering. LPA’s also have yet to link more than one accelerator together and the lasers are limited in just how much plasma they can accelerate through right now (~20 cm max) so the beam energy runs into a wall quite quickly. Aside from R&D on linking LPA’s together in a way similar to RF LINACs there are two other options. The acceleration gradient is higher the denser the plasma, so a move to denser plasmas and shorter wavelength lasers (to penetrate the plasma) could provide higher beam energies in similar distances. The other option is to use clever optical techniques like the so-called “flying focus” to allow for the laser to go through much longer plasmas. This could in theory allow for more efficient LWFA’s and single stage machines that provide ~1 TeV electron beams in meters! The final issue is beam quality. Even the best LPAs will not make beams that have as tight of a beam spot, energy spread, or emittance as a normal RF LINAC. This means they will end up needing some beam “cleaning” if you want to get a lot of range out of them.

LPAs are of engineering interest for very short wavelength X-ray free-electron lasers in a compact format. Currently these facilities require linear electron accelerators many kilometers in length. With an LPA as the electron accelerator, an XFEL able to image molecular structures and chemical reactions as they happen might be shrunken down to a much more compact format. (Sometimes this is discussed under the ultimate promise of the “benchtop” or even “chip-format”(sic!) XFEL)

==== Beam-Plasma Accelerator ====
Plasma accelerators using particle beams for power are very similar to LPAs in principle but use a particle beam rather than a laser. We typically just call these things Beam Plasma Accelerators (BPA) for short. The concept is very similar to the LPA in that a high intensity beam comes in and blasts a void into a plasma, which particles can then be accelerated in. BPA provides some interesting advantages over LPA in that you can use highly efficient superconducting RF LINACs (or synchrotrons if you want) to provide the particles at very high efficiency. They also have little in the way of plasma density limits and the accelerated beam takes on the characteristics of the beam driving the BPA. Thus little if any beam cleaning will be needed after acceleration. The big downside is the need for a relativistic particle beam to drive the effect, and typically one in the 100’s of MeV to low GeV range, depending on the particle used. Thus your final accelerator may be short, but you still won’t be making a handheld particle beam device.

== Accelerator Shapes ==
The accelerator technologies discussed above are the means by which we create situations for particles to gain energy. The accelerators become embedded in a larger structure that produces, directs, accelerates, even stores the particles until they are directed towards their final use.

There are two principle shapes for accelerator assemblies. Ones where the beampath is not closed onto itself, and designs where the beam path is closed on itself in part of the assembly, forming a loop.

=== Linear accelerator ===
Everything from electron generation to acceleration to conditioning is built in a straight line, one after another. This is the Linear Accelerator or LINAC. The main catch here is that there is only one single passage for the particles through the entire accelerator. Re-using accelerator hardware to drive the same particle to higher energies by repeated passage is out. The particles have to reach their desired energy in a single pass. Practically this puts limitations on the energy achievable with heavy particles in a LINAC. The length required gets impractically large. Even for electrons, lengths of hundreds of meters to kilometers may be required, unless advanced accelerators with high gradients are used. This said, every accelerator contains linear sections. Since so much high-performance accelerator technology depends on particles having a minimum velocity, ions must be “pre-conditioned” in a set of linear accelerators before the re-use of an accelerator through looped beam paths can be considered.

LINACs of low energy may be used in many applications. Medicinal they find use in radiation therapy and radiosurgery.

=== Synchrotrons ===
The other shape which has some interesting properties are accelerators where beampaths are looped onto themselves. These are called synchrotrons. Since beam paths are closed onto themselves, particles can pass through the same accelerator unit repeatedly, gaining more energy on every passage. This is advantageous for heavy particles which do not gain much speed on any single passage. They can pass through the same accelerator thousands or even millions of times instead. This is obviously advantageous for facility bulk and mass and allows us to reach energies that would otherwise be impractical. The accelerated beam bunches can also be held in storage, looping like the rotating mass of a flywheel, holding energy. The LHC can store some 700 Megajoules of energy in the recirculating beam after full speed has been reached.

That said, the devil is in the details. When charged particles are bent in circles they emit electromagnetic radiation which scales very rapidly (to the fourth power!) with relativistic gamma factor and with bending radius (inversely to the second power). This phenomenon is called synchrotron radiation. Due to the scaling, high energy particles emit more synchrotron radiation than lower energy ones and lighter particles (like electrons) at those high energies emit much more than heavier particles. In addition, the bigger you can make the bending radius of the synchrotron (often called “the ring”) the less synchrotron radiation you get.

[[File:SynchrVsE.svg|800px|center|thumb|Synchrotron radiation per particle versus particle energy for electrons and protons in a 50 meter in radius ring.]]

[[File:SynchroVsRing.svg|800px|center|thumb|Synchrotron radiation emission per particle for different particles at different energies, over varying bending radii. ]]

Although these numbers look small, keep in mind that this is PER PARTICLE. A typical pulse in an electron accelerator will have more than more than a billion electrons in it! Large synchrotrons like the LHC will have over 300 trillion ions circulating through it! Synchrotron radiation necessarily has to be managed to reduce radiation shielding, the radiation load (and possibility of hardware failure) on the accelerator, and to reach high energies in the first place - you might lose all the energy your accelerators are adding to synchrotron radiation!

Deflecting the beam also requires the application of magnetic fields, and higher degrees of deflection, in a smaller length, at higher particle energies, require stronger and stronger magnetic fields. Eventually multi-Tesla fields from superconductors will be required. And due to action and reaction, those magnets need to be braced against the pressure the beam exerts as it is deflected. Ultimately this leads to the same limitations as with flywheels - the mechanical strength of the braces and static connections that keep the magnets together as a larger structure is the accelerator’s limitation on J/kg held in storage. A further complication: The magnet strength has to be matched to particle energy. As a beam is accelerated, the deflectors have to ramp up their power to keep the degree of deflection constant.

Sometimes, synchrotron radiation from electrons is intentionally induced in a ring-shaped accelerator. Since synchrotron radiation of high-energy electrons tends to be hard x-rays emitted along narrow vectors, it’s a way to get x-rays used in various imaging techniques in material science and molecular physics.

Synchrotrons used for ions also need to respect the limitations of the involved accelerators. Synchrotrons often use radiofrequency accelerators. Since radiofrequency accelerators must match the speed and speed gain of their particles, an ion synchrotron cannot be made up of a single accelerator loop. Instead there are sequential loops connected in sequence, each designed to boost the particle through a specific velocity and energy region so that the next accelerator takes over. A designed example is the LHC. Particles headed for the main LHC actually circulate in two other synchrotrons beforehand, gaining sufficient energy so that the LHCs RF cavities can take over. (Also, before being injected into the first synchrotron ring, the ions must first go through multiple linear accelerators using different technologies to reach a base energy.)

An interesting design consideration: Synchrotrons don’t necessarily have to take the form of a (near-ideal) ring. Funamentally any dipole represents a single point of deflection of a straight path - they're polygons! Any geometric shape that contains a loop, and where any single bending curve keeps within the limits of the synchrotron radiation you are willing to tolerate, works. As such, complicated two- or three-dimensional “folded-up” shapes are also possible formats for a synchrotron.

=== Storage Rings ===
Since synchrotrons have to match their deflection magnet strength to the energy of the particles being accelerated, a single synchrotron track cannot hold more than one small section of beam bunches in storage at a time, and becomes unavailable to accelerate further bunches until it discharges the stored beam. This is disadvantageous for various applications where we may want to store a fully accelerated particle beam “bolt” for a longer time.

Storage rings are separate synchrotrons designed to hold multiple sets of bunches in circulation at a target energy at a time. The bunches are delivered from an accelerating synchrotron (sometimes called a booster ring) and held in circulation in the storage ring. Since there are synchrotron radiation losses, a few accelerator sections must compensate for these.

Storage rings can also be used to condition a stored beam. A common thing to do in storage rings is to “cool” a beam, draining unwanted movement energy out of the particles. Beam cooling may take significant time, which makes it ideal to conduct in a storage ring. The result is a beam with less tendency to drift apart, advantageous for many applications.

== Other parts ==
=== Particle sources ===
Particle sources are assemblies which supply particles to accelerators. Here we can group them into three categories: electron sources, ion sources, and exotic particle sources. The later involve the production of particles by means of gathering them from collision reactions (or the handwavium blackbox of your choice) and feeding them into another accelerator for further use. Positrons for example may be manufactured in beam collisions.

Electron sources require the production of a free stream of electrons. The oldest method here are thermionic sources. A hot piece of material ejects free electrons into a vacuum, where they can be drawn in using electric differentials and forwarded to the accelerator.

Field emission electron sources instead use a strong potential between a needle-like cathode and an anode plate with a hole. The electrons are drawn from the needle and through the anode due to the high charge differential. This method produces a “cooler” beam with less random divergence than thermionic sources.

A further improvement still are photoemission sources. Here the precise application of laser light onto a photosensitive material energizes electrons in the photocathode until they break free. This process is thermodynamically very precise and due to the nature of the laser light, also very controllable. The result is a low-emittance stream of electrons. These may be accelerated using electrostatic attraction with an anode at one end of the vacuum chamber, or directly using an RF field.

Ion beam sources are more complicated. The physics of ion formation are complex, which opens many design choices when creating an ion beam. Generally a plasma must be produced, from which the ions can be extracted.

The plasma can be formed in a variety of ways. Electrons from a local cathode-anode array, plasma frequency heating, the injection of electron beams, the application of laser light. The end results can be as diverse. Some methods only produce weakly charged ions and on many vectors, so with high emittance. Other methods offer greater ion charges or reduced emittance.

=== Magnets ===
While acceleration of charged particles is effected with electric fields, magnetic fields can be used to bend the trajectories of particle beams. As such, electromagnets can be used to bend the trajectories of charged particles in various ways, and in a controllable fashion.

==== Dipole Magnets ====
Dipole electromagnet are used to steer the entire particle beam in one axis. (To steer in two axes, we must either rotate one dipole about the beam axis, or couple two dipoles in sequence.) They can be used to direct a beam inside a particle accelerator, inject and extract beams out of looping constructs by switching on and off at the right moment, and point a beam at a target.

In a dipole magnet, a positive and negative coil oppose each other, embedded in a “choke” assembly often made of steel. Charged particles moving through the magnetic field between the coils experience a deflection in one direction specified by the polarization of the field, and a degree of deflection specific to the field strength.

==== Quadrupole Magnet ====
Dipoles can steer, but they influence all particles equally. As such, they cannot be used to manipulate the beam's shape. The first type of magnet able to influence a particle beam’s shape are quadrupole magnets.

They are assembled from four coils at 90 degree angles, with coils of the same polarity opposing each other. The passing beam is compressed on one axis, and expanded on the other. To focus the beam overall, at least two quadpoles must be arrayed in sequence.

Quadrupoles are the basic and essential “magnetic lens” of particle beams. Without them, charged beams would inevitably defocus and run havoc.

==== Hexapole Magnets ====
Hexapole magnets exist to address one problem of quadrupoles - they focus differently depending on the energy of the particle passing through the magnetic field. The effect is likened by particle accelerator physicists to the different diffraction of light in a prism - and hence the effect is called “chromatic aberration”. Hexapoles are used to correct for this, stronger deflecting higher-energy beams.

==== Integrated optics ====
As the field of particle accelerator science and engineering advances, more complex designs are proposed to handle more complex phenomena inside accelerators. An upcoming field is “integrated optics” - complex magnetic systems with nonlinear behaviors. These could provide more advanced manipulation of electron beams, building upon existing capabilities and adding new ones for shaping a beam.

=== Plasma lenses ===
Plasma lenses are an up-and-coming technology, where very high currents in plasma are used to create very strong magnetic fields. Similar to plasma-based accelerator technology, the fact that charges are already free in plasma is used to circumvent conventional charge breakdown and critical current limitations on solid-matter devices. The resulting magnetic fields can be much stronger and bend beams over shorter distances. Furthermore, by creating complicated profiles in the plasma, elaborate beam manipulations could be performed, more akin to sophisticated optical elements. As well, the charges in the plasma neutralize the particle beam in the region of the plasma lense - affecting its propagation behavior further and in a positive manner, as self-repulsion is neutralized.

The catch is that this is plasma - that means you are dealing with all the complications of plasma physics. Heating of the plasma has been an early challenge to successfully forming and maintaining plasma lenses.

Plasma lenses require an appropriate chamber, possibly active containment to keep the plasma from losing energy to the walls or leaking into other accelerator parts, and the necessary plasma sources, maintenance heating devices and then lense shapers and control sensors. In high-powered plasma lenses there will also be a need to cool the plasma as it heats up from waste heat. Potentially we may manage this by open-cycle cooling - venting the plasma controlled in one direction into open space, rather than have a closed-cycle solution for reconditioning the plasma. This could cause some very cool visuals!

Plasma lenses for particle beams should not be confused with optical plasma lenses for lasers! Those exist as well, but are their own topic.

=== Chicanes ===
Chicanes are assemblies of multiple dipole magnets used to manage the longitudinal dispersion of a particle beam due to different particle energies. Since particles with higher energy move faster, they drift ahead in the beam distribution over time; and particles with less energy increasingly lag behind.

Chicanes deflect high-energy particles onto an outward curving trajectory. Since traversing the curve takes longer, the high-energy particles are effectively delayed, allowing lower-energy particles to catch up. By the end of the chicane assembly, the bunch has been compressed in the longitudinal direction.

Chicanes however, do not remove energy from too-fast particles. That is the responsibility of other structures.

=== Undulators and wigglers ===
Undulators and wigglers are magnetic assemblies used to transform particle energy into electromagnetic emissions, with different properties from the specific implementation of a shared basic principle.

Assembled from stacks of generally dipole magnets (other configurations exist), both undulators and wigglers force charged particles onto “wiggling” trajectories by bending their path left and right. Wigglers force very strong arcs; undulators less so. In both cases, the rapid deflection of the charged particles causes them to shed parts of their energy as electromagnetic emissions. Due to the energy of the particles, these photons are generally highly energetic x-rays.

Undulators may be used to build free-electron laser systems.

They can also be used as a means of cooling the beam. The removal of energy from particles can also suppress unwanted movement modes, allowing accelerator structures to re-add energy only to desired movement modes. Of course, this means the particle beam actively loses energy, and produces x-rays that have to be absorbed (and cooled away.)

=== Beam coolers ===
Beam coolers are a collection of systems and strategies that are used to remove unwanted movements from a particle beam. Undulators and wigglers are one way to do this.

One method set of beam cooling relies on finding particles who are very “hot” with random motion, and either removing the particles from the beam using a kick device, or applying a precise counter-force that acts against the unwanted motion.

If one is dealing with ions, one clever trick is to inject a beam of electrons. Due to the thermodynamics within the particle population, the lighter electrons absorb heat from the heavier ions, taking this unwanted motion energy with them when the electron beam is bent out.

As well, lasers can be used (what can’t lasers be used for!) to apply counter-force on ions and remove unwanted movement in this manner.

Beam coolers are generally not instantaneous devices. They work over time to make a beam better. Often they involve the removal of energy or particles from the stream, requiring accelerators and sources to make up the loss and creating waste heat.

=== Beam recovery ===
Some accelerator systems can actually be run in reverse - they can turn beam energy back into some other energy format and then electricity. This can be used to recover some to most of the energy from a particle beam before disposing of it. This is advantageous both for your energy bills and making a very high-energy beam liable to spray radiation everywhere, easier to absorb in a beam dump.

=== Beam dump ===
A beam dump is a construction where a particle beam can be routed in case it has to be disposed. Beam dumps are designed to safely absorb the particle's energy without leakage of dangeorus radiation or activated materials to the outside enviroment. Common materials are carbon and metals, embedded in an actively cooled assembly.

= Propagation physics =
==Magnetic fields==

Magnetic fields are things of motion. They are created by moving electric charges. And they only affect electric charges that are moving. An electric charge at rest in the presence of only a strong magnetic field will just sit there. But when it starts moving, the magnetic field interacts with it, starts deflecting it.

But not along the field. Oh no. The magnetic field does not point in the direction it will move the charge. No, the charge experiences a force that is perpendicular to the direction of the field. And also that is perpendicular to its own velocity. A charged particle moving straight down a magnetic field line will not experience any force.

On the other hand, if the particle is moving directly perpendicular to the field, the magnetic force will always push it to the side, even as it keeps turning. This results in the particle moving in a circle. The radius of this circle is called the <b>gyroradius</b>, and the rate at which the particle makes circles is called the <b>gyrofrequency</b> (or sometimes <b>cyclotron frequency</b>).
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Circular_motion.png|frameless]]
<td width=255>[[File:magnetic_forces.png|frameless]]
<td width=255>[[File:spiral_around_field_lines.png|frameless]]
<tr>
<td>To get circular motion, you need a force (blue) that is always the same amount but always pushing perpendicular to the direction of motion, toward the center of the circle. A charged particle in a constant magnetic field moving perpendicular to the magnetic field meets these conditions, and moves in a circular path.
<td>A diagram of how a magnetic field (magenta) exerts forces (blue) on a moving particle (with velocity shown in green).
<td>If a particle is moving partially along the magnetic field (magenta), it will drift down the field lines while spiraling around it (black path). Where the magnetic field is stronger (indicated by having the field lines closer together) it will execute tighter, faster circles in its spiral and will drift more slowly than when the field is weaker.
</table>

It can be convenient to split up a particle's velocity into a part parallel to the magnetic field (the <b>longitudinal velocity</b>) and a part perpendicular to the field (the <b>transverse velocity</b>). The longitudinal velocity describes how fast the particle drifts along the magnetic field line. The transverse velocity determines how tightly and how quickly the particle circles around the field lines.

If the gyroradius is much smaller than the extent of the magnetic field, you get a case where the particle approximately cannot cross magnetic field lines. It just spirals around them. On the other hand, if the magnetic field ends before the particle can make a significant portion of a full revolution, then the particle ends up being somewhat deflected from its original trajectory. If the gydroradius is enogh to deflect the particle into a region where the magnetic field has a different strength or direction, then things get complicated!

When a particle is spiraling around the magnetic field while drifting down the field lines into regions where the magnetic field is stronger, it will circle around the field lines in a faster, smaller circle. This increases the kinetic energy in the spiraling motion; this kinetic energy must come from the kinetic energy of the longitudinal motion. If the field becomes too strong, the particle will slow down all the way and come to a stop. Then it will be reflected back, pushed the other way down the field lines toward the region where the field is not as strong. This phenomenon is called magnetic mirroring, and is responsible for the way radiation gets trapped in the Van Allen belts around Earth with the particles bouncing back and forth along the Earth's field lines and getting reflected back and forth at the poles where the field is strongest.

If you want to calculate for yourself what the gyroradius <math>r_c</math> and gyrofrequency <math>f_c</math> are
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
r_c = \frac{\gamma \, m \, v_\perp}{q \, B}
</math></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
f_c = \frac{q \, B}{\gamma \, m}
</math></div>
where <math>B</math> is the strength of the magnetic field, <math>q</math> is the amount of electric charge of the particle, <math>m</math> is the mass of the particle, <math>\gamma</math> is the Lorentz factor, and <math>v_\perp</math> is the part of the velocity that is perpendicular to the magnetic field. To get these formula to work, you need to use a consistent set of units. So for example if you measure the magnetic field in tesla you need to measure the charge in coulombs (not fundamental charges) and the mass in kilograms (not eV).

One consequence of this is that you can bend the beam coming out of your accelerator by placing a uniform magnetic field across the end. So you can steer your beam to where it needs to go. You can also bend the beam inside of your accelerator – to make it go around in a circle, for example.

Another consequence is that if your beam has to pass through regions with magnetic fields in it, it will get deflected. If you want the beam to end up at the right spot, you will need to account for this. If the exact strength or direction of the field are not known, you will get unpredictable deflections to your beam.

If your beam has been neutralized, the magnetic fields will work to bend the positively charged particles one way and the negatively charged particles the other way. If you have enough beam current, these separated charges will attract each other and counteract the magnetic force. If you don't, the beam will split and you'll get positive charges zipping off in one direction and negative charges going another direction. If you needed to neutralize your beam so that it doesn't electrostatically blow itself apart, well now it will be doing just that.

Even perfectly neutral atoms can be torn apart if the atoms are going fast enough and if the magnetic field is strong enough. Atoms are pretty resistant to this, but you can get conditions where it happens.

Magnetic fields affect the path of beams. But beams are made of charged particles and thus can have currents in them. Currents create magnetic fields themselves, which can influence the surrounding fields. Even a neutralized beam forming a plasma, when it encounters a magnetic field, will get currents induced in the plasma that helps to screen its interior from the field. A beam with enough punch can deform the fields around itself rather than having the fields bending it. As a rough rule of thumb, if the energy density of the beam over a regions larger than a gyroradius is more than the energy density of the field over that region, then the beam gets to have its way with the field, and vice versa. Highly focused, high current beams can thus be expected to better deal with ambient magnetic fields than those with less focus or lower current.

==Focusing==

In the previous section, we discussed how a particle beam can be steered by directing it through a uniform magnetic field. We can make this kind of field with a magnetic north pole directly opposite a magnetic south pole (with additional shimming magnets to keep things uniform). Because it has two poles, one north and one south, this kind of magnet is called a <b>dipole magnet</b>.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:dipole_beam_steering.png|frameless]]
</table>

But we want to do more than just steer the beam. we want to be able to expand it, contract it, and focus it to a point. To do this, we need a different kind of configuration. We will put two north poles opposite each other and, and 90 degrees, two south poles opposite each other. With four poles, this is called a <b>quadrupole magnet</b>, or sometimes just a "quad".
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_magnet.png|frameless]]
</table>
At the center of the quad arrangement, the magnetic fields from each side cancel out. there is no field there. A particle going through the center will travel straight through without deflection. However, the farther particles get from the center, the higher the field they encounter and the more they get deflected. A particle slightly off to one side may experience a force that curves it toward the center-line. If the quads are properly shimmed, a particle twice as far away will experience twice the force and twice the deflection from its trajectory back toward the center. This curves all the particles so they meet in one spot.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_focusing.png|frameless]]
</table>
However, there is a catch. Those particles that are offset from the center-line at 90 degrees from these focused trajectories are bent away from the center-line rather than towards it! A quad focuses in one direction and de-focuses in the other.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_defocusing.png|frameless]]
</table>

Is there anything e can do to get an arrangement of magnets that purely focuses the beam? Well yes. We can put another quad arrangement downstream of the first, rotated by 90 degrees. Now consider a set of particles in an arrangement that is de-focused by the first quad. When they get to the second quad, they are further from the center than they were when they went through the first quad. Because the second quad is aligned to focus anything the first quad de-focuses, these particles will be focused; and because they are farther from the center, the focusing deflection will be larger than their previous de-focusing deflection. So the net result is that the particles are focused.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Dual_quad_focusing.png|frameless]]
</table>
For particles with the opposite orientation that are first defocused, these will be defocused by the second quad but because they will have converged they will be going through a region of lower field and will be defocused with a smaller deflection than they were initially focused. This dual quad arrangement will focus any particle that goes through it, regardless of orientation.

For small deflection angles, a dual quad set-up acts like an ideal lens for particles. An ideal lens can focus all the particles diverging from a point onto another point, or into perfectly parallel lines. It can focus particles on perfectly parallel lines onto a point. So now we can use the particle beam to deliver focused particles on our target.

Or, at least we could if all the particles came from a point, or if they traveled on perfectly parallel lines. Unfortunately, they don't. When particles are made, they come from a source with a finite spot size and a finite temperature. The random motion of the particles moving around from their initial thermal motion means that they are never moving perfectly parallel to each other. And when you try to focus them, instead of all converging onto a single point they make a spot of finite size. And if your focusing and beam bending elements are not perfect, you add additional imperfections to the beam that further hider your ability to focus it.

Your ability to focus can be quantified by a measure called <b>emittance</b>. Emittance is measured in units of a length scale times an angle - nominally meters-radians (m<math>\cdot</math>rad) but the technical literature more commonly gives units of millimeters-milliradians (or mm<math>\cdot</math>mrad) or even 2 <math>\pi</math> mm<math>\cdot</math>mrad. The divergence angle of a beam that you try to make parallel will the the emittance divided by the width of the beam. If you focus a beam down so it converges at an angle, the minimum spot size you can get at the focal point is the emittance divided by the beam angle. If you try to focus a beam with emittance <math>\epsilon</math> on a spot a distance <math>R</math> away with an initial beam diameter of <math>D</math>, the spot size <math>S</math> on the target you can achieve is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
S = R \frac{\epsilon}{D}
</math></div>

So to get good focusing, we want low emittance. Is there a way we can lower the meittance beyond what we get from our source? Engineers are pretty bright folks, and they have found some ways to do this. You can, for example, pass a beam through a thin foil. This causes some stopping, slowing the beam down. You can then accelerate the beam back up to its original speed. However, the stopping is along the direction of the beam particle's tranjectory, even if that trajectory is at an angle. Meanwhile, all the acceleration is in the direction of the beam axis. So the net result is to reduce the random motion perpendicular to the beam, and thus reducing its emittance. Similar tricks can be played with magnet set-ups that force particles going through them to emit radiation.

Another method can be used with a relatively high emittance beam, say a beam of protons or ions or muons. Electron beams can be made with substantially lower emittances. So you can send a low emittance electron beam to co-propagate along with your primary particle beam. The beams will interact, with the particles scattering off each other. This exchanges heat between the two beams and so the temperatures, and thus emittances, of the two beams get closer to each other (and, if given enough time, would become equal). If you repeat this often enough, you can get the emittance down to the emittance of the electron beams. This is more practical in beams going around circular accelerators, because they will have time to get cooled over and over.

A final method that has been used is to install sensors to measure the beam fluctuations. For beams going in circles, the time to get a light-speed signal directly across the diameter to the opposite side wil always be less than the time it takes for the particles to go around the curve to that spot. So you can send signals to give electromagnetic kicks to the beam particles to knock them back into alignment when they get to that spot.

It is intriguing to note that these beam cooling techniques can take <i>hours</i> to work! <ref>[https://www.bnl.gov/isd/documents/86783.pdf M. Minty, R. Connolly, C. Liu,T. Summers, and S. Tepikian, "Absolute beam emittance measurements at RHIC using ionization profile monitors", Brookhaven National Laboratory formal report BNL-105970-2014-IR]</ref>

But perhaps the simplest method to reduce emittance is just to speed the particles up. For particles moving at a fraction of light speed <math>\beta</math> and Lorentz factor <math>\gamma</math>, the emittance of particles from the same source with the same temperature and same random motion will be proportional to <math>1/(\beta \, \gamma)</math>. To reflect this, it is convenient to define a normalized emittance <math>\epsilon_n</math> such that for the usual geometric emittance <math>\epsilon</math> that we have been talking about
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
\epsilon_n = \beta \, \gamma \, \epsilon.
</math></div>

Normalized emittance measures makes it convenient to get basic estimates of particle beam performance and compare emittance of different sources and beams. For example, various high-performing electron beams around the world have managed to push their normalized emittances down to about <math>2 \times 10^{-6}</math> m<math>\cdot</math>rad to <math>3 \times 10^{-6}</math> m<math>\cdot</math>rad. Facilities with proton or ion beams seem to have about an order of magnitude higher normalized emittance.
<table>
<tr><td width=200>Machine <td width=300>normalized emittance (m<math>\cdot</math>rad) <td>reference
<tr><td>European XFEL <td><math>2.3\times10^{-6}</math> <td><ref>[https://journals.aps.org/prab/pdf/10.1103/PhysRevAccelBeams.24.110702 T. Hara <i>et al.</i>, "Low-emittance beam injection for a synchrotron radiation source using an X-ray free-electron laser linear accelerator", Physical Review Accelerators and Beams <b>24</b>, 110702 (2021)]</ref>
<tr><td>SLAC <td><math>3\times 10^{-6}</math> <td><ref>[https://accelconf.web.cern.ch/pac97/papers/pdf/4W009.PDF M. Hernandez <i>et al.</i>, "Emittance Measurements for the SLAC Gun Test Facility", 1998 IEEE]</ref>
<tr><td>LHC <td><math>3.75\times 10^{-5}</math> <td><ref>[https://www.lhc-closer.es/taking_a_closer_look_at_lhc/0.beta___emittance Beta & Emittance; Taking a closer look at LHC]</ref>
<tr><td>RHIC (with protons) <td><math>\approx 5\times 10^{-5}</math> <td><ref>[http://www.osti.gov/servlets/purl/910428-6sKfP0/ Zhang, S, and Ptitsyn, V., "Proton Beam Emittance Growth at RHIC", Brookhaven National Laboratory formal report BNL-77563-2007-CP]</ref>
</table>
So as a first estimate for your sci-fi electron guns, you can take <math>\epsilon_n \approx 3\times 10^{-6}</math> m<math>\cdot</math>rad and for proton guns <math>\epsilon_n \approx 4\times 10^{-5}</math> m<math>\cdot</math>rad. Then divide by <math>\beta \, \gamma</math> for your beam to find the geometrical emittance. Which in turn can let you figure out how well you can focus your beam.

==Space charge==

==Energy loss in matter==

Sometimes, a particle beam needs to go through stuff to get it where you want it to go. Stuff like air, for example. Or the tissue in the way of getting the beam to a tumor. Also, sometimes, you want your particle beam to interact with stuff. Like the material you are trying to machine with an ion beam, or the bacteria you are trying to kill on medical equipment. All of this falls under the general realm of the interaction of the beam particles with matter. This is an enormously complex subject, but we can cover many of the basics here.

The primary effect on the beam of any interactions with matter are loss of energy of the beam particles (<b>stopping</b>), random changes to the particle directions that make the beam spread out (<b>straggling</b>), and knocking particles clean out of the beam altogether (<b>scattering</b>).

===Ionization===

The particles coming out of your particle accelerator are charged. When a fast moving charged particle goes through matter, the electric field from the charge pushes and pulls on the various charged particles that make up that matter – mainly electrons and the atomic nuclei around which the electrons orbit. At speeds above approximately 1% of the speed of light, it is primarily light and nimble electrons being affected while the ponderous nuclei just sit there barely perturbed by the passage of the particle. When the passing particle yanks on an electron hard enough, it can pop it clean off of the atom it was attached to, leaving a positively charged ion behind and giving you a free electron that can go and glom on to a previously neutral atom to make a negative ion. This process, unsurprisingly, is called <b>ionization</b>. For most particle beams, ionization is the dominant interaction with matter; although when you get very fast or very slow particles other effects can become more important.

if your particle was not initially charged when it goes into matter, it soon will be. The electric charges of the electrons and nuclei in the matter will do the same thing to the particle that a charged particle would do to them. This will quickly strip off any electrons around a high speed atom, leaving an ion and bare electrons to go through the matter. Only particles that are fundamentally neutral (like neutrons or photons) will not be immediately ionized.

It takes about 1 to 10 eV to ionize an atom. This energy must come from the kinetic energy of the beam particle. Particles from an accelerator have millions to billions of eV. So it takes a lot of ionizations to meaningfully affect the particles. However, one thing about matter is that it has lots of atoms that can be ionized so you do get those lots of ionizations. Because each ionization is so small compared to the total energy this is often approximated as a continuous slowing down of the beam particles. So many individual ionizations tend to average out, leading to all the beam particles being slowed down uniformly.

When the particle is yanking on nearby electrons to tear them out of atoms, those electrons are yanking back. This jitters the beam particle slightly from side to side. Again, these sideways yanks tend to average out. The statistical variations from the average slowing down and side-to-size jerking are what lead to straggling. Protons and ions are heavy, and will be deflected little from their original paths - like bowling balls smashing through matchsticks. These particles tend to go in mostly straight lines. Electrons, on the other hand, have the same mass as the electrons they are knocking about. So electron tracks through matter tend to zig-zag around a lot more.

As the particle whizzes past, it leaves a trail of atomic destruction in its wake - a trail of ions where the atoms are literally torn apart (not the nuclei, just the atoms). These particle tracks can sometimes be seen in detectors like cloud chambers. The faster the particle is going, the less time it has to yank on nearby electrons and so the less ionization it leaves behind. As a result, fast particles are not slowed down much but as they start to slow down they continue to be slowed down faster and faster. A consequence of this is that the particle dumps most of its energy into the material at the end of its track, just before it comes to a stop. This sudden peak in energy deposition is called the <b>Bragg peak</b>.

The electrons that a particle knocks out can end up going pretty fast. So these secondary electrons can produce further ionization in what is called an <b>electron cascade</b>. if an inner core electron is knocked out of an atom, other electrons falling in to fill that empty spot in the atom can release x-rays (<b>x-ray fluorescence</b>) or cause shake-ups in the outer electrons that knock other electrons off the atom (<b>Auger effect</b>).

===Brehmsstrahlung===

A charged particle is surrounded by an electric field. If it is moving, it is also surrounded by a circulating magnetic field. These fields move along with the particle.

If the particle knocks into something, it will slow down and probably change direction. The information about this bump only goes through the electromagnetic field at the speed of light. So a lot of that field that was moving along with the particle might not get the memo in time and keep right on going. Stray electromagnetic fields like this get this self-inductance effect going on where they start making themselves over and over again, allowing them to propagate as freely moving waves of electromagnetic radiation. This EM radiation from a slowing down particle is given the delightful name of <b>bremsstrahlung</b>.

Particles from an accelerator can be moving so fast that they make bremsstrahlung in the x-ray or gamma ray part of the spectrum. If you get very high energy bremsstrahlung gamma rays, the gamma rays can interact with matter by boiling an electron and its antimatter counterpart, a positron, out of the vacuum (using the nearby matter as the necessary momentum sink for the process). These positrons and electrons can themselves be very high energy and can in turn create more bremsstrahlung, and the process can repeat until the particles are no longer high enough energy to continue making each other. This is called an <b>electron-gamma shower</b>.

The x-ray machines used by dentists and radiologists use both bremsstrahlung and x-ray fluorescence that you get when shining an electron beam onto tungsten to make their x-rays.

At very high speeds, bremsstrahlung can be more a important source of energy loss than ionization.

===Electron capture and screening===

When an ion has slowed down to about 1% of the speed of light or less, it is moving slower than the electrons zipping around their atoms in the matter the ion is going through. This allows the electrons to stick on to the ion, now becoming the electrons of the atom that is moving through the material rather than of the material itself. The ion begins to lose its charge, and the rate of ionization goes down. Soon, ionization becomes negligible compared to the beam atom bumping in to other atoms.

===Phonons and displacement===

When an atom or ion moving through a material bumps into another atom, it will cause that other atom to move. This can have one of two primary effects.

First, if the atoms whack into each other hard enough, the atom in the material can be knocked clean out of place. Like a pool ball being thwacked by the cue ball, the struck atom will go zinging through the material. Atoms getting knocked out of their place like this is called <b>displacement</b>. Displacement creates chemical changes to molecules as atoms are removed, and creates defects called <b>vacancies</b> in crystals (which, as you might imagine, is just a missing atom in the crystal lattice). When the atom comes to rest, it can chemically attack other molecules, again changing the chemistry, or create an extra vacancy called an <b>interstitial</b> jammed into the crystal lattice where it is not supposed to be. If the displaced atom is going fast enough, it can whack other atoms and displace them, leading to a <b>displacement cascade</b> like tenpins knocking each other down. As a result, at the end of its track, an ion can leave a region of disordered material and messed up chemistry where the atoms have all crashed into each other like a freeway pileup.

If an atom is displaced near the surface of a material, the displaced atom can shoot out of the surface and escape the material altogether. This is called <b>sputtering</b>.

Second, the beam atom might only deliver a glancing blow. Or the beam particle might be an electron, which is so light that it is usually incapable of displacing atoms. In this case, the struck atom gets kicked to the side a bit and runs into other atoms; maybe it is also pulled back by the chemical bonds of other atoms that it is attached to. This transfers the motion of the atom to other atoms around it in the form of pressure waves (or, in a solid, shear waves). This adds heat to the material without causing direct damage – although as you might imagine, enough heat can also cause localized melting, evaporation, or decomposition. The individual quanta of vibration – fundamental particles of sound, if you want to think of them like that – are called phonons in analogy with the photons that are the fundamental particles of light. Much like ionization, a particle that excites pressure waves slows down gradually and continuously, unlike the sudden changes in speed and direction you get from knock-out displacement.

===Nuclear collisions===

So far, we have only discussed the things that happen when beam particles interact with matter particles by electromagnetic forces. Even atoms knocking into each other to produce phonons or displacement happens through the mutual repulsion of the electrons around the atoms (although this is dressed by other weird effects like how two electrons can't be in the same place at the same time, or how the fields of one electron can affect other nearby electrons that in turn affect the original electrons, leading to odd correlation effects and charge density waves and other fun stuff that keeps condensed matter physicists and quantum chemists busy).

But electromagnetism isn't the only thing you can get going on. below the level of structure of the atom, you have the atomic nucleus. And a direct hit on the nucleus can end up giving very different effects.

The simplest thing that can happen is an <b>elastic collision</b>, where no energy goes in to any internal nuclear state or into breaking up the nucleus. An elastic collision just ends up giving you displacement (or possibly phonons, but if you hit the nucleus you usually hit it hard enough to send it flying).

But beyond that you can cause any number of nuclear reactions. You can leave the nucleus in an excited nuclear state, from which it usually relaxes back down to its original state by emitting gamma rays. You can knock out nuclear particles, such as neutrons, protons, deuterons (a proton stuck to a neutron), tritons (a proton stuck to two neutrons), helions (a neutron stuck to two protons), alpha particles (two protons stuck to two neutrons), or potentially even larger nuclear fragments. If you hit a heavy nucleus like uranium, you can cause it to split apart by fission. And if you hit the nucleus really hard you might just shatter it into many tiny fragments of those protons, neutrons, deuterons, tritons, helions, alpha particles, and heavier nuclear fragments mentioned earlier. To get any of these <b>inelastic collisions</b>, you need to hit the nucleus hard enough. If your beam particle doesn't have enough energy to excite the nucleus or knock particles, these things simply can't happen. As it gets enough energy to excite a particular interaction channel (as they are called), that process becomes possible and as the energy increases the process becomes more and more likely.

A nucleus is only about a femtometer across (<math>1 \times 10^{-15}</math> m). Which makes it only about <math>1 \times 10^{-30}</math> m² in cross sectional area. In the meantime, chemical bonds between atoms are usually about 1/10th of a nanometer (<math>1 \times 10^{-10}</math> m) long. In condensed matter (liquids and solids), this means that you have somewhere about <math>1 \times 10^{30}</math> atoms per cubic meter. A meter thickness of atoms thus presents just about about enough atoms that their nuclei can cover the projected area – in other words, you can expect a particle to go through (very roughly) about a meter of condensed matter before it whacks into a nucleus.

This assumes that a particle that goes through a nucleus interacts with it. This is usually a good assumption for particles that interact by the strong nuclear force, such as protons, neutrons, mesons, and the atomic nuclei of ions. But many particles cannot interact by the strong nuclear force, such as electrons or muons. These are much less likely to interact with an atomic nucleus even if they pass through it, so they will end up going considerably farther between hitting nuclei than the above assumption. But nuclei can interact electromagnetically (they are made up of charged protons, after all, and at a smaller level of charged quarks), so electrons and muons can produce some nuclear interactions; just far less than actual nuclear particles like protons and ions.

A nuclear strike is a discrete event that will remove the impacting particle from the beam and will (except for elastic collisions) probably produce a spray of radiation.

===High energy physics===

As you get to higher and higher energies, you unlock all sorts of new and weird and interesting processes that can happen. Boiling exotic particles out of the vacuum, unifying forces that had been divorced since the big bang, and producing bizarre unstable resonances in the quantum fields. This is what physics build accelerators to measure. From the point of view of getting a beam to its target, or what happens to the target, however, the main effect is mostly that the interacting particle is removed from the beam and a bunch of additional radiation is produced.

==Self pinching==

= Operational Considerations =
== Logistics ==
The logistics of particle beam can be a mixed bag. By itself, any particle accelerator is a complicated piece of machinery full of delicate parts that have to deliver high performance. This may require cryogenic temperatures with associated challenges on the materials and mountings used (insulation, and distortions under temperature changes). When working, particle accelerators invariably generate some radiation across the whole assembly, which irradiates and activates the structure. Some parts may be exposed to significantly higher radiation flux due to their working principles. Often the particle beam is a dynamic system that must be managed with active, closed control loops, involving sensors, modeling and control software, actuators, and the computing and communication hardware that brings it all together. This poses engineering challenges across all disciplines.

A practical particle beam is inherently a counterintuitively sturdy device. If it was too delicate, it would break too much to be useful, though of course this is a sliding scale. Still, while sturdy, this doesn’t eliminate complexity, it manages complexity. The result is a system that is the product of a high-tech civilization. A superior machine shop and well-trained engineers are a must to maintain a particle beam system. Some parts of a particle accelerator may be mundane - others may be custom-machined parts that require their own dedicated industries to produce. That said, the manifold utility of particle accelerators is on your side here. Since there are so many practical applications, it is economic to produce parts for industrial demand. (that said, these may not be the kind of components you want in specific high-performance applications.)

The main thing a particle accelerator requires the most is electricity. Any mature power production facility can supply the power.

However, all particle beams require at the least a source of neutralizing mass. The projection of a charged beam will leave the projecting object oppositely charged. In an atmosphere the natural plasma formed by the beam will ground everything out, but in space the diffuse vacuum will not do the job.

Furthermore particle beams using ions also require a feedstock for their ions to produce from, and in an appropriate format. This can be a gas, a fluid or an easily ionized solid. It can be the feedstock species in its pure form or in an easily split molecule, though every complexity adds to the amount of preparation work that has to be done before the feedstock enters the particle beam.

Don’t worry about the mass of your source or neutralizing ions though. Large particle accelerators, like the LHC in Switzerland, only consume tiny gas bottles worth of feed gas every couple years. The mass of a single 5.56 NATO bullet worth of hydrogen could feed a 10 GW, 10 GeV ion beam for nearly 4 days of firing! You will not be running out of beam feedstock easily.

== Cooling ==
Cooling a particle accelerator can be a serious challenge. Various bits of a particle accelerator love to be supercooled for efficiency or basic working principle - any bit of superconductor used in the accelerator, or magnets, or EM generators, will require cooling to its generally supercool operating temperature.

Even if Standard Temperature and Pressure Superconductors (273 Kelvin, 100 Kilopascal) are available, present superconductors have demonstrated superior performance by being supercooled. That means to achieve even stronger fields, even STP superconductors may be cooled to cryogenic conditions.

Cooling cryogenic systems sucks. Thermodynamics shows itself from its worst side. You can forget running cryogenic radiators - the Stefan-Boltzman Law penalizes the direct radiative cooling of anything cryogenic severely as much as it empowers very high-temperature radiators. This only leaves stages of heat pumps to provide cryogenic fluids that can in turn, cool cryogenic systems. Heat pumps draw power, and for such a giant temperature gradient as we can have here, they will draw a lot of power.

However, we have other options. Cryogenic systems also tend to be highly efficient, so while they do heat up, they do so very slowly and there is little overall energy absorbed. This provides the option of running off of bunkered cryogenic fluid in dewars for the duration of operating the equipment and not running any heat pumps all the time.

Other bits of a particle accelerator system aren’t as required, but really nothing is a fan of high temperatures. Going higher than room temperature can be treated as an oddity. High-performance lasers may not require cryogenic conditions, but they certainly like it cool. Radiofrequency producers of any kind generally fare worse with rising temperatures. That means cooling overall remains a challenge - 300 Kelvin is still not that great of a hot loop temperature for radiators.

== Safety ==
=== Ionizing Radiation ===
There are surprisingly many sources of ionizing radiation in a particle accelerator, besides the obvious threat of the output beam itself. Not all particles are accelerated correctly or in the correct direction. When such particles interact with the structure of the accelerator, they release a shower of secondary ionizing radiation. In electron accelerators and sources, stray electrons may be accelerated and release x-rays from bremsstrahlung as they collide with surrounding matter. The resulting doses can be significant and dangerous. Also, such collisions leave behind residual activation, making the accelerator a radiation hazard for some time even after it has been shut down. Generally, people should not work near the particle accelerator while it is functioning, and radiation barriers isolate the core accelerator from adjacent facilities. (In real-life facilities, you will often see accelerators inside larger buildings being cocooned inside doped concrete plates to shield the rest of the room.)

=== Radiofrequency ===
RF energy is used in many kinds of particle accelerators. This energy may leak from the structures it is generated, contained and used in due to damages or incorrect assembly. As microwave radiation and above is used, the radiation can be readily absorbed by bodies, hydrogen-rich materials or other objects. The end result of this can be rapid heating and thermal effects. Especially in accelerators designed to throughput a lot of energy (say a GW or more), this may become a serious issue if a waveguide is damaged. It’s not recommendable to stand in front of a multi-MW radar either!

=== High-powered electrical and electronic systems ===
The operation of the magnetic systems and accelerators often involves high-powered electrical and electronic systems. High voltages and/or amperes are involved. This poses risks like shock hazards, electrical arcs, and electrical fires. Some electrical systems such as capacitors or inductors may have combustive or explosive failure modes.

=== Flammables ===
Parts of an accelerator or its infrastructure may be manufactured from flammable substances such as hydrocarbons. In the case of an ignition source such as a high-powered beam cutting into a physical substance or an electrical fault, these may catch fire.

=== Lasers ===
Some accelerator systems use high-powered lasers. Laser plasma accelerators use very high-powered pulse lasers to provide pumping power. The safety concerns of lasers for blinding, physical damage, and possible exposure to ionizing radiation as a secondary effect of the beam impact, apply. Some high-powered lasers can also drive electrons which generate strong EMPs, which may damage or destroy other systems in turn. Parts of a laser assembly or laser beam guide may shatter if exposed to overloading conditions.

=== Magnetic systems ===
Parts of a particle accelerator may use strong magnets, up to superconductive magnets with multi-Tesla fields. While the magnets are integrated into chokes that shape the magnetic fields, fringe fields invariably extend beyond the core magnet installation.

Multi-Tesla transient electromagnetic field can affect the human nervous system temporarily, inducing nausea, vertigo, metallic taste and optical phenomena, if the person moves in relation to the magnetic field. Otherwise, the risk of exposure is low. That said, electronic systems that are not shielded properly, such as say, implants, could be adversely affected by the magnetic fields.

The biggest physical risk are reactions of other magnetic objects with the magnets. You can look up cases where metallic objects interacted with the multi-Tesla fields of MRI scanners. Such parts may fly across the room towards the magnet at increasing speeds, and damage anything or injure anyone in their path, possibly ending with the object or person squashed (or impaled!) to the magnet.

Also, large superconductive magnets store not insignificant amounts of energy. If the superconductive property breaks down, the magnet quenches and becomes a resistor. The resistance site heats up as current tries to flow through it, expanding the quench and possibly being heated until it sublimates into gas. This could possibly happen explosively. In such a case, up to many GJ of energy might be unleashed in a conflagration or explosion (subsonic or supersonic), damaging surrounding systems and ejecting shrapnell.

=== Cryogenic cooling systems ===
Particle accelerators often involve cryogenically cooled systems. This means cryogenic fluids like nitrogen and helium are present in the accelerator infrastructure. Such gasses have both a very low temperature and a high ratio of gaseous volume at high temperatures to their liquid volume.

Exposure to the leaked fluid can lead to cryogenic cold damage, injury, and death. Furthermore such gasses are efficient at displacing ambient breathing gasses such as oxygen, so they pose an invisible, and rapidly spreading, suffocation hazard.

Also, if insulation is damaged, ambient heat may leak into the cryogenic circuits, and cryogenic fluid may begin evaporating. This can lead to dramatic rises in internal pressure. Relief valves should trigger to safely direct such pressure spikes away from important places and keep the system from failing catastrophically, but otherwise, pipes and cryogenic storage devices may fail, up to explosively.

=== Beam excursions ===
If the magnets responsible for manipulating the beam break down, the beam may excurse from its designed paths. The beam may become defocused, or maintain its focus. The beam begins cutting into the accelerator structure, evaporating material and depositing energy. This will definitely foul the ultra-high vacuum inside the accelerator, leading to secondary radiation production as the beam collides with gas inside the accelerator. Some sensitive facilities requiring ultra-high vacuum such as some RF cavities may “crash” and become ruined. Further systems may become damaged or destroyed from either physical stress or the radiation pulse. If the beam bunches contain a lot of energy and maybe are heavy particles, the deposition may lead to an energetic local explosion. Alternatively the beam may escape and cut into structure beyond the particle accelerator itself, damaging and destroying equipment and seriously hurting or killing personnel.

In the worst cases, a beam excursion on a very powerful beam may lead to an explosion that destroys the accelerator utterly. Even in a less bad case, the radiation pulse might damage or destroy critical equipment around the accelerator and take the entire facility offline.

=== Beam dumping ===
If any other systems fail, the accelerator probably has to be shut down. This poses a problem - what do you do with the beam energy that is still in the accelerator system?

Sometimes, the beam is designed to leave the accelerator immediately, and shutting down the beam is easy. But in the case of cyclotrons and storage rings, the beam may circulate for a while. What do you do then? You have to dump the beam from the accelerator. Special beam stops are used on research and industrial accelerators, large blocks of dense material able to absorb the beam. During this process the blocks both heat up and become radioactive. For particle beams storing very high energies, this may become an increasing challenge. How do you safely dumb a beam containing many GJ or more? In some cases you may have no other choice but to dump the beam out into the environment. Where and when can you do this safely without threatening third parties?

= Application areas =
(This is a skeletal overview summary)
Particle accelerators have many, many, many areas of application.

In science, the big particle accelerators like the LHC are common knowledge. But electron microscopes also use components of particle accelerators. Many particle accelerators worldwide are used to produce high-energy x-rays in synchrotrons or linear accelerators, imaging samples for material science, chemistry, biochemistry and other areas.

In manufacture, particle beams can be used to cut material, but also weld material. Precision beams can break down gaseous feedstocks to deposit metal in micrometer-sized structures. Directed ion guns are used to implant traces of elements into surfaces for the manufacture of electronic devices.

And in medicine, particle accelerators of ever-greater sophistication find increasing use in cancer treatment and beyond, destroying life-threatening tissues with ever-reduced doses to surrounding healthy tissue.

= Further material =

= Sources =

== Authors ==
Sevoris Doe, Gerrit Bruhaug, Luke Campbell.

== References ==

Particle Accelerators

2022-08-17T12:41:52Z

Sevoris: /* Accelerator physics */

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Particle accelerators are an incredibly useful piece of technology and a diverse set of artifacts produced for many purposes. While particle accelerators are often invoked as weapons of war in science fiction, and have captured the public attention as giant research systems the size of cities, they have an incredible widespread range of applications. There are also many, many ways to build a particle accelerator, depending on what effects you need to achieve and what circumstances you work under.

This article aims to be a general primer on particle accelerators, their physics and a selection of technologies and construction styles. In detail, all particle accelerators are high-tech engineering customized to their circumstances in many, many aspects.

Further pages covering application fields are in the work.

= Accelerator physics =
The simple explanation is that we use electric fields to accelerate charged particles. But from this basic principle of leveraging the electromagnetic forces springs an entire set of accelerator technologies with different performances, trade-offs and characteristics. The most basic particle accelerators are just a big high-voltage source and can still be found in old CRT TV’s and dentist x-ray machines around the world. Things rapidly get more complex from there, but first we need to lay out some terms so we can all have a common conversation.

The first thing to keep in mind with particle accelerators is that we are talking about charged beams here, even if we neutralize them later. So no neutrons, photons, or other innately neutral particles. Your choices range from the simple electron up to anti-uranium and more, but they all must be charged. The particle of choice determines just about everything else about the accelerator and any potential uses. Really heavy particles (like large ions) don’t accelerate easily and take a lot of energy to get relativistic enough for time dilation tricks to take into place. Really light particles (like electrons) accelerate super easily, but can have other radiation problems to worry about.

The second most important thing to know about a given accelerator is the particle energy. This is a measurement of the kinetic energy of any individual electron, ion, whatever in the beam and is measured in a unit called the electron-volt or eV for short. The lower bound for any worthwhile particle accelerator is in the MeV (or million eV) range and we typically want a lot more! Inside of the accelerator we tend to talk about the electric field doing the accelerating (measured in Volts/meter). For singly charged particles (like electrons) this gets really to figure out the energy because of how the eV is defined. We just take the electric field (also called the accelerating gradient) and multiply by the length of the machine! Boom, now you have particle energy! For more complex particles you have to do some scaling based on mass and charge state.

Also a quick trick to help you with deciding if something is relativistic or not is to divide the particle energy <math>E</math> by the mass energy <math>M</math> of the accelerated particle (0.511 MeV for electrons/positrons and about 1 GeV for a proton) and add one. The resulting number is called the Lorentz gamma value and is used all over in accelerator engineering.
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>\gamma = 1+\frac{E}{M}</math></div>
Once you get a value much larger than 1, you have relativistic particles. Once that number is in the hundreds you get an ultra-relativistic beam. Figure 1 helps show the relationship between Lorentz gamma and speed of the particle.

[[File:LorentzGvsSped.svg|900px|thumb|center|Lorentz gamma versus fraction of the speed of light]]

The third most important feature of a particle accelerator is the emittance which is just a fancy term for divergence of the beam and can be thought of similar to the wavelength of lasers. No beam is made perfect and they will naturally want to expand outwards. The smaller the emittance, the longer the beam will stay in a tight spot. Emittance can be altered with a wide variety of beam engineering techniques that take entire books to discuss, but one other cheating thing you can do is drive the beam energy higher. This naturally lowers the emittance by a factor of Lorentz gamma due to the cheating effect of special relativity! Thank god for Einstein!

== Acceleration forces ==
Every accelerator tries to build an electric gradient appropriate for the charge of the particles it is accelerating. The greater the electric gradient, the bigger the amount of work that can be done, the more energy is imparted on a particle as it travels through the gradient, and thus the higher the acceleration. The simplest way to build up a gradient is simply two charged electrodes with a strong potential between them. Many other accelerator methods use electromagnetic fields, in the form of waves confined in electrically conductive structures. In this case we exploit that electromagnetic fields have both an electric and a magnetic component. The magnetic component cannot be used to accelerate particles (only bend their trajectory - which can be exploited for effects like deflection or focusing) but as the electromagnetic waves oscillate, so does an electric field. This naturally creates a charge difference across half of the phase of an electromagnetic wave - this electrical gradient can accelerate particles. Both of these technologies encounter problems with the physical materials asked to contain the electric or electromagnetic energy. Eventually the fields get so intense that solid materials get broken down and an arc is formed. As such, various advanced concepts use clouds of charged particles and plasmas, which are not subject to the material limitations of solid matter.

The strong and weak nuclear force as well as gravity (which are all of the forces we know of at the time of writing) are not suitable for accelerating particles. This is related to the power and range of these forces. The strong and weak nuclear force have extremely short ranges - even if we had a means to artificially shape them like with electromagnets, their range would be too short. Gravity has the range but is incredibly weak. To generate a measly 1 G of acceleration, 9.81 m/s², requires the mass-energy of the entire earth, some 5.97e24 kg of mass, to do the job! The acceleration involved in particle accelerators is much, much, much larger (on the order of 10^13 times larger or more).

{{MascotSays|Interestingly, one source of very high-energy particles are giant gravity wells. Black holes generate conditions around them that can eject particles with enormous energies. The jets of particles that shoot from the north and south poles of many such stellar entities are particle accelerators.}}

If you want to use a force other than electromagnetism to accelerate particles, it’ll have to be a fictional one. The forcefields of various sci-fi properties could be considered, since they can often apply forces to particles. If you propagate a wave of such forcefields down a tube, you could have something quite a lot like many particle accelerator systems!

== The particle bunch ==
Essentially all high energy particle beams are not constant streams of particles, but a time sequence of discrete groups of particles, commonly referred to as “bunches”. These bunches can be described both in their distribution in lateral space to the traveling direction (x and y axis) and longitude (z axis) along the traveling direction, anchored on an origin point (0,0,0) that moves in time.

[[File:ParticleBunch.png|center|900px]]

Bunches can have various profiles. Generally they are somewhat distributed in the lateral direction just because the charged particles have slightly different starting positions and lateral velocities (also called transverse emittance). In longitudinal space, differing acceleration experienced by individual particles as well as dynamics within the bunch scatter them out, with “noses” and “tails” that contain fewer particles. The end results is that most beam profiles follow a gaussian distribution, though there are exceptions.

This bunching behavior allows us to analyze other properties of particle beams very similar to lasers. There is a “pulse frequency” given in Hertz. Each bunch contains a certain energy, which can be large enough that we give it in Joules and not electronvolts. There is an average output energy over a given timespan, given in Watts. We have a beam peak power, given by how quickly a bunch arrives at a target and deposits its energy, which we give in Watts. Related to this we also have a related peak intensity depending on how much area/volume the beam is deposited into, with units of either W/m² or W/m³. If we have charged particles being accelerated, we get a beam voltage, the same way as in a wire. Beams can also have a temperature, the same way as any other group of particles with random motion from their energy, given in Kelvin or eV.

The systems that generate particle beams of course have efficiencies, with the wall plug efficiency (power inserted into the entire system versus power deposited in the particle beam) being the most interesting for first-order engineering. They also have an operating temperature.

== Particles for acceleration ==
With the constraint that our particles must have an electric charge in order to be accelerated with anything but fantastic technology, we can discuss quite a large family of particles, from the conventional to the exotic. They differ in the applications they can be interesting for. Some can be similar, but no two particles behave entirely alike.

=== Electrons ===
The ubiquitous electron is the most common particle to be accelerated and the first one ever pushed up near the speed of light by humans. Due to its excellent charge to mass ratio (the best possible due to the extremely low mass and charge of 1e) electrons are very easy to get up to relativistic speeds. Electron accelerators for radiography and sterilization are often portable by a single person and not much larger than a laptop in size (although they can be much bigger for power output reasons). The ease in which electrons reach relativistic speeds means that electron RF accelerators can be designed with no speed changes assumed (since the electrons are already moving at effectively light speed) and thus tend to be highly efficient and have the highest acceleration gradients achievable. For similar reasons electrons are also able to be accelerated in plasma based accelerators, allowing for even higher acceleration gradients and smaller machines for a given energy. The major downside of electrons is that due to their large charge to mass ratio they will radiate energy like crazy when bent by a magnetic field. Thus ring shaped accelerators are of limited use for electrons since you have to make the ring extremely large to not just lose all of your energy as beams of x-rays coming out of the machine! This very trick is used to make high quality x-ray sources for industrial and research applications.

When electrons hit matter they tend to release their energy as photons in a process called bremsstrahlung, or braking radiation. These extremely high energy photons (roughly ~⅓ the peak energy of the electron on average) then proceed to make electron/positron pairs that then make photons that then make pairs that then make photons… This whole thing is a massive cascade of radiation that ionizes, heats and can even damage on a nuclear level! At high enough energies you will generate every particle known and might even make some new ones!

=== Proton ===
Protons are the next most common particle to be accelerated and are currently used by the absolute largest particle accelerator ever made (the Large Hadron Collider or LHC) as the primary particle of choice. Since they have ~2000X lower charge to mass ratio than electrons, protons are about ~2000X harder to get up to speed. Unlike electrons they are not promptly at light speed so we have to send protons through a series of various accelerators depending on the energy of choice. For example the protons in the LHC the protons go through 5 different particle accelerators plus the extremely low energy stage that first makes the beam of ions! This difficulty in reaching highly relativistic speeds has two other bad side effects. Firstly, if we want to transmit our protons through empty space we will probably need to neutralize them with a beam of electrons.

{{Note|''When shooting extremely high current (kA), relatively low energy (not highly relativistic or in some cases relativistic at all) beams in the atmosphere the higher mass and positive charge of protons gives them many beam propagation advantages over electrons if the beams can be made. In this case no neutralization is required as the atmosphere itself becomes a plasma and neutralizes the beam.''}}
Luckily this is not very hard and will add almost nothing to your comparatively large proton accelerator, but may increase your beam spread a little bit via some neutralization physics. This does mean your beam is technically plasma, which is cool! Second, we effectively cannot use those neat plasma based accelerators for protons since we would need nearly TeV class proton beams to inject into the plasma accelerator and by the time you reach those energies you have basically already done all of the work. Maybe in the future there will be ways around this, but as of right now we are limited! There is one major benefit to this poor charge to mass ratio though, and that is the lack of synchrotron radiation! Ring shaped accelerators are ideal for protons, and basically all ions.

When protons hit matter they can interact with all of the four fundamental forces, although only two of them (Strong Nuclear and Electromagnetism) tend to play big roles. Protons barrel on in causing huge ionization cascades and can also release bremsstrahlung photons at high enough energies. They are also ideal at knocking neutrons, protons, deuterons and more out of the target nuclei which just adds to the fun! Protons also come with a neat trick known as the Bragg peak, where they deposit a large portion of their energy right at the end of their trajectory through a target material. This is a very useful trick for medicine and means you can “tune” the beam to almost ignore large portions of material and primarily heat, ionize, and destroy the area of interest. You can imagine how useful this is for cancer therapy or more nefarious things!

=== Light ions ===
Light ions are a rather fuzzy term but here we will use the definition of any ion heavier than a proton but lighter cobalt. The fuzzy reasoning here is that you get on the other end of the nuclear binding curve (i.e. can’t get net energy from fusion) at iron, so cobalt will then no longer be “light” and has to be made in a different process. Feel free to pick your own cut off! Historically beams up to neon have been experimented with for nuclear fission, nuclear fusion and medical therapy applications. Light ions have even worse charge to mass ratios than protons (Z/A times worse) and typically you will find light ions have about ½ the charge to mass ratio of a proton on average, although the isotope 3He is a notable exception. This means they are even slower to accelerate, although often the exact same machines as protons can be used with minor tweaks. Light ions have similar requirements for neutralization as protons, although their heavier mass can help fight beam bloom for a similar beam speed (not energy since the increased mass means the particles are higher energy for a given speed). The decreased charge to mass ratio also means that synchrotron radiation is even lower so beams can be driven to higher speeds before radiating. Light ions have similar target interaction physics as protons, although the ion itself will break up at very low speeds and add to the radiation environment.

=== Heavy ions ===
Heavy ions are being defined here as cobalt up to whatever unstable element you can jam into your accelerator! Typically heavy ion beams are made of elements like lead (used in some runs on the LHC), gold (used in the Relativistic Heavy Ion Collider) and even californium (used in some runs on the Argonne Tandem Linac Accelerator System). These beams tend to be for pure nuclear physics research, but there has been a long standing interest in heavy ion beams for inertial confinement fusion research and as potential methods of propelling spacecraft. These extremely heavy ions will be even harder to accelerate to ultra high speeds than light ions and much slower for a given energy, but they do provide a lot of momentum to make up for that! They have the same neutralization requirements and same potential upsides for beam bloom as light ions, but will most likely be orders of magnitude slower reaching our targets than the lighter beams. Heavy ions can also be used in the same machines as protons with small modifications and generate almost no synchrotron radiation as a result of their much smaller charge to mass ratio. They also have similar interaction physics to protons and light ions, but will stop in much thinner layers of materials. While electron, proton and light ion beams tend to both mechanically destroy and irradiate targets through-out, heavy ion beams will act much more like lasers and burn in from the outside of the target on in. There will still be a spray of ionizing radiation, but far less of one than an equivalent energy electron, proton, or light ion beam. The much higher momentum per particle, lower radiation concerns and ease of deflection makes heavy ion beams very interesting for beamed propulsion purposes.

=== Macrons ===
Macrons are typically defined as macroscopic particles, such as extremely large molecules or even grains of sand, that are charged up and accelerated via similar technologies as traditional particle accelerators. There are many difficulties with getting macrons up to appreciable speeds due to the extremely low charge to mass ratio, engineering difficulties with getting consistent and good charging, phase matching with accelerating cavities and lack of effective focusing optics. There are so called “dust accelerators” in existence that utilize the first stage of ion accelerators to generate their “beams” and proposals have been made to create better versions using updated technology. Macrons will interact just like a very small and very fast bullet unless they are designed to carry a nuclear payload to boost this output. Antimatter is an assured option for getting energy gain from a macron impacting a target, although there are potentially options for fusion, fission and fission/fusion given the correct sizes (mm scale or bigger), clever target design and the right impact speeds (most likely >>100 km/sec). The various energy gains versus speeds can be seen here:

[[File:IvKmacron.svg|900px|thumb|center]]

It can be seen that antimatter proves to be worth it up near the speed of light, but both fission and fusion stop providing much gain well below the relativistic regime. This also helps explain why heavy ion uranium beams are not going to provide some incredible advantage over any other ion.

=== Antimatter ===

Antimatter is the evil cousin of normal matter and whenever the two meet a release of energy equivalent to their combined mass energy follows. Now this does sound like an amazing option for beams but there are two downsides. First, antimatter is energetically expensive to make! Positrons (antimatter electrons) are the “cheapest” at 1.02 MeV of input energy at a theoretical best, but they only give you that much energy back when hitting a target. Since a typical positron beam can be expected to be equivalent to an electron beam, there is little gain in adding 1.02 MeV per particle to a multiple GeV (or even TeV) beam. You can see this explicitly:

[[File:IvKhBarPostron.svg|900px|thumb|center]]

For anti-protons the math becomes more favorable, but creating these particles is much more difficult and energy consuming. At best anti-protons require ~2 GeV of input energy to create, but in practice >400 GeV per anti-proton has been needed. Luckily significant energy gain is had up to several GeV of beam energy, so if anti-protons can be made at a central facility of some kind and easily stored (neither being easy things) then there may be some advantage to such a beam. Heavier ions are significantly harder to make since fusion with antimatter must be performed to generate those ions. Barring a naturally occurring source being found, antimatter heavier than a proton is most likely not worth using.

{{Note|'' Positrons can potentially fill an interesting niche either as a co-propagating neutralizing beam for electrons or perhaps as a higher performance atmospheric beam than electrons due to the positive charge. In this case the annihilation properties are no what is of interest, rather the positive charge and relative ease of acceleration compared to ions. Since positrons share all properties with electrons except for charge, they can be accelerated just as easily and even co-accelerated with electrons in the same machine.}}

=== Muons ===
Muons are an interesting cousin of electrons and positrons. They are unstable and heavy cousins of electrons, massing about 200X more than electrons and living about 2 microseconds in the lab frame before decaying into an electron (or positron) and a neutrino. If we accelerate them up to relativistic speeds (about 10X easier than protons and the muons are typically born relativistically thanks to how we make them) then the muons can be made to “live” far longer. Muons have long been of interest to accelerator physicists due to their position as an almost ideal intermediary between electrons and protons. The requirement to make them on site has always been an impediment though and muons are currently only used for certain niche experiments. During the planning stages for a potential muon collider it was also noted that a muon collider would also have a serious issue with neutrino radiation! This is due to how neutrinos are more likely to interact with matter at high energies and a muon collider would be generating a LOT of really high energy neutrinos in a ring around the machine. This has been proposed as a sort of super weapon even!

Muons should interact with matter in a similar fashion to electron beams and the beams will probably be treated in a similar manner. They can propagate through the atmosphere quite well in theory and should outperform electron and proton beams for a given beam energy, although at quite a cost… Muons are typically generated with proton beams impacting targets and will create an equal number of positive and negative muons at high energies that must be carefully captured. This would allow for the easy creation of a neutral muon beam though, which may make for a great giant doom beam! Do note that negative muons also can help catalyze nuclear fusion so a very high power beam hitting a fusion fuel tank could produce some rather… Interesting results!

=== Other exotic options ===
There are a wide variety of odd particles that could also be chosen for use in beams given a specific need for them, although generation and utilization of these particles will be quite difficult! Extremely short lived particles like pions, kaons and taus can be generated and accelerated up to extreme speeds to help propagate them far enough to matter, although that will need some very extreme speeds. At the upper end of extreme there are even proposals for accelerating tiny charged blackholes!

= Accelerator technologies =
There are a wide variety of accelerator technologies and here we try to summarize them as best as possible. Different accelerator types may be sequenced one after another. Some particle accelerator technologies depend on the particles fulfilling certain requirements of speed (thus energy) and hence different systems may be employed in sequence. Accelerator systems are described in terms of their efficiency of turning input power into particle energy, their acceleration gradient measured in electric field which translates to volts per meter, and the mass per length meter. An interesting secondary parameter is their operating temperature. Some accelerators can operate well at high temperatures. Others require cold or cryogenic conditions to work well.

== Core accelerator assembly ==
We define the core accelerator assembly as the structure where energy is imparted onto the particles.

=== DC Accelerators ===
For very low beam energies we can utilize large DC electric fields between plates or grids to accelerate particles up to speed. This is the same technique as is used in ion drives and is the most common industrial accelerator used. Every dentist's office and old CRT television uses this acceleration technique. Ion accelerators often also use a DC accelerator to generate the initial beam of ions, while electron beams utilize other techniques. These machines are limited by material breakdown to about 0.5 MV/m accelerating gradient and typically no more than a couple of MeV total particle energy for extremely large machines. The highest energy DC accelerators can reach 40 MeV for proton beams, but this is for machines massing in the 100s of tons and at low total beam powers. Very large DC accelerators are currently the only tested method for accelerating macrons up to 10’s of km/sec.These accelerators are limited by the risk of arcs jumping between the electrodes at too high voltage differentials.

=== Radiofrequency Quadrupole ===
A more advanced first stage accelerator for ion beams is the radiofrequency quadrupole or RFQ. These machines utilize low frequency radiowaves and carefully designed conductive bar structures to efficiently generate high current ion beams up to energies of ~7 MeV in a couple of meters. RFQs were a revolution in ion accelerator physics and caused quite a stir when the USSR first showed them off as they provided a much higher power and higher quality ion source in a smaller package than anything the USA had built! RFQs have also been proposed as methods of efficiently accelerating macrons up to >100 km/sec. These machines can be normal or superconducting and can operate with efficiencies as high as RF LINACs in theory.

=== Cyclotrons and Betatrons ===
One of the earliest ways to accelerate particles up to relativistic speeds is with circular accelerators known as cyclotrons (for ions) and betatrons (for electrons). In Cyclotrons the accelerator is a round chamber which houses two half-circle electrodes roughly shaped like a D (and thus called “dees”) separated by a gap, sandwiched between the poles of a large electromagnet. The magnet creates a static magnetic field which bends the path of charged particles into a circular motion. An oscillating alternative voltage is applied to the electrodes (what we often call RF or radiofrequency) to accelerate the ions. This radiofrequency field’s oscillation is matched so that one cycle of the voltage matches one circuit of the particles. As a consequence when crossing the gap from one electrode to the other, the charged ions experience a voltage gradient and are accelerated. As the particles gain energy and speed, their radius in the magnetic field gets larger. The result is an overall spiraling trajectory from an injection point in the center to an exit point at the rim where the charged particles are bent out of the cyclotron. Cyclotrons use the same voltage gradient many times over, defeating the electrostatic breakdown that limits the single-time electrostatic accelerators. As a consequence, proton energies of up to 70 MeV may be reached. Cyclotrons energies can also be increased by switching from normal magnets and RF resonators to superconductive components. This eliminates various losses from electric resistance and absorption and also allows for much stronger bending magnets, which allows the cyclotron to contain particles up to a higher energy if desired. Superconducting cyclotrons have achieved up to 520 MeV. Still, cyclotrons suffer from poor beam current, pulse rate, and beam quality issues that make them a less than ideal choice. The large magnet required (measured in meters) also does not lend itself to a lightweight weapon and as such, cyclotrons are regulated to niche medical and physics uses.

A close cousin is the betatron, which is used to accelerate electrons. In this case the magnetic field strength is increased as the electrons speed up, which prevents any spiraling trajectories of the particles. Betatrons have been operated up to electrons energies of 300 MeV, but are more typically in the ~10 MeV range. In principle a betatron could also be made superconducting, although magnet ramping may prove to be difficult. Other types of ring shaped accelerators may prove to be more useful, although electrons are heavily limited by synchrotron radiation and it is often better to use linear accelerators for them. For now betatrons remain limited to small (sometimes portable) accelerators for nuclear physics, industry and security work.

=== Radiofrequency Linear Accelerator ===
The real work-horse of the particle accelerator world is the RF linear accelerator. These babies use the same alternating field trick that cyclotrons use, but in a straight line rather than needing to bend the beam around in a circle. This provides a couple of nice advantages over the more traditional cyclotron. First and foremost you get to leave that large, bulky magnet behind which saves you a ton of mass, cooling and power consumption! You also get to now have much higher RF frequencies (1-10 GHz right now compared to the MHz used in cyclotrons), which means higher accelerating gradients. To generate these powerful, high frequency RF waves (also known as microwaves) that drive the accelerator we typically use military radar power supplies or (in more advanced cases) even lower energy particle accelerators! State of the art RF LINAC technology is well into the 100 MV/m range and have no upper limit on beam energy! The Stanford Linear Accelerator reaches 50 GeV and was built back in the 1960’s! There have been traditional RF LINAC designs for machines reaching up to 1 TeV, with the only limit being how much space you have to put these monsters. They do operate best (i.e. with high accelerating gradients and efficiencies) with particles already moving at relativistic speeds, so any ions fed in need to use a series of lower gradients LINACs or other pre-accelerators. Electrons and positrons can naturally be used in LINACs since they accelerate up to relativistic speeds so quickly.

LINACs also have been shown to operate at high average powers (MW scale is now in use) and high pulse energy (10’s of kJ pulses fired at high rep rate) while also maintaining a high quality beam that stays well focused. The two newest advancements have been to push the frequency ever higher for higher acceleration gradients (300+ MV/m has been shown and there are plans for even more) and to switch to superconducting RF accelerating cavities (the name of the actual piece of equipment that does the accelerating) which then dramatically increases the electrical to beam efficiency (80-90% has been shown and even copper waveguides have been operated at 70%). All in all, LINACs beat the tar out of cyclotrons which is why we don’t build cyclotrons anymore. To quote an old Fermilab engineer “cyclotrons are an expensive and bulky way to get a really crappy beam or a cheap way to get a giant chunk of lightly radioactive steel that can never leave your lab”.

LINACs work in detail by shaping the radiofrequency energy in a confining waveguide. The oscillations of the RF energy may be held in space, or allowed to travel down the waveguide over time. Radiofrequency waves have connected magnetic and electric components, which oscillate.

[[File:2022 08 16 GalLib ElectromagneticWave.svg|center|800px]]

Inside the RF accelerator, the electromagnetic radiation is oriented 90 degrees to the particle travel direction, which points the associated electrical field in the travel direction of the particles.The radiofrequency field naturally oscilates along its wave, and the connected electrical and magnetic fields oscilate as well: building up, reaching a peak, and diminishing, inversing, building up, reaching a peak, and diminishing, and the cycle repeats. The detailed behavior depends on the specific cavity.

[[File:CavityWithEM.png|600px|thumb|A resonant RF acceleration cavity and the electric field inside of it, created by resonating RF fields, at one moment in time.]]

Particles of the right charge in the right space at the right time experience an acceleration from the electrical field, being drawn to the opposite electric charge. If they land in the other region, they are instead decelerated. Inbetween, particles experience less to no acceleration. This behavior naturally separates a stream of particles into discrete '''bunches'''.

[[File:CavityWithEMAndParticles.svg|600px|thumb|Bunches being accelerated inside the electric field. ]]

It also creates an implicit limit to what can be effectively accelerated using a specific radiofrequency accelerator design. The particles driven through it have to keep pace with the oscillations of the electric field, reaching the next correct region inside the cavity at that time when the electric field is oriented correctly. If they are too slow to keep inside the accelerating regions, the accelerating action will be ineffective. This requires RF cavities to be matched to the particle's velocity gain under energy imparture and their velocity when entering, being inside, and exiting the accelerator. For electrons and positrons, this isn’t that complicated. Since these lightweight particles accelerate readily, they have few problems “staying in the pocket”. Particles that are slower to gain speed, such as ions and even larger particles, must however be accomodated. This in turn impacts how much acceleration and energy per meter can be fitted into the accelerator in the first place. The acceleration of heavy particles requires specifically designed and sequenced RF accelerators for specific regions of particle speed. This in turn complicates the design of the larger accelerator infrastructure, with multiple discrete stages getting the particles to the energy required so that the next stage in sequence can take over.

RF cavity accelerators are limited in their eV/m capability by the critical current at the containing waveguide. To increase the acceleration provided, more intense electrical fields must be built up. Eventually the electrical charge of the RF waves becomes so strong it can rip electrons out of the cavity walls, causing an arc to form and the cavity material to break down physically.

=== Advanced Accelerator technologies ===
Present accelerator technology caps out at around 250 MeV/m for the most overdriven designs. This poses issues for a variety of applications, where higher energies in shorter distances are desired. In science, "desktop" and "chipset" accelerators are of interest. When militarizing particle beams, some effects only become available at very high energies - but you want something compact enough to turret.

urrent and even near future accelerator isn’t really ready to supply that sort of scale quite yet, so we have to move on to the cutting edge of accelerator physics. There are three big options with sub technologies within these broad fields. We have extreme RF cavities using much higher frequencies, plasma accelerators using lasers for power and plasma accelerators using particle beams for power. Each technology provides a unique series of advantages and disadvantages, but fundamentally they are all a way to make linear accelerators that are much smaller.

==== Dielectric Wakefield Accelerator ====
LINACs using much higher frequencies than traditional LINACs have two fundamental problems to fight. The first is electrical breakdown and the second is generation of those mm-wave (10-100 GHz) or even THz (100-1000+ GHz) sources at sufficient power and efficiency to power a particle accelerator. As the frequency of the driving power goes up the electric field climbs up, which means the accelerating gradient gets larger (which is good) but also means that we run a higher risk of a huge arc forming in the accelerator and breaking it (which is bad). This is often fought by changing the accelerator cavity material from a metal (or superconductor) to a dielectric material like ceramic, or by using a coating of dielectric, which can sometimes earn them the moniker of Dielectric Wake Field Accelerator (DWFA). The issue of creating powerful and efficient enough sources of this higher frequency drive power is being actively pursued for a variety of fields (5G cell signals, beamed power, etc) but is still quite immature compared to the microwave RF sources of traditional LINACs. There are options using essentially "over-clocked" microwave sources, low energy particle beams, and even laser-plasmas. There is still a lot to be done here, but these sorts of problems are being actively worked on there are working accelerators using this technology. There is potential for GV/m or greater acceleration gradiatens and all of the same advantages of traditional RF LINACs (i.e. high powers, high fire rates, high energy pulses, quality beams) should stay the same. The wall plug efficiency is liable to be a bit lower, but that will be highly dependent on the efficiency of the source of these high frequency waves.

==== Laser-Plasma Accelerator ====
Plasma accelerators using lasers for power are a whole world in and of themselves. There are a wide variety of ways to accelerate particles with lasers and plasmas, but we will focus on the most studied technique known as Laser Plasma Accelerators (LPA). LPA accelerators utilize ultrashort (<1psec) and ultraintense (>10^18 W/cm^2) lasers to drive a plasma wake (thus sometimes earning them the name laser wakefield accelerator) and accelerate particles through the resulting charge differential. They have only shown the ability to accelerate electrons and positrons, but any ions that could be injected at the right time and at extremely relativistic speeds could conceivably be accelerated this way. In the operation of a typical LPA the ultraintense laser is fired into a jet of gas (turning it into plasma) or a channel of lightly pre-ionized plasma. The laser then blasts through this plasma like a boat through water and leaves a void behind it (the wake). This void ends up having a HUGE accelerating gradient (100 GV/m) and suffers no risk of electrical breakdown since the plasma is already broken down! A pre-accelerated bunch of particles can be injected into that void at the right time (called the witness bunch) for a BIG acceleration or you can rely on the wake itself spewing electrons out into the void and make a beam out of the plasma itself. The gas or plasma then quickly is reformed and the laser fires as soon as its cooling and charging allows. Keep in mind that even though the beams can hit incredible energies in short distances, the laser itself is not a pocket sized device. These lasers are room sized at minimum and not easy to shrink. The pulse lengths required are also much shorter than most of our hypothetical weapons-grade pulse lasers that may operate in the nanoseconds regime - dual-use may prove difficult. The intensities reached also pose significant issues for the laser hardware. If you don’t spread the energies out, it becomes quite possible you will break parts of your laser from the produced intensities.

Current LPA systems suffer from a couple of problems that are being actively worked on. First and foremost they are incredibly efficiency limited by the laser tech being used. There are plans to move on to much more efficient fiber lasers and eventually make LPA machines that operate in the 25% or greater wall plug efficiency, but that is not currently doable. These lasers also need to be fired at higher rates than the current ~1 Hz, but that is much more of a limit of laser engineering than accelerating engineering. LPA’s also have yet to link more than one accelerator together and the lasers are limited in just how much plasma they can accelerate through right now (~20 cm max) so the beam energy runs into a wall quite quickly. Aside from R&D on linking LPA’s together in a way similar to RF LINACs there are two other options. The acceleration gradient is higher the denser the plasma, so a move to denser plasmas and shorter wavelength lasers (to penetrate the plasma) could provide higher beam energies in similar distances. The other option is to use clever optical techniques like the so-called “flying focus” to allow for the laser to go through much longer plasmas. This could in theory allow for more efficient LWFA’s and single stage machines that provide ~1 TeV electron beams in meters! The final issue is beam quality. Even the best LPAs will not make beams that have as tight of a beam spot, energy spread, or emittance as a normal RF LINAC. This means they will end up needing some beam “cleaning” if you want to get a lot of range out of them.

LPAs are of engineering interest for very short wavelength X-ray free-electron lasers in a compact format. Currently these facilities require linear electron accelerators many kilometers in length. With an LPA as the electron accelerator, an XFEL able to image molecular structures and chemical reactions as they happen might be shrunken down to a much more compact format. (Sometimes this is discussed under the ultimate promise of the “benchtop” or even “chip-format”(sic!) XFEL)

==== Beam-Plasma Accelerator ====
Plasma accelerators using particle beams for power are very similar to LPAs in principle but use a particle beam rather than a laser. We typically just call these things Beam Plasma Accelerators (BPA) for short. The concept is very similar to the LPA in that a high intensity beam comes in and blasts a void into a plasma, which particles can then be accelerated in. BPA provides some interesting advantages over LPA in that you can use highly efficient superconducting RF LINACs (or synchrotrons if you want) to provide the particles at very high efficiency. They also have little in the way of plasma density limits and the accelerated beam takes on the characteristics of the beam driving the BPA. Thus little if any beam cleaning will be needed after acceleration. The big downside is the need for a relativistic particle beam to drive the effect, and typically one in the 100’s of MeV to low GeV range, depending on the particle used. Thus your final accelerator may be short, but you still won’t be making a handheld particle beam device.

== Accelerator Shapes ==
The accelerator technologies discussed above are the means by which we create situations for particles to gain energy. The accelerators become embedded in a larger structure that produces, directs, accelerates, even stores the particles until they are directed towards their final use.

There are two principle shapes for accelerator assemblies. Ones where the beampath is not closed onto itself, and designs where the beam path is closed on itself in part of the assembly, forming a loop.

=== Linear accelerator ===
Everything from electron generation to acceleration to conditioning is built in a straight line, one after another. This is the Linear Accelerator or LINAC. The main catch here is that there is only one single passage for the particles through the entire accelerator. Re-using accelerator hardware to drive the same particle to higher energies by repeated passage is out. The particles have to reach their desired energy in a single pass. Practically this puts limitations on the energy achievable with heavy particles in a LINAC. The length required gets impractically large. Even for electrons, lengths of hundreds of meters to kilometers may be required, unless advanced accelerators with high gradients are used. This said, every accelerator contains linear sections. Since so much high-performance accelerator technology depends on particles having a minimum velocity, ions must be “pre-conditioned” in a set of linear accelerators before the re-use of an accelerator through looped beam paths can be considered.

LINACs of low energy may be used in many applications. Medicinal they find use in radiation therapy and radiosurgery.

=== Synchrotrons ===
The other shape which has some interesting properties are accelerators where beampaths are looped onto themselves. These are called synchrotrons. Since beam paths are closed onto themselves, particles can pass through the same accelerator unit repeatedly, gaining more energy on every passage. This is advantageous for heavy particles which do not gain much speed on any single passage. They can pass through the same accelerator thousands or even millions of times instead. This is obviously advantageous for facility bulk and mass and allows us to reach energies that would otherwise be impractical. The accelerated beam bunches can also be held in storage, looping like the rotating mass of a flywheel, holding energy. The LHC can store some 700 Megajoules of energy in the recirculating beam after full speed has been reached.

That said, the devil is in the details. When charged particles are bent in circles they emit electromagnetic radiation which scales very rapidly (to the fourth power!) with relativistic gamma factor and with bending radius (inversely to the second power). This phenomenon is called synchrotron radiation. Due to the scaling, high energy particles emit more synchrotron radiation than lower energy ones and lighter particles (like electrons) at those high energies emit much more than heavier particles. In addition, the bigger you can make the bending radius of the synchrotron (often called “the ring”) the less synchrotron radiation you get.

[[File:SynchrVsE.svg|800px|center|thumb|Synchrotron radiation per particle versus particle energy for electrons and protons in a 50 meter in radius ring.]]

[[File:SynchroVsRing.svg|800px|center|thumb|Synchrotron radiation emission per particle for different particles at different energies, over varying bending radii. ]]

Although these numbers look small, keep in mind that this is PER PARTICLE. A typical pulse in an electron accelerator will have more than more than a billion electrons in it! Large synchrotrons like the LHC will have over 300 trillion ions circulating through it! Synchrotron radiation necessarily has to be managed to reduce radiation shielding, the radiation load (and possibility of hardware failure) on the accelerator, and to reach high energies in the first place - you might lose all the energy your accelerators are adding to synchrotron radiation!

Deflecting the beam also requires the application of magnetic fields, and higher degrees of deflection, in a smaller length, at higher particle energies, require stronger and stronger magnetic fields. Eventually multi-Tesla fields from superconductors will be required. And due to action and reaction, those magnets need to be braced against the pressure the beam exerts as it is deflected. Ultimately this leads to the same limitations as with flywheels - the mechanical strength of the braces and static connections that keep the magnets together as a larger structure is the accelerator’s limitation on J/kg held in storage. A further complication: The magnet strength has to be matched to particle energy. As a beam is accelerated, the deflectors have to ramp up their power to keep the degree of deflection constant.

Sometimes, synchrotron radiation from electrons is intentionally induced in a ring-shaped accelerator. Since synchrotron radiation of high-energy electrons tends to be hard x-rays emitted along narrow vectors, it’s a way to get x-rays used in various imaging techniques in material science and molecular physics.

Synchrotrons used for ions also need to respect the limitations of the involved accelerators. Synchrotrons often use radiofrequency accelerators. Since radiofrequency accelerators must match the speed and speed gain of their particles, an ion synchrotron cannot be made up of a single accelerator loop. Instead there are sequential loops connected in sequence, each designed to boost the particle through a specific velocity and energy region so that the next accelerator takes over. A designed example is the LHC. Particles headed for the main LHC actually circulate in two other synchrotrons beforehand, gaining sufficient energy so that the LHCs RF cavities can take over. (Also, before being injected into the first synchrotron ring, the ions must first go through multiple linear accelerators using different technologies to reach a base energy.)

An interesting design consideration: Synchrotrons don’t necessarily have to take the form of a (near-ideal) ring. Funamentally any dipole represents a single point of deflection of a straight path - they're polygons! Any geometric shape that contains a loop, and where any single bending curve keeps within the limits of the synchrotron radiation you are willing to tolerate, works. As such, complicated two- or three-dimensional “folded-up” shapes are also possible formats for a synchrotron.

=== Storage Rings ===
Since synchrotrons have to match their deflection magnet strength to the energy of the particles being accelerated, a single synchrotron track cannot hold more than one small section of beam bunches in storage at a time, and becomes unavailable to accelerate further bunches until it discharges the stored beam. This is disadvantageous for various applications where we may want to store a fully accelerated particle beam “bolt” for a longer time.

Storage rings are separate synchrotrons designed to hold multiple sets of bunches in circulation at a target energy at a time. The bunches are delivered from an accelerating synchrotron (sometimes called a booster ring) and held in circulation in the storage ring. Since there are synchrotron radiation losses, a few accelerator sections must compensate for these.

Storage rings can also be used to condition a stored beam. A common thing to do in storage rings is to “cool” a beam, draining unwanted movement energy out of the particles. Beam cooling may take significant time, which makes it ideal to conduct in a storage ring. The result is a beam with less tendency to drift apart, advantageous for many applications.

== Other parts ==
=== Particle sources ===
Particle sources are assemblies which supply particles to accelerators. Here we can group them into three categories: electron sources, ion sources, and exotic particle sources. The later involve the production of particles by means of gathering them from collision reactions (or the handwavium blackbox of your choice) and feeding them into another accelerator for further use. Positrons for example may be manufactured in beam collisions.

Electron sources require the production of a free stream of electrons. The oldest method here are thermionic sources. A hot piece of material ejects free electrons into a vacuum, where they can be drawn in using electric differentials and forwarded to the accelerator.

Field emission electron sources instead use a strong potential between a needle-like cathode and an anode plate with a hole. The electrons are drawn from the needle and through the anode due to the high charge differential. This method produces a “cooler” beam with less random divergence than thermionic sources.

A further improvement still are photoemission sources. Here the precise application of laser light onto a photosensitive material energizes electrons in the photocathode until they break free. This process is thermodynamically very precise and due to the nature of the laser light, also very controllable. The result is a low-emittance stream of electrons. These may be accelerated using electrostatic attraction with an anode at one end of the vacuum chamber, or directly using an RF field.

Ion beam sources are more complicated. The physics of ion formation are complex, which opens many design choices when creating an ion beam. Generally a plasma must be produced, from which the ions can be extracted.

The plasma can be formed in a variety of ways. Electrons from a local cathode-anode array, plasma frequency heating, the injection of electron beams, the application of laser light. The end results can be as diverse. Some methods only produce weakly charged ions and on many vectors, so with high emittance. Other methods offer greater ion charges or reduced emittance.

=== Magnets ===
While acceleration of charged particles is effected with electric fields, magnetic fields can be used to bend the trajectories of particle beams. As such, electromagnets can be used to bend the trajectories of charged particles in various ways, and in a controllable fashion.

==== Dipole Magnets ====
Dipole electromagnet are used to steer the entire particle beam in one axis. (To steer in two axes, we must either rotate one dipole about the beam axis, or couple two dipoles in sequence.) They can be used to direct a beam inside a particle accelerator, inject and extract beams out of looping constructs by switching on and off at the right moment, and point a beam at a target.

In a dipole magnet, a positive and negative coil oppose each other, embedded in a “choke” assembly often made of steel. Charged particles moving through the magnetic field between the coils experience a deflection in one direction specified by the polarization of the field, and a degree of deflection specific to the field strength.

==== Quadrupole Magnet ====
Dipoles can steer, but they influence all particles equally. As such, they cannot be used to manipulate the beam's shape. The first type of magnet able to influence a particle beam’s shape are quadrupole magnets.

They are assembled from four coils at 90 degree angles, with coils of the same polarity opposing each other. The passing beam is compressed on one axis, and expanded on the other. To focus the beam overall, at least two quadpoles must be arrayed in sequence.

Quadrupoles are the basic and essential “magnetic lens” of particle beams. Without them, charged beams would inevitably defocus and run havoc.

==== Hexapole Magnets ====
Hexapole magnets exist to address one problem of quadrupoles - they focus differently depending on the energy of the particle passing through the magnetic field. The effect is likened by particle accelerator physicists to the different diffraction of light in a prism - and hence the effect is called “chromatic aberration”. Hexapoles are used to correct for this, stronger deflecting higher-energy beams.

==== Integrated optics ====
As the field of particle accelerator science and engineering advances, more complex designs are proposed to handle more complex phenomena inside accelerators. An upcoming field is “integrated optics” - complex magnetic systems with nonlinear behaviors. These could provide more advanced manipulation of electron beams, building upon existing capabilities and adding new ones for shaping a beam.

=== Plasma lenses ===
Plasma lenses are an up-and-coming technology, where very high currents in plasma are used to create very strong magnetic fields. Similar to plasma-based accelerator technology, the fact that charges are already free in plasma is used to circumvent conventional charge breakdown and critical current limitations on solid-matter devices. The resulting magnetic fields can be much stronger and bend beams over shorter distances. Furthermore, by creating complicated profiles in the plasma, elaborate beam manipulations could be performed, more akin to sophisticated optical elements. As well, the charges in the plasma neutralize the particle beam in the region of the plasma lense - affecting its propagation behavior further and in a positive manner, as self-repulsion is neutralized.

The catch is that this is plasma - that means you are dealing with all the complications of plasma physics. Heating of the plasma has been an early challenge to successfully forming and maintaining plasma lenses.

Plasma lenses require an appropriate chamber, possibly active containment to keep the plasma from losing energy to the walls or leaking into other accelerator parts, and the necessary plasma sources, maintenance heating devices and then lense shapers and control sensors. In high-powered plasma lenses there will also be a need to cool the plasma as it heats up from waste heat. Potentially we may manage this by open-cycle cooling - venting the plasma controlled in one direction into open space, rather than have a closed-cycle solution for reconditioning the plasma. This could cause some very cool visuals!

Plasma lenses for particle beams should not be confused with optical plasma lenses for lasers! Those exist as well, but are their own topic.

=== Chicanes ===
Chicanes are assemblies of multiple dipole magnets used to manage the longitudinal dispersion of a particle beam due to different particle energies. Since particles with higher energy move faster, they drift ahead in the beam distribution over time; and particles with less energy increasingly lag behind.

Chicanes deflect high-energy particles onto an outward curving trajectory. Since traversing the curve takes longer, the high-energy particles are effectively delayed, allowing lower-energy particles to catch up. By the end of the chicane assembly, the bunch has been compressed in the longitudinal direction.

Chicanes however, do not remove energy from too-fast particles. That is the responsibility of other structures.

=== Undulators and wigglers ===
Undulators and wigglers are magnetic assemblies used to transform particle energy into electromagnetic emissions, with different properties from the specific implementation of a shared basic principle.

Assembled from stacks of generally dipole magnets (other configurations exist), both undulators and wigglers force charged particles onto “wiggling” trajectories by bending their path left and right. Wigglers force very strong arcs; undulators less so. In both cases, the rapid deflection of the charged particles causes them to shed parts of their energy as electromagnetic emissions. Due to the energy of the particles, these photons are generally highly energetic x-rays.

Undulators may be used to build free-electron laser systems.

They can also be used as a means of cooling the beam. The removal of energy from particles can also suppress unwanted movement modes, allowing accelerator structures to re-add energy only to desired movement modes. Of course, this means the particle beam actively loses energy, and produces x-rays that have to be absorbed (and cooled away.)

=== Beam coolers ===
Beam coolers are a collection of systems and strategies that are used to remove unwanted movements from a particle beam. Undulators and wigglers are one way to do this.

One method set of beam cooling relies on finding particles who are very “hot” with random motion, and either removing the particles from the beam using a kick device, or applying a precise counter-force that acts against the unwanted motion.

If one is dealing with ions, one clever trick is to inject a beam of electrons. Due to the thermodynamics within the particle population, the lighter electrons absorb heat from the heavier ions, taking this unwanted motion energy with them when the electron beam is bent out.

As well, lasers can be used (what can’t lasers be used for!) to apply counter-force on ions and remove unwanted movement in this manner.

Beam coolers are generally not instantaneous devices. They work over time to make a beam better. Often they involve the removal of energy or particles from the stream, requiring accelerators and sources to make up the loss and creating waste heat.

=== Beam recovery ===
Some accelerator systems can actually be run in reverse - they can turn beam energy back into some other energy format and then electricity. This can be used to recover some to most of the energy from a particle beam before disposing of it. This is advantageous both for your energy bills and making a very high-energy beam liable to spray radiation everywhere, easier to absorb in a beam dump.

=== Beam dump ===
A beam dump is a construction where a particle beam can be routed in case it has to be disposed. Beam dumps are designed to safely absorb the particle's energy without leakage of dangeorus radiation or activated materials to the outside enviroment. Common materials are carbon and metals, embedded in an actively cooled assembly.

= Propagation physics =
==Magnetic fields==

Magnetic fields are things of motion. They are created by moving electric charges. And they only affect electric charges that are moving. An electric charge at rest in the presence of only a strong magnetic field will just sit there. But when it starts moving, the magnetic field interacts with it, starts deflecting it.

But not along the field. Oh no. The magnetic field does not point in the direction it will move the charge. No, the charge experiences a force that is perpendicular to the direction of the field. And also that is perpendicular to its own velocity. A charged particle moving straight down a magnetic field line will not experience any force.

On the other hand, if the particle is moving directly perpendicular to the field, the magnetic force will always push it to the side, even as it keeps turning. This results in the particle moving in a circle. The radius of this circle is called the <b>gyroradius</b>, and the rate at which the particle makes circles is called the <b>gyrofrequency</b> (or sometimes <b>cyclotron frequency</b>).
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Circular_motion.png|frameless]]
<td width=255>[[File:magnetic_forces.png|frameless]]
<td width=255>[[File:spiral_around_field_lines.png|frameless]]
<tr>
<td>To get circular motion, you need a force (blue) that is always the same amount but always pushing perpendicular to the direction of motion, toward the center of the circle. A charged particle in a constant magnetic field moving perpendicular to the magnetic field meets these conditions, and moves in a circular path.
<td>A diagram of how a magnetic field (magenta) exerts forces (blue) on a moving particle (with velocity shown in green).
<td>If a particle is moving partially along the magnetic field (magenta), it will drift down the field lines while spiraling around it (black path). Where the magnetic field is stronger (indicated by having the field lines closer together) it will execute tighter, faster circles in its spiral and will drift more slowly than when the field is weaker.
</table>

It can be convenient to split up a particle's velocity into a part parallel to the magnetic field (the <b>longitudinal velocity</b>) and a part perpendicular to the field (the <b>transverse velocity</b>). The longitudinal velocity describes how fast the particle drifts along the magnetic field line. The transverse velocity determines how tightly and how quickly the particle circles around the field lines.

If the gyroradius is much smaller than the extent of the magnetic field, you get a case where the particle approximately cannot cross magnetic field lines. It just spirals around them. On the other hand, if the magnetic field ends before the particle can make a significant portion of a full revolution, then the particle ends up being somewhat deflected from its original trajectory. If the gydroradius is enogh to deflect the particle into a region where the magnetic field has a different strength or direction, then things get complicated!

When a particle is spiraling around the magnetic field while drifting down the field lines into regions where the magnetic field is stronger, it will circle around the field lines in a faster, smaller circle. This increases the kinetic energy in the spiraling motion; this kinetic energy must come from the kinetic energy of the longitudinal motion. If the field becomes too strong, the particle will slow down all the way and come to a stop. Then it will be reflected back, pushed the other way down the field lines toward the region where the field is not as strong. This phenomenon is called magnetic mirroring, and is responsible for the way radiation gets trapped in the Van Allen belts around Earth with the particles bouncing back and forth along the Earth's field lines and getting reflected back and forth at the poles where the field is strongest.

If you want to calculate for yourself what the gyroradius <math>r_c</math> and gyrofrequency <math>f_c</math> are
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
r_c = \frac{\gamma \, m \, v_\perp}{q \, B}
</math></div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
f_c = \frac{q \, B}{\gamma \, m}
</math></div>
where <math>B</math> is the strength of the magnetic field, <math>q</math> is the amount of electric charge of the particle, <math>m</math> is the mass of the particle, <math>\gamma</math> is the Lorentz factor, and <math>v_\perp</math> is the part of the velocity that is perpendicular to the magnetic field. To get these formula to work, you need to use a consistent set of units. So for example if you measure the magnetic field in tesla you need to measure the charge in coulombs (not fundamental charges) and the mass in kilograms (not eV).

One consequence of this is that you can bend the beam coming out of your accelerator by placing a uniform magnetic field across the end. So you can steer your beam to where it needs to go. You can also bend the beam inside of your accelerator – to make it go around in a circle, for example.

Another consequence is that if your beam has to pass through regions with magnetic fields in it, it will get deflected. If you want the beam to end up at the right spot, you will need to account for this. If the exact strength or direction of the field are not known, you will get unpredictable deflections to your beam.

If your beam has been neutralized, the magnetic fields will work to bend the positively charged particles one way and the negatively charged particles the other way. If you have enough beam current, these separated charges will attract each other and counteract the magnetic force. If you don't, the beam will split and you'll get positive charges zipping off in one direction and negative charges going another direction. If you needed to neutralize your beam so that it doesn't electrostatically blow itself apart, well now it will be doing just that.

Even perfectly neutral atoms can be torn apart if the atoms are going fast enough and if the magnetic field is strong enough. Atoms are pretty resistant to this, but you can get conditions where it happens.

Magnetic fields affect the path of beams. But beams are made of charged particles and thus can have currents in them. Currents create magnetic fields themselves, which can influence the surrounding fields. Even a neutralized beam forming a plasma, when it encounters a magnetic field, will get currents induced in the plasma that helps to screen its interior from the field. A beam with enough punch can deform the fields around itself rather than having the fields bending it. As a rough rule of thumb, if the energy density of the beam over a regions larger than a gyroradius is more than the energy density of the field over that region, then the beam gets to have its way with the field, and vice versa. Highly focused, high current beams can thus be expected to better deal with ambient magnetic fields than those with less focus or lower current.

==Focusing==

In the previous section, we discussed how a particle beam can be steered by directing it through a uniform magnetic field. We can make this kind of field with a magnetic north pole directly opposite a magnetic south pole (with additional shimming magnets to keep things uniform). Because it has two poles, one north and one south, this kind of magnet is called a <b>dipole magnet</b>.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:dipole_beam_steering.png|frameless]]
</table>

But we want to do more than just steer the beam. we want to be able to expand it, contract it, and focus it to a point. To do this, we need a different kind of configuration. We will put two north poles opposite each other and, and 90 degrees, two south poles opposite each other. With four poles, this is called a <b>quadrupole magnet</b>, or sometimes just a "quad".
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_magnet.png|frameless]]
</table>
At the center of the quad arrangement, the magnetic fields from each side cancel out. there is no field there. A particle going through the center will travel straight through without deflection. However, the farther particles get from the center, the higher the field they encounter and the more they get deflected. A particle slightly off to one side may experience a force that curves it toward the center-line. If the quads are properly shimmed, a particle twice as far away will experience twice the force and twice the deflection from its trajectory back toward the center. This curves all the particles so they meet in one spot.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_focusing.png|frameless]]
</table>
However, there is a catch. Those particles that are offset from the center-line at 90 degrees from these focused trajectories are bent away from the center-line rather than towards it! A quad focuses in one direction and de-focuses in the other.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:quad_defocusing.png|frameless]]
</table>

Is there anything e can do to get an arrangement of magnets that purely focuses the beam? Well yes. We can put another quad arrangement downstream of the first, rotated by 90 degrees. Now consider a set of particles in an arrangement that is de-focused by the first quad. When they get to the second quad, they are further from the center than they were when they went through the first quad. Because the second quad is aligned to focus anything the first quad de-focuses, these particles will be focused; and because they are farther from the center, the focusing deflection will be larger than their previous de-focusing deflection. So the net result is that the particles are focused.
<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td width=255>[[File:Dual_quad_focusing.png|frameless]]
</table>
For particles with the opposite orientation that are first defocused, these will be defocused by the second quad but because they will have converged they will be going through a region of lower field and will be defocused with a smaller deflection than they were initially focused. This dual quad arrangement will focus any particle that goes through it, regardless of orientation.

For small deflection angles, a dual quad set-up acts like an ideal lens for particles. An ideal lens can focus all the particles diverging from a point onto another point, or into perfectly parallel lines. It can focus particles on perfectly parallel lines onto a point. So now we can use the particle beam to deliver focused particles on our target.

Or, at least we could if all the particles came from a point, or if they traveled on perfectly parallel lines. Unfortunately, they don't. When particles are made, they come from a source with a finite spot size and a finite temperature. The random motion of the particles moving around from their initial thermal motion means that they are never moving perfectly parallel to each other. And when you try to focus them, instead of all converging onto a single point they make a spot of finite size. And if your focusing and beam bending elements are not perfect, you add additional imperfections to the beam that further hider your ability to focus it.

Your ability to focus can be quantified by a measure called <b>emittance</b>. Emittance is measured in units of a length scale times an angle - nominally meters-radians (m<math>\cdot</math>rad) but the technical literature more commonly gives units of millimeters-milliradians (or mm<math>\cdot</math>mrad) or even 2 <math>\pi</math> mm<math>\cdot</math>mrad. The divergence angle of a beam that you try to make parallel will the the emittance divided by the width of the beam. If you focus a beam down so it converges at an angle, the minimum spot size you can get at the focal point is the emittance divided by the beam angle. If you try to focus a beam with emittance <math>\epsilon</math> on a spot a distance <math>R</math> away with an initial beam diameter of <math>D</math>, the spot size <math>S</math> on the target you can achieve is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
S = R \frac{\epsilon}{D}
</math></div>

So to get good focusing, we want low emittance. Is there a way we can lower the meittance beyond what we get from our source? Engineers are pretty bright folks, and they have found some ways to do this. You can, for example, pass a beam through a thin foil. This causes some stopping, slowing the beam down. You can then accelerate the beam back up to its original speed. However, the stopping is along the direction of the beam particle's tranjectory, even if that trajectory is at an angle. Meanwhile, all the acceleration is in the direction of the beam axis. So the net result is to reduce the random motion perpendicular to the beam, and thus reducing its emittance. Similar tricks can be played with magnet set-ups that force particles going through them to emit radiation.

Another method can be used with a relatively high emittance beam, say a beam of protons or ions or muons. Electron beams can be made with substantially lower emittances. So you can send a low emittance electron beam to co-propagate along with your primary particle beam. The beams will interact, with the particles scattering off each other. This exchanges heat between the two beams and so the temperatures, and thus emittances, of the two beams get closer to each other (and, if given enough time, would become equal). If you repeat this often enough, you can get the emittance down to the emittance of the electron beams. This is more practical in beams going around circular accelerators, because they will have time to get cooled over and over.

A final method that has been used is to install sensors to measure the beam fluctuations. For beams going in circles, the time to get a light-speed signal directly across the diameter to the opposite side wil always be less than the time it takes for the particles to go around the curve to that spot. So you can send signals to give electromagnetic kicks to the beam particles to knock them back into alignment when they get to that spot.

It is intriguing to note that these beam cooling techniques can take <i>hours</i> to work! <ref>[https://www.bnl.gov/isd/documents/86783.pdf M. Minty, R. Connolly, C. Liu,T. Summers, and S. Tepikian, "Absolute beam emittance measurements at RHIC using ionization profile monitors", Brookhaven National Laboratory formal report BNL-105970-2014-IR]</ref>

But perhaps the simplest method to reduce emittance is just to speed the particles up. For particles moving at a fraction of light speed <math>\beta</math> and Lorentz factor <math>\gamma</math>, the emittance of particles from the same source with the same temperature and same random motion will be proportional to <math>1/(\beta \, \gamma)</math>. To reflect this, it is convenient to define a normalized emittance <math>\epsilon_n</math> such that for the usual geometric emittance <math>\epsilon</math> that we have been talking about
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
\epsilon_n = \beta \, \gamma \, \epsilon.
</math></div>

Normalized emittance measures makes it convenient to get basic estimates of particle beam performance and compare emittance of different sources and beams. For example, various high-performing electron beams around the world have managed to push their normalized emittances down to about <math>2 \times 10^{-6}</math> m<math>\cdot</math>rad to <math>3 \times 10^{-6}</math> m<math>\cdot</math>rad. Facilities with proton or ion beams seem to have about an order of magnitude higher normalized emittance.
<table>
<tr><td width=200>Machine <td width=300>normalized emittance (m<math>\cdot</math>rad) <td>reference
<tr><td>European XFEL <td><math>2.3\times10^{-6}</math> <td><ref>[https://journals.aps.org/prab/pdf/10.1103/PhysRevAccelBeams.24.110702 T. Hara <i>et al.</i>, "Low-emittance beam injection for a synchrotron radiation source using an X-ray free-electron laser linear accelerator", Physical Review Accelerators and Beams <b>24</b>, 110702 (2021)]</ref>
<tr><td>SLAC <td><math>3\times 10^{-6}</math> <td><ref>[https://accelconf.web.cern.ch/pac97/papers/pdf/4W009.PDF M. Hernandez <i>et al.</i>, "Emittance Measurements for the SLAC Gun Test Facility", 1998 IEEE]</ref>
<tr><td>LHC <td><math>3.75\times 10^{-5}</math> <td><ref>[https://www.lhc-closer.es/taking_a_closer_look_at_lhc/0.beta___emittance Beta & Emittance; Taking a closer look at LHC]</ref>
<tr><td>RHIC (with protons) <td><math>\approx 5\times 10^{-5}</math> <td><ref>[http://www.osti.gov/servlets/purl/910428-6sKfP0/ Zhang, S, and Ptitsyn, V., "Proton Beam Emittance Growth at RHIC", Brookhaven National Laboratory formal report BNL-77563-2007-CP]</ref>
</table>
So as a first estimate for your sci-fi electron guns, you can take <math>\epsilon_n \approx 3\times 10^{-6}</math> m<math>\cdot</math>rad and for proton guns <math>\epsilon_n \approx 4\times 10^{-5}</math> m<math>\cdot</math>rad. Then divide by <math>\beta \, \gamma</math> for your beam to find the geometrical emittance. Which in turn can let you figure out how well you can focus your beam.

==Space charge==

==Energy loss in matter==

Sometimes, a particle beam needs to go through stuff to get it where you want it to go. Stuff like air, for example. Or the tissue in the way of getting the beam to a tumor. Also, sometimes, you want your particle beam to interact with stuff. Like the material you are trying to machine with an ion beam, or the bacteria you are trying to kill on medical equipment. All of this falls under the general realm of the interaction of the beam particles with matter. This is an enormously complex subject, but we can cover many of the basics here.

The primary effect on the beam of any interactions with matter are loss of energy of the beam particles (<b>stopping</b>), random changes to the particle directions that make the beam spread out (<b>straggling</b>), and knocking particles clean out of the beam altogether (<b>scattering</b>).

===Ionization===

The particles coming out of your particle accelerator are charged. When a fast moving charged particle goes through matter, the electric field from the charge pushes and pulls on the various charged particles that make up that matter – mainly electrons and the atomic nuclei around which the electrons orbit. At speeds above approximately 1% of the speed of light, it is primarily light and nimble electrons being affected while the ponderous nuclei just sit there barely perturbed by the passage of the particle. When the passing particle yanks on an electron hard enough, it can pop it clean off of the atom it was attached to, leaving a positively charged ion behind and giving you a free electron that can go and glom on to a previously neutral atom to make a negative ion. This process, unsurprisingly, is called <b>ionization</b>. For most particle beams, ionization is the dominant interaction with matter; although when you get very fast or very slow particles other effects can become more important.

if your particle was not initially charged when it goes into matter, it soon will be. The electric charges of the electrons and nuclei in the matter will do the same thing to the particle that a charged particle would do to them. This will quickly strip off any electrons around a high speed atom, leaving an ion and bare electrons to go through the matter. Only particles that are fundamentally neutral (like neutrons or photons) will not be immediately ionized.

It takes about 1 to 10 eV to ionize an atom. This energy must come from the kinetic energy of the beam particle. Particles from an accelerator have millions to billions of eV. So it takes a lot of ionizations to meaningfully affect the particles. However, one thing about matter is that it has lots of atoms that can be ionized so you do get those lots of ionizations. Because each ionization is so small compared to the total energy this is often approximated as a continuous slowing down of the beam particles. So many individual ionizations tend to average out, leading to all the beam particles being slowed down uniformly.

When the particle is yanking on nearby electrons to tear them out of atoms, those electrons are yanking back. This jitters the beam particle slightly from side to side. Again, these sideways yanks tend to average out. The statistical variations from the average slowing down and side-to-size jerking are what lead to straggling. Protons and ions are heavy, and will be deflected little from their original paths - like bowling balls smashing through matchsticks. These particles tend to go in mostly straight lines. Electrons, on the other hand, have the same mass as the electrons they are knocking about. So electron tracks through matter tend to zig-zag around a lot more.

As the particle whizzes past, it leaves a trail of atomic destruction in its wake - a trail of ions where the atoms are literally torn apart (not the nuclei, just the atoms). These particle tracks can sometimes be seen in detectors like cloud chambers. The faster the particle is going, the less time it has to yank on nearby electrons and so the less ionization it leaves behind. As a result, fast particles are not slowed down much but as they start to slow down they continue to be slowed down faster and faster. A consequence of this is that the particle dumps most of its energy into the material at the end of its track, just before it comes to a stop. This sudden peak in energy deposition is called the <b>Bragg peak</b>.

The electrons that a particle knocks out can end up going pretty fast. So these secondary electrons can produce further ionization in what is called an <b>electron cascade</b>. if an inner core electron is knocked out of an atom, other electrons falling in to fill that empty spot in the atom can release x-rays (<b>x-ray fluorescence</b>) or cause shake-ups in the outer electrons that knock other electrons off the atom (<b>Auger effect</b>).

===Brehmsstrahlung===

A charged particle is surrounded by an electric field. If it is moving, it is also surrounded by a circulating magnetic field. These fields move along with the particle.

If the particle knocks into something, it will slow down and probably change direction. The information about this bump only goes through the electromagnetic field at the speed of light. So a lot of that field that was moving along with the particle might not get the memo in time and keep right on going. Stray electromagnetic fields like this get this self-inductance effect going on where they start making themselves over and over again, allowing them to propagate as freely moving waves of electromagnetic radiation. This EM radiation from a slowing down particle is given the delightful name of <b>bremsstrahlung</b>.

Particles from an accelerator can be moving so fast that they make bremsstrahlung in the x-ray or gamma ray part of the spectrum. If you get very high energy bremsstrahlung gamma rays, the gamma rays can interact with matter by boiling an electron and its antimatter counterpart, a positron, out of the vacuum (using the nearby matter as the necessary momentum sink for the process). These positrons and electrons can themselves be very high energy and can in turn create more bremsstrahlung, and the process can repeat until the particles are no longer high enough energy to continue making each other. This is called an <b>electron-gamma shower</b>.

The x-ray machines used by dentists and radiologists use both bremsstrahlung and x-ray fluorescence that you get when shining an electron beam onto tungsten to make their x-rays.

At very high speeds, bremsstrahlung can be more a important source of energy loss than ionization.

===Electron capture and screening===

When an ion has slowed down to about 1% of the speed of light or less, it is moving slower than the electrons zipping around their atoms in the matter the ion is going through. This allows the electrons to stick on to the ion, now becoming the electrons of the atom that is moving through the material rather than of the material itself. The ion begins to lose its charge, and the rate of ionization goes down. Soon, ionization becomes negligible compared to the beam atom bumping in to other atoms.

===Phonons and displacement===

When an atom or ion moving through a material bumps into another atom, it will cause that other atom to move. This can have one of two primary effects.

First, if the atoms whack into each other hard enough, the atom in the material can be knocked clean out of place. Like a pool ball being thwacked by the cue ball, the struck atom will go zinging through the material. Atoms getting knocked out of their place like this is called <b>displacement</b>. Displacement creates chemical changes to molecules as atoms are removed, and creates defects called <b>vacancies</b> in crystals (which, as you might imagine, is just a missing atom in the crystal lattice). When the atom comes to rest, it can chemically attack other molecules, again changing the chemistry, or create an extra vacancy called an <b>interstitial</b> jammed into the crystal lattice where it is not supposed to be. If the displaced atom is going fast enough, it can whack other atoms and displace them, leading to a <b>displacement cascade</b> like tenpins knocking each other down. As a result, at the end of its track, an ion can leave a region of disordered material and messed up chemistry where the atoms have all crashed into each other like a freeway pileup.

If an atom is displaced near the surface of a material, the displaced atom can shoot out of the surface and escape the material altogether. This is called <b>sputtering</b>.

Second, the beam atom might only deliver a glancing blow. Or the beam particle might be an electron, which is so light that it is usually incapable of displacing atoms. In this case, the struck atom gets kicked to the side a bit and runs into other atoms; maybe it is also pulled back by the chemical bonds of other atoms that it is attached to. This transfers the motion of the atom to other atoms around it in the form of pressure waves (or, in a solid, shear waves). This adds heat to the material without causing direct damage – although as you might imagine, enough heat can also cause localized melting, evaporation, or decomposition. The individual quanta of vibration – fundamental particles of sound, if you want to think of them like that – are called phonons in analogy with the photons that are the fundamental particles of light. Much like ionization, a particle that excites pressure waves slows down gradually and continuously, unlike the sudden changes in speed and direction you get from knock-out displacement.

===Nuclear collisions===

So far, we have only discussed the things that happen when beam particles interact with matter particles by electromagnetic forces. Even atoms knocking into each other to produce phonons or displacement happens through the mutual repulsion of the electrons around the atoms (although this is dressed by other weird effects like how two electrons can't be in the same place at the same time, or how the fields of one electron can affect other nearby electrons that in turn affect the original electrons, leading to odd correlation effects and charge density waves and other fun stuff that keeps condensed matter physicists and quantum chemists busy).

But electromagnetism isn't the only thing you can get going on. below the level of structure of the atom, you have the atomic nucleus. And a direct hit on the nucleus can end up giving very different effects.

The simplest thing that can happen is an <b>elastic collision</b>, where no energy goes in to any internal nuclear state or into breaking up the nucleus. An elastic collision just ends up giving you displacement (or possibly phonons, but if you hit the nucleus you usually hit it hard enough to send it flying).

But beyond that you can cause any number of nuclear reactions. You can leave the nucleus in an excited nuclear state, from which it usually relaxes back down to its original state by emitting gamma rays. You can knock out nuclear particles, such as neutrons, protons, deuterons (a proton stuck to a neutron), tritons (a proton stuck to two neutrons), helions (a neutron stuck to two protons), alpha particles (two protons stuck to two neutrons), or potentially even larger nuclear fragments. If you hit a heavy nucleus like uranium, you can cause it to split apart by fission. And if you hit the nucleus really hard you might just shatter it into many tiny fragments of those protons, neutrons, deuterons, tritons, helions, alpha particles, and heavier nuclear fragments mentioned earlier. To get any of these <b>inelastic collisions</b>, you need to hit the nucleus hard enough. If your beam particle doesn't have enough energy to excite the nucleus or knock particles, these things simply can't happen. As it gets enough energy to excite a particular interaction channel (as they are called), that process becomes possible and as the energy increases the process becomes more and more likely.

A nucleus is only about a femtometer across (<math>1 \times 10^{-15}</math> m). Which makes it only about <math>1 \times 10^{-30}</math> m² in cross sectional area. In the meantime, chemical bonds between atoms are usually about 1/10th of a nanometer (<math>1 \times 10^{-10}</math> m) long. In condensed matter (liquids and solids), this means that you have somewhere about <math>1 \times 10^{30}</math> atoms per cubic meter. A meter thickness of atoms thus presents just about about enough atoms that their nuclei can cover the projected area – in other words, you can expect a particle to go through (very roughly) about a meter of condensed matter before it whacks into a nucleus.

This assumes that a particle that goes through a nucleus interacts with it. This is usually a good assumption for particles that interact by the strong nuclear force, such as protons, neutrons, mesons, and the atomic nuclei of ions. But many particles cannot interact by the strong nuclear force, such as electrons or muons. These are much less likely to interact with an atomic nucleus even if they pass through it, so they will end up going considerably farther between hitting nuclei than the above assumption. But nuclei can interact electromagnetically (they are made up of charged protons, after all, and at a smaller level of charged quarks), so electrons and muons can produce some nuclear interactions; just far less than actual nuclear particles like protons and ions.

A nuclear strike is a discrete event that will remove the impacting particle from the beam and will (except for elastic collisions) probably produce a spray of radiation.

===High energy physics===

As you get to higher and higher energies, you unlock all sorts of new and weird and interesting processes that can happen. Boiling exotic particles out of the vacuum, unifying forces that had been divorced since the big bang, and producing bizarre unstable resonances in the quantum fields. This is what physics build accelerators to measure. From the point of view of getting a beam to its target, or what happens to the target, however, the main effect is mostly that the interacting particle is removed from the beam and a bunch of additional radiation is produced.

==Self pinching==

= Operational Considerations =
== Logistics ==
The logistics of particle beam can be a mixed bag. By itself, any particle accelerator is a complicated piece of machinery full of delicate parts that have to deliver high performance. This may require cryogenic temperatures with associated challenges on the materials and mountings used (insulation, and distortions under temperature changes). When working, particle accelerators invariably generate some radiation across the whole assembly, which irradiates and activates the structure. Some parts may be exposed to significantly higher radiation flux due to their working principles. Often the particle beam is a dynamic system that must be managed with active, closed control loops, involving sensors, modeling and control software, actuators, and the computing and communication hardware that brings it all together. This poses engineering challenges across all disciplines.

A practical particle beam is inherently a counterintuitively sturdy device. If it was too delicate, it would break too much to be useful, though of course this is a sliding scale. Still, while sturdy, this doesn’t eliminate complexity, it manages complexity. The result is a system that is the product of a high-tech civilization. A superior machine shop and well-trained engineers are a must to maintain a particle beam system. Some parts of a particle accelerator may be mundane - others may be custom-machined parts that require their own dedicated industries to produce. That said, the manifold utility of particle accelerators is on your side here. Since there are so many practical applications, it is economic to produce parts for industrial demand. (that said, these may not be the kind of components you want in specific high-performance applications.)

The main thing a particle accelerator requires the most is electricity. Any mature power production facility can supply the power.

However, all particle beams require at the least a source of neutralizing mass. The projection of a charged beam will leave the projecting object oppositely charged. In an atmosphere the natural plasma formed by the beam will ground everything out, but in space the diffuse vacuum will not do the job.

Furthermore particle beams using ions also require a feedstock for their ions to produce from, and in an appropriate format. This can be a gas, a fluid or an easily ionized solid. It can be the feedstock species in its pure form or in an easily split molecule, though every complexity adds to the amount of preparation work that has to be done before the feedstock enters the particle beam.

Don’t worry about the mass of your source or neutralizing ions though. Large particle accelerators, like the LHC in Switzerland, only consume tiny gas bottles worth of feed gas every couple years. The mass of a single 5.56 NATO bullet worth of hydrogen could feed a 10 GW, 10 GeV ion beam for nearly 4 days of firing! You will not be running out of beam feedstock easily.

== Cooling ==
Cooling a particle accelerator can be a serious challenge. Various bits of a particle accelerator love to be supercooled for efficiency or basic working principle - any bit of superconductor used in the accelerator, or magnets, or EM generators, will require cooling to its generally supercool operating temperature.

Even if Standard Temperature and Pressure Superconductors (273 Kelvin, 100 Kilopascal) are available, present superconductors have demonstrated superior performance by being supercooled. That means to achieve even stronger fields, even STP superconductors may be cooled to cryogenic conditions.

Cooling cryogenic systems sucks. Thermodynamics shows itself from its worst side. You can forget running cryogenic radiators - the Stefan-Boltzman Law penalizes the direct radiative cooling of anything cryogenic severely as much as it empowers very high-temperature radiators. This only leaves stages of heat pumps to provide cryogenic fluids that can in turn, cool cryogenic systems. Heat pumps draw power, and for such a giant temperature gradient as we can have here, they will draw a lot of power.

However, we have other options. Cryogenic systems also tend to be highly efficient, so while they do heat up, they do so very slowly and there is little overall energy absorbed. This provides the option of running off of bunkered cryogenic fluid in dewars for the duration of operating the equipment and not running any heat pumps all the time.

Other bits of a particle accelerator system aren’t as required, but really nothing is a fan of high temperatures. Going higher than room temperature can be treated as an oddity. High-performance lasers may not require cryogenic conditions, but they certainly like it cool. Radiofrequency producers of any kind generally fare worse with rising temperatures. That means cooling overall remains a challenge - 300 Kelvin is still not that great of a hot loop temperature for radiators.

== Safety ==
=== Ionizing Radiation ===
There are surprisingly many sources of ionizing radiation in a particle accelerator, besides the obvious threat of the output beam itself. Not all particles are accelerated correctly or in the correct direction. When such particles interact with the structure of the accelerator, they release a shower of secondary ionizing radiation. In electron accelerators and sources, stray electrons may be accelerated and release x-rays from bremsstrahlung as they collide with surrounding matter. The resulting doses can be significant and dangerous. Also, such collisions leave behind residual activation, making the accelerator a radiation hazard for some time even after it has been shut down. Generally, people should not work near the particle accelerator while it is functioning, and radiation barriers isolate the core accelerator from adjacent facilities. (In real-life facilities, you will often see accelerators inside larger buildings being cocooned inside doped concrete plates to shield the rest of the room.)

=== Radiofrequency ===
RF energy is used in many kinds of particle accelerators. This energy may leak from the structures it is generated, contained and used in due to damages or incorrect assembly. As microwave radiation and above is used, the radiation can be readily absorbed by bodies, hydrogen-rich materials or other objects. The end result of this can be rapid heating and thermal effects. Especially in accelerators designed to throughput a lot of energy (say a GW or more), this may become a serious issue if a waveguide is damaged. It’s not recommendable to stand in front of a multi-MW radar either!

=== High-powered electrical and electronic systems ===
The operation of the magnetic systems and accelerators often involves high-powered electrical and electronic systems. High voltages and/or amperes are involved. This poses risks like shock hazards, electrical arcs, and electrical fires. Some electrical systems such as capacitors or inductors may have combustive or explosive failure modes.

=== Flammables ===
Parts of an accelerator or its infrastructure may be manufactured from flammable substances such as hydrocarbons. In the case of an ignition source such as a high-powered beam cutting into a physical substance or an electrical fault, these may catch fire.

=== Lasers ===
Some accelerator systems use high-powered lasers. Laser plasma accelerators use very high-powered pulse lasers to provide pumping power. The safety concerns of lasers for blinding, physical damage, and possible exposure to ionizing radiation as a secondary effect of the beam impact, apply. Some high-powered lasers can also drive electrons which generate strong EMPs, which may damage or destroy other systems in turn. Parts of a laser assembly or laser beam guide may shatter if exposed to overloading conditions.

=== Magnetic systems ===
Parts of a particle accelerator may use strong magnets, up to superconductive magnets with multi-Tesla fields. While the magnets are integrated into chokes that shape the magnetic fields, fringe fields invariably extend beyond the core magnet installation.

Multi-Tesla transient electromagnetic field can affect the human nervous system temporarily, inducing nausea, vertigo, metallic taste and optical phenomena, if the person moves in relation to the magnetic field. Otherwise, the risk of exposure is low. That said, electronic systems that are not shielded properly, such as say, implants, could be adversely affected by the magnetic fields.

The biggest physical risk are reactions of other magnetic objects with the magnets. You can look up cases where metallic objects interacted with the multi-Tesla fields of MRI scanners. Such parts may fly across the room towards the magnet at increasing speeds, and damage anything or injure anyone in their path, possibly ending with the object or person squashed (or impaled!) to the magnet.

Also, large superconductive magnets store not insignificant amounts of energy. If the superconductive property breaks down, the magnet quenches and becomes a resistor. The resistance site heats up as current tries to flow through it, expanding the quench and possibly being heated until it sublimates into gas. This could possibly happen explosively. In such a case, up to many GJ of energy might be unleashed in a conflagration or explosion (subsonic or supersonic), damaging surrounding systems and ejecting shrapnell.

=== Cryogenic cooling systems ===
Particle accelerators often involve cryogenically cooled systems. This means cryogenic fluids like nitrogen and helium are present in the accelerator infrastructure. Such gasses have both a very low temperature and a high ratio of gaseous volume at high temperatures to their liquid volume.

Exposure to the leaked fluid can lead to cryogenic cold damage, injury, and death. Furthermore such gasses are efficient at displacing ambient breathing gasses such as oxygen, so they pose an invisible, and rapidly spreading, suffocation hazard.

Also, if insulation is damaged, ambient heat may leak into the cryogenic circuits, and cryogenic fluid may begin evaporating. This can lead to dramatic rises in internal pressure. Relief valves should trigger to safely direct such pressure spikes away from important places and keep the system from failing catastrophically, but otherwise, pipes and cryogenic storage devices may fail, up to explosively.

=== Beam excursions ===
If the magnets responsible for manipulating the beam break down, the beam may excurse from its designed paths. The beam may become defocused, or maintain its focus. The beam begins cutting into the accelerator structure, evaporating material and depositing energy. This will definitely foul the ultra-high vacuum inside the accelerator, leading to secondary radiation production as the beam collides with gas inside the accelerator. Some sensitive facilities requiring ultra-high vacuum such as some RF cavities may “crash” and become ruined. Further systems may become damaged or destroyed from either physical stress or the radiation pulse. If the beam bunches contain a lot of energy and maybe are heavy particles, the deposition may lead to an energetic local explosion. Alternatively the beam may escape and cut into structure beyond the particle accelerator itself, damaging and destroying equipment and seriously hurting or killing personnel.

In the worst cases, a beam excursion on a very powerful beam may lead to an explosion that destroys the accelerator utterly. Even in a less bad case, the radiation pulse might damage or destroy critical equipment around the accelerator and take the entire facility offline.

=== Beam dumping ===
If any other systems fail, the accelerator probably has to be shut down. This poses a problem - what do you do with the beam energy that is still in the accelerator system?

Sometimes, the beam is designed to leave the accelerator immediately, and shutting down the beam is easy. But in the case of cyclotrons and storage rings, the beam may circulate for a while. What do you do then? You have to dump the beam from the accelerator. Special beam stops are used on research and industrial accelerators, large blocks of dense material able to absorb the beam. During this process the blocks both heat up and become radioactive. For particle beams storing very high energies, this may become an increasing challenge. How do you safely dumb a beam containing many GJ or more? In some cases you may have no other choice but to dump the beam out into the environment. Where and when can you do this safely without threatening third parties?

= Application areas =
(This is a skeletal overview summary)
Particle accelerators have many, many, many areas of application.

In science, the big particle accelerators like the LHC are common knowledge. But electron microscopes also use components of particle accelerators. Many particle accelerators worldwide are used to produce high-energy x-rays in synchrotrons or linear accelerators, imaging samples for material science, chemistry, biochemistry and other areas.

In manufacture, particle beams can be used to cut material, but also weld material. Precision beams can break down gaseous feedstocks to deposit metal in micrometer-sized structures. Directed ion guns are used to implant traces of elements into surfaces for the manufacture of electronic devices.

And in medicine, particle accelerators of ever-greater sophistication find increasing use in cancer treatment and beyond, destroying life-threatening tissues with ever-reduced doses to surrounding healthy tissue.

= Further material =

= Sources =

Authors: Sevoris Doe, Gerrit Bruhaug. Check reading by Luke Campbell.