Galactic Library - User contributions [en]

Electromagnetic guns

2026-11-03T06:36:12Z

Lwcamp: /* Gauss guns */

[[File:Naval_Electromagnetic_Railgun.png|thumb|General Atomics Electromagnetic Railgun prototype.]]
We can use electromagnetism to make things move. The most familiar way to do this for many of us is to use a rotary electric motor, which turns electric power into the rotary motion of a shaft. This can be use to spin a drill bit or a saw or turn the wheels on a car. But not all electric motors are rotary in nature. Any rotary motor can be unrolled, so to speak, to turn it into a linear electric motor. Now you are using electric power to move an object back and forth. If you only move it forth, and you move it forth very quickly, and you don't bother to catch the forward moving part when it leaves the motor, you have a gun.

== Applications ==

=== High speed ===

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Hypervelocity_projectile_impact_test.png|684 px|frameless]]
<tr>
<td width=684>Test of a railgun-fired hypervelocity projectile penetrating a series of metal plates. The shot is moving from right to left. A video of test firing is available [https://www.youtube.com/watch?v=O2QqOvFMG_A here].
</table>

In a conventional firearm, the propulsion of a projectile becomes increasingly inefficient as the projectile moves at speeds close to the speed of sound in the propellant gas. An electromagnetic gun, lacking propellant gas, does not have this problem. This makes electromagnetic guns contenders for launching high speed projectiles, potentially attaining speeds that would otherwise require an exotic and complicated chemical propellant firearm like a light gas gun. For comparison, artillery will tend to get to around 0.6 to 1 km/s and a tank firing an anti-tank APFSDS dart might reach speeds approaching 2 km/s. Meanwhile, experimental railgun programs for military weapons development commonly reach speeds of 2 km/s to 2.5 km/s<ref name="Tatake1994">S. G. Tatake, K. J. Daniel, K. R. Rao, A. A. Ghosh, and I. I. Khan, "Railgun", Defense Science Journal, Vol 44, No 3, July 1994, pp 257-262 https://web.archive.org/web/20171111205554/http://publications.drdo.gov.in/ojs/index.php/dsj/article/view/4179/2439</ref><ref name="Zielinski1996">A. E. Zielinski, M. D. Werst, J. R. Kitzmiller, "Rapid Fire Railgun For The Cannon Caliber Electromagnetic Gun System", 8th Electromagnetic Launch Symposium, April 1997 https://repositories.lib.utexas.edu/items/6e9f0b8e-2e21-4bba-a42d-c4e664af0e1b , A. E. Zielinski and M. D. Werst, "Cannon Caliber Electromagnetic Launcher", IEEE Transactions on Magnetics, Vol. 33, No. 1, January 1997, pages 630-635 DOI: [https://ui.adsabs.harvard.edu/link_gateway/1997ITM....33..630Z/doi:10.1109/20.560087 10.1109/20.560087] Bibcode:[https://ui.adsabs.harvard.edu/abs/1997ITM....33..630Z 1997ITM....33..630Z].</ref><ref>Brendan Borrell, "Electromagnetic Railgun Blasts Off: A supersonic bullet is fired with a record-breaking 10 megajoules of muzzle energy", MIT Technology Review, February 6 2008 https://www.technologyreview.com/2008/02/06/128211/electromagnetic-railgun-blasts-off/</ref><ref>"Navy Evaluating Second Electromagnetic Railgun Innovative Naval Prototype ", Office of Naval research, Oct. 9, 2012 https://web.archive.org/web/20121012235510/http://www.onr.navy.mil/en/Media-Center/Press-Releases/2012/Electromagnetic-Railgun-General-Atomics-Prototype.aspx</ref><ref>Allen McDuffee, "Navy’s New Railgun Can Hurl a Shell Over 5,000 MPH", Wired, April 9, 2014 https://web.archive.org/web/20170401112308/https://www.wired.com/2014/04/electromagnetic-railgun-launcher/</ref><ref name="wired2010">Spencer Ackerman, "Video: Navy’s Mach 8 Railgun Obliterates Record", Wired, December 10, 2010 https://web.archive.org/web/20140111212221/http://www.wired.com/dangerroom/2010/12/video-navys-mach-8-railgun-obliterates-record/</ref>. Speeds of 3, 3.5, and even 6 km/s are often touted, and some experimental railguns have launched solid (albeit plastic) projectiles at up to 10 km/s<ref name="Parker_1989">J. V. Parker, "Why plasma armature railguns don't work (and what can be done about it)", IEEE Transactions on Magnetics, Vol. 25, No. 1, pages 418-424, January 1989</ref>, but for practical systems near to being fielded such speeds have been stubbornly aspirational rather than actual.

At speeds of more than about 2 to 3 km/s, the dynamic pressures of impact are so much higher than the material strength of anything involved in the collision that both the projectile and the target can be considered to act as a fluid. Mechanical strength ceases to be relevant. A hypervelocity dart impacting at these speeds will penetrate as if it were a jet of liquid of the same length and density. In any such impact, the materials will splash, violently, throwing fragments and debris. This violence will create a blast that can gouge craters and damage nearby materials, equipment, and personnel.

At around 3 km/s, the kinetic energy of a projectile will match the energy released by the detonation of the same mass of TNT. This means that a separate warhead is not needed. The energy liberated by the projectile slamming into a target at more than 3 km/s will produce a bigger explosion than if it were filled with explosives – particularly because artillery shells need to be built extra sturdy to survive launch leaving relatively little space for the warhead. It is worthwhile to note that penetration doesn't increase when speeds get over 2 km/s. A faster speed can flatten a larger area but if your goal is punching through armor any energy used getting faster than 2 km/s at the target is energy wasted. Note that this is speed at the target. Because a projectile launched in atmosphere will suffer from aerodynamic drag in flight, when shooting at distant targets you may need higher speeds at the muzzle to get the desired terminal performance.

As an example, the M795 155mm artillery shell contains 10.8 kg of TNT as explosive filler, giving it an explosive energy yield of approximately 45 MJ. The shell itself has a mass of 46.8 kg<ref>"M795 Projectile 155mm High Explosive HE", GlobalSecurity.org https://www.globalsecurity.org/military/systems/munitions/m795.htm</ref>. The same energy could be achieved without explosives if the shell impacted at a speed in excess of 1.4 km/s.

It is worth considering that for use on a planet, it doesn't really do much good to get an electromagnetic gun beyond about 2 to 2.5 km/s. At those speeds, the projectile will hit hard enough to explode anyway and, as already mentioned, is where the penetration is going to be maximized. If you go faster, you will lose more energy early on to aerodynamic drag if the projectile has to go through an atmosphere (and some designs, such as railguns, will have more problems with rail wear). For a given energy-per-shot budget, it will often be advantageous to keep the projectile speed at 2.5 km/s or less but increase the mass to match your energy output. Neglecting drag, a projectile launched at 2 km/s can go 200 km; one launched at 2.5 km/s can go over 300 km. When you include drag, the distance decreases. The more massive the projectile, the less drag will affect its trajectory and the farther it will go – giving yet another reason to favor mass over additional speed. The hypervelocity projectile program<ref name="BAE HyperVelocity Projectile">https://www.baesystems.com/en-media/uploadFile/20210404062224/1434555443512.pdf</ref> gives a range of 185 km for a railgun-launched HVP dart; the railgun this was designed for is commonly reported to shoot projectiles at around 2 km/s, suggesting that for big naval guns like this and narrow aerodynamic darts, drag will not have too much of an effect.

In space, there is no aerodynamic drag to worry about. Meanwhile, typical relative speeds between spacecraft range from several km/s to several tens of km/s, potentially much more for spacecraft boasting advanced fusion thrusters, beam-driven spacecraft, or plasma sail spacecraft. This means that the problem is not getting the impact to occur at high speeds – you get that for free. Rather, the main problem is getting your projectile to intercept your target. If your target and you are separating at a higher speed than your gun's projectiles go, you won't hit. If your target can thrust to get more speed than your projectile before it arrives, it also will not hit. Because space is big, even a very fast projectile can take a long time to reach its target. If the projectile is not guided, this means that any maneuvering on the part of the target will cause the projectile to miss. This is why serious consideration of electromagnetic guns for space warfare will design around shooting guided rocket projectiles that can home in on a target.

Sometimes it is useful to have a gun that can shoot projectiles really fast for purposes other than shooting other people or their war machines. Electromagnetic guns can be used for the field of hypervelocity impact science, which is useful for developing protection and mitigation schemes for satellites and other spacecraft against micrometeorites. Some fusion energy schemes rely on impact fusion, where you collide a hypervelocity projectile with a stationary target, at least one of which contains fusion fuel. These sorts of studies often require impact speeds well in excess of 5 km/s. While the standard go-to instrument for reaching these speeds are light gas guns, some electromagnetic guns have been developed that can get things going this fast and faster.

No method of acceleration can ever be 100% efficient. Ohmic heating, eddy currents, time varying mechanical strain as the projectile is subject to changing magnetic forces during launch; these and others will all dissipate some energy in the projectile. This additional energy will cause the projectile to heat up. When the kinetic energy of the projectile is so much higher than the energy needed to completely vaporize it, even small inefficiencies can lead to a large enough temperature spike to disable, warp, or melt the projectile. This means that there is an upper limit to the speed of a launched projectile before the projectile is destroyed. The particular speed limit depends on the implementation – see the description of the electromagnetic rocket gun below for a design that uses this waste heat to its advantage to go even faster!

A projectile going at 2 km/s or more will explode. If it is a dense dart, it might punch through a considerable thickness of material in the process, but it will produce a significant blast in the process. Common media depictions of hypervelocity guns leaving nice neat holes with no collateral damage to nearby objects are not accurate.

At high enough speed, a projectile will not be able to go through ordinary air without being ablated away. Space debris encountering the tenuous upper atmosphere of Earth's mesosphere, generally at speeds of 7 km/s or higher, get heated by the ram compression of the air in front of them sufficiently that they are nearly or entirely consumed by ablation. In the dense air of Earth's lower atmosphere, the effect would be even more immediate and severe. You can expect that any projectile moving at around 5 km/s or faster will suffer rapid ablation and might not reach its target.

=== Launch assist ===

Building rockets to launch stuff into orbit is hard. One of the things that makes it hard is that you need to lift the propellant you'll need for later on in the burn in your initial stages of ascent. This is all encapsulated in the [https://en.wikipedia.org/wiki/Tsiolkovsky_rocket_equation Tsiolkovsky rocket equation] showing how your initial launch mass grows exponentially with the final speed you need
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
M0 = Mf exp(Δv / ve)
</div>
where Mf is all the stuff you want to get delivered to orbit, Δv is how much speed you need to add to your rocket from the launchpad to get it into orbit, ve is the speed of the exhaust coming out the back of your rocket, and M0 is the total mass of your rocket including Mf but also all the extra propellant you need to use to boost the rocket up to that Δv speed. For something like a [[Hydrolox_Engine|hydrolox rocket engine]] with ve ≈ 4 km/s, and Δv to reach low Earth orbit (including atmospheric losses) something like 8 km/s, you'd need something like six times the mass of your rocket payload in propellant.

All this means that any little boost you can give to cut down on Δv can give substantial savings on launch weight and cost. If you could shoot your rocket out of a railgun at 2 km/s, you could cut the amount of fuel nearly in half. These kinds of arguments have motivated many designs for launch guns to shoot stuff toward space. This does not necessarily have to be a railgun, it could be a coilgun or even a conventional chemical combustion gun.

There are limits to this idea that go beyond merely the technological speed limits of an electromagnetic gun. The most severe is that such a space gun can never launch something directly into orbit. An orbit is an ellipse with one focus on the center of the planet. This means that any orbital ballistic trajectory which starts from the ground will intersect the ground again before a full orbit is completed. To get around this, you will need to perform additional thrust maneuvers above the ground – a trick that can't be done directly with a ground-based launch gun and generally relies on rockets. A space gun can still give some initial speed that will eventually help get the payload into orbit, it can give a boost that will loft the rocket up above the thick part of the atmosphere where the rocket will work more efficiently and the majority of the atmospheric drag is past, and it could potentially launch stuff on escape trajectories that leave the planet forever.

<table class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<tr>
<td>[[File:Keplerian_orbit_ground_impulse.png|533 px|frameless]]
<tr>
<td width=533>Launch with an impulse delivered only at ground level puts a payload on an orbital trajectory that will always intersect the ground. It may be a way to get from Cape Canaveral to Boliva (as shown) but can't put you in an orbit that lets you stay in space.
</table>

== Common features of electromagnetic guns ==

There are many different kinds of electric motors. They all have gun versions. Different kinds of electromagnetic guns have different advantages and disadvantages. But there many features that are similar about them as well.

=== Charging equipment ===

An electromagnetic gun requires high power to be delivered in a very short pulse, on the order of a millisecond long. For artillery or naval canons, this can result in an instantaneous power of tens to hundreds of gigawatts! Unless the electromagnetic gun is directly plugged in to the full electrical output of a major regional power generating station (and you are willing to cause blackouts when it fires), you won't be able to directly deliver that kind of power from your main power supply. Instead, you will need to gradually build up energy over time, [[Energy_Storage|storing it in some kind of equipment]] that can deliver the stored energy in a very fast pulse. Common ways to do this include charging up [[Energy_Storage#Capacitors|capacitor banks]] or spinning up a [[Energy_Storage#Flywheels|compulsator]] (basically a flywheel attached to a generator). At least one research program used a massive inductor to store the energy<ref name="RashleighMarshall1978">S. C. Rashleigh and R. A. Marshall, "Electromagnetic acceleration of macroparticles to high velocities", Journal of Applied Physics 49, 2540-2542 (1978)</ref>, along the same lines [[Superconductive_Magnetic_Energy_Storage|superconductive inductors]] might be useful in this regards if you have the tech for them.

In most cases, the energy accumulator system will store enough energy for one shot. Your rate of fire will depend on the time it takes for your primary power supply to build up enough energy for one shot.

Another option are [[Energy_Storage#Explosively_pumped_flux_compression_generator|explosively pumped flux compression generators]], which store the energy as high explosives and give a one-time surge of power when the explosives are detonated to drive your generator. Because the generator and associated equipment do not usually survive this process, it can be a rather expensive method to power your gun. Combined with the need to carry a magazine that could potentially explode, this starts to eat into many of the potential advantages of electromagnetic guns over conventional guns.

Flywheels, capacitors, and inductors all give a sudden surge of power at the beginning that decays over time. An explosively pumped flux compression generator, on the other hand, delivers a pulse that ramps up from zero to maximum power right at the end of the pulse. However, what you usually want is a specific pulse shape for your particular engineering design – for example, a constant power level for the duration of the shot. This is accomplished with a pulse forming network, which are inductors and capacitors connected in series (or certain kinds of transmission lines) to give the desired pulse profile.

=== Self forces ===

The same magnetic forces that push on the projectile will also push outward on the accelerating machinery of the gun (it's barrel, if you will). In order to keep the barrel from bursting, it will be important that it is built with enough structural reinforcement to hold it together. This means, sadly, that you don't have railguns that look like tuning forks … the outward pressure on the tines would bend them apart from each other, breaking electrical contact with the armature and its projectile, and likely permanently damaging the gun. Instead, the rails will be solidly held together with a strong support case to keep them from bending apart. In a similar fashion, other electromagnetic guns will also need strong, reinforced barrels to resist the forces of launch. The articles on [[Superconductive_Magnetic_Energy_Storage | Superconducting magnetic energy storage]] and [[Plasma_Guns | plasma guns]] discuss this physics in greater detail.

=== Recoil ===

[[File:Comparison_of_the_recoil_of_conventional_and_electromagnetic_cannon.png|thumb|A comparison of the recoil impulse from a powder gun (green), electromagnetic gun (red), and a powder gun with a muzzle brake (blue) for a constant-energy launch (varying the mass of the projectile with speed to keep the total kinetic energy constant). Data from <ref>Edward M. Schmidt, "Comparison of the recoil of conventional and electromagnetic cannon", Shock and Vibration 8 (2001) 141–145 https://doi.org/10.1155/2001/590948</ref>]]Newton's third law of motion stipulates that whenever you push on something, it pushes back on you just as much. In a closed system, you would get no net movement. If something gets pushed away, the rest of the system recoils back in the opposite direction.

In a conventional firearm this recoil comes from the hot high pressure gas pushing the bullet down the barrel. The pressure of the gas pushing back on the breach face gives the gun its kick. In electromagnetic guns, this recoil force comes from the same interactions of currents and fields in the gun that push back on the gun as pushes out the projectile. In a railgun, the current in a high field might push the projectile one way, but that current loop must close somewhere and the field will push back on the rest of the current loop in the gun. In an induction coilgun, the electromagnet in the barrel that pushes on the induced electromagnet in the projectile is in turn pushed back by that same induced electromagnet. A ferromagnetic coilgun, the same interaction occurs in the electromagnet in the stator and the permanent magnets in the armature.

This impulse will be similar in magnitude to the recoil produced by a chemical propellant firearm with the same ballistics, with one important difference. A considerable portion of the recoil from a chemical propellant firearm comes from the hot gases jetting out of the barrel, acting like a rocket. The amount differs depending on the interior ballistics of the firearm but a contribution of very roughly 30% to the recoil is typical. The electromagnetic gun will lack these propellant gases, so that the recoil will be somewhat less than that of an unmodified firearm. However, the same reason it lacks this additional recoil does not allow an electromagnetic gun to use a muzzle brake, which can reduce the recoil of a firearm even further.

Electromagnetic guns are commonly envisioned shooting projectiles at high speeds. From the section on [[Recoil#Effects_of_Recoil | the math and physics of recoil]], we can see that this will reduce the recoil impulse on the gun, and thus the effects of recoil on firing. However, some electromagnetic guns use a shell, sabot, or armature that is separate from the primary projectile. This will add mass to the accelerated stuff coming out the end of the gun, which will increase both recoil and the energy needed for the shot. As a consequence, it is beneficial to reduce the mass of this extra stuff as much as possible compared to the projectile.

=== Ammunition ===

For an electromagnetic gun, the projectile and sabot (if present) make up the entirely of the ammunition. There's no powder, no casing, no primer. This is expected to make electromagnetic gun ammunition rather more compact than that for equivalent conventional firearms – one claim<ref name="Adams2003">D. Adams, U.S. Navy, "Naval Rail Guns Are Revolutionary", U.S. Naval Institute Proceedings. 129 (2): 34. (February 2003) https://web.archive.org/web/20070708054858/http://edusworld.org/ew/ficheros/2004/railguns.pdf</ref> has a railgun able to store over six times the number of shots in the same volume as conventional ammunition.

=== Safety ===

One advantage suggested for electromagnetic guns is that without the need for reactive propellants, ships and ammunition storage warehouses will be significantly less hazardous if hit by enemy fire. In operation, an electromagnetic gun might only store enough energy for one shot in volatile fast-discharge energy storage devices such as capacitors. The fast discharge energy storage would be recharged by a generator between shots. On a vehicle, this could be the same generator used for electric traction, thus allowing a unified power supply system for the drive train and weapons. While hits to the fuel or generator are still possible, this hazard is inherent to the operation of the vehicle and is present whether the vehicle is armed with an electromagnetic gun or not.

=== Shot consistency ===

In modern artillery, variance in timing of ignition and burning rate of the powder can produce differences in muzzle speed and delays of launch of the projectile. This can result in decreased accuracy. These will not be present in many designs of electromagnetic guns, resulting in more consistent exterior ballistics and improved accuracy.

=== Cost considerations ===

One of the motivations for recent naval railgun development is the low cost per shot of railguns compared to missile systems<ref name="Adams2003"></ref>. A guided railgun projectile from a naval cannon might cost $25,000<ref>Kris Osborn "Navy Rail Gun Showing Promise", Defensetech, January 16 2014, https://web.archive.org/web/20140118235547/http://defensetech.org/2014/01/16/navy-rail-gun-showing-promise/</ref>, compared to $500,000 to $1,5000,000 for conventional missiles. The argument is that a large naval railgun could strike targets within 300 km at a rate of perhaps six to ten rounds per minute; because 85% of the world’s population resides within 300 km of shore<ref>W. A. Walls, W. F. Weldon, S. B. Pratap, M. Palmer, and D. Adams, "Application of Electromagnetic Guns to Future Naval Platforms", IEEE Transactions on Magnetics, January 1999, pp. 262-67</ref> a ship could deliver strikes to most targets at a fraction of the cost. The same general arguments are likely to apply to any other electromagnetic gun.

== Kinds of electromagnetic guns ==

<ref>Henry Kolm, Kevin Fine, Fred Williams and Peter Mongeau, " Overview of Electromagnetic Guns: Electromagnetic Guns, Launchers and Reaction Engines", 1980 https://coilgun.info/theorymath/electroguns.htm</ref>

=== Railguns ===
[[File:Railgun_projectile_2.png|thumb|Railgun rails and projectile.]]
[[Railguns|Main article]].

The best known and most mature electromagnetic gun is the railgun. It is the unrolled equivalent of a homopolar rotary motor (usually, one that uses its current to drive an electromagnet rather than using permanent magnets for the magnetic field). Two rails guide a conductive sabot enclosing the projectile. A high current up one rail, across the sabot, and down the other rail produces a strong magnetic field that pushes the current in the sabot, launching the projectile. There is enough to say about railguns that we [[Railguns|gave them their own page]]. Go check it out for the details, benefits, and limitations of these guns.

=== Coilguns ===

The various kinds of brushless electric motors give you coilguns when they are unrolled into a line. Coilguns have the major benefit that they do not require electrical contact – or indeed, contact of any kind – to accelerate their projectiles. A coilgun can use levitated projectiles, using foil bearings or magnetic levitation. This allows the coilgun to eliminate the barrel wear that plagues many other kinds of high speed electromagnetic guns.

Compared to railguns, coilguns have another benefit: each electromagnet can be energized locally only when needed, thus avoiding the issue of the barrel holding magnetic energy everywhere along the path that the projectile has already traveled. This allows induction coilguns to potentially be more efficient than the standard railgun design.

A coilgun projectile could be spun up with a rotary electric motor in the breach before firing, allowing spin stabilization that is difficult with many other kinds of electromagnetic gun.

The main limit on coilguns is their complexity. It will require precise switching of the electromagnets so as to generate the specific field shape needed for acceleration.

==== Ferromagnetic coilguns ====

A ferromagnetic coilgun is what you get when you unroll a brushless synchronous electric motor into a line. The projectile must be made of a magnetic material. A series of electromagnets are energized in sequence along the barrel, switching on as the projectile approaches as a traveling magnetic wave pulse sweeping up the barrel. This attracts the magnetic projectile, accelerating it down the barrel.

Ferromagnetic coilguns are limited by the saturation field of the ferromagnet in the projectile. For a current I, losses from electrical resistance and stored magnetic fields scale as I². If the field in the bore is well less than the saturation field, the kinetic energy also increases as I² and it is possible to get good efficiency. However, for reasonable performance as a weapon, generally fields are needed that are well above the saturation field, in which case the kinetic energy scales as I and the efficiency quickly becomes atrociously low.

==== Induction coilguns ====

An induction coilgun is what you get when you unroll a brushless asynchronous electric motor into a line. The projectile must be made of a conductive material or enclosed in a conductive sabot. The projectile does not have to be magnetic – in fact, it should not be; materials such as iron should be avoided in the driver shell and if present should only be in a core region shielded from the accelerating field by an outer layer of suitable skin depth to attenuate the field.

To understand how an induction coilgun works, first we discuss one of the fundamental properties of electromagnetism. When a magnetic field changes, it induces an electric field circulating around it. A convenient way to think of this is that for any closed loop:
<ul>
<li>A change in the net amount of magnetic field going through the loop (the magnetic flux) creates an electric field around the loop proportional to the change in magnetic flux.
<li>If the loop is conductive, the electric field drives a current flow in the loop, called an eddy current.
<li>Currents create a magnetic field circulating around them.
<li>The magnetic flux going through the loop created by the driven currents opposes the initial change in magnetic flux from the original magnetic field.
</ul>
A consequence of this is that for a perfect conductor, the flux through any closed loop surrounded by the conductor will never change. Because the interior of any object has a closed loop of the material of the object around it, the flux through a perfectly conductive material will always stay the same. If the conductor starts out with zero flux in its interior, any changing magnetic field will induce currents in the conductor that act to cancel the change in flux. (We'll get to imperfect conductors in just a moment! This is to get the basic idea across.)
<ul>
<li>Because the conductor has current circulating around its perimeter, it creates an electromagnet. As with any magnet, it can be attracted toward or repelled from another magnet with a non-zero magnetic field gradient.
<li>So to make a coilgun, use a stator electromagnet to ramp up a magnetic field in the vicinity of a conductive armature. This induces another electromagnet in the armature, which will be attracted to the stator electromagnet from their mutual magnetic fields.
</ul>
As with the ferromagnetic coilgun, a series of electromagnets are energized along the bore to create a traveling magnetic wave moving up the barrel just in front of the armature. The electromagnets pull the armature towards them via this induction process.

Induction does not saturate, such that induction coilguns can potentially be used at very high speeds where ferromagnetic coilguns would have too low of efficiency to be competitive.

A conductor that is not a perfect conductor will initially screen the magnetic field from its interior with the field inside the conductor falling off with a characteristic length scale called the skin depth. At a small fraction of the skin depth from the surface, the magnetic field will be at nearly its full strength – there is not enough circulating current outside that depth to create enough of an electromagnet to cancel the field. But at several skin depths deep, the magnetic field will be almost entirely screened by the circulating current and will fall to near zero and the deeper you go the closer to zero the field gets. (For the mathematically inclined, the function is B(x) = B0 exp[-x/λ] where lambda is the skin depth, B0 is the external field strength, and B(x) is the field strength at distance x.) However, because the conductor is not perfect the current will be reduced over time by electrical resistance. As the current turns into heat, there is not enough current to continue cancelling the field and the magnetic flux begins to migrate into the conductor.

For many practical purposes, superconductors are perfect conductors. Like regular conductors, a magnetic field will penetrate into them with a characteristic skin depth called the London penetration depth. Unlike regular conductors, the superconductor suffers no electrical resistance so the flux never penetrates deeper into the superconductor over time. Unlike other conductors, superconductors expel all magnetic fields from their interior at the moment they form (the Meissner effect) so any superconductive region deeper than several London depths will always have no magnetic field in it. There are some kinds of superconductor that allow narrow threads of non-superconductive regions to pass through them that contain magnetic flux if the outside field gets high enough (so-called Type-II superconductors), while others continue to expel external fields until the fields get too high and turn the superconductivity off (at the critical field).

For a coilgun acting like a conventional gun, the field will ramp up and then down over a time scale of milliseconds or less. For these conditions, the skin depth of good conductors like copper or aluminum will be a few millimeters. Meanwhile, the London depth of superconductors is generally less than a micrometer.

For an armature made of normal conductors, as the flux migrates into the conductor the driving coils become less effective. A solution is to let the position of the driving coil "slip" further down the length of the armature to fresher conductor where the magnetic flux has not yet penetrated. In this sense the armature moves ahead of the magnetic wave propagating up the barrel. This is entirely analogous to the slip of a rotary brushless asynchronous induction motor between the angle of the rotating magnetic field and the angle of the armature. Another solution is to keep the driving coil in the same position relative to the armature but to switch the direction of the magnet, called current reversal. Now the flux frozen in to the conductor of the armature helps to attract it even harder to the switched direction magnet. Again, this is analogous to the rotary induction motor with the change in phase of the field at any given angle relative to the armature. These two mechanisms can be used together.

An induction coilgun can launch a solid continuous slug of metal or other conductor. This has the disadvantage that the driving fields and currents are concentrated within about a skin depth into the driving jacket. For projectiles thicker than a few millimeters, this concentrates the induced current in a thin section at the surface, leading to increased resistive heating and possible melting of the driver jacket. This can be avoided by using a wound coil for the driver jacket, which more uniformly distributes the current through the depth of the projectile<ref>I. R. Shokair, M. Cowan, R. J. Kaye, and B. M. Marder, "performance of an Induction Coil Gun", report SAND93-1358 (1993) https://digital.library.unt.edu/ark:/67531/metadc1280251/</ref>. For smallarms with bullet diameters of only a few millimeters, the skin depth may be large enough to allow acceleration over much of the volume of the projectile without using windings in the projectile.

Coilguns can in principle be quite efficient. Estimates of room temperature aluminum armatures with low launch pressures and large sizes suggest efficiencies in excess of 80%, while higher pressure (~0.2 GPa) and more moderate sized (10-20 cm bore diameters) coilguns might reach 40% to 50% efficiency at turning electrical energy into the kinetic energy of the armature and projectile<ref name="Cowan_1993">M. Cowan, E. C. Cnare, B. W. Duggin, R. J. Kaye, B. M. Marder, I. IL Shokair, "The Continuing Challenge of Electromagnetic Launch", https://www.osti.gov/servlets/purl/10177176</ref>. These efficiencies will improve with improved electrical conductivity of the armature, which could be achieved either by cooling the aluminum to cryogenic temperatures before launch or by using materials with higher room temperature conductivity. This could include copper (with roughly a factor of two higher conductivity), exotic future materials, or superconductors.

The upper limit to the speed of an induction coilgun using ordinary conductors on the projectile is where the resistive heating from the induced currents melts the conductor in the projectile or sabot. One study<ref name="Winterberg EMRG">F. Winterberg, "The electromagnetic rocket gun", Acta Astronautica Vol. 12, No. 3, pp. 155-161, 1985</ref> suggests this will happen for speeds of more than approximately 20 km/s. This limit does not apply to superconductors, as long as the driving field is kept below the superconductor's critical field.

==== Quench guns ====

This induction coilgun design uses a series of superconducting coils along the barrel. Before launch, each coil is energized with a persistent supercurrent, making the barrel a superconducting solenoid of independent coils. The projectile either has a superconductive jacket or a superconductive sabot. To launch the projectile, each coil starting with that closest to the projectile is quenched in turn; quenching is the name for when a superconductor is made non-superconductive. This creates the traveling magnetic wave that sweeps the projectile along with it, as the field ramping down induces currents in the projectile that attract it to the quenching loop's falling field.

Quench guns are curious in that the barrel itself acts as the energy accumulator, and all the energy for launch is stored in the energized coils.

=== Helical railguns ===
[[File:Helical_railgun.png|thumb|Diagram of the helical coil, rails, and projectile of a helical railgun, including the internal electrical connections of the projectile.]]
[[Helical_railguns|Main article]].

[[Helical_railguns|Helical railguns]] are a hybrid between a railgun and a coilgun. They are essentially a brushed synchronous electric motor that has been rolled out into a straight line. Like a railgun, the projectile slides on rails that deliver current to flow through the rails. Like an induction coilgun, acceleration is due to the forces between an electromagnet in the projectile and and electromagnet in the barrel.

=== Centrifugal guns ===

Centrifugal guns are the ones that break all the rules. Rather than a linear electric motor, they use a rotary motor. While the motor does not have to be electrical, they can be made to work with an electric rotary motor.

The centrifuge gun has a rotary section that mechanically holds a number of bullets. As the rotor is spun up, the bullets will be moving in a circle at high speed. When ready to be fired, the bullet is released and flies straight tangentially to its former circle of motion. In this case, the energy accumulator is the rotation of the motor armature, using electric energy to build up rotational kinetic energy that can is suddenly discharged by releasing the projectile. The internal stresses come not from electromagnetic self-force, but from the centrifugal forces of the spinning rotor.

=== Electromagnetic rocket guns ===

One solution to the problem of resistive heating of the projectile giving a cap to attainable speed is to fill the projectile with a propellant fluid<ref name="Winterberg EMRG"></ref>. As the projectile is heated up, the heat is conducted to the fluid which vaporizes and is ejected out the back, forming a rocket. The projectile is now driven by both its rocketing coolant vapor and the electromagnetic driving fields. Further, the rocket exhaust can in turn support currents, either in the form of an electric arc between the rails for a railgun or a circular arc around the jet in the case of an induction coilgun. This further heats the propellant, and the same magnetic forces that are accelerating the projectile also accelerate the propellant into the projectile, forming a situation somewhat similar to a [[Railguns#Plasma_armature_railguns|plasma armature railgun]]. The projectile, now cooled and accelerated by both rocket thrust, the electromagnetic driving forces, and being pushed forward by the electromagnetically accelerated rocket exhaust, allows the projectile to reach considerable speed. It is estimated<ref name="Winterberg EMRG"></ref> that final speeds of a few hundred km/s could be attained.

=== Gauss guns ===

One often hears of a kind of device called a "Gauss gun". Exactly what a Gauss gun is differs depending on the speaker and is often only vaguely described. Some will state with certainty that a Gauss gun is a certain type of device, and yet there is no standard for what it is, little historical precedent for the use of the term, and its use in language has varied over the years.

The name implies that a Gauss gun is somehow associated with Johann Carl Friedrich Gauss (1777 – 1855), a German mathematician and scientist whose wide-ranging interests and penetrating genius has led to his name being associated with many different things in both mathematics (Gaussian functions, Gaussian integration, Gauss's theorem for vector fields; among many others) and physics (Gauss's law of electromagnetism is of particular note). Gauss was known for his exacting work on the measurement of the geomagnetic field and work on electromagnetism including an unsuccessful attempt to unify electricity and magnetism. This painstaking effort led to a unit of magnetism in the cgs system to be named for him; the gauss (1 gauss = 10-4 tesla in the mks system). What Gauss was not known for was inventing any kind of magnetic or electromagnetic weapon. So we find no real clue here, except that perhaps people were trying to associate a nebulously magnetic gun with a polymath who was famous for his work on magnetism.

The earliest examples this author is aware of for the term Gauss gun (or Gauss rifle or Gauss pistol) date back to the 1970's, in the tabletop role playing game Traveller and in the novel The Stainless Steel Rat's Revenge by Harry Harrison. This will be updated as new information comes to light. Later, the term appears in various other works, notably many science fiction role playing games and video games. The descriptions of their mechanisms and physical principles are typically very vague. The original Traveller description reads "the gauss rifle generates an electromagnetic field along the length of the barrel which accelerates a [...] needle bullet [...]."<ref>Game Designer's Workshop, "Traveller: Science-Fiction Adventure in the Far Future – Book 4: Mercenary" (1978)</ref> Descriptions commonly include non-descript mentions of "magnetic field", "electromagnetic field", "magnetic impulse", or "electromagnetic impulse" <ref>David Pulver, "GURPS Ultra-Tech", Steve Jackson Games, 1989; David Pulver, "GURPS Ultra-Tech 2", Steve Jackson Games, 1997; David Pulver and Kenneth Peters, "GURPS Fourth Edition Ultra-Tech", Steve Jackson Games, 2006</ref><ref>"Traveller: The New Era", GDW, 1993</ref>. Occasionally, the description seems to imply some kind of coilgun or railgun – "electromagnetic coils" or electromagnetic rails", sometimes both in the same document<ref>Traveller SRD: Equipment https://www.traveller-srd.com/core-rules/equipment/</ref>. The Traveller: The New Era supplement Fire, Fusion, & Steel states "These weapons use electric energy to power electromagnets which accelerate a projectile down the barrel of the gun", and then shows a diagram that explicitly shows how it works – which turns out to be what the Galactic Library calls a [[Helical_railguns|helical railgun]]<ref>"Traveller The New Era: Fire, Fusion, & Steel Technical Architecture" GDW, 1993</ref>. In most cases, the Gauss gun is said to fire projectiles at notably high speed (presumably higher speed than a modern rifle) and commonly launches projectiles that are described using terms like "needle", "dart", or "sliver"; indicating a high aspect ratio bullet perhaps like smaller versions of the APFSDS long rod anti-tank penetrator rounds fired from modern tank cannons.

More recently, there has been a trend to use the term Gauss gun for a physics demonstration involving a high field permanent magnet attracting a steel ball down a track from one side, while the other side has a stack of steel balls stuck to the magnet. When the first ball rolls down the track it is attracted to the magnet so that it hits it at considerably higher speed than that which it was first sent rolling. The impact is then transferred through the stack of balls in the same manner as a Newton's cradle<ref>Chris Schulz, "How Newton's Cradles Work", HowStuffWorks, https://science.howstuffworks.com/innovation/inventions/newtons-cradle.htm</ref>, and because the last ball in the stack is only weakly bound it is launched at considerably higher speed than the initial speed of the first ball<ref>Arsène Chemin, Pauline Besserve, Aude Caussarieu, Nicolas Taberlet, and Nicolas Plihon, "Magnetic cannon: The physics of the Gauss rifle", Am. J. Phys. 85, 495–502 (2017), https://pubs.aip.org/aapt/ajp/article-abstract/85/7/495/1039674/Magnetic-cannon-The-physics-of-the-Gauss-rifle?redirectedFrom=fulltext</ref>. The earliest reference this author has been able to find so far dates to a toy from around 2002 called a Gaussian gun, although it is claimed that the idea came from Japan so the name may go back even farther <ref>Grand Illusions toy shop, the Gaussian Gizmo, https://www.grand-illusions.com/collections/magnetic-toys/products/gaussian-gun</ref><ref>alien7 "Gaussian Gun", https://www.youtube.com/watch?v=Z7CyPtF0ChA</ref>. This device seems to have been referred to as a Gaussian gun up until about 2007 before the term Gauss gun started to be used<ref>sdense, Gaussian Gun, https://www.youtube.com/watch?v=BxkZaYYSHFs (2007)</ref><ref>robgallagher, The Worlds First Real Gauss Rifle, https://www.youtube.com/watch?v=QPciBQnZw3c (2008)</ref>. Again, this will be updated as new information is found.

== Credit ==

Author: Luke Campbell

== References ==

[[Category:Warfare]][[Category:Physics & Engineering]][[Category:Physics & Math & Engineering]][[Category:Engineering]][[Category:Military Technology]]

Dielectric Breakdown

2026-11-01T23:47:10Z

Lwcamp:

An electron is happily minding its own business floating freely around in the air when suddenly a high power laser beam comes by. How rude! The electric field of the laser beam starts tugging on the electric charge of the electron, accelerating it to faster and faster speeds. Soon, the electron has more than enough energy to knock out other electrons that are stuck on to atoms. Now the electric field starts tugging on these newly freed electrons, too, and they start knocking off more free electrons of their own. Soon, you get a runaway process where the air turns into an electron soup. This electron soup is called a plasma, and the air has just experienced dielectric breakdown.

You need to get a couple things right in order to get dielectric breakdown. First, if the electrons bump into air molecules before they gain enough energy to knock out additional electrons, then they just get scattered in random directions and need to be re-accelerated by the field. So too dense of air or not strong enough laser light means free electrons just bounce off air molecules without ever starting the breakdown cascade.

Second, light is a wave, and it has a frequency. This means the electric field periodically turns around and pulls the electrons the other way. All that energy they’d been getting from the field is now being drained away. If the laser light’s frequency is high enough, the electrons just jiggle up and down slightly without ever getting enough energy to start the breakdown process.

As you might have guessed by now, this process is not good if it happens part-way along your beam when you are trying to shoot a bad guy. Ripping off all those electrons takes a lot of energy, energy that comes from your beam! Not only that, but plasma is hot, low density, and has funky electrical properties that turn it into a lens. The light it doesn’t absorb gets scattered out of your beam and away from your target. And finally, if the wavelength of your beam is longer than somewhere between 5 and 2 μm (65 to 150 THz or 0.3 to 0.7 eV, in sea level density air and depending on how much ionization occurs), the plasma just directly absorbs your beam, using your beam’s energy to heat itself up even more so it can absorb and scatter your beam further. Not nice, plasma! Are you working for my target, perhaps?

But will your design for a laser beam death annihilation ray suffer from breakdown? Here’s a way to get a quick estimate.
<ref name=Radziemski_1989>Leon J. Radziemski and David A. Cremers, Ed., "Laser-Induced Plasmas and Applications", Marcel Dekker, New York, 1989.</ref>
<ref name=Ali_1983>A. W. Ali, “On Laser Air Breakdown, Threshold Power and Laser Generated Channel Length”, NRL Memorandum Report 5187 AD-A133 2.11, September 13, 1983.</ref>
Find the threshold intensity ITh in watts per square centimeter
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="text-align: center; vertical-align: middle; margin-top:0.5em; margin-bottom:0.5em; line-height: 2em;" align="center" cellpadding="0" cellspacing="0">
<tr>
<td rowspan="2" nowrap="nowrap"> ITh, CW = 0.31  
<td rowspan="2" nowrap="nowrap"> [
<td nowrap="nowrap"> 1 m
<td rowspan="2" nowrap="nowrap"> (
<td nowrap="nowrap"> 105 Pa
<td rowspan="2" nowrap="nowrap"> )
<td nowrap="nowrap"> 0.6
<td rowspan="2" nowrap="nowrap"> ]
<td nowrap="nowrap"> 2
<tr>
<td style="border-top:solid 1px black;">λ
<td style="border-top:solid 1px black;">p
<td> 
<td> 
</table>
<table style="text-align: center; vertical-align: middle; margin-top:0.5em; margin-bottom:0.5em; line-height: 2em;" align="center" cellpadding="0" cellspacing="0">
<tr>
<td rowspan="2" nowrap="nowrap"> ITh, pulse = 6.36×10-5  
<td rowspan="2" nowrap="nowrap"> (
<td nowrap="nowrap"> 1 m
<td rowspan="2" nowrap="nowrap"> )
<td nowrap="nowrap"> 2
<td rowspan="2" nowrap="nowrap"> (
<td nowrap="nowrap"> 105 Pa
<td rowspan="2" nowrap="nowrap"> )
<td nowrap="nowrap"> 0.6
<tr>
<td style="border-top:solid 1px black;">λ
<td> 
<td style="border-top:solid 1px black;">p
<td> 
</table>
</div>
where P is the beam power, E is the beam energy, t is the beam duration, p is the air pressure, and λ is the beam wavelength;
and take ITh as the larger of these two. If
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="text-align: center; vertical-align: middle; margin-top:0.5em; margin-bottom:0.5em; line-height: 2em;" align="center" cellpadding="0" cellspacing="0">
<tr>
<td rowspan="2" nowrap="nowrap"> I =  
<td nowrap="nowrap"> 4 P
<td rowspan="2" nowrap="nowrap">   =  
<td nowrap="nowrap"> 4 E
<tr>
<td style="border-top:solid 1px black;">S2
<td style="border-top:solid 1px black;">tS2
<td> 
<td> 
</table>
is close to ITh, there’s a risk you’ll start a cascade breakdown, If it is much larger than ITh, you will almost certainly cause an air breakdown. If it is much less, you’re safe. Go ahead and blaze away!

A curious effect can occur if you start dielectric breakdown. The plasma spark initially grows by absorbing beam energy and conducting that energy to the surrounding air. But once it is big enough, the plasma can shield its back part from the incoming laser while the front region continues to be heated and grow. The result? You launch a wave of plasma back down the beam toward the laser emitter! For focused beams, this plasma wave can only go so far as the beam is still focused enough to sustain the plasma, so you don't risk blowing yourself up when you shoot.

==Credit==
Author: Luke Campbell

==References==

[[Category:Lasers]][[Category:Physics & Engineering]][[Category:Physics & Math & Engineering]]

Dielectric Breakdown

2026-11-01T23:42:01Z

Lwcamp:

An electron is happily minding its own business floating freely around in the air when suddenly a high power laser beam comes by. How rude! The electric field of the laser beam starts tugging on the electric charge of the electron, accelerating it to faster and faster speeds. Soon, the electron has more than enough energy to knock out other electrons that are stuck on to atoms. Now the electric field starts tugging on these newly freed electrons, too, and they start knocking off more free electrons of their own. Soon, you get a runaway process where the air turns into an electron soup. This electron soup is called a plasma, and the air has just experienced dielectric breakdown.

You need to get a couple things right in order to get dielectric breakdown. First, if the electrons bump into air molecules before they gain enough energy to knock out additional electrons, then they just get scattered in random directions and need to be re-accelerated by the field. So too dense of air or not strong enough laser light means free electrons just bounce off air molecules without ever starting the breakdown cascade.

Second, light is a wave, and it has a frequency. This means the electric field periodically turns around and pulls the electrons the other way. All that energy they’d been getting from the field is now being drained away. If the laser light’s frequency is high enough, the electrons just jiggle up and down slightly without ever getting enough energy to start the breakdown process.

As you might have guessed by now, this process is not good if it happens part-way along your beam when you are trying to shoot a bad guy. Ripping off all those electrons takes a lot of energy, energy that comes from your beam! Not only that, but plasma is hot, low density, and has funky electrical properties that turn it into a lens. The light it doesn’t absorb gets scattered out of your beam and away from your target. And finally, if the wavelength of your beam is longer than somewhere between 5 and 2 μm (65 to 150 THz or 0.3 to 0.7 eV, in sea level density air and depending on how much ionization occurs), the plasma just directly absorbs your beam, using your beam’s energy to heat itself up even more so it can absorb and scatter your beam further. Not nice, plasma! Are you working for my target, perhaps?

But will your design for a laser beam death annihilation ray suffer from breakdown? Here’s a way to get a quick estimate.
<ref name=Radziemski_1989>Leon J. Radziemski and David A. Cremers, Ed., "Laser-Induced Plasmas and Applications", Marcel Dekker, New York, 1989.</ref>
<ref name=Ali_1983>A. W. Ali, “On Laser Air Breakdown, Threshold Power and Laser Generated Channel Length”, NRL Memorandum Report 5187 AD-A133 2.11, September 13, 1983.</ref>
Find the threshold intensity ITh in watts per square centimeter
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="text-align: center; vertical-align: middle; margin-top:0.5em; margin-bottom:0.5em; line-height: 2em;" align="center" cellpadding="0" cellspacing="0">
<tr>
<td rowspan="2" nowrap="nowrap"> ITh, CW = 0.31  
<td rowspan="2" nowrap="nowrap"> [
<td nowrap="nowrap"> 1 m
<td rowspan="2" nowrap="nowrap"> (
<td nowrap="nowrap"> 105 Pa
<td rowspan="2" nowrap="nowrap"> )
<td nowrap="nowrap"> 0.6
<td rowspan="2" nowrap="nowrap"> ]
<td nowrap="nowrap"> 2
<tr>
<td style="border-top:solid 1px black;">λ
<td style="border-top:solid 1px black;">p
<td> 
<td> 
</table>
<table style="text-align: center; vertical-align: middle; margin-top:0.5em; margin-bottom:0.5em; line-height: 2em;" align="center" cellpadding="0" cellspacing="0">
<tr>
<td rowspan="2" nowrap="nowrap"> ITh, pulse = 6.36×10-5  
<td rowspan="2" nowrap="nowrap"> [
<td rowspan="2" nowrap="nowrap"> (
<td nowrap="nowrap"> 1 m
<td rowspan="2" nowrap="nowrap"> )
<td nowrap="nowrap"> 2
<td rowspan="2" nowrap="nowrap"> (
<td nowrap="nowrap"> 105 Pa
<td rowspan="2" nowrap="nowrap"> )
<td nowrap="nowrap"> 0.6
<td rowspan="2" nowrap="nowrap"> ]
<tr>
<td style="border-top:solid 1px black;">λ
<td> 
<td style="border-top:solid 1px black;">p
<td> 
</table>
</div>
where <math>P</math> is the beam power, <math>E</math> is the beam energy, <math>t</math> is the beam duration, <math>p</math> is the air pressure, and <math>\lambda</math> is the beam wavelength;
and take <math>I_{Th}</math> as the larger of these two. If
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
I = \frac{4 \, P}{S^2} = \frac{4 \, E}{t S^2}
</math></div>
is close to <math>I_{Th}</math>, there’s a risk you’ll start a cascade breakdown, If it is much larger than <math>I_{Th}</math>, you will almost certainly cause an air breakdown. If it is much less, you’re safe. Go ahead and blaze away!

A curious effect can occur if you start dielectric breakdown. The plasma spark initially grows by absorbing beam energy and conducting that energy to the surrounding air. But once it is big enough, the plasma can shield its back part from the incoming laser while the front region continues to be heated and grow. The result? You launch a wave of plasma back down the beam toward the laser emitter! For focused beams, this plasma wave can only go so far as the beam is still focused enough to sustain the plasma, so you don't risk blowing yourself up when you shoot.

==Credit==
Author: Luke Campbell

==References==

[[Category:Lasers]][[Category:Physics & Engineering]][[Category:Physics & Math & Engineering]]

Particle Beam Weapons

2026-11-01T21:27:24Z

Lwcamp: /* Separation of neutral beams */

{{PageConstructionNotice}}

''For a general overview on particle accelerator technology, see [[Particle Accelerators]]

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.

::: Z = 1.32×102 (γ2-1)3/4 r / √I<ref name=USPAS2016>Martin Reiser, "Theory and Design of Charged Particle Beams", 2008, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim</ref>

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

::: Z = β γ c r √[m/T]

Where Z is the doubling distance like before, r is the initial beam radius, T is the beam temperature in eV, m is the ion mass and β, c, and γ 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.

== Defenses ==

Sure, it's great to have a great big particle cannon to blast your enemies. But what if you are the one being shot at? In that case, it's best to figure out ways to keep from being blown to kingdom come.

=== Armor ===

The simplest way to defend against the high energy particles of a particle beam is to pile a whole bunch of stuff up and put it in the way of the beam. Particle beams may produce extremely penetrating radiation, but enough stuff will block most of that radiation and may allow you to survive. The details depend on what kind of particle is being thrown at you, and are explained below. However, a general rule of thumb is that you probably want about 10 tons of material per square meter of surface you want to armor in order to get sufficient protection.

==== Electrons ====

At the very high energy of an electron accelerated by a weapons-grade particle beam, the primary interaction with matter will be the emission of bremsstrahlung photons in the gamma-ray range as the electron slams into particles of matter. These high energy photons themselves primarily interact by the pair production mechanism – the photon disappears, but leaves behind a highly energetic positron and electron. These two particles are themselves energetic enough to produce further bremsstrahlung, and so the process continues. You get the formation of an electron-gamma shower (also called an electromagnetic cascade), which only stops after either the particles have penetrated through the target or the initial energy of the electron has been spread through so many shower particles (including all the electrons, positrons, and photons) that no particle has enough individual energy to pair produce or to be likely to bremsstrahlung again before it is stopped by other mechanisms.

To stop electrons, you'll want the atoms in your armor to be as heavy as you can practically get them. High atomic number both encourages bremsstrahlung and pair production, giving a double whammy for stopping that cascade. However, occasionally one of those electrons, positrons, or gammas will hit a nucleus and knock out neutrons. So you probably want a back-up layer of several tens of centimeters of hydrogen rich material such as plastic or water mixed with neutron absorbing materials like boron, lithium, cadmium, or gadolinium. High density polyethylene loaded with 5% boron by weight is a typical go-to neutron blocker.

The actual methods to calculate the penetration of the electromagnetic cascade is given are the box below. However, for some examples:
<ul>
<li> A 10 GeV beam incident on thick iron (or steel) will stop cascading at about 9 cm, will deposit half of its energy in the first 12 cm, and will have dumped 98% of its energy within 35 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 2.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 100 GeV beam incident on thick iron (or steel) will stop cascading at about 13 cm, will deposit half of its energy in the first 16 cm, and will have dumped 98% of its energy within 47 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 3.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick iron (or steel) will stop cascading at about 17 cm, will deposit half of its energy in the first 20 cm, and will have dumped 98% of its energy within 59 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 4.6 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick tungsten will stop cascading at about 3.5 cm, will deposit half of its energy in the first 4 cm, and will have dumped 98% of its energy within 12 cm. 90% of the cascade will be contained within about 0.93 cm and 95% of the cascade will be contained within 1.86 cm of the beam axis. You will get 98% protection with a mass thickness of 2.3 tons/m2 and an additional 1/e reduction for each further 0.13 tons/m2 beyond that.
<li> A 1 TeV beam incident on water will stop cascading at about 290 cm, will deposit half of its energy in the first 340 cm, and will have dumped 98% of its energy within 1030 cm. 90% of the cascade will be contained within about 11 cm and 95% of the cascade will be contained within 22 cm of the beam axis. You will get 98% protection with a mass thickness of 10.3 tons/m2 and an additional 1/e reduction for each further 0.72 tons/m2 beyond that.
</ul>
Note that for sufficiently intense particle beams, 98% protection may not be enough. You might need to throw on several more tons per square meter to be sure.

<table border=1>
<tr><td> How Far it Goes:
(This section follows methods presented in Leroy and Rancoita<ref>C. Leroy and P-G Rancoita, "Physics of cascading shower generation and propagation in matter: principles of high-energy, ultrahigh-energy and compensating calorimetry", Reports on Progress in Physics 63 (2000) 505–606. DOI: 10.1088/0034-4885/63/4/202</ref>.)

When trying to estimate the extent of the irradiated region of the target, it is helpful to know the radiation length X0 in the material. This is the distance that an electron will travel, on average, before emitting a bremsstrahlung photon. Conveniently, it is also the distance a high energy photon will travel through the material before pair producing. The radiation length can be found from
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   = 4 α  
<td nowrap="nowrap"> NA
<td rowspan="2" nowrap="nowrap" align="right">   Z (Z+ζ) re2 ln
<td rowspan="2" nowrap="nowrap" align="right"> [
<td nowrap="nowrap"> 183
<td rowspan="2" nowrap="nowrap" align="right"> ]
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> A
<td style="border-top:solid 1px black;"> Z1/3
</table>
</div>
where α = 1/137 is the fine structure constant, NA = 6.02214076×1023 is Avogadro's number (or one mole), A is the atomic weight of the atoms in the material (usually expressed in grams per mole), Z is the atomic number (nuclear charge) of the atoms, and re = 2.8179403205×10-13 cm is the classical electron radius. ζ is a constant of approximately 1.3. This gives the thickness of material in terms of the areal density (mass of material per unit area perpendicular to the beam) – if you use A in g/mol and re in cm the result will be in cm2/g and X0 will be in g/cm2. If your target is made up of more than one kind of atom, you can get the total radiation length by finding the radiation length of each atom of type i, X0i and using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   =  
<td rowspan="2" nowrap="nowrap" align="right"> Σi  
<td nowrap="nowrap"> fi
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> X0i
</table>
</div>
where fi is the fraction by weight of the atom type i. Once you have the radiation length in terms of areal density, you can easily convert it to distance by dividing by the density ρ. If X0i is in g/cm2 and density is in g/cm3, then X0i/ρ will be in cm.

We will also need to know the critical energy εc. This is the energy where an electron is losing energy as rapidly to ionization as it is to radiation. That is, it is the energy at which the cascade stops multiplying, even though the individual particles already made can continue traveling for some distance. The critical energy can be estimated using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> εc = B  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> Z X0
<td rowspan="2" nowrap="nowrap" align="right"> )
<td nowrap="nowrap"> h
<tr>
<td style="border-top:solid 1px black;"> A
<td>  
</table>
</div>
where B = 2.66 MeV and h = 1.11.

We can now describe the distribution of radiation in the target material. The depth tmax where the cascade stops multiplying (in g/cm2) for an incident electron of energy E (in MeV) is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tmax = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> - 1
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
Similarly, the depth where half the energy of the beam has been deposited (in g/cm2) is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tcg = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> + 0.37
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
As usual, you can convert to a depth in cm by dividing by the material density. 98% of the beam energy will be deposited within a distance of &cong; 3 tcg. Beyond this, the beam energy will continue to fall off with thickness t approximately as an exponential exp[-t/λ], where the characteristic attenuation length λ &cong; 2 X0.

The width of a cascade, where 90% of the energy is contained, is approximately given by
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> RM =  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> EM
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
where EM &cong; 21.2 MeV. 95% of the radiation will be contained within a distance of 2 RM from the beam axis.
</table>

==== Protons and Ions ====

<table align=right><tr><td>
[[File:Regolith_Shielding.png|350 px|frameless]]
<tr><td width=350>
Relative effect of particle radiation (compared to no shielding) from galactic cosmic rays behind different thicknesses of water, aluminum, and lunar regolith<ref name="Slaba2022">Tony C. Slaba, "Radiation Shielding for Lunar Missions: Regolith Considerations", LSIC Crosstalk 7/18/2022 https://lsic.jhuapl.edu/uploadedDocs/focus-files/1604-E&C%20+%20EE%20Monthly%20Meeting%20-%202022%2007%20July_Presentation%20-%20NASA%20Slaba.pdf</ref>.
</table>
Protons and ions are grouped together because ions are made of protons and neutrons and when you accelerate these things for a weapon it is the protons and neutrons that are going to be doing the heavy lifting for blasting your target. This is because protons and neutrons are both about 2000 times more massive than electrons so any electrons along for the ride will contribute negligibly to the energy. Protons and neutrons together are called nuclear particles, and both behave quite similarly in nuclear collisions.

Nuclear particles are themselves made of quarks and can interact via the strong nuclear force. At the energies appropriate for weapons use, essentially all of the way protons or ions are blocked is through [[Particle_Accelerators#Nuclear_collisions|direct collisions with the atomic nuclei]] of the armor material. One or more quarks in the projectile will slam into quarks in the target nucleus and stretch them away from the rest of the quarks in the nuclear particles. Physics tells us that you can't have isolated quarks, so the "string" attaching the quark to its neighbors stretches and stretches until it snaps. Snapping releases so much energy that more quarks are made that clump together into all sorts of other nuclear particles, both exotic and familiar. This process is called a hadronic jet. The only particles produced in these jets long-lived enough to travel any noticeable distance through the armor material will be protons, neutrons, the charged pi mesons (or pions), and some of the K mesons (or kaons). The particles too short lived to go far enough to matter will decay into other particles, and these particles are often very high energy gamma rays. Much like with electrons, the long-lived secondary particles can be sufficiently energetic to smash into other nuclei and cause more hadronic jets (or, if they are low enough in energy, just knock more nuclear particles out of target nuclei) and this produces a similar shower or cascade. But in addition, the armor nuclei that were hit will be very hot after the impact from all of the energy given to them from the incident particles. These nuclei will cool off by evaporation – but instead of evaporating away molecules, they evaporate off nuclear particles from the nucleus. These evaporated particles are mostly neutrons, so any armor against proton or ion beam strikes will produce quite a bit of neutron radiation.

Eventually, the original particle's energy will be distributed among so many other particles that the charged particles will come to a stop by making ion tracks rather than from colliding with more nuclei, and the cascade halts. But you still have all the neutrons to worry about. And the gamma rays as well, which are high enough energy to kick off an electromagnetic cascade in the same way that electrons do.

Unlike electrons, which have reasonable approximations of their cascade behavior from relatively simple equations, the cascades of protons and ions usually require sophisticated simulations or experiments to get the resulting distribution of radiation. However, we can get a reasonable idea of their general behavior by comparing them to [[Space_Radiation#Shielding_Against_Cosmic_Rays|cosmic ray particles interacting with shielding]]. Cosmic rays are energetic protons and ions, after all. From this we see that a relatively small amount of armor – in the range of 1 to 3 tons per square meter – can actually make the radiation dose worse. The incident particles will knock off so many secondary particles in their shower that the dose is multiplied from so many tracks passing through the target. Hydrogen rich materials such as water and plastics work the best against proton and ion beam particles; heavy elements are the worst and produce the most secondaries. You could probably obtain adequate radiation protection from most incident beams with about 10 tons per square meter of hydrogen rich material as armor. However, you will want to add neutron absorbing elements to your hydrogen rich armor, such as lithium, boron, cadmium, or gadolinium. Further, in order to deal with the gamma rays you will want a layer of at least several centimeters of heavy elements, such as tungsten or lead, behind the hydrogen rich particle armor.

==== Structural Considerations ====

When your armor material absorbs a radiation beam, the energy of that radiation turns into heat inside the armor where the radiation was absorbed. If the beam pulse has enough energy as was sufficiently tightly focused, this heat can damage your armor. Your armor may melt in places, or thermal expansion could cause it to crack or warp. Perhaps the worst thing it could do is vaporize or flash into a plasma, because the pressure from the vapor or plasma can be sufficient to plastically deform the armor. This leaves a hole along the beam path where the expanding gas or plasma blasted out a long, deep crater. This means a beam that fires rapid shots spaced maybe milliseconds apart can have subsequent shots go down the same hole so your armor will not protect against them. Each shot will dig out a deeper hole until either your armor is penetrated or the attacking beam runs out of shots in its pulse train.

A consequence of this is that when designing your armor you may wish to use refractory materials with a high structural strength (like tungsten), or that is otherwise resistant to being blasted apart by its own evaporated remains.

=== Magnetic fields ===

[[Particle_Accelerators#Magnetic_fields|A charged particle crossing a magnetic field will bend so as to spiral around the field lines]]. If you try to shoot your particle beam through a magnetic field, it will bend away from the direction you shot it. This can make it hard to hit your target. Unfortunately for you, there are magnetic fields everywhere – around planets, and even trapped in the solar wind as it crosses the solar system.

As an example, the Earth has a magnetic field strength of about 5×10-5 tesla near its surface, and falling off farther away. If you are shooting a 1 TeV electron beam past the Earth, your beam is likely to bend by several milliradians (with the exact deflection depending on how close you are skimming the Earth). If you are in a medium Earth orbit and shooting at another target also in a medium Earth orbit on a trajectory that goes near the Earth, then without correcting for the magnetic field your beam will likely be off-target by tens to hundreds of kilometers.

For another example, the interplanetary magnetic field is about 10-9 tesla at Earth's distance from the sun. If you shoot that same 1 TeV electron beam at a target 0.1 light seconds away, you are likely to be off by a couple hundred meters.

<table border=1>
<tr><td>
The radius of curvature of the bend of a particle in a magnetic field is called the gyroradius rc. For a particle with a charge q, mass m, and kinetic energy E, find the relativistic momentum p
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
p = √[E2 - m2 c4] / c
</div>
To convert from energy units of eV to joules, multiply by 1.60×10-19. An electron has a mass of 9.11×10-31 kg, a proton of 1.67×10-27 kg, and an ion of 1.66×10-27×A kg, where A is the atomic weight of the ion. The speed of light c = 3.00×108 m/s.

In a magnetic field of strength B with an angle θ between the field direction and the direction of motion
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rc = |p sin(θ) / (q B)|
</div>
For electrons, q = -e, where the fundamental charge e = 1.60×10-19 coulomb. For a proton, q = +e. For an ion, multiply e by the charge state of the ion.

If the field acts over a distance d that is much smaller than the gyroradius, the deflection angle φ is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = d / rc
</div>
The deflection x at a target at a range R for small φ (φ &Lt; 1) will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
x = φ R.
</div>

Note that d will always be less than or equal to x. Also note that if d is not smaller than twice the gyroradius the beam will never reach the target because it will completely spiral around the field lines before it reaches the target.
</table>

So, it is clear that as the defender it will be helpful to hang out in areas with high magnetic fields, and keep some distance between you and the attacker. Meanwhile, the attacker will benefit from being closer where small deflections in the beam trajectory don't translate into large excursions from where the beam is aimed (and also because there will be less space filled with fields between the two of you). But the attacker can also try to compensate for known fields. If you happen to know that the magnetic fields are between you and your target, you can adjust your fire so that the deflection of the field bends your beam onto the target. This is not a simple task. It is easy enough to measure the field strength and direction where you are, but predicting how it changes over the path of your shot does not have an obvious solution. You can also try to gain intelligence on likely battle sites before you ever fight on them – for example, getting good data on the structure of the fields of a planet using spies and technical intelligence during peacetime (heck, on Earth this information is published for free in academic journals and by government agencies such as NOAA). Planetary magnetospheres are also affected by the solar wind, so different solar weather will change the fields you have to shoot through. But space weather forecasts and predictive simulations of the solar wind interacting with a planetary magnetosphere can give you a good idea of what the field structure is like. But even with all that, there are changes in the fields that happen on a scale of around a second ... anyone who has ever seen the auroras dancing across the sky in the polar night can see this. Compensating for these changes will be a significant challenge.

What would really help is being able to see the path of your own beam. This would allow you to see how far off you were from your target and in what direction, and then adjust fire to walk your shots onto the target. Unfortunately, particle beams in space are invisible. There might be ways to track them, but if so we don't know what they are.

==== Synchrotron radiation ====

When charged particle trajectories are bent, they emit [[Particle_Accelerators#Synchrotron_radiation|synchrotron radiation]]. This can cause the particle beam to lose energy, but if the bend is too tight it can also generate high energy gamma rays that ignore magnetic fields and directly strike the target. Synchrotron radiation can generally be ignored for proton or ion beams, but can be significant for electron beams.

<table border=1><tr><td>
For a beam particle with energy E and mass m, the synchrotron radiation is concentrated in a forward-directed cone with an opening angle of approximately 1/γ, where γ = 1 + E/(m c2). (Interestingly, the radiation intensity peaks an at angle of 1/(2 γ) and vanishes directly on-axis.) Because the beam is bending, you will get emission along the plane along which the beam is being bent. The energy of the photons produced during synchrotron emission is approximately<ref name="Jackson_Electrodynamics">J. D. Jackson, "Classical Electrodynamics: Second Edition", Wiley (1962)</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Eγ &cong; 3 γ3 c &hbar; / rc
</div>
where &hbar; = 1.05×10-34 J s (if you want the energy in joules) or &hbar; = 6.58×10-16 eV s (if you want the energy in eV). The photon energy will not exceed the energy of the incident particles, if you get a photon energy higher than the energy of the incident particles it just means that the synchrotron radiation spectrum extends up to the incident particle energy.

The fraction of the energy (or power) of the beam turned into synchrotron radiation will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = [1/(4 π ε0)] [4 π / 3] [q² / rc] β2 γ4 [φ / (2 π)] / E
</div>
where β = √[1 - 1/γ2] is the particle's speed measured as a fraction of the speed of light, and 1/(4 π ε0) = 8.99×109 is the Coulomb constant. For highly relativistic electrons (like all practical electron beam weapons emit), β &cong; 1 and we can use
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = 8.85×10-2 [E (GeV)3 [φ / (2 π)] / (rc (m))]
</div>
This fraction cannot be more than 1. If f gets close to one, it means that our original assumption that the beam particles mostly keep their original speed no longer holds, and the beam will lose most or almost all of its energy. Instead, when the beam is being deflected it will only make it through an angle of φ/f and will irradiate a region of length x/f starting at where the beam would have been incident if not for the magnets and extending in the direction the beam was deflected.
</table>

For example, a 1 TeV electron beam will have γ &cong; 2,000,000. So if it's bent by a magnetic field the synchtrotron radiation will spread out in a cone of about 5×10-7 radians. If this beam was shooting at a target 0.1 light seconds away from where it was bent, the radiation would arrive at the target over a region with a radius of about 15 meters with a maximum intensity at about 7.5 meters and no radiation directly along the path the beam was pointing while it was being bent. If the beam was bent by the interplanetary magnetic field from our earlier example, it would produce light with photon energies of around 1.3 eV (near infrared) and the lost energy would be a few parts in a hundred trillion. If it was instead deflected by the Earth's magnetic field you might expect a peak photon energy of tens of keV – in the hard x-ray part of the spectrum – and the fraction of energy in the synchrotron radiation would be somewhere in the low parts in 100,000 or less.

==== Separation of neutral beams ====

Magnetic fields only affect charged objects. So one way to overcome magnetic defenses or environments is to use beams that have no net charge. For example, a proton beam could be neutralized with a co-propagating electron beam with an equal number of protons and electrons. Or, potentially, you could fire an ion beam and then neutralize the ions after acceleration to get a beam of neutral atoms.

But, the magnetic fields are still tugging at the charged particles making up the beam as they go past. This will pull the electrons one way and the protons or ions the other way. As the particles are pulled in opposite directions you get a net positive charge on one side of the beam and a net negative charge on the other side (called polarization). This sets up an electric field inside the beam that acts to pull the particles back together. If the particles are going fast enough and the magnetic field is strong enough, the magnetic force can still separate the charges and will split the beam in two, one with the positive charges and one with the negative charges. But if the particles are not that fast or the magnetic field is not that strong this polarization of the beam will keep it together and let it fly right through the magnetic field.

<table border=1><tr><td>
Let's figure out how to analyze this problem. For definiteness, we will have a beam with an electric current of the positive species of I and an electric current of electrons of -I, all moving in the z direction. The beam radius will be R. The particle speed will be v with the usual relativistic parameters β = v/c and γ = √[1/(1-β2)]. If you know the energy per particle E of the positive species and the mass m of the positive species, γ = 1 + E/(m c2) and then β = √[1 - 1/γ2].

Let's assume that the beam is going through a magnetic field of strength B at an angle θ to the field direction. The quantity B sin(θ) is the component of the magnetic field perpendicular to the direction of the beam.

The charge density ρ of the positive component of the beam will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρL = I / (π v R2).
</div>

It is easiest to analyze the problem in the rest frame of the particles in the beam. Now the particles are at rest. Particles at rest are not affected by a magnetic field, but the change of reference frame Lorentz transforms the initial magnetic field into a combination of electric and magnetic fields in the rest frame. We can ignore the magnetic field, but the electric field is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔A = γ v B sin(θ)
</div>
in a direction perpendicular to both v and B. For our purposes, assume the electric field points in the x direction.

At the same time, relativistic length contraction increases the charge density in the rest frame
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρ = γ ρL.
</div>

Let r be the vector from the beam axis to the point inside the beam. It will be useful to separate this into its Cartesian coordinates. If x and y are the unit vectors in the x and y directions, respectively, then
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
r = x x + y y
</div>

If we consider only the positive component of the beam, from [https://en.wikipedia.org/wiki/Gauss%27s_law Gauss's law] it will have an electric field inside of the beam at a distance r from the center axis of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔+ = ρ r / (2 ε0) = (ρ/(2 ε0)) (x x + y y)
</div>
where ε0 = 8.85×10-12 is the vacuum permittivity. Now let's displace the electrons in the beam by a distance δ in the x direction. By itself, this electron beam will have an electric field of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔- = -(ρ/(2 ε0)) ((x-δ) x + y y)
</div>
By the principle of superposition, the electric field inside the beam with both components is the sum of the electric fields of the separate components including the external field
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔B = [(ρ/(2 ε0)) δ + γ v B sin(θ)]x
</div>
Because the charges are free to move, they will do so as long as a field is applied to them. Thus, the beam becomes stable once the charges separate enough that the field internal to the beam is zero. Therefore
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
δ = -2 ε0 γ v B sin(θ) / ρ = -2 π ε0 v2 R2 B sin(θ) / I
</div>
If |δ| > 2 |R|, the beam will fully separate and each component will go its own way. If δ is small compared to R, the beam's polarization suffices to hold it together in the overlap region. However, where the positive and negative charges do not overlap the electric fields no longer cancel, and the external electric field carries away the outer polarized part of the beam. This gives us a rough threshold for which a magnetic field will quickly separate a neutral beam into its charged components and bend them away from their target
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr><td> B sin(θ) < I / (π ε0 v2 R) <td>       <td> Beam is partially eroded.
<tr><td> B sin(θ) ≥ I / (π ε0 v2 R) <td> <td> Beam promptly separates.
</table>
</div>
For a partially eroding beam, the polarized region that was peeled off by the field still acts to screen the remaining neutral portion, at least to within this approximation, leaving the still-neutral beam to hit the target. The fraction of the beam that is charged and stripped away by the field is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="text-align: center; vertical-align: middle; margin-top:0.5em; margin-bottom:0.5em; line-height: 2em;" align="center" cellpadding="0" cellspacing="0">
<tr>
<td nowrap="nowrap"> δ √(R2 - δ2/4)
<td rowspan="2" nowrap="nowrap">   +  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap">   asin
<td rowspan="2" nowrap="nowrap"> [
<td nowrap="nowrap"> δ
<td rowspan="2" nowrap="nowrap"> ]
<tr>
<td style="border-top:solid 1px black;">π R2
<td style="border-top:solid 1px black;">π
<td style="border-top:solid 1px black;">2 R
</table>
</div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Neutralized_beam_in_B_field.png|5140 px|frameless]]
<tr><td width=700>
A profile of the beam charge distribution of a neutralized beam in a magnetic field. Negative charge density is shown in yellow, positive charge density in cyan, with neutral charge density in light green. The magnetic field is vertical, the electric field in the beam reference frame is shown in magenta including both the polarization field and the Lorentz transformed electric field resulting from the magnetic field. As positive charges are pulled in the direction of the field and negative charges opposite to the field direction, the polarized boundary will be pulled off the beam by the electric field even though the electric field is zero in nearly all the beam interior. This expanding polarization layer will continue to largely screen the still-neutral portion, however; but the extent to which the screening is only partial would require more sophisticated analysis methods than used here. The material removed by the field will expand to a maximum of the deflection distance of a uniform charge based on the magnetic field by the time it reaches the target.
</table>
</div>
<tr><td>
For a beam of neutral atoms, let a be the radius of the atom and let Eb be the binding energy of the least tightly bound electron to the atom. An external electric field in excess of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
|𝓔A| > Eb / (e a)
</div>
will promptly ionize the atom, ripping away its electron. Here, e = 1.60×10-19 C is the magnitude of the electric charge of the electron. For fields close to but a bit weaker than this, the electron can still rapidly pop off due to quantum tunneling<ref>Ph. Durand and I. Paidarová, “Ionization of the hydrogen atom: from weak to strong static electric fields,” The European Physical Journal D, Vol. 26, pg. 253-259, 2003.</ref>. In general, expect that for fields greater in magnitude than (very approximately) Eb / (10 e a) the electron will be quickly removed, while for fields weaker than this the electrons will stay on their atoms.

It is useful to note that Eb/e is just the binding energy in units of eV. The most weakly bound electrons are usually bound to their atom with about 10 eV of energy. Atoms are typically about 10-10 m in radius. Thus, you can expect fields of around 1010 to 1011 V/m to directly ionize atoms. If we call this threshold value 𝓔T, and using the Lorentz transform of a magnetic field to an electric field in the rest frame of the atom, then a beam will be ionized when passing through a magnetic field with strength greater than
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B sin(θ) > 𝓔T / (γ v)
</div>
</table>

Example: Let's consider a 1 GeV neutralized proton beam from a synchrotron, with a beam current of 200 kA in each bunch (incidentally, if a bunch is 5 cm long this means that each bunch packs 38 kJ worth of energy). The speed of the bunch is 87% the speed of light, or 262 million m/s, with a γ of 2.07. Consider when the beam's focus takes it to a radius of 5 cm (or 0.05 m, for our calculation). Plugging this in, we find that the protons in the bunch will separate from the electrons in a field with a transverse component greater than about 2.1 T, and the bunch will pass through the field with only minimal losses if the transverse component is much smaller than this.

Example: We'll take the same 1 GeV beam, but this time recombine the electrons with protons to make hydrogen atoms. An electron is bound to a hydrogen atom with an energy of 13.6 eV (the Ryberg constant) and the radius of the electron orbital in a hydrogen atom is 0.529×10-10 m (the Bohr radius). This gives a threshold electric field for prompt ionization of hydrogen of 2.57×1011 V/m. We will get this with a transverse component of the magnetic field of 474 T. Above this field the atoms promptly ionize and electrons are stripped away from the protons. For fields higher than roughly 1/10 of this – so about 40 to 50 T – you may still get significant ionization by quantum tunneling; lower than this your losses from the beam due to field-induced ionization are likely negligible.

==== Magnetic shields ====

What happens when we can't rely on magnetic fields in the environment to protect us? One idea is to produce powerful artificial magnetic fields around our spacecraft to deflect incoming particle beams. This is very similar in concept to using active magnetic shielding to protect against space radiation – [[Space_Radiation#Magnetic_Shielding|check out our article on space radiation for more]].
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr>
<td width=100>
[[File:Unconfined_FRC_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:racetrack_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:Magnetic_shielding_Halback_Array.png|200 px|frameless]]
<tr><td colspan=3>
Some potential configurations for magnetic shields. The currents generating the fields are in green, the magnetic fields themselves are shown in magenta<ref name="Battiston2012">R. Battiston, W.J. Burger, V. Calvelli, R. Musenich, V. Choutko, V.I. Datskov, A. Della Torre, F. Venditti,
C. Gargiulo, G. Laurenti, S. Lucidi, S. Harrison, and R. Meinke, "ARSSEM Active Radiation Shield for Space Exploration Missions", Final Report ESTEC Contract N° 4200023087/10/NL/AF : “Superconductive Magnet for Radiation Shielding of Human Spacecraft” (2012) https://arxiv.org/abs/1209.1907 https://www.researchgate.net/publication/265945847_Active_Radiation_Shield_for_Space_Exploration_Missions</ref><ref name="Desiati2022">Paolo Desiati and Elena D'Onghia, "CREW HaT: A Magnetic Shielding System for Space Habitats", arXiv:2209.13624 [physics.space-ph] https://doi.org/10.48550/arXiv.2209.13624</ref>.
</table>
</div>

One immediate issue is that you want to keep very high magnetic fields out of the crew area. Fields in excess of 0.5 T can start to be unpleasant, and it is recommended to keep the field below 0.2 T for continuous exposure.

Strong magnets can suffer from high self-forces acting to tear them apart. Most shielding designs will have multiple magnetic coils, which will exert high forces and torques on each other. This will result in the need for strong structural bracing to keep the magnets intact and in place. Such strong structural designs will impose significant extra mass on the spacecraft.

Consider a magnetic field configuration that can keep radiation out of a certain region of space. There is no trajectory of a charged radiation particle up to a certain energy that will allow those particles to enter the protected region. But the dynamics of charged particles in a magnetic field are the same whether going forward or backward in time. If there is no way for a particle to get in, by this time reversal symmetry any particle which is in that forbidden region can never escape. Consequently any field configuration that is a magnetic trap is also a magnetic shield; any field configuration that is a magnetic shield is also a magnetic trap. A spacecraft with a magnetic shield may end up surrounded by strong radiation belts.

Using a magnet to bend a weaponized electron beam very often ends up with the beam being bent so hard that it dumps most of its energy in the form of synchrotron radiation at gamma ray energies. Because gamma rays are not charged, they will not be deflected by the field. Thus, if you would have been hit by the electron beam you will be hit instead by the gamma rays.

As an example, consider a 1 TeV electron beam. We'll surround our spacecraft with a 10 T magnetic field in a 100 meter radius region just outside our spacecraft. This will deflect the beam by 30 meters, which is enough to make it miss. But we are not saved, because the beam dumps nearly all of its energy into synchrotron radiation with an energy spectrum of gamma rays up to 13 GeV. This radiation will hit us in a strip about 2.4 meters long and not much more than the width of the original beam with essentially all of the energy of the original beam.

==== Magnet drones ====

If a magnetic shield around your spacecraft will just end up killing you from synchrotron radiation, what if you put the magnetic shield on a drone and put that drone between yourself and the enemy trying to shoot you? This lets you use a much weaker magnet because you don't need nearly so high of a deflection angle to make the shot miss. This means that less beam energy will turn into synchrotron radiation that could kill you instead and will spread what synchrotron radiation does occur out over a much wider area when it reaches your spacecraft, hopefully so that most of it just misses your spacecraft.

It does mean that your enemy will start shooting your drones so that they can eventually take a shot at you. But hopefully by that time you've already cleared out their drones and cored their spacecraft with your own synchrocannon or UREB.

==== Plasma magnets ====

Usually, to make a really big magnetic field you would run current through a really big conductive loop. But really big things are heavy and expensive. One way around this could be to use the ambient space plasma to carry the electric current rather than any kind of wire. In order to make this current, you use the same mechanism used in a rotary induction electric motor: you generate a rotating magnetic field and induce a current in the conductive loop (in this case, the space plasma). How do you make a rotating magnetic field? The method proposed<ref name="Slough2006">John Slough, "The Plasma Magnet", NASA Institute for Advanced Concepts Phase II Final Report (2006) https://www.niac.usra.edu/files/studies/final_report/917Slough.pdf</ref> is to use an antenna of four much smaller current loops as electromagnets at 90° to each other, and sinusoidally vary the current through the loops with a respective 90° phase difference between them. When this is done, a circular current is induced in the space plasma and the self-force of a magnetic field on its generating current inflates the current loop to much larger radii. You end up with a current sheet in the plane of the rotating field's rotation circulating about the antenna nearly synchronously with the field rotation. This current then creates a normal dipolar magnetic field as with any current loop. The current sheet and the resulting magnetic field expand until the magnetic field's pressure equals that of the ambient space plasma. This gives an enormous region of magnetic field from a relatively small and lightweight system.

The system proposed in the initial study would use an antenna approximately 10 meters in radius and produce a region of field approximately 30 km in radius at Earth's orbit. The field inside would not be constant - at the edges it would be 50 nT, giving the right pressure to keep out the solar wind at Earth's orbit. A circulating current sheet produces a field that falls off as 1/r with distance, meaning that adjacent to the antenna the field would be about 0.15 mT. Using the methods in the box below, we can estimate that this particular design as part of a spacecraft's shields will not deflect a TeV electron beam by more than a few centimeters, which is usually insufficient to force a miss. However, it could potentially be useful for a drone magnet, where the much larger distance between the magnetic field and the target spacecraft would allow a small deflection angle to give perhaps tens or low hundreds of meters change in beam location near the target at near light second ranges. The relatively low field and gradual bend of the incident beam suggests that synchrotron radiation will not be an issue; however this does not exclude more powerful plasma magnet designs causing significant synchrotron radiation effects.

<table border=1><tr><td>
Calculating the deflection from a spatially varying magnetic field affected by the solar wind is no trivial task. However, we can make an approximation that both should be within about an order of magnitude of the true value and also represent an upper limit to the deflection angle; although it does neglect the influence of the driving rotating magnetic field which could in some cases increase the deflection.

We will take R0 as the radius of the antenna. B0 is the field strength at the antenna. Rm is the distance at which the magnetic influence stops because the magnetic pressure equals the space plasma pressure. Bm is the field strength at Rm. According to Slough<ref name="Slough2006"></ref>, the magnetic field strength B(r) at any distance from the antenna center r between R0 and Rm is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B(r) = B0 R0 / r.
</div>
Therefore, once we know Bm and Rm, B0 is completely determined
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B0 = Bm Rm / R0.
</div>

The force on a particle with charge q passing perpendicularly through a magnetic field B(r) at a speed v is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
F(r) = q v B(r)
</div>
For an electron beam, v is so close to c as to make no difference. If the particle is moving down the y direction and the field is in the z direction, the force will be in the x direction. We assume that the deflection will be small, so we can choose a straight-line path in the y direction offset from the center by an amount x. The plasma bubble radius is Rm, so the particle will enter at ymax = √[Rm2-x2] and leave at ymin = -√[Rm2-x2].

The momentum change the particle experiences from this force is the time integral of the force over its path. Because of reflection symmetry about y = 0, we only need to integrate from ymax to 0 and then multiply by 2
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] dt F(r)
<tr> <td> <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] (dy/v) F(r)
<tr> <td> <td> = <td align="left">2 q B(0) R0 [Integral from 0 to √[Rm2-x2]] dy 1/√[y2 + x2]
<tr> <td> <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/x]
</table>
</div>
This holds for |x| > R0. But the expression for the field is not valid for r < R0 and the integral will give nonsense and diverging results if we evaluate for |x| < R0. In this case, we can put the integral lower bound at r = R0, or y = √[R02-x2]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/(√[R02-x2] + R0)]
</table>
</div>

The deflection angle is now estimated as
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = px / p
</div>
where the total momentum of the particle p can be found from the methods in the first box in the Magnetic Fields section.

The deflection for a shot aimed right at the spacecraft at the position of the spacecraft can be estimated by setting x = 0 and only taking the first half of the integral (so ignore the factor of 2)
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
q B(0) R0 Rm ln[Rm/R0] / p
</div>
This is an additional overestimate because it fails to account for how the deflection builds up over distance during the integration, but it should still give something of an order-of-magnitude and upper limit estimate of what kind of deflections might be expected.
</table>

=== Electric fields ===

If electric fields can accelerate charged particles, maybe they can keep the charged particles in a weaponized particle beam away from your spacecraft? To completely block a particle beam, you would need a voltage on your spacecraft equal to the particle energy in eV of the particles in the beam. That is, if a 1 GeV proton beam is coming at you, you'd need an electric potential of a billion volts to stop that beam. This leads to three immediate problems.

The first is that particles of the opposite charge will be attracted to your spacecraft. Because space isn't really empty but rather a rarefied plasma (called the solar wind in a solar system or the interstellar medium out between stars), there will be plenty of particles of the opposite charge around to attract. And they will hit with an energy in eV equal to the voltage on your spacecraft. In our example of defending against a 1 GeV proton beam with a gigavolt field, the electrons from the solar wind will come in and hit with an energy of 1 GeV each. This will generate dangerous radiation.

The second problem is a consequence of the first. Each of those GeV electrons from our example need to get their energy from somewhere, which turns out to be from the electric field we are using as shielding. You will need to make up for this energy loss with power from your electrical system, which gives a constant drain on your power. Some estimates put the electrical effort to maintain a several hundred megavolt shield in space at many tens of gigawatts<ref name="Mechmann2019">Claire Mechmann, "Analysis of Proposed Active Radiation Shielding Design Concept for Spacecraft" (2019) Thesis, College of Engineering and Science of Florida Institute of Technology</ref>.

And the third problem is similar to one for magnetic fields. Your electrodes will experience high self forces and forces between separate electrodes that will try to tear your shield generator apart. Consequently you will need heavy structural reinforcement in order for your shield to work.

There are potential ways to mitigate some of these problems. Interested readers are directed to [[Space_Radiation#Electrostatic_Shielding|the active shielding section in our page on space radiation]] for more of the juicy details; there's no need to repeat the same stuff here.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Elctrostatic_active_shielding.png|200 px|frameless]]
<td>
[[File:Electrostatic_active_shielding_2.png|200 px|frameless]]
<tr><td colspan=2>
Proposed designs for electrostatic shields to protect from positively charged particles<ref name="Tripathi2006">Ram K. Tripathi, John W. Wilson, and Robert C. Youngquist, "Electrostatic Active Radiation Shielding - Revisited", 2006 IEEE Aerospace Conference, Big Sky, MT, USA, 2006, pp. 9 pp.-, doi: 10.1109/AERO.2006.1655760.</ref><ref name="Chowdhury2023">Rajarshi Pal Chowdhury, Luke A. Stegeman, Matthew L. Lund, Dan Fry, Stojan Madzunkov, and Amir A. Bahadori, "Hybrid methods of radiation shielding against deep-space radiation", Life Sciences in Space Research, Volume 38, 2023, Pages 67-78, ISSN 2214-5524, https://doi.org/10.1016/j.lssr.2023.04.004.</ref>.
</table>
</div>

== Damage ==

Particle beams are, first and foremost, devices that produce beams of [[Nuclear_radiation|ionizing radiation]]. The radiation of weaponizable particle beams is extremely penetrating; thus of the [[Particle_Accelerators#Energy_loss_in_matter|interaction mechanisms listed for radiation from particle beams]], we mostly need to worry about bremsstrahlung and nuclear collisions and these produce extensive secondary showers of radiation that increase the penetration of the radiation exposure. Because of this, any target hit by a particle beam weapon will be irradiated, and radiation dose will always be one of the primary damage mechanisms of a particle beam weapon strike. Further, this radiation will be distributed along a track passing deeply through the material of the target, allowing it to affect critical systems through heavy armor, structural material, robust machine components, and passive material such as fuel load, coolant, or cargo. As the radiation deposits its energy into the target material, that material will be heated. If the target is heated sufficiently, it may experience [[Beam-Target_Interactions|thermal or mechanical]] damage from melting, cooking, combustion, vapor explosions, and warping or cracking due to thermal expansion.

The discussion of the radiation penetration into armor also applies to the same radiation penetrating through the target. So the formulas and rules of thumb for estimating the depth of radiation penetration into armor can also be used to estimate how far into a tank or spacecraft a particle beam strike delivers effects.

In addition to the radiation and thermal effects, a particle beam can also produce a sharp electromagnetic pulse that can damage semiconductor electronics.

== Hitting Your Target ==

It doesn't do much to have the most horrifically powerful death ray if it can't hit. What are the challenges for directing your beam of radioactive fury onto your enemies?

We have already mentioned the [[Particle_Beam_Weapons#Magnetic_fields|challenges of stray magnetic field]] that are part of the ambient environment messing with your targeting. This will be a challenge when you are launching beams of charged particles. And we've discussed some of the means of getting around this, like using neutralized beams.

But another issue that will be a challenge at very long ranges is dealing with [[Jitter|jitter]]. A weapon made out of a linac will be very long, and so it will be rather wibbly. This could make its jitter problems worse. A synchrotron might not be so badly affected; and for any future hypothetical high energy wakefield accelerators, no one really knows what they'll be like. There are certainly tricks you can play to reduce jitter (see the previously linked jitter page), but at ranges approaching a light second it's certainly plausible that it will be difficult to land a hit.

== 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, Gerrit Bruhaug, and Luke Campbell for writing

Gerrit Bruhaug and Luke Campbell for scientific research, math wrangling and advise

ATriffleMind and Luke W. Campbell for proof reading.

==References==

[[Category:Beams]][[Category:Physics & Engineering‏‎]][[Category:Engineering‏‎]][[Category:Military Technology]][[Category:Warfare‏‎]]

Particle Beam Weapons

2026-10-29T19:53:20Z

Lwcamp: /* Separation of neutral beams */

{{PageConstructionNotice}}

''For a general overview on particle accelerator technology, see [[Particle Accelerators]]

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.

::: Z = 1.32×102 (γ2-1)3/4 r / √I<ref name=USPAS2016>Martin Reiser, "Theory and Design of Charged Particle Beams", 2008, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim</ref>

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

::: Z = β γ c r √[m/T]

Where Z is the doubling distance like before, r is the initial beam radius, T is the beam temperature in eV, m is the ion mass and β, c, and γ 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.

== Defenses ==

Sure, it's great to have a great big particle cannon to blast your enemies. But what if you are the one being shot at? In that case, it's best to figure out ways to keep from being blown to kingdom come.

=== Armor ===

The simplest way to defend against the high energy particles of a particle beam is to pile a whole bunch of stuff up and put it in the way of the beam. Particle beams may produce extremely penetrating radiation, but enough stuff will block most of that radiation and may allow you to survive. The details depend on what kind of particle is being thrown at you, and are explained below. However, a general rule of thumb is that you probably want about 10 tons of material per square meter of surface you want to armor in order to get sufficient protection.

==== Electrons ====

At the very high energy of an electron accelerated by a weapons-grade particle beam, the primary interaction with matter will be the emission of bremsstrahlung photons in the gamma-ray range as the electron slams into particles of matter. These high energy photons themselves primarily interact by the pair production mechanism – the photon disappears, but leaves behind a highly energetic positron and electron. These two particles are themselves energetic enough to produce further bremsstrahlung, and so the process continues. You get the formation of an electron-gamma shower (also called an electromagnetic cascade), which only stops after either the particles have penetrated through the target or the initial energy of the electron has been spread through so many shower particles (including all the electrons, positrons, and photons) that no particle has enough individual energy to pair produce or to be likely to bremsstrahlung again before it is stopped by other mechanisms.

To stop electrons, you'll want the atoms in your armor to be as heavy as you can practically get them. High atomic number both encourages bremsstrahlung and pair production, giving a double whammy for stopping that cascade. However, occasionally one of those electrons, positrons, or gammas will hit a nucleus and knock out neutrons. So you probably want a back-up layer of several tens of centimeters of hydrogen rich material such as plastic or water mixed with neutron absorbing materials like boron, lithium, cadmium, or gadolinium. High density polyethylene loaded with 5% boron by weight is a typical go-to neutron blocker.

The actual methods to calculate the penetration of the electromagnetic cascade is given are the box below. However, for some examples:
<ul>
<li> A 10 GeV beam incident on thick iron (or steel) will stop cascading at about 9 cm, will deposit half of its energy in the first 12 cm, and will have dumped 98% of its energy within 35 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 2.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 100 GeV beam incident on thick iron (or steel) will stop cascading at about 13 cm, will deposit half of its energy in the first 16 cm, and will have dumped 98% of its energy within 47 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 3.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick iron (or steel) will stop cascading at about 17 cm, will deposit half of its energy in the first 20 cm, and will have dumped 98% of its energy within 59 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 4.6 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick tungsten will stop cascading at about 3.5 cm, will deposit half of its energy in the first 4 cm, and will have dumped 98% of its energy within 12 cm. 90% of the cascade will be contained within about 0.93 cm and 95% of the cascade will be contained within 1.86 cm of the beam axis. You will get 98% protection with a mass thickness of 2.3 tons/m2 and an additional 1/e reduction for each further 0.13 tons/m2 beyond that.
<li> A 1 TeV beam incident on water will stop cascading at about 290 cm, will deposit half of its energy in the first 340 cm, and will have dumped 98% of its energy within 1030 cm. 90% of the cascade will be contained within about 11 cm and 95% of the cascade will be contained within 22 cm of the beam axis. You will get 98% protection with a mass thickness of 10.3 tons/m2 and an additional 1/e reduction for each further 0.72 tons/m2 beyond that.
</ul>
Note that for sufficiently intense particle beams, 98% protection may not be enough. You might need to throw on several more tons per square meter to be sure.

<table border=1>
<tr><td> How Far it Goes:
(This section follows methods presented in Leroy and Rancoita<ref>C. Leroy and P-G Rancoita, "Physics of cascading shower generation and propagation in matter: principles of high-energy, ultrahigh-energy and compensating calorimetry", Reports on Progress in Physics 63 (2000) 505–606. DOI: 10.1088/0034-4885/63/4/202</ref>.)

When trying to estimate the extent of the irradiated region of the target, it is helpful to know the radiation length X0 in the material. This is the distance that an electron will travel, on average, before emitting a bremsstrahlung photon. Conveniently, it is also the distance a high energy photon will travel through the material before pair producing. The radiation length can be found from
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   = 4 α  
<td nowrap="nowrap"> NA
<td rowspan="2" nowrap="nowrap" align="right">   Z (Z+ζ) re2 ln
<td rowspan="2" nowrap="nowrap" align="right"> [
<td nowrap="nowrap"> 183
<td rowspan="2" nowrap="nowrap" align="right"> ]
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> A
<td style="border-top:solid 1px black;"> Z1/3
</table>
</div>
where α = 1/137 is the fine structure constant, NA = 6.02214076×1023 is Avogadro's number (or one mole), A is the atomic weight of the atoms in the material (usually expressed in grams per mole), Z is the atomic number (nuclear charge) of the atoms, and re = 2.8179403205×10-13 cm is the classical electron radius. ζ is a constant of approximately 1.3. This gives the thickness of material in terms of the areal density (mass of material per unit area perpendicular to the beam) – if you use A in g/mol and re in cm the result will be in cm2/g and X0 will be in g/cm2. If your target is made up of more than one kind of atom, you can get the total radiation length by finding the radiation length of each atom of type i, X0i and using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   =  
<td rowspan="2" nowrap="nowrap" align="right"> Σi  
<td nowrap="nowrap"> fi
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> X0i
</table>
</div>
where fi is the fraction by weight of the atom type i. Once you have the radiation length in terms of areal density, you can easily convert it to distance by dividing by the density ρ. If X0i is in g/cm2 and density is in g/cm3, then X0i/ρ will be in cm.

We will also need to know the critical energy εc. This is the energy where an electron is losing energy as rapidly to ionization as it is to radiation. That is, it is the energy at which the cascade stops multiplying, even though the individual particles already made can continue traveling for some distance. The critical energy can be estimated using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> εc = B  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> Z X0
<td rowspan="2" nowrap="nowrap" align="right"> )
<td nowrap="nowrap"> h
<tr>
<td style="border-top:solid 1px black;"> A
<td>  
</table>
</div>
where B = 2.66 MeV and h = 1.11.

We can now describe the distribution of radiation in the target material. The depth tmax where the cascade stops multiplying (in g/cm2) for an incident electron of energy E (in MeV) is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tmax = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> - 1
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
Similarly, the depth where half the energy of the beam has been deposited (in g/cm2) is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tcg = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> + 0.37
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
As usual, you can convert to a depth in cm by dividing by the material density. 98% of the beam energy will be deposited within a distance of &cong; 3 tcg. Beyond this, the beam energy will continue to fall off with thickness t approximately as an exponential exp[-t/λ], where the characteristic attenuation length λ &cong; 2 X0.

The width of a cascade, where 90% of the energy is contained, is approximately given by
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> RM =  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> EM
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
where EM &cong; 21.2 MeV. 95% of the radiation will be contained within a distance of 2 RM from the beam axis.
</table>

==== Protons and Ions ====

<table align=right><tr><td>
[[File:Regolith_Shielding.png|350 px|frameless]]
<tr><td width=350>
Relative effect of particle radiation (compared to no shielding) from galactic cosmic rays behind different thicknesses of water, aluminum, and lunar regolith<ref name="Slaba2022">Tony C. Slaba, "Radiation Shielding for Lunar Missions: Regolith Considerations", LSIC Crosstalk 7/18/2022 https://lsic.jhuapl.edu/uploadedDocs/focus-files/1604-E&C%20+%20EE%20Monthly%20Meeting%20-%202022%2007%20July_Presentation%20-%20NASA%20Slaba.pdf</ref>.
</table>
Protons and ions are grouped together because ions are made of protons and neutrons and when you accelerate these things for a weapon it is the protons and neutrons that are going to be doing the heavy lifting for blasting your target. This is because protons and neutrons are both about 2000 times more massive than electrons so any electrons along for the ride will contribute negligibly to the energy. Protons and neutrons together are called nuclear particles, and both behave quite similarly in nuclear collisions.

Nuclear particles are themselves made of quarks and can interact via the strong nuclear force. At the energies appropriate for weapons use, essentially all of the way protons or ions are blocked is through [[Particle_Accelerators#Nuclear_collisions|direct collisions with the atomic nuclei]] of the armor material. One or more quarks in the projectile will slam into quarks in the target nucleus and stretch them away from the rest of the quarks in the nuclear particles. Physics tells us that you can't have isolated quarks, so the "string" attaching the quark to its neighbors stretches and stretches until it snaps. Snapping releases so much energy that more quarks are made that clump together into all sorts of other nuclear particles, both exotic and familiar. This process is called a hadronic jet. The only particles produced in these jets long-lived enough to travel any noticeable distance through the armor material will be protons, neutrons, the charged pi mesons (or pions), and some of the K mesons (or kaons). The particles too short lived to go far enough to matter will decay into other particles, and these particles are often very high energy gamma rays. Much like with electrons, the long-lived secondary particles can be sufficiently energetic to smash into other nuclei and cause more hadronic jets (or, if they are low enough in energy, just knock more nuclear particles out of target nuclei) and this produces a similar shower or cascade. But in addition, the armor nuclei that were hit will be very hot after the impact from all of the energy given to them from the incident particles. These nuclei will cool off by evaporation – but instead of evaporating away molecules, they evaporate off nuclear particles from the nucleus. These evaporated particles are mostly neutrons, so any armor against proton or ion beam strikes will produce quite a bit of neutron radiation.

Eventually, the original particle's energy will be distributed among so many other particles that the charged particles will come to a stop by making ion tracks rather than from colliding with more nuclei, and the cascade halts. But you still have all the neutrons to worry about. And the gamma rays as well, which are high enough energy to kick off an electromagnetic cascade in the same way that electrons do.

Unlike electrons, which have reasonable approximations of their cascade behavior from relatively simple equations, the cascades of protons and ions usually require sophisticated simulations or experiments to get the resulting distribution of radiation. However, we can get a reasonable idea of their general behavior by comparing them to [[Space_Radiation#Shielding_Against_Cosmic_Rays|cosmic ray particles interacting with shielding]]. Cosmic rays are energetic protons and ions, after all. From this we see that a relatively small amount of armor – in the range of 1 to 3 tons per square meter – can actually make the radiation dose worse. The incident particles will knock off so many secondary particles in their shower that the dose is multiplied from so many tracks passing through the target. Hydrogen rich materials such as water and plastics work the best against proton and ion beam particles; heavy elements are the worst and produce the most secondaries. You could probably obtain adequate radiation protection from most incident beams with about 10 tons per square meter of hydrogen rich material as armor. However, you will want to add neutron absorbing elements to your hydrogen rich armor, such as lithium, boron, cadmium, or gadolinium. Further, in order to deal with the gamma rays you will want a layer of at least several centimeters of heavy elements, such as tungsten or lead, behind the hydrogen rich particle armor.

==== Structural Considerations ====

When your armor material absorbs a radiation beam, the energy of that radiation turns into heat inside the armor where the radiation was absorbed. If the beam pulse has enough energy as was sufficiently tightly focused, this heat can damage your armor. Your armor may melt in places, or thermal expansion could cause it to crack or warp. Perhaps the worst thing it could do is vaporize or flash into a plasma, because the pressure from the vapor or plasma can be sufficient to plastically deform the armor. This leaves a hole along the beam path where the expanding gas or plasma blasted out a long, deep crater. This means a beam that fires rapid shots spaced maybe milliseconds apart can have subsequent shots go down the same hole so your armor will not protect against them. Each shot will dig out a deeper hole until either your armor is penetrated or the attacking beam runs out of shots in its pulse train.

A consequence of this is that when designing your armor you may wish to use refractory materials with a high structural strength (like tungsten), or that is otherwise resistant to being blasted apart by its own evaporated remains.

=== Magnetic fields ===

[[Particle_Accelerators#Magnetic_fields|A charged particle crossing a magnetic field will bend so as to spiral around the field lines]]. If you try to shoot your particle beam through a magnetic field, it will bend away from the direction you shot it. This can make it hard to hit your target. Unfortunately for you, there are magnetic fields everywhere – around planets, and even trapped in the solar wind as it crosses the solar system.

As an example, the Earth has a magnetic field strength of about 5×10-5 tesla near its surface, and falling off farther away. If you are shooting a 1 TeV electron beam past the Earth, your beam is likely to bend by several milliradians (with the exact deflection depending on how close you are skimming the Earth). If you are in a medium Earth orbit and shooting at another target also in a medium Earth orbit on a trajectory that goes near the Earth, then without correcting for the magnetic field your beam will likely be off-target by tens to hundreds of kilometers.

For another example, the interplanetary magnetic field is about 10-9 tesla at Earth's distance from the sun. If you shoot that same 1 TeV electron beam at a target 0.1 light seconds away, you are likely to be off by a couple hundred meters.

<table border=1>
<tr><td>
The radius of curvature of the bend of a particle in a magnetic field is called the gyroradius rc. For a particle with a charge q, mass m, and kinetic energy E, find the relativistic momentum p
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
p = √[E2 - m2 c4] / c
</div>
To convert from energy units of eV to joules, multiply by 1.60×10-19. An electron has a mass of 9.11×10-31 kg, a proton of 1.67×10-27 kg, and an ion of 1.66×10-27×A kg, where A is the atomic weight of the ion. The speed of light c = 3.00×108 m/s.

In a magnetic field of strength B with an angle θ between the field direction and the direction of motion
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rc = |p sin(θ) / (q B)|
</div>
For electrons, q = -e, where the fundamental charge e = 1.60×10-19 coulomb. For a proton, q = +e. For an ion, multiply e by the charge state of the ion.

If the field acts over a distance d that is much smaller than the gyroradius, the deflection angle φ is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = d / rc
</div>
The deflection x at a target at a range R for small φ (φ &Lt; 1) will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
x = φ R.
</div>

Note that d will always be less than or equal to x. Also note that if d is not smaller than twice the gyroradius the beam will never reach the target because it will completely spiral around the field lines before it reaches the target.
</table>

So, it is clear that as the defender it will be helpful to hang out in areas with high magnetic fields, and keep some distance between you and the attacker. Meanwhile, the attacker will benefit from being closer where small deflections in the beam trajectory don't translate into large excursions from where the beam is aimed (and also because there will be less space filled with fields between the two of you). But the attacker can also try to compensate for known fields. If you happen to know that the magnetic fields are between you and your target, you can adjust your fire so that the deflection of the field bends your beam onto the target. This is not a simple task. It is easy enough to measure the field strength and direction where you are, but predicting how it changes over the path of your shot does not have an obvious solution. You can also try to gain intelligence on likely battle sites before you ever fight on them – for example, getting good data on the structure of the fields of a planet using spies and technical intelligence during peacetime (heck, on Earth this information is published for free in academic journals and by government agencies such as NOAA). Planetary magnetospheres are also affected by the solar wind, so different solar weather will change the fields you have to shoot through. But space weather forecasts and predictive simulations of the solar wind interacting with a planetary magnetosphere can give you a good idea of what the field structure is like. But even with all that, there are changes in the fields that happen on a scale of around a second ... anyone who has ever seen the auroras dancing across the sky in the polar night can see this. Compensating for these changes will be a significant challenge.

What would really help is being able to see the path of your own beam. This would allow you to see how far off you were from your target and in what direction, and then adjust fire to walk your shots onto the target. Unfortunately, particle beams in space are invisible. There might be ways to track them, but if so we don't know what they are.

==== Synchrotron radiation ====

When charged particle trajectories are bent, they emit [[Particle_Accelerators#Synchrotron_radiation|synchrotron radiation]]. This can cause the particle beam to lose energy, but if the bend is too tight it can also generate high energy gamma rays that ignore magnetic fields and directly strike the target. Synchrotron radiation can generally be ignored for proton or ion beams, but can be significant for electron beams.

<table border=1><tr><td>
For a beam particle with energy E and mass m, the synchrotron radiation is concentrated in a forward-directed cone with an opening angle of approximately 1/γ, where γ = 1 + E/(m c2). (Interestingly, the radiation intensity peaks an at angle of 1/(2 γ) and vanishes directly on-axis.) Because the beam is bending, you will get emission along the plane along which the beam is being bent. The energy of the photons produced during synchrotron emission is approximately<ref name="Jackson_Electrodynamics">J. D. Jackson, "Classical Electrodynamics: Second Edition", Wiley (1962)</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Eγ &cong; 3 γ3 c &hbar; / rc
</div>
where &hbar; = 1.05×10-34 J s (if you want the energy in joules) or &hbar; = 6.58×10-16 eV s (if you want the energy in eV). The photon energy will not exceed the energy of the incident particles, if you get a photon energy higher than the energy of the incident particles it just means that the synchrotron radiation spectrum extends up to the incident particle energy.

The fraction of the energy (or power) of the beam turned into synchrotron radiation will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = [1/(4 π ε0)] [4 π / 3] [q² / rc] β2 γ4 [φ / (2 π)] / E
</div>
where β = √[1 - 1/γ2] is the particle's speed measured as a fraction of the speed of light, and 1/(4 π ε0) = 8.99×109 is the Coulomb constant. For highly relativistic electrons (like all practical electron beam weapons emit), β &cong; 1 and we can use
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = 8.85×10-2 [E (GeV)3 [φ / (2 π)] / (rc (m))]
</div>
This fraction cannot be more than 1. If f gets close to one, it means that our original assumption that the beam particles mostly keep their original speed no longer holds, and the beam will lose most or almost all of its energy. Instead, when the beam is being deflected it will only make it through an angle of φ/f and will irradiate a region of length x/f starting at where the beam would have been incident if not for the magnets and extending in the direction the beam was deflected.
</table>

For example, a 1 TeV electron beam will have γ &cong; 2,000,000. So if it's bent by a magnetic field the synchtrotron radiation will spread out in a cone of about 5×10-7 radians. If this beam was shooting at a target 0.1 light seconds away from where it was bent, the radiation would arrive at the target over a region with a radius of about 15 meters with a maximum intensity at about 7.5 meters and no radiation directly along the path the beam was pointing while it was being bent. If the beam was bent by the interplanetary magnetic field from our earlier example, it would produce light with photon energies of around 1.3 eV (near infrared) and the lost energy would be a few parts in a hundred trillion. If it was instead deflected by the Earth's magnetic field you might expect a peak photon energy of tens of keV – in the hard x-ray part of the spectrum – and the fraction of energy in the synchrotron radiation would be somewhere in the low parts in 100,000 or less.

==== Separation of neutral beams ====

Magnetic fields only affect charged objects. So one way to overcome magnetic defenses or environments is to use beams that have no net charge. For example, a proton beam could be neutralized with a co-propagating electron beam with an equal number of protons and electrons. Or, potentially, you could fire an ion beam and then neutralize the ions after acceleration to get a beam of neutral atoms.

But, the magnetic fields are still tugging at the charged particles making up the beam as they go past. This will pull the electrons one way and the protons or ions the other way. As the particles are pulled in opposite directions you get a net positive charge on one side of the beam and a net negative charge on the other side (called polarization). This sets up an electric field inside the beam that acts to pull the particles back together. If the particles are going fast enough and the magnetic field is strong enough, the magnetic force can still separate the charges and will split the beam in two, one with the positive charges and one with the negative charges. But if the particles are not that fast or the magnetic field is not that strong this polarization of the beam will keep it together and let it fly right through the magnetic field.

<table border=1><tr><td>
Let's figure out how to analyze this problem. For definiteness, we will have a beam with an electric current of the positive species of I and an electric current of electrons of -I, all moving in the z direction. The beam radius will be R. The particle speed will be v with the usual relativistic parameters β = v/c and γ = √[1/(1-β2)]. If you know the energy per particle E of the positive species and the mass m of the positive species, γ = 1 + E/(m c2) and then β = √[1 - 1/γ2].

Let's assume that the beam is going through a magnetic field of strength B at an angle θ to the field direction. The quantity B sin(θ) is the component of the magnetic field perpendicular to the direction of the beam.

The charge density ρ of the positive component of the beam will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρL = I / (π v R2).
</div>

It is easiest to analyze the problem in the rest frame of the particles in the beam. Now the particles are at rest. Particles at rest are not affected by a magnetic field, but the change of reference frame Lorentz transforms the initial magnetic field into a combination of electric and magnetic fields in the rest frame. We can ignore the magnetic field, but the electric field is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔A = γ v B sin(θ)
</div>
in a direction perpendicular to both v and B. For our purposes, assume the electric field points in the x direction.

At the same time, relativistic length contraction increases the charge density in the rest frame
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρ = γ ρL.
</div>

Let r be the vector from the beam axis to the point inside the beam. It will be useful to separate this into its Cartesian coordinates. If x and y are the unit vectors in the x and y directions, respectively, then
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
r = x x + y y
</div>

If we consider only the positive component of the beam, from [https://en.wikipedia.org/wiki/Gauss%27s_law Gauss's law] it will have an electric field inside of the beam at a distance r from the center axis of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔+ = ρ r / (2 ε0) = (ρ/(2 ε0)) (x x + y y)
</div>
where ε0 = 8.85×10-12 is the vacuum permittivity. Now let's displace the electrons in the beam by a distance δ in the x direction. By itself, this electron beam will have an electric field of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔- = -(ρ/(2 ε0)) ((x-δ) x + y y)
</div>
By the principle of superposition, the electric field inside the beam with both components is the sum of the electric fields of the separate components including the external field
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔B = [(ρ/(2 ε0)) δ + γ v B sin(θ)]x
</div>
Because the charges are free to move, they will do so as long as a field is applied to them. Thus, the beam becomes stable once the charges separate enough that the field internal to the beam is zero. Therefore
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
δ = -2 ε0 γ v B sin(θ) / ρ = -2 π ε0 v2 R2 B sin(θ) / I
</div>
If |δ| > 2 |R|, the beam will fully separate and each component will go its own way. If δ is small compared to R, the beam's polarization suffices to hold it together in the overlap region. However, where the positive and negative charges do not overlap the electric fields no longer cancel, and the external electric field carries away the outer polarized part of the beam. This gives us a rough threshold for which a magnetic field will quickly separate a neutral beam into its charged components and bend them away from their target
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr><td> B sin(θ) < I / (π ε0 v2 R) <td>       <td> Beam is partially eroded.
<tr><td> B sin(θ) ≥ I / (π ε0 v2 R) <td> <td> Beam promptly separates.
</table>
</div>
For a partially eroding beam, the polarized region that was peeled off by the field still acts to screen the remaining neutral portion, at least to within this approximation, leaving the still-neutral beam to hit the target. The fraction of the beam that is charged and stripped away by the field is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="text-align: center; vertical-align: middle; margin-top:0.5em; margin-bottom:0.5em; line-height: 2em;" align="center" cellpadding="0" cellspacing="0">
<tr>
<td nowrap="nowrap"> δ √(R2 - δ2/4)
<td rowspan="2" nowrap="nowrap">   +  
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap">   asin
<td rowspan="2" nowrap="nowrap"> [
<td nowrap="nowrap"> δ
<td rowspan="2" nowrap="nowrap"> ]
<tr>
<td style="border-top:solid 1px black;">π R2
<td style="border-top:solid 1px black;">π
<td style="border-top:solid 1px black;">2 R
</table>
</div>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Neutralized_beam_in_B_field.png|5140 px|frameless]]
<tr><td width=700>
A profile of the beam charge distribution of a neutralized beam in a magnetic field. Negative charge density is shown in yellow, positive charge density in cyan, with neutral charge density in light green. The magnetic field is vertical, the electric field in the beam reference frame is shown in magenta including both the polarization field and the Lorentz transformed electric field resulting from the magnetic field. As positive charges are pulled in the direction of the field and negative charges opposite to the field direction, the polarized boundary will be pulled off the beam by the electric field even though the electric field is zero in nearly all the beam interior. This expanding polarization layer will continue to largely screen the still-neutral portion, however; but the extent to which the screening is only partial would require more sophisticated analysis methods than used here. The material removed by the field will expand to a maximum of the deflection distance of a uniform charge based on the magnetic field by the time it reaches the target.
</table>
</div>
<tr><td>
For a beam of neutral atoms, let a be the radius of the atom and let Eb be the binding energy of the least tightly bound electron to the atom. An external electric field in excess of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
|𝓔A| > Eb / (e a)
</div>
will promptly ionize the atom, ripping away its electron. Here, e = 1.60×10-19 C is the magnitude of the electric charge of the electron. For fields close to but a bit weaker than this, the electron can still rapidly pop off due to quantum tunneling<ref>Ph. Durand and I. Paidarová, “Ionization of the hydrogen atom: from weak to strong static electric fields,” The European Physical Journal D, Vol. 26, pg. 253-259, 2003.</ref>. In general, expect that for fields greater in magnitude than (very approximately) Eb / (10 e a) the electron will be quickly removed, while for fields weaker than this the electrons will stay on their atoms.

It is useful to note that Eb/e is just the binding energy in units of eV. The most weakly bound electrons are usually bound to their atom with about 10 eV of energy. Atoms are typically about 10-10 m in radius. Thus, you can expect fields of around 1010 to 1011 V/m to directly ionize atoms. If we call this threshold value 𝓔T, then a beam will be ionized when passing through a field with strength greater than
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B sin(θ) > 𝓔T / (γ v)
</div>
</table>

Example: Let's consider a 1 GeV neutralized proton beam from a synchrotron, with a beam current of 200 kA in each bunch (incidentally, if a bunch is 5 cm long this means that each bunch packs 38 kJ worth of energy). The speed of the bunch is 87% the speed of light, or 262 million m/s, with a γ of 2.07. Consider when the beam's focus takes it to a radius of 5 cm (or 0.05 m, for our calculation). Plugging this in, we find that the protons in the bunch will separate from the electrons in a field with a transverse component greater than about 2.1 T, and the bunch will pass through the field with only minimal losses if the transverse component is much smaller than this.

Example: We'll take the same 1 GeV beam, but this time recombine the electrons with protons to make hydrogen atoms. An electron is bound to a hydrogen atom with an energy of 13.6 eV (the Ryberg constant) and the radius of the electron orbital in a hydrogen atom is 0.529×10-10 m (the Bohr radius). This gives a threshold electric field for prompt ionization of hydrogen of 2.57×1011 V/m. We will get this with a transverse component of the magnetic field of 474 T. Above this field the atoms promptly ionize and electrons are stripped away from the protons. For fields higher than roughly 1/10 of this – so about 40 to 50 T – you may still get significant ionization by quantum tunneling; lower than this your losses from the beam due to field-induced ionization are likely negligible.

==== Magnetic shields ====

What happens when we can't rely on magnetic fields in the environment to protect us? One idea is to produce powerful artificial magnetic fields around our spacecraft to deflect incoming particle beams. This is very similar in concept to using active magnetic shielding to protect against space radiation – [[Space_Radiation#Magnetic_Shielding|check out our article on space radiation for more]].
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr>
<td width=100>
[[File:Unconfined_FRC_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:racetrack_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:Magnetic_shielding_Halback_Array.png|200 px|frameless]]
<tr><td colspan=3>
Some potential configurations for magnetic shields. The currents generating the fields are in green, the magnetic fields themselves are shown in magenta<ref name="Battiston2012">R. Battiston, W.J. Burger, V. Calvelli, R. Musenich, V. Choutko, V.I. Datskov, A. Della Torre, F. Venditti,
C. Gargiulo, G. Laurenti, S. Lucidi, S. Harrison, and R. Meinke, "ARSSEM Active Radiation Shield for Space Exploration Missions", Final Report ESTEC Contract N° 4200023087/10/NL/AF : “Superconductive Magnet for Radiation Shielding of Human Spacecraft” (2012) https://arxiv.org/abs/1209.1907 https://www.researchgate.net/publication/265945847_Active_Radiation_Shield_for_Space_Exploration_Missions</ref><ref name="Desiati2022">Paolo Desiati and Elena D'Onghia, "CREW HaT: A Magnetic Shielding System for Space Habitats", arXiv:2209.13624 [physics.space-ph] https://doi.org/10.48550/arXiv.2209.13624</ref>.
</table>
</div>

One immediate issue is that you want to keep very high magnetic fields out of the crew area. Fields in excess of 0.5 T can start to be unpleasant, and it is recommended to keep the field below 0.2 T for continuous exposure.

Strong magnets can suffer from high self-forces acting to tear them apart. Most shielding designs will have multiple magnetic coils, which will exert high forces and torques on each other. This will result in the need for strong structural bracing to keep the magnets intact and in place. Such strong structural designs will impose significant extra mass on the spacecraft.

Consider a magnetic field configuration that can keep radiation out of a certain region of space. There is no trajectory of a charged radiation particle up to a certain energy that will allow those particles to enter the protected region. But the dynamics of charged particles in a magnetic field are the same whether going forward or backward in time. If there is no way for a particle to get in, by this time reversal symmetry any particle which is in that forbidden region can never escape. Consequently any field configuration that is a magnetic trap is also a magnetic shield; any field configuration that is a magnetic shield is also a magnetic trap. A spacecraft with a magnetic shield may end up surrounded by strong radiation belts.

Using a magnet to bend a weaponized electron beam very often ends up with the beam being bent so hard that it dumps most of its energy in the form of synchrotron radiation at gamma ray energies. Because gamma rays are not charged, they will not be deflected by the field. Thus, if you would have been hit by the electron beam you will be hit instead by the gamma rays.

As an example, consider a 1 TeV electron beam. We'll surround our spacecraft with a 10 T magnetic field in a 100 meter radius region just outside our spacecraft. This will deflect the beam by 30 meters, which is enough to make it miss. But we are not saved, because the beam dumps nearly all of its energy into synchrotron radiation with an energy spectrum of gamma rays up to 13 GeV. This radiation will hit us in a strip about 2.4 meters long and not much more than the width of the original beam with essentially all of the energy of the original beam.

==== Magnet drones ====

If a magnetic shield around your spacecraft will just end up killing you from synchrotron radiation, what if you put the magnetic shield on a drone and put that drone between yourself and the enemy trying to shoot you? This lets you use a much weaker magnet because you don't need nearly so high of a deflection angle to make the shot miss. This means that less beam energy will turn into synchrotron radiation that could kill you instead and will spread what synchrotron radiation does occur out over a much wider area when it reaches your spacecraft, hopefully so that most of it just misses your spacecraft.

It does mean that your enemy will start shooting your drones so that they can eventually take a shot at you. But hopefully by that time you've already cleared out their drones and cored their spacecraft with your own synchrocannon or UREB.

==== Plasma magnets ====

Usually, to make a really big magnetic field you would run current through a really big conductive loop. But really big things are heavy and expensive. One way around this could be to use the ambient space plasma to carry the electric current rather than any kind of wire. In order to make this current, you use the same mechanism used in a rotary induction electric motor: you generate a rotating magnetic field and induce a current in the conductive loop (in this case, the space plasma). How do you make a rotating magnetic field? The method proposed<ref name="Slough2006">John Slough, "The Plasma Magnet", NASA Institute for Advanced Concepts Phase II Final Report (2006) https://www.niac.usra.edu/files/studies/final_report/917Slough.pdf</ref> is to use an antenna of four much smaller current loops as electromagnets at 90° to each other, and sinusoidally vary the current through the loops with a respective 90° phase difference between them. When this is done, a circular current is induced in the space plasma and the self-force of a magnetic field on its generating current inflates the current loop to much larger radii. You end up with a current sheet in the plane of the rotating field's rotation circulating about the antenna nearly synchronously with the field rotation. This current then creates a normal dipolar magnetic field as with any current loop. The current sheet and the resulting magnetic field expand until the magnetic field's pressure equals that of the ambient space plasma. This gives an enormous region of magnetic field from a relatively small and lightweight system.

The system proposed in the initial study would use an antenna approximately 10 meters in radius and produce a region of field approximately 30 km in radius at Earth's orbit. The field inside would not be constant - at the edges it would be 50 nT, giving the right pressure to keep out the solar wind at Earth's orbit. A circulating current sheet produces a field that falls off as 1/r with distance, meaning that adjacent to the antenna the field would be about 0.15 mT. Using the methods in the box below, we can estimate that this particular design as part of a spacecraft's shields will not deflect a TeV electron beam by more than a few centimeters, which is usually insufficient to force a miss. However, it could potentially be useful for a drone magnet, where the much larger distance between the magnetic field and the target spacecraft would allow a small deflection angle to give perhaps tens or low hundreds of meters change in beam location near the target at near light second ranges. The relatively low field and gradual bend of the incident beam suggests that synchrotron radiation will not be an issue; however this does not exclude more powerful plasma magnet designs causing significant synchrotron radiation effects.

<table border=1><tr><td>
Calculating the deflection from a spatially varying magnetic field affected by the solar wind is no trivial task. However, we can make an approximation that both should be within about an order of magnitude of the true value and also represent an upper limit to the deflection angle; although it does neglect the influence of the driving rotating magnetic field which could in some cases increase the deflection.

We will take R0 as the radius of the antenna. B0 is the field strength at the antenna. Rm is the distance at which the magnetic influence stops because the magnetic pressure equals the space plasma pressure. Bm is the field strength at Rm. According to Slough<ref name="Slough2006"></ref>, the magnetic field strength B(r) at any distance from the antenna center r between R0 and Rm is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B(r) = B0 R0 / r.
</div>
Therefore, once we know Bm and Rm, B0 is completely determined
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B0 = Bm Rm / R0.
</div>

The force on a particle with charge q passing perpendicularly through a magnetic field B(r) at a speed v is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
F(r) = q v B(r)
</div>
For an electron beam, v is so close to c as to make no difference. If the particle is moving down the y direction and the field is in the z direction, the force will be in the x direction. We assume that the deflection will be small, so we can choose a straight-line path in the y direction offset from the center by an amount x. The plasma bubble radius is Rm, so the particle will enter at ymax = √[Rm2-x2] and leave at ymin = -√[Rm2-x2].

The momentum change the particle experiences from this force is the time integral of the force over its path. Because of reflection symmetry about y = 0, we only need to integrate from ymax to 0 and then multiply by 2
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] dt F(r)
<tr> <td> <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] (dy/v) F(r)
<tr> <td> <td> = <td align="left">2 q B(0) R0 [Integral from 0 to √[Rm2-x2]] dy 1/√[y2 + x2]
<tr> <td> <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/x]
</table>
</div>
This holds for |x| > R0. But the expression for the field is not valid for r < R0 and the integral will give nonsense and diverging results if we evaluate for |x| < R0. In this case, we can put the integral lower bound at r = R0, or y = √[R02-x2]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/(√[R02-x2] + R0)]
</table>
</div>

The deflection angle is now estimated as
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = px / p
</div>
where the total momentum of the particle p can be found from the methods in the first box in the Magnetic Fields section.

The deflection for a shot aimed right at the spacecraft at the position of the spacecraft can be estimated by setting x = 0 and only taking the first half of the integral (so ignore the factor of 2)
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
q B(0) R0 Rm ln[Rm/R0] / p
</div>
This is an additional overestimate because it fails to account for how the deflection builds up over distance during the integration, but it should still give something of an order-of-magnitude and upper limit estimate of what kind of deflections might be expected.
</table>

=== Electric fields ===

If electric fields can accelerate charged particles, maybe they can keep the charged particles in a weaponized particle beam away from your spacecraft? To completely block a particle beam, you would need a voltage on your spacecraft equal to the particle energy in eV of the particles in the beam. That is, if a 1 GeV proton beam is coming at you, you'd need an electric potential of a billion volts to stop that beam. This leads to three immediate problems.

The first is that particles of the opposite charge will be attracted to your spacecraft. Because space isn't really empty but rather a rarefied plasma (called the solar wind in a solar system or the interstellar medium out between stars), there will be plenty of particles of the opposite charge around to attract. And they will hit with an energy in eV equal to the voltage on your spacecraft. In our example of defending against a 1 GeV proton beam with a gigavolt field, the electrons from the solar wind will come in and hit with an energy of 1 GeV each. This will generate dangerous radiation.

The second problem is a consequence of the first. Each of those GeV electrons from our example need to get their energy from somewhere, which turns out to be from the electric field we are using as shielding. You will need to make up for this energy loss with power from your electrical system, which gives a constant drain on your power. Some estimates put the electrical effort to maintain a several hundred megavolt shield in space at many tens of gigawatts<ref name="Mechmann2019">Claire Mechmann, "Analysis of Proposed Active Radiation Shielding Design Concept for Spacecraft" (2019) Thesis, College of Engineering and Science of Florida Institute of Technology</ref>.

And the third problem is similar to one for magnetic fields. Your electrodes will experience high self forces and forces between separate electrodes that will try to tear your shield generator apart. Consequently you will need heavy structural reinforcement in order for your shield to work.

There are potential ways to mitigate some of these problems. Interested readers are directed to [[Space_Radiation#Electrostatic_Shielding|the active shielding section in our page on space radiation]] for more of the juicy details; there's no need to repeat the same stuff here.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Elctrostatic_active_shielding.png|200 px|frameless]]
<td>
[[File:Electrostatic_active_shielding_2.png|200 px|frameless]]
<tr><td colspan=2>
Proposed designs for electrostatic shields to protect from positively charged particles<ref name="Tripathi2006">Ram K. Tripathi, John W. Wilson, and Robert C. Youngquist, "Electrostatic Active Radiation Shielding - Revisited", 2006 IEEE Aerospace Conference, Big Sky, MT, USA, 2006, pp. 9 pp.-, doi: 10.1109/AERO.2006.1655760.</ref><ref name="Chowdhury2023">Rajarshi Pal Chowdhury, Luke A. Stegeman, Matthew L. Lund, Dan Fry, Stojan Madzunkov, and Amir A. Bahadori, "Hybrid methods of radiation shielding against deep-space radiation", Life Sciences in Space Research, Volume 38, 2023, Pages 67-78, ISSN 2214-5524, https://doi.org/10.1016/j.lssr.2023.04.004.</ref>.
</table>
</div>

== Damage ==

Particle beams are, first and foremost, devices that produce beams of [[Nuclear_radiation|ionizing radiation]]. The radiation of weaponizable particle beams is extremely penetrating; thus of the [[Particle_Accelerators#Energy_loss_in_matter|interaction mechanisms listed for radiation from particle beams]], we mostly need to worry about bremsstrahlung and nuclear collisions and these produce extensive secondary showers of radiation that increase the penetration of the radiation exposure. Because of this, any target hit by a particle beam weapon will be irradiated, and radiation dose will always be one of the primary damage mechanisms of a particle beam weapon strike. Further, this radiation will be distributed along a track passing deeply through the material of the target, allowing it to affect critical systems through heavy armor, structural material, robust machine components, and passive material such as fuel load, coolant, or cargo. As the radiation deposits its energy into the target material, that material will be heated. If the target is heated sufficiently, it may experience [[Beam-Target_Interactions|thermal or mechanical]] damage from melting, cooking, combustion, vapor explosions, and warping or cracking due to thermal expansion.

The discussion of the radiation penetration into armor also applies to the same radiation penetrating through the target. So the formulas and rules of thumb for estimating the depth of radiation penetration into armor can also be used to estimate how far into a tank or spacecraft a particle beam strike delivers effects.

In addition to the radiation and thermal effects, a particle beam can also produce a sharp electromagnetic pulse that can damage semiconductor electronics.

== Hitting Your Target ==

It doesn't do much to have the most horrifically powerful death ray if it can't hit. What are the challenges for directing your beam of radioactive fury onto your enemies?

We have already mentioned the [[Particle_Beam_Weapons#Magnetic_fields|challenges of stray magnetic field]] that are part of the ambient environment messing with your targeting. This will be a challenge when you are launching beams of charged particles. And we've discussed some of the means of getting around this, like using neutralized beams.

But another issue that will be a challenge at very long ranges is dealing with [[Jitter|jitter]]. A weapon made out of a linac will be very long, and so it will be rather wibbly. This could make its jitter problems worse. A synchrotron might not be so badly affected; and for any future hypothetical high energy wakefield accelerators, no one really knows what they'll be like. There are certainly tricks you can play to reduce jitter (see the previously linked jitter page), but at ranges approaching a light second it's certainly plausible that it will be difficult to land a hit.

== 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, Gerrit Bruhaug, and Luke Campbell for writing

Gerrit Bruhaug and Luke Campbell for scientific research, math wrangling and advise

ATriffleMind and Luke W. Campbell for proof reading.

==References==

[[Category:Beams]][[Category:Physics & Engineering‏‎]][[Category:Engineering‏‎]][[Category:Military Technology]][[Category:Warfare‏‎]]

Particle Beam Weapons

2026-10-28T00:53:06Z

Lwcamp: /* Plasma magnets */

{{PageConstructionNotice}}

''For a general overview on particle accelerator technology, see [[Particle Accelerators]]

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.

::: Z = 1.32×102 (γ2-1)3/4 r / √I<ref name=USPAS2016>Martin Reiser, "Theory and Design of Charged Particle Beams", 2008, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim</ref>

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

::: Z = β γ c r √[m/T]

Where Z is the doubling distance like before, r is the initial beam radius, T is the beam temperature in eV, m is the ion mass and β, c, and γ 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.

== Defenses ==

Sure, it's great to have a great big particle cannon to blast your enemies. But what if you are the one being shot at? In that case, it's best to figure out ways to keep from being blown to kingdom come.

=== Armor ===

The simplest way to defend against the high energy particles of a particle beam is to pile a whole bunch of stuff up and put it in the way of the beam. Particle beams may produce extremely penetrating radiation, but enough stuff will block most of that radiation and may allow you to survive. The details depend on what kind of particle is being thrown at you, and are explained below. However, a general rule of thumb is that you probably want about 10 tons of material per square meter of surface you want to armor in order to get sufficient protection.

==== Electrons ====

At the very high energy of an electron accelerated by a weapons-grade particle beam, the primary interaction with matter will be the emission of bremsstrahlung photons in the gamma-ray range as the electron slams into particles of matter. These high energy photons themselves primarily interact by the pair production mechanism – the photon disappears, but leaves behind a highly energetic positron and electron. These two particles are themselves energetic enough to produce further bremsstrahlung, and so the process continues. You get the formation of an electron-gamma shower (also called an electromagnetic cascade), which only stops after either the particles have penetrated through the target or the initial energy of the electron has been spread through so many shower particles (including all the electrons, positrons, and photons) that no particle has enough individual energy to pair produce or to be likely to bremsstrahlung again before it is stopped by other mechanisms.

To stop electrons, you'll want the atoms in your armor to be as heavy as you can practically get them. High atomic number both encourages bremsstrahlung and pair production, giving a double whammy for stopping that cascade. However, occasionally one of those electrons, positrons, or gammas will hit a nucleus and knock out neutrons. So you probably want a back-up layer of several tens of centimeters of hydrogen rich material such as plastic or water mixed with neutron absorbing materials like boron, lithium, cadmium, or gadolinium. High density polyethylene loaded with 5% boron by weight is a typical go-to neutron blocker.

The actual methods to calculate the penetration of the electromagnetic cascade is given are the box below. However, for some examples:
<ul>
<li> A 10 GeV beam incident on thick iron (or steel) will stop cascading at about 9 cm, will deposit half of its energy in the first 12 cm, and will have dumped 98% of its energy within 35 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 2.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 100 GeV beam incident on thick iron (or steel) will stop cascading at about 13 cm, will deposit half of its energy in the first 16 cm, and will have dumped 98% of its energy within 47 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 3.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick iron (or steel) will stop cascading at about 17 cm, will deposit half of its energy in the first 20 cm, and will have dumped 98% of its energy within 59 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 4.6 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick tungsten will stop cascading at about 3.5 cm, will deposit half of its energy in the first 4 cm, and will have dumped 98% of its energy within 12 cm. 90% of the cascade will be contained within about 0.93 cm and 95% of the cascade will be contained within 1.86 cm of the beam axis. You will get 98% protection with a mass thickness of 2.3 tons/m2 and an additional 1/e reduction for each further 0.13 tons/m2 beyond that.
<li> A 1 TeV beam incident on water will stop cascading at about 290 cm, will deposit half of its energy in the first 340 cm, and will have dumped 98% of its energy within 1030 cm. 90% of the cascade will be contained within about 11 cm and 95% of the cascade will be contained within 22 cm of the beam axis. You will get 98% protection with a mass thickness of 10.3 tons/m2 and an additional 1/e reduction for each further 0.72 tons/m2 beyond that.
</ul>
Note that for sufficiently intense particle beams, 98% protection may not be enough. You might need to throw on several more tons per square meter to be sure.

<table border=1>
<tr><td> How Far it Goes:
(This section follows methods presented in Leroy and Rancoita<ref>C. Leroy and P-G Rancoita, "Physics of cascading shower generation and propagation in matter: principles of high-energy, ultrahigh-energy and compensating calorimetry", Reports on Progress in Physics 63 (2000) 505–606. DOI: 10.1088/0034-4885/63/4/202</ref>.)

When trying to estimate the extent of the irradiated region of the target, it is helpful to know the radiation length X0 in the material. This is the distance that an electron will travel, on average, before emitting a bremsstrahlung photon. Conveniently, it is also the distance a high energy photon will travel through the material before pair producing. The radiation length can be found from
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   = 4 α  
<td nowrap="nowrap"> NA
<td rowspan="2" nowrap="nowrap" align="right">   Z (Z+ζ) re2 ln
<td rowspan="2" nowrap="nowrap" align="right"> [
<td nowrap="nowrap"> 183
<td rowspan="2" nowrap="nowrap" align="right"> ]
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> A
<td style="border-top:solid 1px black;"> Z1/3
</table>
</div>
where α = 1/137 is the fine structure constant, NA = 6.02214076×1023 is Avogadro's number (or one mole), A is the atomic weight of the atoms in the material (usually expressed in grams per mole), Z is the atomic number (nuclear charge) of the atoms, and re = 2.8179403205×10-13 cm is the classical electron radius. ζ is a constant of approximately 1.3. This gives the thickness of material in terms of the areal density (mass of material per unit area perpendicular to the beam) – if you use A in g/mol and re in cm the result will be in cm2/g and X0 will be in g/cm2. If your target is made up of more than one kind of atom, you can get the total radiation length by finding the radiation length of each atom of type i, X0i and using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   =  
<td rowspan="2" nowrap="nowrap" align="right"> Σi  
<td nowrap="nowrap"> fi
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> X0i
</table>
</div>
where fi is the fraction by weight of the atom type i. Once you have the radiation length in terms of areal density, you can easily convert it to distance by dividing by the density ρ. If X0i is in g/cm2 and density is in g/cm3, then X0i/ρ will be in cm.

We will also need to know the critical energy εc. This is the energy where an electron is losing energy as rapidly to ionization as it is to radiation. That is, it is the energy at which the cascade stops multiplying, even though the individual particles already made can continue traveling for some distance. The critical energy can be estimated using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> εc = B  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> Z X0
<td rowspan="2" nowrap="nowrap" align="right"> )
<td nowrap="nowrap"> h
<tr>
<td style="border-top:solid 1px black;"> A
<td>  
</table>
</div>
where B = 2.66 MeV and h = 1.11.

We can now describe the distribution of radiation in the target material. The depth tmax where the cascade stops multiplying (in g/cm2) for an incident electron of energy E (in MeV) is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tmax = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> - 1
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
Similarly, the depth where half the energy of the beam has been deposited (in g/cm2) is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tcg = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> + 0.37
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
As usual, you can convert to a depth in cm by dividing by the material density. 98% of the beam energy will be deposited within a distance of &cong; 3 tcg. Beyond this, the beam energy will continue to fall off with thickness t approximately as an exponential exp[-t/λ], where the characteristic attenuation length λ &cong; 2 X0.

The width of a cascade, where 90% of the energy is contained, is approximately given by
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> RM =  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> EM
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
where EM &cong; 21.2 MeV. 95% of the radiation will be contained within a distance of 2 RM from the beam axis.
</table>

==== Protons and Ions ====

<table align=right><tr><td>
[[File:Regolith_Shielding.png|350 px|frameless]]
<tr><td width=350>
Relative effect of particle radiation (compared to no shielding) from galactic cosmic rays behind different thicknesses of water, aluminum, and lunar regolith<ref name="Slaba2022">Tony C. Slaba, "Radiation Shielding for Lunar Missions: Regolith Considerations", LSIC Crosstalk 7/18/2022 https://lsic.jhuapl.edu/uploadedDocs/focus-files/1604-E&C%20+%20EE%20Monthly%20Meeting%20-%202022%2007%20July_Presentation%20-%20NASA%20Slaba.pdf</ref>.
</table>
Protons and ions are grouped together because ions are made of protons and neutrons and when you accelerate these things for a weapon it is the protons and neutrons that are going to be doing the heavy lifting for blasting your target. This is because protons and neutrons are both about 2000 times more massive than electrons so any electrons along for the ride will contribute negligibly to the energy. Protons and neutrons together are called nuclear particles, and both behave quite similarly in nuclear collisions.

Nuclear particles are themselves made of quarks and can interact via the strong nuclear force. At the energies appropriate for weapons use, essentially all of the way protons or ions are blocked is through [[Particle_Accelerators#Nuclear_collisions|direct collisions with the atomic nuclei]] of the armor material. One or more quarks in the projectile will slam into quarks in the target nucleus and stretch them away from the rest of the quarks in the nuclear particles. Physics tells us that you can't have isolated quarks, so the "string" attaching the quark to its neighbors stretches and stretches until it snaps. Snapping releases so much energy that more quarks are made that clump together into all sorts of other nuclear particles, both exotic and familiar. This process is called a hadronic jet. The only particles produced in these jets long-lived enough to travel any noticeable distance through the armor material will be protons, neutrons, the charged pi mesons (or pions), and some of the K mesons (or kaons). The particles too short lived to go far enough to matter will decay into other particles, and these particles are often very high energy gamma rays. Much like with electrons, the long-lived secondary particles can be sufficiently energetic to smash into other nuclei and cause more hadronic jets (or, if they are low enough in energy, just knock more nuclear particles out of target nuclei) and this produces a similar shower or cascade. But in addition, the armor nuclei that were hit will be very hot after the impact from all of the energy given to them from the incident particles. These nuclei will cool off by evaporation – but instead of evaporating away molecules, they evaporate off nuclear particles from the nucleus. These evaporated particles are mostly neutrons, so any armor against proton or ion beam strikes will produce quite a bit of neutron radiation.

Eventually, the original particle's energy will be distributed among so many other particles that the charged particles will come to a stop by making ion tracks rather than from colliding with more nuclei, and the cascade halts. But you still have all the neutrons to worry about. And the gamma rays as well, which are high enough energy to kick off an electromagnetic cascade in the same way that electrons do.

Unlike electrons, which have reasonable approximations of their cascade behavior from relatively simple equations, the cascades of protons and ions usually require sophisticated simulations or experiments to get the resulting distribution of radiation. However, we can get a reasonable idea of their general behavior by comparing them to [[Space_Radiation#Shielding_Against_Cosmic_Rays|cosmic ray particles interacting with shielding]]. Cosmic rays are energetic protons and ions, after all. From this we see that a relatively small amount of armor – in the range of 1 to 3 tons per square meter – can actually make the radiation dose worse. The incident particles will knock off so many secondary particles in their shower that the dose is multiplied from so many tracks passing through the target. Hydrogen rich materials such as water and plastics work the best against proton and ion beam particles; heavy elements are the worst and produce the most secondaries. You could probably obtain adequate radiation protection from most incident beams with about 10 tons per square meter of hydrogen rich material as armor. However, you will want to add neutron absorbing elements to your hydrogen rich armor, such as lithium, boron, cadmium, or gadolinium. Further, in order to deal with the gamma rays you will want a layer of at least several centimeters of heavy elements, such as tungsten or lead, behind the hydrogen rich particle armor.

==== Structural Considerations ====

When your armor material absorbs a radiation beam, the energy of that radiation turns into heat inside the armor where the radiation was absorbed. If the beam pulse has enough energy as was sufficiently tightly focused, this heat can damage your armor. Your armor may melt in places, or thermal expansion could cause it to crack or warp. Perhaps the worst thing it could do is vaporize or flash into a plasma, because the pressure from the vapor or plasma can be sufficient to plastically deform the armor. This leaves a hole along the beam path where the expanding gas or plasma blasted out a long, deep crater. This means a beam that fires rapid shots spaced maybe milliseconds apart can have subsequent shots go down the same hole so your armor will not protect against them. Each shot will dig out a deeper hole until either your armor is penetrated or the attacking beam runs out of shots in its pulse train.

A consequence of this is that when designing your armor you may wish to use refractory materials with a high structural strength (like tungsten), or that is otherwise resistant to being blasted apart by its own evaporated remains.

=== Magnetic fields ===

[[Particle_Accelerators#Magnetic_fields|A charged particle crossing a magnetic field will bend so as to spiral around the field lines]]. If you try to shoot your particle beam through a magnetic field, it will bend away from the direction you shot it. This can make it hard to hit your target. Unfortunately for you, there are magnetic fields everywhere – around planets, and even trapped in the solar wind as it crosses the solar system.

As an example, the Earth has a magnetic field strength of about 5×10-5 tesla near its surface, and falling off farther away. If you are shooting a 1 TeV electron beam past the Earth, your beam is likely to bend by several milliradians (with the exact deflection depending on how close you are skimming the Earth). If you are in a medium Earth orbit and shooting at another target also in a medium Earth orbit on a trajectory that goes near the Earth, then without correcting for the magnetic field your beam will likely be off-target by tens to hundreds of kilometers.

For another example, the interplanetary magnetic field is about 10-9 tesla at Earth's distance from the sun. If you shoot that same 1 TeV electron beam at a target 0.1 light seconds away, you are likely to be off by a couple hundred meters.

<table border=1>
<tr><td>
The radius of curvature of the bend of a particle in a magnetic field is called the gyroradius rc. For a particle with a charge q, mass m, and kinetic energy E, find the relativistic momentum p
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
p = √[E2 - m2 c4] / c
</div>
To convert from energy units of eV to joules, multiply by 1.60×10-19. An electron has a mass of 9.11×10-31 kg, a proton of 1.67×10-27 kg, and an ion of 1.66×10-27×A kg, where A is the atomic weight of the ion. The speed of light c = 3.00×108 m/s.

In a magnetic field of strength B with an angle θ between the field direction and the direction of motion
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rc = |p sin(θ) / (q B)|
</div>
For electrons, q = -e, where the fundamental charge e = 1.60×10-19 coulomb. For a proton, q = +e. For an ion, multiply e by the charge state of the ion.

If the field acts over a distance d that is much smaller than the gyroradius, the deflection angle φ is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = d / rc
</div>
The deflection x at a target at a range R for small φ (φ &Lt; 1) will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
x = φ R.
</div>

Note that d will always be less than or equal to x. Also note that if d is not smaller than twice the gyroradius the beam will never reach the target because it will completely spiral around the field lines before it reaches the target.
</table>

So, it is clear that as the defender it will be helpful to hang out in areas with high magnetic fields, and keep some distance between you and the attacker. Meanwhile, the attacker will benefit from being closer where small deflections in the beam trajectory don't translate into large excursions from where the beam is aimed (and also because there will be less space filled with fields between the two of you). But the attacker can also try to compensate for known fields. If you happen to know that the magnetic fields are between you and your target, you can adjust your fire so that the deflection of the field bends your beam onto the target. This is not a simple task. It is easy enough to measure the field strength and direction where you are, but predicting how it changes over the path of your shot does not have an obvious solution. You can also try to gain intelligence on likely battle sites before you ever fight on them – for example, getting good data on the structure of the fields of a planet using spies and technical intelligence during peacetime (heck, on Earth this information is published for free in academic journals and by government agencies such as NOAA). Planetary magnetospheres are also affected by the solar wind, so different solar weather will change the fields you have to shoot through. But space weather forecasts and predictive simulations of the solar wind interacting with a planetary magnetosphere can give you a good idea of what the field structure is like. But even with all that, there are changes in the fields that happen on a scale of around a second ... anyone who has ever seen the auroras dancing across the sky in the polar night can see this. Compensating for these changes will be a significant challenge.

What would really help is being able to see the path of your own beam. This would allow you to see how far off you were from your target and in what direction, and then adjust fire to walk your shots onto the target. Unfortunately, particle beams in space are invisible. There might be ways to track them, but if so we don't know what they are.

==== Synchrotron radiation ====

When charged particle trajectories are bent, they emit [[Particle_Accelerators#Synchrotron_radiation|synchrotron radiation]]. This can cause the particle beam to lose energy, but if the bend is too tight it can also generate high energy gamma rays that ignore magnetic fields and directly strike the target. Synchrotron radiation can generally be ignored for proton or ion beams, but can be significant for electron beams.

<table border=1><tr><td>
For a beam particle with energy E and mass m, the synchrotron radiation is concentrated in a forward-directed cone with an opening angle of approximately 1/γ, where γ = 1 + E/(m c2). (Interestingly, the radiation intensity peaks an at angle of 1/(2 γ) and vanishes directly on-axis.) Because the beam is bending, you will get emission along the plane along which the beam is being bent. The energy of the photons produced during synchrotron emission is approximately<ref name="Jackson_Electrodynamics">J. D. Jackson, "Classical Electrodynamics: Second Edition", Wiley (1962)</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Eγ &cong; 3 γ3 c &hbar; / rc
</div>
where &hbar; = 1.05×10-34 J s (if you want the energy in joules) or &hbar; = 6.58×10-16 eV s (if you want the energy in eV). The photon energy will not exceed the energy of the incident particles, if you get a photon energy higher than the energy of the incident particles it just means that the synchrotron radiation spectrum extends up to the incident particle energy.

The fraction of the energy (or power) of the beam turned into synchrotron radiation will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = [1/(4 π ε0)] [4 π / 3] [q² / rc] β2 γ4 [φ / (2 π)] / E
</div>
where β = √[1 - 1/γ2] is the particle's speed measured as a fraction of the speed of light, and 1/(4 π ε0) = 8.99×109 is the Coulomb constant. For highly relativistic electrons (like all practical electron beam weapons emit), β &cong; 1 and we can use
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = 8.85×10-2 [E (GeV)3 [φ / (2 π)] / (rc (m))]
</div>
This fraction cannot be more than 1. If f gets close to one, it means that our original assumption that the beam particles mostly keep their original speed no longer holds, and the beam will lose most or almost all of its energy. Instead, when the beam is being deflected it will only make it through an angle of φ/f and will irradiate a region of length x/f starting at where the beam would have been incident if not for the magnets and extending in the direction the beam was deflected.
</table>

For example, a 1 TeV electron beam will have γ &cong; 2,000,000. So if it's bent by a magnetic field the synchtrotron radiation will spread out in a cone of about 5×10-7 radians. If this beam was shooting at a target 0.1 light seconds away from where it was bent, the radiation would arrive at the target over a region with a radius of about 15 meters with a maximum intensity at about 7.5 meters and no radiation directly along the path the beam was pointing while it was being bent. If the beam was bent by the interplanetary magnetic field from our earlier example, it would produce light with photon energies of around 1.3 eV (near infrared) and the lost energy would be a few parts in a hundred trillion. If it was instead deflected by the Earth's magnetic field you might expect a peak photon energy of tens of keV – in the hard x-ray part of the spectrum – and the fraction of energy in the synchrotron radiation would be somewhere in the low parts in 100,000 or less.

==== Separation of neutral beams ====

Magnetic fields only affect charged objects. So one way to overcome magnetic defenses or environments is to use beams that have no net charge. For example, a proton beam could be neutralized with a co-propagating electron beam with an equal number of protons and electrons. Or, potentially, you could fire an ion beam and then neutralize the ions after acceleration to get a beam of neutral atoms.

But, the magnetic fields are still tugging at the charged particles making up the beam as they go past. This will pull the electrons one way and the protons or ions the other way. As the particles are pulled in opposite directions you get a net positive charge on one side of the beam and a net negative charge on the other side (called polarization). This sets up an electric field inside the beam that acts to pull the particles back together. If the particles are going fast enough and the magnetic field is strong enough, the magnetic force can still separate the charges and will split the beam in two, one with the positive charges and one with the negative charges. But if the particles are not that fast or the magnetic field is not that strong this polarization of the beam will keep it together and let it fly right through the magnetic field.

<table border=1><tr><td>
Let's figure out how to analyze this problem. For definiteness, we will have a beam with an electric current of the positive species of I and an electric current of electrons of -I, all moving in the z direction. The beam radius will be R. The particle speed will be v with the usual relativistic parameters β = v/c and γ = √[1/(1-β2)]. If you know the energy per particle E of the positive species and the mass m of the positive species, γ = 1 + E/(m c2) and then β = √[1 - 1/γ2].

Let's assume that the beam is going through a magnetic field of strength B at an angle θ to the field direction. The quantity B sin(θ) is the component of the magnetic field perpendicular to the direction of the beam.

The charge density ρ of the positive component of the beam will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρL = I / (π v R2).
</div>

It is easiest to analyze the problem in the rest frame of the particles in the beam. Now the particles are at rest. Particles at rest are not affected by a magnetic field, but the change of reference frame Lorentz transforms the initial magnetic field into a combination of electric and magnetic fields in the rest frame. We can ignore the magnetic field, but the electric field is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔A = γ v B sin(θ)
</div>
in a direction perpendicular to both v and B. For our purposes, assume the electric field points in the x direction.

At the same time, relativistic length contraction increases the charge density in the rest frame
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρ = γ ρL.
</div>

Let r be the vector from the beam axis to the point inside the beam. It will be useful to separate this into its Cartesian coordinates. If x and y are the unit vectors in the x and y directions, respectively, then
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
r = x x + y y
</div>

If we consider only the positive component of the beam, from [https://en.wikipedia.org/wiki/Gauss%27s_law Gauss's law] it will have an electric field inside of the beam at a distance r from the center axis of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔+ = ρ r / (2 ε0) = (ρ/(2 ε0)) (x x + y y)
</div>
where ε0 = 8.85×10-12 is the vacuum permittivity. Now let's displace the electrons in the beam by a distance δ in the x direction. By itself, this electron beam will have an electric field of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔- = -(ρ/(2 ε0)) ((x-δ) x + y y)
</div>
By the principle of superposition, the electric field inside the beam with both components is the sum of the electric fields of the separate components including the external field
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔B = [(ρ/(2 ε0)) δ + γ v B sin(θ)]x
</div>
Because the charges are free to move, they will do so as long as a field is applied to them. Thus, the beam becomes stable once the charges separate enough that the field internal to the beam is zero. Therefore
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
δ = -2 ε0 γ v B sin(θ) / ρ = -2 π ε0 v2 R2 B sin(θ) / I
</div>
If |δ| > 2 |R|, the beam will fully separate and each component will go its own way. If δ is small compared to R, the beam's polarization suffices to hold it together in the overlap region. However, where the positive and negative charges do not overlap the electric fields no longer cancel, and the external electric field carries away the outer polarized part of the beam. This gives us a rough threshold for which a magnetic field will quickly separate a neutral beam into its charged components and bend them away from their target
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr><td> B sin(θ) < I / (π ε0 v2 R) <td>       <td> Beam is partially eroded.
<tr><td> B sin(θ) ≥ I / (π ε0 v2 R) <td> <td> Beam promptly separates.
</table>
</div>
For a partially eroding beam, the polarized region that was peeled off by the field still acts to screen the remaining neutral portion, at least to within this approximation, leaving the still-neutral beam to hit the target.
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Neutralized_beam_in_B_field.png|5140 px|frameless]]
<tr><td width=700>
A profile of the beam charge distribution of a neutralized beam in a magnetic field. Negative charge density is shown in yellow, positive charge density in cyan, with neutral charge density in light green. The magnetic field is vertical, the electric field in the beam reference frame is shown in magenta including both the polarization field and the Lorentz transformed electric field resulting from the magnetic field. As positive charges are pulled in the direction of the field and negative charges opposite to the field direction, the polarized boundary will be pulled off the beam by the electric field even though the electric field is zero in nearly all the beam interior. This expanding polarization layer will continue to largely screen the still-neutral portion, however; but the extent to which the screening is only partial would require more sophisticated analysis methods than used here. The material removed by the field will expand to a maximum of the deflection distance of a uniform charge based on the magnetic field by the time it reaches the target.
</table>
</div>
<tr><td>
For a beam of neutral atoms, let a be the radius of the atom and let Eb be the binding energy of the least tightly bound electron to the atom. An external electric field in excess of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
|𝓔A| > Eb / (e a)
</div>
will promptly ionize the atom, ripping away its electron. Here, e = 1.60×10-19 C is the magnitude of the electric charge of the electron. For fields close to but a bit weaker than this, the electron can still rapidly pop off due to quantum tunneling<ref>Ph. Durand and I. Paidarová, “Ionization of the hydrogen atom: from weak to strong static electric fields,” The European Physical Journal D, Vol. 26, pg. 253-259, 2003.</ref>. In general, expect that for fields greater in magnitude than (very approximately) Eb / (10 e a) the electron will be quickly removed, while for fields weaker than this the electrons will stay on their atoms.

It is useful to note that Eb/e is just the binding energy in units of eV. The most weakly bound electrons are usually bound to their atom with about 10 eV of energy. Atoms are typically about 10-10 m in radius. Thus, you can expect fields of around 1010 to 1011 V/m to directly ionize atoms. If we call this threshold value 𝓔T, then a beam will be ionized when passing through a field with strength greater than
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B sin(θ) > 𝓔T / (γ v)
</div>
</table>

Example: Let's consider a 1 GeV neutralized proton beam from a synchrotron, with a beam current of 200 kA in each bunch (incidentally, if a bunch is 5 cm long this means that each bunch packs 38 kJ worth of energy). The speed of the bunch is 87% the speed of light, or 262 million m/s, with a γ of 2.07. Consider when the beam's focus takes it to a radius of 5 cm (or 0.05 m, for our calculation). Plugging this in, we find that the protons in the bunch will separate from the electrons in a field with a transverse component greater than about 2.1 T, and the bunch will pass through the field with only minimal losses if the transverse component is much smaller than this.

Example: We'll take the same 1 GeV beam, but this time recombine the electrons with protons to make hydrogen atoms. An electron is bound to a hydrogen atom with an energy of 13.6 eV (the Ryberg constant) and the radius of the electron orbital in a hydrogen atom is 0.529×10-10 m (the Bohr radius). This gives a threshold electric field for prompt ionization of hydrogen of 2.57×1011 V/m. We will get this with a transverse component of the magnetic field of 474 T. Above this field the atoms promptly ionize and electrons are stripped away from the protons. For fields higher than roughly 1/10 of this – so about 40 to 50 T – you may still get significant ionization by quantum tunneling; lower than this your losses from the beam due to field-induced ionization are likely negligible.

==== Magnetic shields ====

What happens when we can't rely on magnetic fields in the environment to protect us? One idea is to produce powerful artificial magnetic fields around our spacecraft to deflect incoming particle beams. This is very similar in concept to using active magnetic shielding to protect against space radiation – [[Space_Radiation#Magnetic_Shielding|check out our article on space radiation for more]].
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr>
<td width=100>
[[File:Unconfined_FRC_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:racetrack_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:Magnetic_shielding_Halback_Array.png|200 px|frameless]]
<tr><td colspan=3>
Some potential configurations for magnetic shields. The currents generating the fields are in green, the magnetic fields themselves are shown in magenta<ref name="Battiston2012">R. Battiston, W.J. Burger, V. Calvelli, R. Musenich, V. Choutko, V.I. Datskov, A. Della Torre, F. Venditti,
C. Gargiulo, G. Laurenti, S. Lucidi, S. Harrison, and R. Meinke, "ARSSEM Active Radiation Shield for Space Exploration Missions", Final Report ESTEC Contract N° 4200023087/10/NL/AF : “Superconductive Magnet for Radiation Shielding of Human Spacecraft” (2012) https://arxiv.org/abs/1209.1907 https://www.researchgate.net/publication/265945847_Active_Radiation_Shield_for_Space_Exploration_Missions</ref><ref name="Desiati2022">Paolo Desiati and Elena D'Onghia, "CREW HaT: A Magnetic Shielding System for Space Habitats", arXiv:2209.13624 [physics.space-ph] https://doi.org/10.48550/arXiv.2209.13624</ref>.
</table>
</div>

One immediate issue is that you want to keep very high magnetic fields out of the crew area. Fields in excess of 0.5 T can start to be unpleasant, and it is recommended to keep the field below 0.2 T for continuous exposure.

Strong magnets can suffer from high self-forces acting to tear them apart. Most shielding designs will have multiple magnetic coils, which will exert high forces and torques on each other. This will result in the need for strong structural bracing to keep the magnets intact and in place. Such strong structural designs will impose significant extra mass on the spacecraft.

Consider a magnetic field configuration that can keep radiation out of a certain region of space. There is no trajectory of a charged radiation particle up to a certain energy that will allow those particles to enter the protected region. But the dynamics of charged particles in a magnetic field are the same whether going forward or backward in time. If there is no way for a particle to get in, by this time reversal symmetry any particle which is in that forbidden region can never escape. Consequently any field configuration that is a magnetic trap is also a magnetic shield; any field configuration that is a magnetic shield is also a magnetic trap. A spacecraft with a magnetic shield may end up surrounded by strong radiation belts.

Using a magnet to bend a weaponized electron beam very often ends up with the beam being bent so hard that it dumps most of its energy in the form of synchrotron radiation at gamma ray energies. Because gamma rays are not charged, they will not be deflected by the field. Thus, if you would have been hit by the electron beam you will be hit instead by the gamma rays.

As an example, consider a 1 TeV electron beam. We'll surround our spacecraft with a 10 T magnetic field in a 100 meter radius region just outside our spacecraft. This will deflect the beam by 30 meters, which is enough to make it miss. But we are not saved, because the beam dumps nearly all of its energy into synchrotron radiation with an energy spectrum of gamma rays up to 13 GeV. This radiation will hit us in a strip about 2.4 meters long and not much more than the width of the original beam with essentially all of the energy of the original beam.

==== Magnet drones ====

If a magnetic shield around your spacecraft will just end up killing you from synchrotron radiation, what if you put the magnetic shield on a drone and put that drone between yourself and the enemy trying to shoot you? This lets you use a much weaker magnet because you don't need nearly so high of a deflection angle to make the shot miss. This means that less beam energy will turn into synchrotron radiation that could kill you instead and will spread what synchrotron radiation does occur out over a much wider area when it reaches your spacecraft, hopefully so that most of it just misses your spacecraft.

It does mean that your enemy will start shooting your drones so that they can eventually take a shot at you. But hopefully by that time you've already cleared out their drones and cored their spacecraft with your own synchrocannon or UREB.

==== Plasma magnets ====

Usually, to make a really big magnetic field you would run current through a really big conductive loop. But really big things are heavy and expensive. One way around this could be to use the ambient space plasma to carry the electric current rather than any kind of wire. In order to make this current, you use the same mechanism used in a rotary induction electric motor: you generate a rotating magnetic field and induce a current in the conductive loop (in this case, the space plasma). How do you make a rotating magnetic field? The method proposed<ref name="Slough2006">John Slough, "The Plasma Magnet", NASA Institute for Advanced Concepts Phase II Final Report (2006) https://www.niac.usra.edu/files/studies/final_report/917Slough.pdf</ref> is to use an antenna of four much smaller current loops as electromagnets at 90° to each other, and sinusoidally vary the current through the loops with a respective 90° phase difference between them. When this is done, a circular current is induced in the space plasma and the self-force of a magnetic field on its generating current inflates the current loop to much larger radii. You end up with a current sheet in the plane of the rotating field's rotation circulating about the antenna nearly synchronously with the field rotation. This current then creates a normal dipolar magnetic field as with any current loop. The current sheet and the resulting magnetic field expand until the magnetic field's pressure equals that of the ambient space plasma. This gives an enormous region of magnetic field from a relatively small and lightweight system.

The system proposed in the initial study would use an antenna approximately 10 meters in radius and produce a region of field approximately 30 km in radius at Earth's orbit. The field inside would not be constant - at the edges it would be 50 nT, giving the right pressure to keep out the solar wind at Earth's orbit. A circulating current sheet produces a field that falls off as 1/r with distance, meaning that adjacent to the antenna the field would be about 0.15 mT. Using the methods in the box below, we can estimate that this particular design as part of a spacecraft's shields will not deflect a TeV electron beam by more than a few centimeters, which is usually insufficient to force a miss. However, it could potentially be useful for a drone magnet, where the much larger distance between the magnetic field and the target spacecraft would allow a small deflection angle to give perhaps tens or low hundreds of meters change in beam location near the target at near light second ranges. The relatively low field and gradual bend of the incident beam suggests that synchrotron radiation will not be an issue; however this does not exclude more powerful plasma magnet designs causing significant synchrotron radiation effects.

<table border=1><tr><td>
Calculating the deflection from a spatially varying magnetic field affected by the solar wind is no trivial task. However, we can make an approximation that both should be within about an order of magnitude of the true value and also represent an upper limit to the deflection angle; although it does neglect the influence of the driving rotating magnetic field which could in some cases increase the deflection.

We will take R0 as the radius of the antenna. B0 is the field strength at the antenna. Rm is the distance at which the magnetic influence stops because the magnetic pressure equals the space plasma pressure. Bm is the field strength at Rm. According to Slough<ref name="Slough2006"></ref>, the magnetic field strength B(r) at any distance from the antenna center r between R0 and Rm is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B(r) = B0 R0 / r.
</div>
Therefore, once we know Bm and Rm, B0 is completely determined
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B0 = Bm Rm / R0.
</div>

The force on a particle with charge q passing perpendicularly through a magnetic field B(r) at a speed v is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
F(r) = q v B(r)
</div>
For an electron beam, v is so close to c as to make no difference. If the particle is moving down the y direction and the field is in the z direction, the force will be in the x direction. We assume that the deflection will be small, so we can choose a straight-line path in the y direction offset from the center by an amount x. The plasma bubble radius is Rm, so the particle will enter at ymax = √[Rm2-x2] and leave at ymin = -√[Rm2-x2].

The momentum change the particle experiences from this force is the time integral of the force over its path. Because of reflection symmetry about y = 0, we only need to integrate from ymax to 0 and then multiply by 2
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] dt F(r)
<tr> <td> <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] (dy/v) F(r)
<tr> <td> <td> = <td align="left">2 q B(0) R0 [Integral from 0 to √[Rm2-x2]] dy 1/√[y2 + x2]
<tr> <td> <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/x]
</table>
</div>
This holds for |x| > R0. But the expression for the field is not valid for r < R0 and the integral will give nonsense and diverging results if we evaluate for |x| < R0. In this case, we can put the integral lower bound at r = R0, or y = √[R02-x2]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/(√[R02-x2] + R0)]
</table>
</div>

The deflection angle is now estimated as
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = px / p
</div>
where the total momentum of the particle p can be found from the methods in the first box in the Magnetic Fields section.

The deflection for a shot aimed right at the spacecraft at the position of the spacecraft can be estimated by setting x = 0 and only taking the first half of the integral (so ignore the factor of 2)
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
q B(0) R0 Rm ln[Rm/R0] / p
</div>
This is an additional overestimate because it fails to account for how the deflection builds up over distance during the integration, but it should still give something of an order-of-magnitude and upper limit estimate of what kind of deflections might be expected.
</table>

=== Electric fields ===

If electric fields can accelerate charged particles, maybe they can keep the charged particles in a weaponized particle beam away from your spacecraft? To completely block a particle beam, you would need a voltage on your spacecraft equal to the particle energy in eV of the particles in the beam. That is, if a 1 GeV proton beam is coming at you, you'd need an electric potential of a billion volts to stop that beam. This leads to three immediate problems.

The first is that particles of the opposite charge will be attracted to your spacecraft. Because space isn't really empty but rather a rarefied plasma (called the solar wind in a solar system or the interstellar medium out between stars), there will be plenty of particles of the opposite charge around to attract. And they will hit with an energy in eV equal to the voltage on your spacecraft. In our example of defending against a 1 GeV proton beam with a gigavolt field, the electrons from the solar wind will come in and hit with an energy of 1 GeV each. This will generate dangerous radiation.

The second problem is a consequence of the first. Each of those GeV electrons from our example need to get their energy from somewhere, which turns out to be from the electric field we are using as shielding. You will need to make up for this energy loss with power from your electrical system, which gives a constant drain on your power. Some estimates put the electrical effort to maintain a several hundred megavolt shield in space at many tens of gigawatts<ref name="Mechmann2019">Claire Mechmann, "Analysis of Proposed Active Radiation Shielding Design Concept for Spacecraft" (2019) Thesis, College of Engineering and Science of Florida Institute of Technology</ref>.

And the third problem is similar to one for magnetic fields. Your electrodes will experience high self forces and forces between separate electrodes that will try to tear your shield generator apart. Consequently you will need heavy structural reinforcement in order for your shield to work.

There are potential ways to mitigate some of these problems. Interested readers are directed to [[Space_Radiation#Electrostatic_Shielding|the active shielding section in our page on space radiation]] for more of the juicy details; there's no need to repeat the same stuff here.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Elctrostatic_active_shielding.png|200 px|frameless]]
<td>
[[File:Electrostatic_active_shielding_2.png|200 px|frameless]]
<tr><td colspan=2>
Proposed designs for electrostatic shields to protect from positively charged particles<ref name="Tripathi2006">Ram K. Tripathi, John W. Wilson, and Robert C. Youngquist, "Electrostatic Active Radiation Shielding - Revisited", 2006 IEEE Aerospace Conference, Big Sky, MT, USA, 2006, pp. 9 pp.-, doi: 10.1109/AERO.2006.1655760.</ref><ref name="Chowdhury2023">Rajarshi Pal Chowdhury, Luke A. Stegeman, Matthew L. Lund, Dan Fry, Stojan Madzunkov, and Amir A. Bahadori, "Hybrid methods of radiation shielding against deep-space radiation", Life Sciences in Space Research, Volume 38, 2023, Pages 67-78, ISSN 2214-5524, https://doi.org/10.1016/j.lssr.2023.04.004.</ref>.
</table>
</div>

== Damage ==

Particle beams are, first and foremost, devices that produce beams of [[Nuclear_radiation|ionizing radiation]]. The radiation of weaponizable particle beams is extremely penetrating; thus of the [[Particle_Accelerators#Energy_loss_in_matter|interaction mechanisms listed for radiation from particle beams]], we mostly need to worry about bremsstrahlung and nuclear collisions and these produce extensive secondary showers of radiation that increase the penetration of the radiation exposure. Because of this, any target hit by a particle beam weapon will be irradiated, and radiation dose will always be one of the primary damage mechanisms of a particle beam weapon strike. Further, this radiation will be distributed along a track passing deeply through the material of the target, allowing it to affect critical systems through heavy armor, structural material, robust machine components, and passive material such as fuel load, coolant, or cargo. As the radiation deposits its energy into the target material, that material will be heated. If the target is heated sufficiently, it may experience [[Beam-Target_Interactions|thermal or mechanical]] damage from melting, cooking, combustion, vapor explosions, and warping or cracking due to thermal expansion.

The discussion of the radiation penetration into armor also applies to the same radiation penetrating through the target. So the formulas and rules of thumb for estimating the depth of radiation penetration into armor can also be used to estimate how far into a tank or spacecraft a particle beam strike delivers effects.

In addition to the radiation and thermal effects, a particle beam can also produce a sharp electromagnetic pulse that can damage semiconductor electronics.

== Hitting Your Target ==

It doesn't do much to have the most horrifically powerful death ray if it can't hit. What are the challenges for directing your beam of radioactive fury onto your enemies?

We have already mentioned the [[Particle_Beam_Weapons#Magnetic_fields|challenges of stray magnetic field]] that are part of the ambient environment messing with your targeting. This will be a challenge when you are launching beams of charged particles. And we've discussed some of the means of getting around this, like using neutralized beams.

But another issue that will be a challenge at very long ranges is dealing with [[Jitter|jitter]]. A weapon made out of a linac will be very long, and so it will be rather wibbly. This could make its jitter problems worse. A synchrotron might not be so badly affected; and for any future hypothetical high energy wakefield accelerators, no one really knows what they'll be like. There are certainly tricks you can play to reduce jitter (see the previously linked jitter page), but at ranges approaching a light second it's certainly plausible that it will be difficult to land a hit.

== 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, Gerrit Bruhaug, and Luke Campbell for writing

Gerrit Bruhaug and Luke Campbell for scientific research, math wrangling and advise

ATriffleMind and Luke W. Campbell for proof reading.

==References==

[[Category:Beams]][[Category:Physics & Engineering‏‎]][[Category:Engineering‏‎]][[Category:Military Technology]][[Category:Warfare‏‎]]

Particle Beam Weapons

2026-10-27T23:57:00Z

Lwcamp: /* Hitting Your Target */

{{PageConstructionNotice}}

''For a general overview on particle accelerator technology, see [[Particle Accelerators]]

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.

::: Z = 1.32×102 (γ2-1)3/4 r / √I<ref name=USPAS2016>Martin Reiser, "Theory and Design of Charged Particle Beams", 2008, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim</ref>

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

::: Z = β γ c r √[m/T]

Where Z is the doubling distance like before, r is the initial beam radius, T is the beam temperature in eV, m is the ion mass and β, c, and γ 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.

== Defenses ==

Sure, it's great to have a great big particle cannon to blast your enemies. But what if you are the one being shot at? In that case, it's best to figure out ways to keep from being blown to kingdom come.

=== Armor ===

The simplest way to defend against the high energy particles of a particle beam is to pile a whole bunch of stuff up and put it in the way of the beam. Particle beams may produce extremely penetrating radiation, but enough stuff will block most of that radiation and may allow you to survive. The details depend on what kind of particle is being thrown at you, and are explained below. However, a general rule of thumb is that you probably want about 10 tons of material per square meter of surface you want to armor in order to get sufficient protection.

==== Electrons ====

At the very high energy of an electron accelerated by a weapons-grade particle beam, the primary interaction with matter will be the emission of bremsstrahlung photons in the gamma-ray range as the electron slams into particles of matter. These high energy photons themselves primarily interact by the pair production mechanism – the photon disappears, but leaves behind a highly energetic positron and electron. These two particles are themselves energetic enough to produce further bremsstrahlung, and so the process continues. You get the formation of an electron-gamma shower (also called an electromagnetic cascade), which only stops after either the particles have penetrated through the target or the initial energy of the electron has been spread through so many shower particles (including all the electrons, positrons, and photons) that no particle has enough individual energy to pair produce or to be likely to bremsstrahlung again before it is stopped by other mechanisms.

To stop electrons, you'll want the atoms in your armor to be as heavy as you can practically get them. High atomic number both encourages bremsstrahlung and pair production, giving a double whammy for stopping that cascade. However, occasionally one of those electrons, positrons, or gammas will hit a nucleus and knock out neutrons. So you probably want a back-up layer of several tens of centimeters of hydrogen rich material such as plastic or water mixed with neutron absorbing materials like boron, lithium, cadmium, or gadolinium. High density polyethylene loaded with 5% boron by weight is a typical go-to neutron blocker.

The actual methods to calculate the penetration of the electromagnetic cascade is given are the box below. However, for some examples:
<ul>
<li> A 10 GeV beam incident on thick iron (or steel) will stop cascading at about 9 cm, will deposit half of its energy in the first 12 cm, and will have dumped 98% of its energy within 35 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 2.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 100 GeV beam incident on thick iron (or steel) will stop cascading at about 13 cm, will deposit half of its energy in the first 16 cm, and will have dumped 98% of its energy within 47 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 3.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick iron (or steel) will stop cascading at about 17 cm, will deposit half of its energy in the first 20 cm, and will have dumped 98% of its energy within 59 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 4.6 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick tungsten will stop cascading at about 3.5 cm, will deposit half of its energy in the first 4 cm, and will have dumped 98% of its energy within 12 cm. 90% of the cascade will be contained within about 0.93 cm and 95% of the cascade will be contained within 1.86 cm of the beam axis. You will get 98% protection with a mass thickness of 2.3 tons/m2 and an additional 1/e reduction for each further 0.13 tons/m2 beyond that.
<li> A 1 TeV beam incident on water will stop cascading at about 290 cm, will deposit half of its energy in the first 340 cm, and will have dumped 98% of its energy within 1030 cm. 90% of the cascade will be contained within about 11 cm and 95% of the cascade will be contained within 22 cm of the beam axis. You will get 98% protection with a mass thickness of 10.3 tons/m2 and an additional 1/e reduction for each further 0.72 tons/m2 beyond that.
</ul>
Note that for sufficiently intense particle beams, 98% protection may not be enough. You might need to throw on several more tons per square meter to be sure.

<table border=1>
<tr><td> How Far it Goes:
(This section follows methods presented in Leroy and Rancoita<ref>C. Leroy and P-G Rancoita, "Physics of cascading shower generation and propagation in matter: principles of high-energy, ultrahigh-energy and compensating calorimetry", Reports on Progress in Physics 63 (2000) 505–606. DOI: 10.1088/0034-4885/63/4/202</ref>.)

When trying to estimate the extent of the irradiated region of the target, it is helpful to know the radiation length X0 in the material. This is the distance that an electron will travel, on average, before emitting a bremsstrahlung photon. Conveniently, it is also the distance a high energy photon will travel through the material before pair producing. The radiation length can be found from
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   = 4 α  
<td nowrap="nowrap"> NA
<td rowspan="2" nowrap="nowrap" align="right">   Z (Z+ζ) re2 ln
<td rowspan="2" nowrap="nowrap" align="right"> [
<td nowrap="nowrap"> 183
<td rowspan="2" nowrap="nowrap" align="right"> ]
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> A
<td style="border-top:solid 1px black;"> Z1/3
</table>
</div>
where α = 1/137 is the fine structure constant, NA = 6.02214076×1023 is Avogadro's number (or one mole), A is the atomic weight of the atoms in the material (usually expressed in grams per mole), Z is the atomic number (nuclear charge) of the atoms, and re = 2.8179403205×10-13 cm is the classical electron radius. ζ is a constant of approximately 1.3. This gives the thickness of material in terms of the areal density (mass of material per unit area perpendicular to the beam) – if you use A in g/mol and re in cm the result will be in cm2/g and X0 will be in g/cm2. If your target is made up of more than one kind of atom, you can get the total radiation length by finding the radiation length of each atom of type i, X0i and using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   =  
<td rowspan="2" nowrap="nowrap" align="right"> Σi  
<td nowrap="nowrap"> fi
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> X0i
</table>
</div>
where fi is the fraction by weight of the atom type i. Once you have the radiation length in terms of areal density, you can easily convert it to distance by dividing by the density ρ. If X0i is in g/cm2 and density is in g/cm3, then X0i/ρ will be in cm.

We will also need to know the critical energy εc. This is the energy where an electron is losing energy as rapidly to ionization as it is to radiation. That is, it is the energy at which the cascade stops multiplying, even though the individual particles already made can continue traveling for some distance. The critical energy can be estimated using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> εc = B  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> Z X0
<td rowspan="2" nowrap="nowrap" align="right"> )
<td nowrap="nowrap"> h
<tr>
<td style="border-top:solid 1px black;"> A
<td>  
</table>
</div>
where B = 2.66 MeV and h = 1.11.

We can now describe the distribution of radiation in the target material. The depth tmax where the cascade stops multiplying (in g/cm2) for an incident electron of energy E (in MeV) is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tmax = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> - 1
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
Similarly, the depth where half the energy of the beam has been deposited (in g/cm2) is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tcg = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> + 0.37
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
As usual, you can convert to a depth in cm by dividing by the material density. 98% of the beam energy will be deposited within a distance of &cong; 3 tcg. Beyond this, the beam energy will continue to fall off with thickness t approximately as an exponential exp[-t/λ], where the characteristic attenuation length λ &cong; 2 X0.

The width of a cascade, where 90% of the energy is contained, is approximately given by
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> RM =  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> EM
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
where EM &cong; 21.2 MeV. 95% of the radiation will be contained within a distance of 2 RM from the beam axis.
</table>

==== Protons and Ions ====

<table align=right><tr><td>
[[File:Regolith_Shielding.png|350 px|frameless]]
<tr><td width=350>
Relative effect of particle radiation (compared to no shielding) from galactic cosmic rays behind different thicknesses of water, aluminum, and lunar regolith<ref name="Slaba2022">Tony C. Slaba, "Radiation Shielding for Lunar Missions: Regolith Considerations", LSIC Crosstalk 7/18/2022 https://lsic.jhuapl.edu/uploadedDocs/focus-files/1604-E&C%20+%20EE%20Monthly%20Meeting%20-%202022%2007%20July_Presentation%20-%20NASA%20Slaba.pdf</ref>.
</table>
Protons and ions are grouped together because ions are made of protons and neutrons and when you accelerate these things for a weapon it is the protons and neutrons that are going to be doing the heavy lifting for blasting your target. This is because protons and neutrons are both about 2000 times more massive than electrons so any electrons along for the ride will contribute negligibly to the energy. Protons and neutrons together are called nuclear particles, and both behave quite similarly in nuclear collisions.

Nuclear particles are themselves made of quarks and can interact via the strong nuclear force. At the energies appropriate for weapons use, essentially all of the way protons or ions are blocked is through [[Particle_Accelerators#Nuclear_collisions|direct collisions with the atomic nuclei]] of the armor material. One or more quarks in the projectile will slam into quarks in the target nucleus and stretch them away from the rest of the quarks in the nuclear particles. Physics tells us that you can't have isolated quarks, so the "string" attaching the quark to its neighbors stretches and stretches until it snaps. Snapping releases so much energy that more quarks are made that clump together into all sorts of other nuclear particles, both exotic and familiar. This process is called a hadronic jet. The only particles produced in these jets long-lived enough to travel any noticeable distance through the armor material will be protons, neutrons, the charged pi mesons (or pions), and some of the K mesons (or kaons). The particles too short lived to go far enough to matter will decay into other particles, and these particles are often very high energy gamma rays. Much like with electrons, the long-lived secondary particles can be sufficiently energetic to smash into other nuclei and cause more hadronic jets (or, if they are low enough in energy, just knock more nuclear particles out of target nuclei) and this produces a similar shower or cascade. But in addition, the armor nuclei that were hit will be very hot after the impact from all of the energy given to them from the incident particles. These nuclei will cool off by evaporation – but instead of evaporating away molecules, they evaporate off nuclear particles from the nucleus. These evaporated particles are mostly neutrons, so any armor against proton or ion beam strikes will produce quite a bit of neutron radiation.

Eventually, the original particle's energy will be distributed among so many other particles that the charged particles will come to a stop by making ion tracks rather than from colliding with more nuclei, and the cascade halts. But you still have all the neutrons to worry about. And the gamma rays as well, which are high enough energy to kick off an electromagnetic cascade in the same way that electrons do.

Unlike electrons, which have reasonable approximations of their cascade behavior from relatively simple equations, the cascades of protons and ions usually require sophisticated simulations or experiments to get the resulting distribution of radiation. However, we can get a reasonable idea of their general behavior by comparing them to [[Space_Radiation#Shielding_Against_Cosmic_Rays|cosmic ray particles interacting with shielding]]. Cosmic rays are energetic protons and ions, after all. From this we see that a relatively small amount of armor – in the range of 1 to 3 tons per square meter – can actually make the radiation dose worse. The incident particles will knock off so many secondary particles in their shower that the dose is multiplied from so many tracks passing through the target. Hydrogen rich materials such as water and plastics work the best against proton and ion beam particles; heavy elements are the worst and produce the most secondaries. You could probably obtain adequate radiation protection from most incident beams with about 10 tons per square meter of hydrogen rich material as armor. However, you will want to add neutron absorbing elements to your hydrogen rich armor, such as lithium, boron, cadmium, or gadolinium. Further, in order to deal with the gamma rays you will want a layer of at least several centimeters of heavy elements, such as tungsten or lead, behind the hydrogen rich particle armor.

==== Structural Considerations ====

When your armor material absorbs a radiation beam, the energy of that radiation turns into heat inside the armor where the radiation was absorbed. If the beam pulse has enough energy as was sufficiently tightly focused, this heat can damage your armor. Your armor may melt in places, or thermal expansion could cause it to crack or warp. Perhaps the worst thing it could do is vaporize or flash into a plasma, because the pressure from the vapor or plasma can be sufficient to plastically deform the armor. This leaves a hole along the beam path where the expanding gas or plasma blasted out a long, deep crater. This means a beam that fires rapid shots spaced maybe milliseconds apart can have subsequent shots go down the same hole so your armor will not protect against them. Each shot will dig out a deeper hole until either your armor is penetrated or the attacking beam runs out of shots in its pulse train.

A consequence of this is that when designing your armor you may wish to use refractory materials with a high structural strength (like tungsten), or that is otherwise resistant to being blasted apart by its own evaporated remains.

=== Magnetic fields ===

[[Particle_Accelerators#Magnetic_fields|A charged particle crossing a magnetic field will bend so as to spiral around the field lines]]. If you try to shoot your particle beam through a magnetic field, it will bend away from the direction you shot it. This can make it hard to hit your target. Unfortunately for you, there are magnetic fields everywhere – around planets, and even trapped in the solar wind as it crosses the solar system.

As an example, the Earth has a magnetic field strength of about 5×10-5 tesla near its surface, and falling off farther away. If you are shooting a 1 TeV electron beam past the Earth, your beam is likely to bend by several milliradians (with the exact deflection depending on how close you are skimming the Earth). If you are in a medium Earth orbit and shooting at another target also in a medium Earth orbit on a trajectory that goes near the Earth, then without correcting for the magnetic field your beam will likely be off-target by tens to hundreds of kilometers.

For another example, the interplanetary magnetic field is about 10-9 tesla at Earth's distance from the sun. If you shoot that same 1 TeV electron beam at a target 0.1 light seconds away, you are likely to be off by a couple hundred meters.

<table border=1>
<tr><td>
The radius of curvature of the bend of a particle in a magnetic field is called the gyroradius rc. For a particle with a charge q, mass m, and kinetic energy E, find the relativistic momentum p
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
p = √[E2 - m2 c4] / c
</div>
To convert from energy units of eV to joules, multiply by 1.60×10-19. An electron has a mass of 9.11×10-31 kg, a proton of 1.67×10-27 kg, and an ion of 1.66×10-27×A kg, where A is the atomic weight of the ion. The speed of light c = 3.00×108 m/s.

In a magnetic field of strength B with an angle θ between the field direction and the direction of motion
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rc = |p sin(θ) / (q B)|
</div>
For electrons, q = -e, where the fundamental charge e = 1.60×10-19 coulomb. For a proton, q = +e. For an ion, multiply e by the charge state of the ion.

If the field acts over a distance d that is much smaller than the gyroradius, the deflection angle φ is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = d / rc
</div>
The deflection x at a target at a range R for small φ (φ &Lt; 1) will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
x = φ R.
</div>

Note that d will always be less than or equal to x. Also note that if d is not smaller than twice the gyroradius the beam will never reach the target because it will completely spiral around the field lines before it reaches the target.
</table>

So, it is clear that as the defender it will be helpful to hang out in areas with high magnetic fields, and keep some distance between you and the attacker. Meanwhile, the attacker will benefit from being closer where small deflections in the beam trajectory don't translate into large excursions from where the beam is aimed (and also because there will be less space filled with fields between the two of you). But the attacker can also try to compensate for known fields. If you happen to know that the magnetic fields are between you and your target, you can adjust your fire so that the deflection of the field bends your beam onto the target. This is not a simple task. It is easy enough to measure the field strength and direction where you are, but predicting how it changes over the path of your shot does not have an obvious solution. You can also try to gain intelligence on likely battle sites before you ever fight on them – for example, getting good data on the structure of the fields of a planet using spies and technical intelligence during peacetime (heck, on Earth this information is published for free in academic journals and by government agencies such as NOAA). Planetary magnetospheres are also affected by the solar wind, so different solar weather will change the fields you have to shoot through. But space weather forecasts and predictive simulations of the solar wind interacting with a planetary magnetosphere can give you a good idea of what the field structure is like. But even with all that, there are changes in the fields that happen on a scale of around a second ... anyone who has ever seen the auroras dancing across the sky in the polar night can see this. Compensating for these changes will be a significant challenge.

What would really help is being able to see the path of your own beam. This would allow you to see how far off you were from your target and in what direction, and then adjust fire to walk your shots onto the target. Unfortunately, particle beams in space are invisible. There might be ways to track them, but if so we don't know what they are.

==== Synchrotron radiation ====

When charged particle trajectories are bent, they emit [[Particle_Accelerators#Synchrotron_radiation|synchrotron radiation]]. This can cause the particle beam to lose energy, but if the bend is too tight it can also generate high energy gamma rays that ignore magnetic fields and directly strike the target. Synchrotron radiation can generally be ignored for proton or ion beams, but can be significant for electron beams.

<table border=1><tr><td>
For a beam particle with energy E and mass m, the synchrotron radiation is concentrated in a forward-directed cone with an opening angle of approximately 1/γ, where γ = 1 + E/(m c2). (Interestingly, the radiation intensity peaks an at angle of 1/(2 γ) and vanishes directly on-axis.) Because the beam is bending, you will get emission along the plane along which the beam is being bent. The energy of the photons produced during synchrotron emission is approximately<ref name="Jackson_Electrodynamics">J. D. Jackson, "Classical Electrodynamics: Second Edition", Wiley (1962)</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Eγ &cong; 3 γ3 c &hbar; / rc
</div>
where &hbar; = 1.05×10-34 J s (if you want the energy in joules) or &hbar; = 6.58×10-16 eV s (if you want the energy in eV). The photon energy will not exceed the energy of the incident particles, if you get a photon energy higher than the energy of the incident particles it just means that the synchrotron radiation spectrum extends up to the incident particle energy.

The fraction of the energy (or power) of the beam turned into synchrotron radiation will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = [1/(4 π ε0)] [4 π / 3] [q² / rc] β2 γ4 [φ / (2 π)] / E
</div>
where β = √[1 - 1/γ2] is the particle's speed measured as a fraction of the speed of light, and 1/(4 π ε0) = 8.99×109 is the Coulomb constant. For highly relativistic electrons (like all practical electron beam weapons emit), β &cong; 1 and we can use
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = 8.85×10-2 [E (GeV)3 [φ / (2 π)] / (rc (m))]
</div>
This fraction cannot be more than 1. If f gets close to one, it means that our original assumption that the beam particles mostly keep their original speed no longer holds, and the beam will lose most or almost all of its energy. Instead, when the beam is being deflected it will only make it through an angle of φ/f and will irradiate a region of length x/f starting at where the beam would have been incident if not for the magnets and extending in the direction the beam was deflected.
</table>

For example, a 1 TeV electron beam will have γ &cong; 2,000,000. So if it's bent by a magnetic field the synchtrotron radiation will spread out in a cone of about 5×10-7 radians. If this beam was shooting at a target 0.1 light seconds away from where it was bent, the radiation would arrive at the target over a region with a radius of about 15 meters with a maximum intensity at about 7.5 meters and no radiation directly along the path the beam was pointing while it was being bent. If the beam was bent by the interplanetary magnetic field from our earlier example, it would produce light with photon energies of around 1.3 eV (near infrared) and the lost energy would be a few parts in a hundred trillion. If it was instead deflected by the Earth's magnetic field you might expect a peak photon energy of tens of keV – in the hard x-ray part of the spectrum – and the fraction of energy in the synchrotron radiation would be somewhere in the low parts in 100,000 or less.

==== Separation of neutral beams ====

Magnetic fields only affect charged objects. So one way to overcome magnetic defenses or environments is to use beams that have no net charge. For example, a proton beam could be neutralized with a co-propagating electron beam with an equal number of protons and electrons. Or, potentially, you could fire an ion beam and then neutralize the ions after acceleration to get a beam of neutral atoms.

But, the magnetic fields are still tugging at the charged particles making up the beam as they go past. This will pull the electrons one way and the protons or ions the other way. As the particles are pulled in opposite directions you get a net positive charge on one side of the beam and a net negative charge on the other side (called polarization). This sets up an electric field inside the beam that acts to pull the particles back together. If the particles are going fast enough and the magnetic field is strong enough, the magnetic force can still separate the charges and will split the beam in two, one with the positive charges and one with the negative charges. But if the particles are not that fast or the magnetic field is not that strong this polarization of the beam will keep it together and let it fly right through the magnetic field.

<table border=1><tr><td>
Let's figure out how to analyze this problem. For definiteness, we will have a beam with an electric current of the positive species of I and an electric current of electrons of -I, all moving in the z direction. The beam radius will be R. The particle speed will be v with the usual relativistic parameters β = v/c and γ = √[1/(1-β2)]. If you know the energy per particle E of the positive species and the mass m of the positive species, γ = 1 + E/(m c2) and then β = √[1 - 1/γ2].

Let's assume that the beam is going through a magnetic field of strength B at an angle θ to the field direction. The quantity B sin(θ) is the component of the magnetic field perpendicular to the direction of the beam.

The charge density ρ of the positive component of the beam will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρL = I / (π v R2).
</div>

It is easiest to analyze the problem in the rest frame of the particles in the beam. Now the particles are at rest. Particles at rest are not affected by a magnetic field, but the change of reference frame Lorentz transforms the initial magnetic field into a combination of electric and magnetic fields in the rest frame. We can ignore the magnetic field, but the electric field is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔A = γ v B sin(θ)
</div>
in a direction perpendicular to both v and B. For our purposes, assume the electric field points in the x direction.

At the same time, relativistic length contraction increases the charge density in the rest frame
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρ = γ ρL.
</div>

Let r be the vector from the beam axis to the point inside the beam. It will be useful to separate this into its Cartesian coordinates. If x and y are the unit vectors in the x and y directions, respectively, then
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
r = x x + y y
</div>

If we consider only the positive component of the beam, from [https://en.wikipedia.org/wiki/Gauss%27s_law Gauss's law] it will have an electric field inside of the beam at a distance r from the center axis of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔+ = ρ r / (2 ε0) = (ρ/(2 ε0)) (x x + y y)
</div>
where ε0 = 8.85×10-12 is the vacuum permittivity. Now let's displace the electrons in the beam by a distance δ in the x direction. By itself, this electron beam will have an electric field of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔- = -(ρ/(2 ε0)) ((x-δ) x + y y)
</div>
By the principle of superposition, the electric field inside the beam with both components is the sum of the electric fields of the separate components including the external field
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔B = [(ρ/(2 ε0)) δ + γ v B sin(θ)]x
</div>
Because the charges are free to move, they will do so as long as a field is applied to them. Thus, the beam becomes stable once the charges separate enough that the field internal to the beam is zero. Therefore
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
δ = -2 ε0 γ v B sin(θ) / ρ = -2 π ε0 v2 R2 B sin(θ) / I
</div>
If |δ| > 2 |R|, the beam will fully separate and each component will go its own way. If δ is small compared to R, the beam's polarization suffices to hold it together in the overlap region. However, where the positive and negative charges do not overlap the electric fields no longer cancel, and the external electric field carries away the outer polarized part of the beam. This gives us a rough threshold for which a magnetic field will quickly separate a neutral beam into its charged components and bend them away from their target
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr><td> B sin(θ) < I / (π ε0 v2 R) <td>       <td> Beam is partially eroded.
<tr><td> B sin(θ) ≥ I / (π ε0 v2 R) <td> <td> Beam promptly separates.
</table>
</div>
For a partially eroding beam, the polarized region that was peeled off by the field still acts to screen the remaining neutral portion, at least to within this approximation, leaving the still-neutral beam to hit the target.
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Neutralized_beam_in_B_field.png|5140 px|frameless]]
<tr><td width=700>
A profile of the beam charge distribution of a neutralized beam in a magnetic field. Negative charge density is shown in yellow, positive charge density in cyan, with neutral charge density in light green. The magnetic field is vertical, the electric field in the beam reference frame is shown in magenta including both the polarization field and the Lorentz transformed electric field resulting from the magnetic field. As positive charges are pulled in the direction of the field and negative charges opposite to the field direction, the polarized boundary will be pulled off the beam by the electric field even though the electric field is zero in nearly all the beam interior. This expanding polarization layer will continue to largely screen the still-neutral portion, however; but the extent to which the screening is only partial would require more sophisticated analysis methods than used here. The material removed by the field will expand to a maximum of the deflection distance of a uniform charge based on the magnetic field by the time it reaches the target.
</table>
</div>
<tr><td>
For a beam of neutral atoms, let a be the radius of the atom and let Eb be the binding energy of the least tightly bound electron to the atom. An external electric field in excess of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
|𝓔A| > Eb / (e a)
</div>
will promptly ionize the atom, ripping away its electron. Here, e = 1.60×10-19 C is the magnitude of the electric charge of the electron. For fields close to but a bit weaker than this, the electron can still rapidly pop off due to quantum tunneling<ref>Ph. Durand and I. Paidarová, “Ionization of the hydrogen atom: from weak to strong static electric fields,” The European Physical Journal D, Vol. 26, pg. 253-259, 2003.</ref>. In general, expect that for fields greater in magnitude than (very approximately) Eb / (10 e a) the electron will be quickly removed, while for fields weaker than this the electrons will stay on their atoms.

It is useful to note that Eb/e is just the binding energy in units of eV. The most weakly bound electrons are usually bound to their atom with about 10 eV of energy. Atoms are typically about 10-10 m in radius. Thus, you can expect fields of around 1010 to 1011 V/m to directly ionize atoms. If we call this threshold value 𝓔T, then a beam will be ionized when passing through a field with strength greater than
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B sin(θ) > 𝓔T / (γ v)
</div>
</table>

Example: Let's consider a 1 GeV neutralized proton beam from a synchrotron, with a beam current of 200 kA in each bunch (incidentally, if a bunch is 5 cm long this means that each bunch packs 38 kJ worth of energy). The speed of the bunch is 87% the speed of light, or 262 million m/s, with a γ of 2.07. Consider when the beam's focus takes it to a radius of 5 cm (or 0.05 m, for our calculation). Plugging this in, we find that the protons in the bunch will separate from the electrons in a field with a transverse component greater than about 2.1 T, and the bunch will pass through the field with only minimal losses if the transverse component is much smaller than this.

Example: We'll take the same 1 GeV beam, but this time recombine the electrons with protons to make hydrogen atoms. An electron is bound to a hydrogen atom with an energy of 13.6 eV (the Ryberg constant) and the radius of the electron orbital in a hydrogen atom is 0.529×10-10 m (the Bohr radius). This gives a threshold electric field for prompt ionization of hydrogen of 2.57×1011 V/m. We will get this with a transverse component of the magnetic field of 474 T. Above this field the atoms promptly ionize and electrons are stripped away from the protons. For fields higher than roughly 1/10 of this – so about 40 to 50 T – you may still get significant ionization by quantum tunneling; lower than this your losses from the beam due to field-induced ionization are likely negligible.

==== Magnetic shields ====

What happens when we can't rely on magnetic fields in the environment to protect us? One idea is to produce powerful artificial magnetic fields around our spacecraft to deflect incoming particle beams. This is very similar in concept to using active magnetic shielding to protect against space radiation – [[Space_Radiation#Magnetic_Shielding|check out our article on space radiation for more]].
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr>
<td width=100>
[[File:Unconfined_FRC_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:racetrack_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:Magnetic_shielding_Halback_Array.png|200 px|frameless]]
<tr><td colspan=3>
Some potential configurations for magnetic shields. The currents generating the fields are in green, the magnetic fields themselves are shown in magenta<ref name="Battiston2012">R. Battiston, W.J. Burger, V. Calvelli, R. Musenich, V. Choutko, V.I. Datskov, A. Della Torre, F. Venditti,
C. Gargiulo, G. Laurenti, S. Lucidi, S. Harrison, and R. Meinke, "ARSSEM Active Radiation Shield for Space Exploration Missions", Final Report ESTEC Contract N° 4200023087/10/NL/AF : “Superconductive Magnet for Radiation Shielding of Human Spacecraft” (2012) https://arxiv.org/abs/1209.1907 https://www.researchgate.net/publication/265945847_Active_Radiation_Shield_for_Space_Exploration_Missions</ref><ref name="Desiati2022">Paolo Desiati and Elena D'Onghia, "CREW HaT: A Magnetic Shielding System for Space Habitats", arXiv:2209.13624 [physics.space-ph] https://doi.org/10.48550/arXiv.2209.13624</ref>.
</table>
</div>

One immediate issue is that you want to keep very high magnetic fields out of the crew area. Fields in excess of 0.5 T can start to be unpleasant, and it is recommended to keep the field below 0.2 T for continuous exposure.

Strong magnets can suffer from high self-forces acting to tear them apart. Most shielding designs will have multiple magnetic coils, which will exert high forces and torques on each other. This will result in the need for strong structural bracing to keep the magnets intact and in place. Such strong structural designs will impose significant extra mass on the spacecraft.

Consider a magnetic field configuration that can keep radiation out of a certain region of space. There is no trajectory of a charged radiation particle up to a certain energy that will allow those particles to enter the protected region. But the dynamics of charged particles in a magnetic field are the same whether going forward or backward in time. If there is no way for a particle to get in, by this time reversal symmetry any particle which is in that forbidden region can never escape. Consequently any field configuration that is a magnetic trap is also a magnetic shield; any field configuration that is a magnetic shield is also a magnetic trap. A spacecraft with a magnetic shield may end up surrounded by strong radiation belts.

Using a magnet to bend a weaponized electron beam very often ends up with the beam being bent so hard that it dumps most of its energy in the form of synchrotron radiation at gamma ray energies. Because gamma rays are not charged, they will not be deflected by the field. Thus, if you would have been hit by the electron beam you will be hit instead by the gamma rays.

As an example, consider a 1 TeV electron beam. We'll surround our spacecraft with a 10 T magnetic field in a 100 meter radius region just outside our spacecraft. This will deflect the beam by 30 meters, which is enough to make it miss. But we are not saved, because the beam dumps nearly all of its energy into synchrotron radiation with an energy spectrum of gamma rays up to 13 GeV. This radiation will hit us in a strip about 2.4 meters long and not much more than the width of the original beam with essentially all of the energy of the original beam.

==== Magnet drones ====

If a magnetic shield around your spacecraft will just end up killing you from synchrotron radiation, what if you put the magnetic shield on a drone and put that drone between yourself and the enemy trying to shoot you? This lets you use a much weaker magnet because you don't need nearly so high of a deflection angle to make the shot miss. This means that less beam energy will turn into synchrotron radiation that could kill you instead and will spread what synchrotron radiation does occur out over a much wider area when it reaches your spacecraft, hopefully so that most of it just misses your spacecraft.

It does mean that your enemy will start shooting your drones so that they can eventually take a shot at you. But hopefully by that time you've already cleared out their drones and cored their spacecraft with your own synchrocannon or UREB.

==== Plasma magnets ====

Usually, to make a really big magnetic field you would run current through a really big conductive loop. But really big things are heavy and expensive. One way around this could be to use the ambient space plasma to carry the electric current rather than any kind of wire. In order to make this current, you use the same mechanism used in a rotary induction electric motor: you generate a rotating magnetic field and induce a current in the conductive loop (in this case, the space plasma). How do you make a rotating magnetic field? The method proposed<ref name="Slough2006">John Slough, "The Plasma Magnet", NASA Institute for Advanced Concepts Phase II Final Report (2006) https://www.niac.usra.edu/files/studies/final_report/917Slough.pdf</ref> is to use an antenna of four much smaller current loops as electromagnets at 90° to each other, and sinusoidally vary the current through the loops with a respective 90° phase difference between them. When this is done, a circular current is induced in the space plasma and the self-force of a magnetic field on its generating current inflates the current loop to much larger radii. You end up with a current sheet in the plane of the rotating field's rotation circulating about the antenna nearly synchronously with the field rotation. This current then creates a normal dipolar magnetic field as with any current loop. The current sheet and the resulting magnetic field expand until the magnetic field's pressure equals that of the ambient space plasma. This gives an enormous region of magnetic field from a relatively small and lightweight system.

The system proposed in the initial study would use an antenna approximately 10 meters in radius and produce a region of field approximately 30 km in radius at Earth's orbit. The field inside would not be constant - at the edges it would be 50 nT, giving the right pressure to keep out the solar wind at Earth's orbit. A circulating current sheet produces a field that falls off as 1/r with distance, meaning that adjacent to the antenna the field would be about 0.15 mT. Using the methods in the box below, we can estimate that this particular design as part of a spacecraft's shields will not deflect a TeV electron beam by more than a few centimeters, which is usually insufficient to force a miss. However, it could potentially be useful for a drone magnet, where the much larger distance between the magnetic field and the target spacecraft would allow a small deflection angle to give perhaps tens or low hundreds of meters change in beam location near the target at near light second ranges. The relatively low field and gradual bend of the incident beam suggests that synchrotron radiation will not be an issue; however this does not exclude more powerful plasma magnet designs causing significant synchrotron radiation effects.

<table border=1><tr><td>
Calculating the deflection from a spatially varying magnetic field affected by the solar wind is no trivial task. However, we can make an approximation that both should be within about an order of magnitude of the true value and also represent an upper limit to the deflection angle; although it does neglect the influence of the driving rotating magnetic field which could in some cases increase the deflection.

We will take R0 as the radius of the antenna. B0 is the field strength at the antenna. Rm is the distance at which the magnetic influence stops because the magnetic pressure equals the space plasma pressure. Bm is the field strength at Rm. According to Slough<ref name="Slough2006"></ref>, the magnetic field strength B(r) at any distance from the antenna center r between R0 and Rm is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B(r) = B0 R0 / r.
</div>
Therefore, once we know Bm and Rm, B0 is completely determined
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B0 = Bm Rm / R0.
</div>

The force on a particle with charge q passing perpendicularly through a magnetic field B(r) at a speed v is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
F(r) = q v B(r)
</div>
For an electron beam, v is so close to c as to make no difference. If the particle is moving down the y direction and the field is in the z direction, the force will be in the x direction. We assume that the deflection will be small, so we can choose a straight-line path in the y direction offset from the center by an amount x. The plasma bubble radius is Rm, so the particle will enter at ymax = √[Rm2-x2] and leave at ymin = -√[Rm2-x2].

The momentum change the particle experiences from this force is the time integral of the force over its path. Because of reflection symmetry about y = 0, we only need to integrate from ymax to 0 and then multiply by 2
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] dt F(r)
<tr> <td> <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] (dy/v) F(r)
<tr> <td> <td> = <td align="left">2 q B(0) R0 [Integral from 0 to √[Rm2-x2]] dy 1/√[y2 + x2]
<tr> <td> <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/x]
</table>
</div>
This holds for |x| > R0. But the expression for the field is not valid for r < R0 and the integral will give nonsense and diverging results if we evaluate for |x| < R0. In this case, we can put the integral lower bound at r = R0, or y = √[R02-x2]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/(√[R02-x2] + R0)]
</table>
</div>

The deflection angle is now estimated as
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = px / p
</div>
where the total momentum of the particle p can be found from the methods in the first box in the Magnetic Fields section.

The deflection for a shot aimed right at the spacecraft at the position of the spacecraft can be estimated by setting x = 0 and only taking the first half of the integral (so ignore the factor of 2)
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
q B(0) R0 ln[Rm/R0] / (p Rm)
</div>
This is an additional overestimate because it fails to account for how the deflection builds up over distance during the integration, but it should still give something of an order-of-magnitude and upper limit estimate of what kind of deflections might be expected.
</table>

=== Electric fields ===

If electric fields can accelerate charged particles, maybe they can keep the charged particles in a weaponized particle beam away from your spacecraft? To completely block a particle beam, you would need a voltage on your spacecraft equal to the particle energy in eV of the particles in the beam. That is, if a 1 GeV proton beam is coming at you, you'd need an electric potential of a billion volts to stop that beam. This leads to three immediate problems.

The first is that particles of the opposite charge will be attracted to your spacecraft. Because space isn't really empty but rather a rarefied plasma (called the solar wind in a solar system or the interstellar medium out between stars), there will be plenty of particles of the opposite charge around to attract. And they will hit with an energy in eV equal to the voltage on your spacecraft. In our example of defending against a 1 GeV proton beam with a gigavolt field, the electrons from the solar wind will come in and hit with an energy of 1 GeV each. This will generate dangerous radiation.

The second problem is a consequence of the first. Each of those GeV electrons from our example need to get their energy from somewhere, which turns out to be from the electric field we are using as shielding. You will need to make up for this energy loss with power from your electrical system, which gives a constant drain on your power. Some estimates put the electrical effort to maintain a several hundred megavolt shield in space at many tens of gigawatts<ref name="Mechmann2019">Claire Mechmann, "Analysis of Proposed Active Radiation Shielding Design Concept for Spacecraft" (2019) Thesis, College of Engineering and Science of Florida Institute of Technology</ref>.

And the third problem is similar to one for magnetic fields. Your electrodes will experience high self forces and forces between separate electrodes that will try to tear your shield generator apart. Consequently you will need heavy structural reinforcement in order for your shield to work.

There are potential ways to mitigate some of these problems. Interested readers are directed to [[Space_Radiation#Electrostatic_Shielding|the active shielding section in our page on space radiation]] for more of the juicy details; there's no need to repeat the same stuff here.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Elctrostatic_active_shielding.png|200 px|frameless]]
<td>
[[File:Electrostatic_active_shielding_2.png|200 px|frameless]]
<tr><td colspan=2>
Proposed designs for electrostatic shields to protect from positively charged particles<ref name="Tripathi2006">Ram K. Tripathi, John W. Wilson, and Robert C. Youngquist, "Electrostatic Active Radiation Shielding - Revisited", 2006 IEEE Aerospace Conference, Big Sky, MT, USA, 2006, pp. 9 pp.-, doi: 10.1109/AERO.2006.1655760.</ref><ref name="Chowdhury2023">Rajarshi Pal Chowdhury, Luke A. Stegeman, Matthew L. Lund, Dan Fry, Stojan Madzunkov, and Amir A. Bahadori, "Hybrid methods of radiation shielding against deep-space radiation", Life Sciences in Space Research, Volume 38, 2023, Pages 67-78, ISSN 2214-5524, https://doi.org/10.1016/j.lssr.2023.04.004.</ref>.
</table>
</div>

== Damage ==

Particle beams are, first and foremost, devices that produce beams of [[Nuclear_radiation|ionizing radiation]]. The radiation of weaponizable particle beams is extremely penetrating; thus of the [[Particle_Accelerators#Energy_loss_in_matter|interaction mechanisms listed for radiation from particle beams]], we mostly need to worry about bremsstrahlung and nuclear collisions and these produce extensive secondary showers of radiation that increase the penetration of the radiation exposure. Because of this, any target hit by a particle beam weapon will be irradiated, and radiation dose will always be one of the primary damage mechanisms of a particle beam weapon strike. Further, this radiation will be distributed along a track passing deeply through the material of the target, allowing it to affect critical systems through heavy armor, structural material, robust machine components, and passive material such as fuel load, coolant, or cargo. As the radiation deposits its energy into the target material, that material will be heated. If the target is heated sufficiently, it may experience [[Beam-Target_Interactions|thermal or mechanical]] damage from melting, cooking, combustion, vapor explosions, and warping or cracking due to thermal expansion.

The discussion of the radiation penetration into armor also applies to the same radiation penetrating through the target. So the formulas and rules of thumb for estimating the depth of radiation penetration into armor can also be used to estimate how far into a tank or spacecraft a particle beam strike delivers effects.

In addition to the radiation and thermal effects, a particle beam can also produce a sharp electromagnetic pulse that can damage semiconductor electronics.

== Hitting Your Target ==

It doesn't do much to have the most horrifically powerful death ray if it can't hit. What are the challenges for directing your beam of radioactive fury onto your enemies?

We have already mentioned the [[Particle_Beam_Weapons#Magnetic_fields|challenges of stray magnetic field]] that are part of the ambient environment messing with your targeting. This will be a challenge when you are launching beams of charged particles. And we've discussed some of the means of getting around this, like using neutralized beams.

But another issue that will be a challenge at very long ranges is dealing with [[Jitter|jitter]]. A weapon made out of a linac will be very long, and so it will be rather wibbly. This could make its jitter problems worse. A synchrotron might not be so badly affected; and for any future hypothetical high energy wakefield accelerators, no one really knows what they'll be like. There are certainly tricks you can play to reduce jitter (see the previously linked jitter page), but at ranges approaching a light second it's certainly plausible that it will be difficult to land a hit.

== 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, Gerrit Bruhaug, and Luke Campbell for writing

Gerrit Bruhaug and Luke Campbell for scientific research, math wrangling and advise

ATriffleMind and Luke W. Campbell for proof reading.

==References==

[[Category:Beams]][[Category:Physics & Engineering‏‎]][[Category:Engineering‏‎]][[Category:Military Technology]][[Category:Warfare‏‎]]

Particle Beam Weapons

2026-10-27T23:55:48Z

Lwcamp: /* Damage */

{{PageConstructionNotice}}

''For a general overview on particle accelerator technology, see [[Particle Accelerators]]

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.

::: Z = 1.32×102 (γ2-1)3/4 r / √I<ref name=USPAS2016>Martin Reiser, "Theory and Design of Charged Particle Beams", 2008, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim</ref>

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

::: Z = β γ c r √[m/T]

Where Z is the doubling distance like before, r is the initial beam radius, T is the beam temperature in eV, m is the ion mass and β, c, and γ 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.

== Defenses ==

Sure, it's great to have a great big particle cannon to blast your enemies. But what if you are the one being shot at? In that case, it's best to figure out ways to keep from being blown to kingdom come.

=== Armor ===

The simplest way to defend against the high energy particles of a particle beam is to pile a whole bunch of stuff up and put it in the way of the beam. Particle beams may produce extremely penetrating radiation, but enough stuff will block most of that radiation and may allow you to survive. The details depend on what kind of particle is being thrown at you, and are explained below. However, a general rule of thumb is that you probably want about 10 tons of material per square meter of surface you want to armor in order to get sufficient protection.

==== Electrons ====

At the very high energy of an electron accelerated by a weapons-grade particle beam, the primary interaction with matter will be the emission of bremsstrahlung photons in the gamma-ray range as the electron slams into particles of matter. These high energy photons themselves primarily interact by the pair production mechanism – the photon disappears, but leaves behind a highly energetic positron and electron. These two particles are themselves energetic enough to produce further bremsstrahlung, and so the process continues. You get the formation of an electron-gamma shower (also called an electromagnetic cascade), which only stops after either the particles have penetrated through the target or the initial energy of the electron has been spread through so many shower particles (including all the electrons, positrons, and photons) that no particle has enough individual energy to pair produce or to be likely to bremsstrahlung again before it is stopped by other mechanisms.

To stop electrons, you'll want the atoms in your armor to be as heavy as you can practically get them. High atomic number both encourages bremsstrahlung and pair production, giving a double whammy for stopping that cascade. However, occasionally one of those electrons, positrons, or gammas will hit a nucleus and knock out neutrons. So you probably want a back-up layer of several tens of centimeters of hydrogen rich material such as plastic or water mixed with neutron absorbing materials like boron, lithium, cadmium, or gadolinium. High density polyethylene loaded with 5% boron by weight is a typical go-to neutron blocker.

The actual methods to calculate the penetration of the electromagnetic cascade is given are the box below. However, for some examples:
<ul>
<li> A 10 GeV beam incident on thick iron (or steel) will stop cascading at about 9 cm, will deposit half of its energy in the first 12 cm, and will have dumped 98% of its energy within 35 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 2.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 100 GeV beam incident on thick iron (or steel) will stop cascading at about 13 cm, will deposit half of its energy in the first 16 cm, and will have dumped 98% of its energy within 47 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 3.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick iron (or steel) will stop cascading at about 17 cm, will deposit half of its energy in the first 20 cm, and will have dumped 98% of its energy within 59 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 4.6 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick tungsten will stop cascading at about 3.5 cm, will deposit half of its energy in the first 4 cm, and will have dumped 98% of its energy within 12 cm. 90% of the cascade will be contained within about 0.93 cm and 95% of the cascade will be contained within 1.86 cm of the beam axis. You will get 98% protection with a mass thickness of 2.3 tons/m2 and an additional 1/e reduction for each further 0.13 tons/m2 beyond that.
<li> A 1 TeV beam incident on water will stop cascading at about 290 cm, will deposit half of its energy in the first 340 cm, and will have dumped 98% of its energy within 1030 cm. 90% of the cascade will be contained within about 11 cm and 95% of the cascade will be contained within 22 cm of the beam axis. You will get 98% protection with a mass thickness of 10.3 tons/m2 and an additional 1/e reduction for each further 0.72 tons/m2 beyond that.
</ul>
Note that for sufficiently intense particle beams, 98% protection may not be enough. You might need to throw on several more tons per square meter to be sure.

<table border=1>
<tr><td> How Far it Goes:
(This section follows methods presented in Leroy and Rancoita<ref>C. Leroy and P-G Rancoita, "Physics of cascading shower generation and propagation in matter: principles of high-energy, ultrahigh-energy and compensating calorimetry", Reports on Progress in Physics 63 (2000) 505–606. DOI: 10.1088/0034-4885/63/4/202</ref>.)

When trying to estimate the extent of the irradiated region of the target, it is helpful to know the radiation length X0 in the material. This is the distance that an electron will travel, on average, before emitting a bremsstrahlung photon. Conveniently, it is also the distance a high energy photon will travel through the material before pair producing. The radiation length can be found from
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   = 4 α  
<td nowrap="nowrap"> NA
<td rowspan="2" nowrap="nowrap" align="right">   Z (Z+ζ) re2 ln
<td rowspan="2" nowrap="nowrap" align="right"> [
<td nowrap="nowrap"> 183
<td rowspan="2" nowrap="nowrap" align="right"> ]
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> A
<td style="border-top:solid 1px black;"> Z1/3
</table>
</div>
where α = 1/137 is the fine structure constant, NA = 6.02214076×1023 is Avogadro's number (or one mole), A is the atomic weight of the atoms in the material (usually expressed in grams per mole), Z is the atomic number (nuclear charge) of the atoms, and re = 2.8179403205×10-13 cm is the classical electron radius. ζ is a constant of approximately 1.3. This gives the thickness of material in terms of the areal density (mass of material per unit area perpendicular to the beam) – if you use A in g/mol and re in cm the result will be in cm2/g and X0 will be in g/cm2. If your target is made up of more than one kind of atom, you can get the total radiation length by finding the radiation length of each atom of type i, X0i and using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   =  
<td rowspan="2" nowrap="nowrap" align="right"> Σi  
<td nowrap="nowrap"> fi
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> X0i
</table>
</div>
where fi is the fraction by weight of the atom type i. Once you have the radiation length in terms of areal density, you can easily convert it to distance by dividing by the density ρ. If X0i is in g/cm2 and density is in g/cm3, then X0i/ρ will be in cm.

We will also need to know the critical energy εc. This is the energy where an electron is losing energy as rapidly to ionization as it is to radiation. That is, it is the energy at which the cascade stops multiplying, even though the individual particles already made can continue traveling for some distance. The critical energy can be estimated using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> εc = B  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> Z X0
<td rowspan="2" nowrap="nowrap" align="right"> )
<td nowrap="nowrap"> h
<tr>
<td style="border-top:solid 1px black;"> A
<td>  
</table>
</div>
where B = 2.66 MeV and h = 1.11.

We can now describe the distribution of radiation in the target material. The depth tmax where the cascade stops multiplying (in g/cm2) for an incident electron of energy E (in MeV) is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tmax = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> - 1
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
Similarly, the depth where half the energy of the beam has been deposited (in g/cm2) is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tcg = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> + 0.37
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
As usual, you can convert to a depth in cm by dividing by the material density. 98% of the beam energy will be deposited within a distance of &cong; 3 tcg. Beyond this, the beam energy will continue to fall off with thickness t approximately as an exponential exp[-t/λ], where the characteristic attenuation length λ &cong; 2 X0.

The width of a cascade, where 90% of the energy is contained, is approximately given by
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> RM =  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> EM
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
where EM &cong; 21.2 MeV. 95% of the radiation will be contained within a distance of 2 RM from the beam axis.
</table>

==== Protons and Ions ====

<table align=right><tr><td>
[[File:Regolith_Shielding.png|350 px|frameless]]
<tr><td width=350>
Relative effect of particle radiation (compared to no shielding) from galactic cosmic rays behind different thicknesses of water, aluminum, and lunar regolith<ref name="Slaba2022">Tony C. Slaba, "Radiation Shielding for Lunar Missions: Regolith Considerations", LSIC Crosstalk 7/18/2022 https://lsic.jhuapl.edu/uploadedDocs/focus-files/1604-E&C%20+%20EE%20Monthly%20Meeting%20-%202022%2007%20July_Presentation%20-%20NASA%20Slaba.pdf</ref>.
</table>
Protons and ions are grouped together because ions are made of protons and neutrons and when you accelerate these things for a weapon it is the protons and neutrons that are going to be doing the heavy lifting for blasting your target. This is because protons and neutrons are both about 2000 times more massive than electrons so any electrons along for the ride will contribute negligibly to the energy. Protons and neutrons together are called nuclear particles, and both behave quite similarly in nuclear collisions.

Nuclear particles are themselves made of quarks and can interact via the strong nuclear force. At the energies appropriate for weapons use, essentially all of the way protons or ions are blocked is through [[Particle_Accelerators#Nuclear_collisions|direct collisions with the atomic nuclei]] of the armor material. One or more quarks in the projectile will slam into quarks in the target nucleus and stretch them away from the rest of the quarks in the nuclear particles. Physics tells us that you can't have isolated quarks, so the "string" attaching the quark to its neighbors stretches and stretches until it snaps. Snapping releases so much energy that more quarks are made that clump together into all sorts of other nuclear particles, both exotic and familiar. This process is called a hadronic jet. The only particles produced in these jets long-lived enough to travel any noticeable distance through the armor material will be protons, neutrons, the charged pi mesons (or pions), and some of the K mesons (or kaons). The particles too short lived to go far enough to matter will decay into other particles, and these particles are often very high energy gamma rays. Much like with electrons, the long-lived secondary particles can be sufficiently energetic to smash into other nuclei and cause more hadronic jets (or, if they are low enough in energy, just knock more nuclear particles out of target nuclei) and this produces a similar shower or cascade. But in addition, the armor nuclei that were hit will be very hot after the impact from all of the energy given to them from the incident particles. These nuclei will cool off by evaporation – but instead of evaporating away molecules, they evaporate off nuclear particles from the nucleus. These evaporated particles are mostly neutrons, so any armor against proton or ion beam strikes will produce quite a bit of neutron radiation.

Eventually, the original particle's energy will be distributed among so many other particles that the charged particles will come to a stop by making ion tracks rather than from colliding with more nuclei, and the cascade halts. But you still have all the neutrons to worry about. And the gamma rays as well, which are high enough energy to kick off an electromagnetic cascade in the same way that electrons do.

Unlike electrons, which have reasonable approximations of their cascade behavior from relatively simple equations, the cascades of protons and ions usually require sophisticated simulations or experiments to get the resulting distribution of radiation. However, we can get a reasonable idea of their general behavior by comparing them to [[Space_Radiation#Shielding_Against_Cosmic_Rays|cosmic ray particles interacting with shielding]]. Cosmic rays are energetic protons and ions, after all. From this we see that a relatively small amount of armor – in the range of 1 to 3 tons per square meter – can actually make the radiation dose worse. The incident particles will knock off so many secondary particles in their shower that the dose is multiplied from so many tracks passing through the target. Hydrogen rich materials such as water and plastics work the best against proton and ion beam particles; heavy elements are the worst and produce the most secondaries. You could probably obtain adequate radiation protection from most incident beams with about 10 tons per square meter of hydrogen rich material as armor. However, you will want to add neutron absorbing elements to your hydrogen rich armor, such as lithium, boron, cadmium, or gadolinium. Further, in order to deal with the gamma rays you will want a layer of at least several centimeters of heavy elements, such as tungsten or lead, behind the hydrogen rich particle armor.

==== Structural Considerations ====

When your armor material absorbs a radiation beam, the energy of that radiation turns into heat inside the armor where the radiation was absorbed. If the beam pulse has enough energy as was sufficiently tightly focused, this heat can damage your armor. Your armor may melt in places, or thermal expansion could cause it to crack or warp. Perhaps the worst thing it could do is vaporize or flash into a plasma, because the pressure from the vapor or plasma can be sufficient to plastically deform the armor. This leaves a hole along the beam path where the expanding gas or plasma blasted out a long, deep crater. This means a beam that fires rapid shots spaced maybe milliseconds apart can have subsequent shots go down the same hole so your armor will not protect against them. Each shot will dig out a deeper hole until either your armor is penetrated or the attacking beam runs out of shots in its pulse train.

A consequence of this is that when designing your armor you may wish to use refractory materials with a high structural strength (like tungsten), or that is otherwise resistant to being blasted apart by its own evaporated remains.

=== Magnetic fields ===

[[Particle_Accelerators#Magnetic_fields|A charged particle crossing a magnetic field will bend so as to spiral around the field lines]]. If you try to shoot your particle beam through a magnetic field, it will bend away from the direction you shot it. This can make it hard to hit your target. Unfortunately for you, there are magnetic fields everywhere – around planets, and even trapped in the solar wind as it crosses the solar system.

As an example, the Earth has a magnetic field strength of about 5×10-5 tesla near its surface, and falling off farther away. If you are shooting a 1 TeV electron beam past the Earth, your beam is likely to bend by several milliradians (with the exact deflection depending on how close you are skimming the Earth). If you are in a medium Earth orbit and shooting at another target also in a medium Earth orbit on a trajectory that goes near the Earth, then without correcting for the magnetic field your beam will likely be off-target by tens to hundreds of kilometers.

For another example, the interplanetary magnetic field is about 10-9 tesla at Earth's distance from the sun. If you shoot that same 1 TeV electron beam at a target 0.1 light seconds away, you are likely to be off by a couple hundred meters.

<table border=1>
<tr><td>
The radius of curvature of the bend of a particle in a magnetic field is called the gyroradius rc. For a particle with a charge q, mass m, and kinetic energy E, find the relativistic momentum p
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
p = √[E2 - m2 c4] / c
</div>
To convert from energy units of eV to joules, multiply by 1.60×10-19. An electron has a mass of 9.11×10-31 kg, a proton of 1.67×10-27 kg, and an ion of 1.66×10-27×A kg, where A is the atomic weight of the ion. The speed of light c = 3.00×108 m/s.

In a magnetic field of strength B with an angle θ between the field direction and the direction of motion
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rc = |p sin(θ) / (q B)|
</div>
For electrons, q = -e, where the fundamental charge e = 1.60×10-19 coulomb. For a proton, q = +e. For an ion, multiply e by the charge state of the ion.

If the field acts over a distance d that is much smaller than the gyroradius, the deflection angle φ is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = d / rc
</div>
The deflection x at a target at a range R for small φ (φ &Lt; 1) will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
x = φ R.
</div>

Note that d will always be less than or equal to x. Also note that if d is not smaller than twice the gyroradius the beam will never reach the target because it will completely spiral around the field lines before it reaches the target.
</table>

So, it is clear that as the defender it will be helpful to hang out in areas with high magnetic fields, and keep some distance between you and the attacker. Meanwhile, the attacker will benefit from being closer where small deflections in the beam trajectory don't translate into large excursions from where the beam is aimed (and also because there will be less space filled with fields between the two of you). But the attacker can also try to compensate for known fields. If you happen to know that the magnetic fields are between you and your target, you can adjust your fire so that the deflection of the field bends your beam onto the target. This is not a simple task. It is easy enough to measure the field strength and direction where you are, but predicting how it changes over the path of your shot does not have an obvious solution. You can also try to gain intelligence on likely battle sites before you ever fight on them – for example, getting good data on the structure of the fields of a planet using spies and technical intelligence during peacetime (heck, on Earth this information is published for free in academic journals and by government agencies such as NOAA). Planetary magnetospheres are also affected by the solar wind, so different solar weather will change the fields you have to shoot through. But space weather forecasts and predictive simulations of the solar wind interacting with a planetary magnetosphere can give you a good idea of what the field structure is like. But even with all that, there are changes in the fields that happen on a scale of around a second ... anyone who has ever seen the auroras dancing across the sky in the polar night can see this. Compensating for these changes will be a significant challenge.

What would really help is being able to see the path of your own beam. This would allow you to see how far off you were from your target and in what direction, and then adjust fire to walk your shots onto the target. Unfortunately, particle beams in space are invisible. There might be ways to track them, but if so we don't know what they are.

==== Synchrotron radiation ====

When charged particle trajectories are bent, they emit [[Particle_Accelerators#Synchrotron_radiation|synchrotron radiation]]. This can cause the particle beam to lose energy, but if the bend is too tight it can also generate high energy gamma rays that ignore magnetic fields and directly strike the target. Synchrotron radiation can generally be ignored for proton or ion beams, but can be significant for electron beams.

<table border=1><tr><td>
For a beam particle with energy E and mass m, the synchrotron radiation is concentrated in a forward-directed cone with an opening angle of approximately 1/γ, where γ = 1 + E/(m c2). (Interestingly, the radiation intensity peaks an at angle of 1/(2 γ) and vanishes directly on-axis.) Because the beam is bending, you will get emission along the plane along which the beam is being bent. The energy of the photons produced during synchrotron emission is approximately<ref name="Jackson_Electrodynamics">J. D. Jackson, "Classical Electrodynamics: Second Edition", Wiley (1962)</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Eγ &cong; 3 γ3 c &hbar; / rc
</div>
where &hbar; = 1.05×10-34 J s (if you want the energy in joules) or &hbar; = 6.58×10-16 eV s (if you want the energy in eV). The photon energy will not exceed the energy of the incident particles, if you get a photon energy higher than the energy of the incident particles it just means that the synchrotron radiation spectrum extends up to the incident particle energy.

The fraction of the energy (or power) of the beam turned into synchrotron radiation will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = [1/(4 π ε0)] [4 π / 3] [q² / rc] β2 γ4 [φ / (2 π)] / E
</div>
where β = √[1 - 1/γ2] is the particle's speed measured as a fraction of the speed of light, and 1/(4 π ε0) = 8.99×109 is the Coulomb constant. For highly relativistic electrons (like all practical electron beam weapons emit), β &cong; 1 and we can use
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = 8.85×10-2 [E (GeV)3 [φ / (2 π)] / (rc (m))]
</div>
This fraction cannot be more than 1. If f gets close to one, it means that our original assumption that the beam particles mostly keep their original speed no longer holds, and the beam will lose most or almost all of its energy. Instead, when the beam is being deflected it will only make it through an angle of φ/f and will irradiate a region of length x/f starting at where the beam would have been incident if not for the magnets and extending in the direction the beam was deflected.
</table>

For example, a 1 TeV electron beam will have γ &cong; 2,000,000. So if it's bent by a magnetic field the synchtrotron radiation will spread out in a cone of about 5×10-7 radians. If this beam was shooting at a target 0.1 light seconds away from where it was bent, the radiation would arrive at the target over a region with a radius of about 15 meters with a maximum intensity at about 7.5 meters and no radiation directly along the path the beam was pointing while it was being bent. If the beam was bent by the interplanetary magnetic field from our earlier example, it would produce light with photon energies of around 1.3 eV (near infrared) and the lost energy would be a few parts in a hundred trillion. If it was instead deflected by the Earth's magnetic field you might expect a peak photon energy of tens of keV – in the hard x-ray part of the spectrum – and the fraction of energy in the synchrotron radiation would be somewhere in the low parts in 100,000 or less.

==== Separation of neutral beams ====

Magnetic fields only affect charged objects. So one way to overcome magnetic defenses or environments is to use beams that have no net charge. For example, a proton beam could be neutralized with a co-propagating electron beam with an equal number of protons and electrons. Or, potentially, you could fire an ion beam and then neutralize the ions after acceleration to get a beam of neutral atoms.

But, the magnetic fields are still tugging at the charged particles making up the beam as they go past. This will pull the electrons one way and the protons or ions the other way. As the particles are pulled in opposite directions you get a net positive charge on one side of the beam and a net negative charge on the other side (called polarization). This sets up an electric field inside the beam that acts to pull the particles back together. If the particles are going fast enough and the magnetic field is strong enough, the magnetic force can still separate the charges and will split the beam in two, one with the positive charges and one with the negative charges. But if the particles are not that fast or the magnetic field is not that strong this polarization of the beam will keep it together and let it fly right through the magnetic field.

<table border=1><tr><td>
Let's figure out how to analyze this problem. For definiteness, we will have a beam with an electric current of the positive species of I and an electric current of electrons of -I, all moving in the z direction. The beam radius will be R. The particle speed will be v with the usual relativistic parameters β = v/c and γ = √[1/(1-β2)]. If you know the energy per particle E of the positive species and the mass m of the positive species, γ = 1 + E/(m c2) and then β = √[1 - 1/γ2].

Let's assume that the beam is going through a magnetic field of strength B at an angle θ to the field direction. The quantity B sin(θ) is the component of the magnetic field perpendicular to the direction of the beam.

The charge density ρ of the positive component of the beam will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρL = I / (π v R2).
</div>

It is easiest to analyze the problem in the rest frame of the particles in the beam. Now the particles are at rest. Particles at rest are not affected by a magnetic field, but the change of reference frame Lorentz transforms the initial magnetic field into a combination of electric and magnetic fields in the rest frame. We can ignore the magnetic field, but the electric field is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔A = γ v B sin(θ)
</div>
in a direction perpendicular to both v and B. For our purposes, assume the electric field points in the x direction.

At the same time, relativistic length contraction increases the charge density in the rest frame
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρ = γ ρL.
</div>

Let r be the vector from the beam axis to the point inside the beam. It will be useful to separate this into its Cartesian coordinates. If x and y are the unit vectors in the x and y directions, respectively, then
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
r = x x + y y
</div>

If we consider only the positive component of the beam, from [https://en.wikipedia.org/wiki/Gauss%27s_law Gauss's law] it will have an electric field inside of the beam at a distance r from the center axis of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔+ = ρ r / (2 ε0) = (ρ/(2 ε0)) (x x + y y)
</div>
where ε0 = 8.85×10-12 is the vacuum permittivity. Now let's displace the electrons in the beam by a distance δ in the x direction. By itself, this electron beam will have an electric field of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔- = -(ρ/(2 ε0)) ((x-δ) x + y y)
</div>
By the principle of superposition, the electric field inside the beam with both components is the sum of the electric fields of the separate components including the external field
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔B = [(ρ/(2 ε0)) δ + γ v B sin(θ)]x
</div>
Because the charges are free to move, they will do so as long as a field is applied to them. Thus, the beam becomes stable once the charges separate enough that the field internal to the beam is zero. Therefore
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
δ = -2 ε0 γ v B sin(θ) / ρ = -2 π ε0 v2 R2 B sin(θ) / I
</div>
If |δ| > 2 |R|, the beam will fully separate and each component will go its own way. If δ is small compared to R, the beam's polarization suffices to hold it together in the overlap region. However, where the positive and negative charges do not overlap the electric fields no longer cancel, and the external electric field carries away the outer polarized part of the beam. This gives us a rough threshold for which a magnetic field will quickly separate a neutral beam into its charged components and bend them away from their target
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr><td> B sin(θ) < I / (π ε0 v2 R) <td>       <td> Beam is partially eroded.
<tr><td> B sin(θ) ≥ I / (π ε0 v2 R) <td> <td> Beam promptly separates.
</table>
</div>
For a partially eroding beam, the polarized region that was peeled off by the field still acts to screen the remaining neutral portion, at least to within this approximation, leaving the still-neutral beam to hit the target.
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Neutralized_beam_in_B_field.png|5140 px|frameless]]
<tr><td width=700>
A profile of the beam charge distribution of a neutralized beam in a magnetic field. Negative charge density is shown in yellow, positive charge density in cyan, with neutral charge density in light green. The magnetic field is vertical, the electric field in the beam reference frame is shown in magenta including both the polarization field and the Lorentz transformed electric field resulting from the magnetic field. As positive charges are pulled in the direction of the field and negative charges opposite to the field direction, the polarized boundary will be pulled off the beam by the electric field even though the electric field is zero in nearly all the beam interior. This expanding polarization layer will continue to largely screen the still-neutral portion, however; but the extent to which the screening is only partial would require more sophisticated analysis methods than used here. The material removed by the field will expand to a maximum of the deflection distance of a uniform charge based on the magnetic field by the time it reaches the target.
</table>
</div>
<tr><td>
For a beam of neutral atoms, let a be the radius of the atom and let Eb be the binding energy of the least tightly bound electron to the atom. An external electric field in excess of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
|𝓔A| > Eb / (e a)
</div>
will promptly ionize the atom, ripping away its electron. Here, e = 1.60×10-19 C is the magnitude of the electric charge of the electron. For fields close to but a bit weaker than this, the electron can still rapidly pop off due to quantum tunneling<ref>Ph. Durand and I. Paidarová, “Ionization of the hydrogen atom: from weak to strong static electric fields,” The European Physical Journal D, Vol. 26, pg. 253-259, 2003.</ref>. In general, expect that for fields greater in magnitude than (very approximately) Eb / (10 e a) the electron will be quickly removed, while for fields weaker than this the electrons will stay on their atoms.

It is useful to note that Eb/e is just the binding energy in units of eV. The most weakly bound electrons are usually bound to their atom with about 10 eV of energy. Atoms are typically about 10-10 m in radius. Thus, you can expect fields of around 1010 to 1011 V/m to directly ionize atoms. If we call this threshold value 𝓔T, then a beam will be ionized when passing through a field with strength greater than
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B sin(θ) > 𝓔T / (γ v)
</div>
</table>

Example: Let's consider a 1 GeV neutralized proton beam from a synchrotron, with a beam current of 200 kA in each bunch (incidentally, if a bunch is 5 cm long this means that each bunch packs 38 kJ worth of energy). The speed of the bunch is 87% the speed of light, or 262 million m/s, with a γ of 2.07. Consider when the beam's focus takes it to a radius of 5 cm (or 0.05 m, for our calculation). Plugging this in, we find that the protons in the bunch will separate from the electrons in a field with a transverse component greater than about 2.1 T, and the bunch will pass through the field with only minimal losses if the transverse component is much smaller than this.

Example: We'll take the same 1 GeV beam, but this time recombine the electrons with protons to make hydrogen atoms. An electron is bound to a hydrogen atom with an energy of 13.6 eV (the Ryberg constant) and the radius of the electron orbital in a hydrogen atom is 0.529×10-10 m (the Bohr radius). This gives a threshold electric field for prompt ionization of hydrogen of 2.57×1011 V/m. We will get this with a transverse component of the magnetic field of 474 T. Above this field the atoms promptly ionize and electrons are stripped away from the protons. For fields higher than roughly 1/10 of this – so about 40 to 50 T – you may still get significant ionization by quantum tunneling; lower than this your losses from the beam due to field-induced ionization are likely negligible.

==== Magnetic shields ====

What happens when we can't rely on magnetic fields in the environment to protect us? One idea is to produce powerful artificial magnetic fields around our spacecraft to deflect incoming particle beams. This is very similar in concept to using active magnetic shielding to protect against space radiation – [[Space_Radiation#Magnetic_Shielding|check out our article on space radiation for more]].
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr>
<td width=100>
[[File:Unconfined_FRC_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:racetrack_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:Magnetic_shielding_Halback_Array.png|200 px|frameless]]
<tr><td colspan=3>
Some potential configurations for magnetic shields. The currents generating the fields are in green, the magnetic fields themselves are shown in magenta<ref name="Battiston2012">R. Battiston, W.J. Burger, V. Calvelli, R. Musenich, V. Choutko, V.I. Datskov, A. Della Torre, F. Venditti,
C. Gargiulo, G. Laurenti, S. Lucidi, S. Harrison, and R. Meinke, "ARSSEM Active Radiation Shield for Space Exploration Missions", Final Report ESTEC Contract N° 4200023087/10/NL/AF : “Superconductive Magnet for Radiation Shielding of Human Spacecraft” (2012) https://arxiv.org/abs/1209.1907 https://www.researchgate.net/publication/265945847_Active_Radiation_Shield_for_Space_Exploration_Missions</ref><ref name="Desiati2022">Paolo Desiati and Elena D'Onghia, "CREW HaT: A Magnetic Shielding System for Space Habitats", arXiv:2209.13624 [physics.space-ph] https://doi.org/10.48550/arXiv.2209.13624</ref>.
</table>
</div>

One immediate issue is that you want to keep very high magnetic fields out of the crew area. Fields in excess of 0.5 T can start to be unpleasant, and it is recommended to keep the field below 0.2 T for continuous exposure.

Strong magnets can suffer from high self-forces acting to tear them apart. Most shielding designs will have multiple magnetic coils, which will exert high forces and torques on each other. This will result in the need for strong structural bracing to keep the magnets intact and in place. Such strong structural designs will impose significant extra mass on the spacecraft.

Consider a magnetic field configuration that can keep radiation out of a certain region of space. There is no trajectory of a charged radiation particle up to a certain energy that will allow those particles to enter the protected region. But the dynamics of charged particles in a magnetic field are the same whether going forward or backward in time. If there is no way for a particle to get in, by this time reversal symmetry any particle which is in that forbidden region can never escape. Consequently any field configuration that is a magnetic trap is also a magnetic shield; any field configuration that is a magnetic shield is also a magnetic trap. A spacecraft with a magnetic shield may end up surrounded by strong radiation belts.

Using a magnet to bend a weaponized electron beam very often ends up with the beam being bent so hard that it dumps most of its energy in the form of synchrotron radiation at gamma ray energies. Because gamma rays are not charged, they will not be deflected by the field. Thus, if you would have been hit by the electron beam you will be hit instead by the gamma rays.

As an example, consider a 1 TeV electron beam. We'll surround our spacecraft with a 10 T magnetic field in a 100 meter radius region just outside our spacecraft. This will deflect the beam by 30 meters, which is enough to make it miss. But we are not saved, because the beam dumps nearly all of its energy into synchrotron radiation with an energy spectrum of gamma rays up to 13 GeV. This radiation will hit us in a strip about 2.4 meters long and not much more than the width of the original beam with essentially all of the energy of the original beam.

==== Magnet drones ====

If a magnetic shield around your spacecraft will just end up killing you from synchrotron radiation, what if you put the magnetic shield on a drone and put that drone between yourself and the enemy trying to shoot you? This lets you use a much weaker magnet because you don't need nearly so high of a deflection angle to make the shot miss. This means that less beam energy will turn into synchrotron radiation that could kill you instead and will spread what synchrotron radiation does occur out over a much wider area when it reaches your spacecraft, hopefully so that most of it just misses your spacecraft.

It does mean that your enemy will start shooting your drones so that they can eventually take a shot at you. But hopefully by that time you've already cleared out their drones and cored their spacecraft with your own synchrocannon or UREB.

==== Plasma magnets ====

Usually, to make a really big magnetic field you would run current through a really big conductive loop. But really big things are heavy and expensive. One way around this could be to use the ambient space plasma to carry the electric current rather than any kind of wire. In order to make this current, you use the same mechanism used in a rotary induction electric motor: you generate a rotating magnetic field and induce a current in the conductive loop (in this case, the space plasma). How do you make a rotating magnetic field? The method proposed<ref name="Slough2006">John Slough, "The Plasma Magnet", NASA Institute for Advanced Concepts Phase II Final Report (2006) https://www.niac.usra.edu/files/studies/final_report/917Slough.pdf</ref> is to use an antenna of four much smaller current loops as electromagnets at 90° to each other, and sinusoidally vary the current through the loops with a respective 90° phase difference between them. When this is done, a circular current is induced in the space plasma and the self-force of a magnetic field on its generating current inflates the current loop to much larger radii. You end up with a current sheet in the plane of the rotating field's rotation circulating about the antenna nearly synchronously with the field rotation. This current then creates a normal dipolar magnetic field as with any current loop. The current sheet and the resulting magnetic field expand until the magnetic field's pressure equals that of the ambient space plasma. This gives an enormous region of magnetic field from a relatively small and lightweight system.

The system proposed in the initial study would use an antenna approximately 10 meters in radius and produce a region of field approximately 30 km in radius at Earth's orbit. The field inside would not be constant - at the edges it would be 50 nT, giving the right pressure to keep out the solar wind at Earth's orbit. A circulating current sheet produces a field that falls off as 1/r with distance, meaning that adjacent to the antenna the field would be about 0.15 mT. Using the methods in the box below, we can estimate that this particular design as part of a spacecraft's shields will not deflect a TeV electron beam by more than a few centimeters, which is usually insufficient to force a miss. However, it could potentially be useful for a drone magnet, where the much larger distance between the magnetic field and the target spacecraft would allow a small deflection angle to give perhaps tens or low hundreds of meters change in beam location near the target at near light second ranges. The relatively low field and gradual bend of the incident beam suggests that synchrotron radiation will not be an issue; however this does not exclude more powerful plasma magnet designs causing significant synchrotron radiation effects.

<table border=1><tr><td>
Calculating the deflection from a spatially varying magnetic field affected by the solar wind is no trivial task. However, we can make an approximation that both should be within about an order of magnitude of the true value and also represent an upper limit to the deflection angle; although it does neglect the influence of the driving rotating magnetic field which could in some cases increase the deflection.

We will take R0 as the radius of the antenna. B0 is the field strength at the antenna. Rm is the distance at which the magnetic influence stops because the magnetic pressure equals the space plasma pressure. Bm is the field strength at Rm. According to Slough<ref name="Slough2006"></ref>, the magnetic field strength B(r) at any distance from the antenna center r between R0 and Rm is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B(r) = B0 R0 / r.
</div>
Therefore, once we know Bm and Rm, B0 is completely determined
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B0 = Bm Rm / R0.
</div>

The force on a particle with charge q passing perpendicularly through a magnetic field B(r) at a speed v is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
F(r) = q v B(r)
</div>
For an electron beam, v is so close to c as to make no difference. If the particle is moving down the y direction and the field is in the z direction, the force will be in the x direction. We assume that the deflection will be small, so we can choose a straight-line path in the y direction offset from the center by an amount x. The plasma bubble radius is Rm, so the particle will enter at ymax = √[Rm2-x2] and leave at ymin = -√[Rm2-x2].

The momentum change the particle experiences from this force is the time integral of the force over its path. Because of reflection symmetry about y = 0, we only need to integrate from ymax to 0 and then multiply by 2
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] dt F(r)
<tr> <td> <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] (dy/v) F(r)
<tr> <td> <td> = <td align="left">2 q B(0) R0 [Integral from 0 to √[Rm2-x2]] dy 1/√[y2 + x2]
<tr> <td> <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/x]
</table>
</div>
This holds for |x| > R0. But the expression for the field is not valid for r < R0 and the integral will give nonsense and diverging results if we evaluate for |x| < R0. In this case, we can put the integral lower bound at r = R0, or y = √[R02-x2]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/(√[R02-x2] + R0)]
</table>
</div>

The deflection angle is now estimated as
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = px / p
</div>
where the total momentum of the particle p can be found from the methods in the first box in the Magnetic Fields section.

The deflection for a shot aimed right at the spacecraft at the position of the spacecraft can be estimated by setting x = 0 and only taking the first half of the integral (so ignore the factor of 2)
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
q B(0) R0 ln[Rm/R0] / (p Rm)
</div>
This is an additional overestimate because it fails to account for how the deflection builds up over distance during the integration, but it should still give something of an order-of-magnitude and upper limit estimate of what kind of deflections might be expected.
</table>

=== Electric fields ===

If electric fields can accelerate charged particles, maybe they can keep the charged particles in a weaponized particle beam away from your spacecraft? To completely block a particle beam, you would need a voltage on your spacecraft equal to the particle energy in eV of the particles in the beam. That is, if a 1 GeV proton beam is coming at you, you'd need an electric potential of a billion volts to stop that beam. This leads to three immediate problems.

The first is that particles of the opposite charge will be attracted to your spacecraft. Because space isn't really empty but rather a rarefied plasma (called the solar wind in a solar system or the interstellar medium out between stars), there will be plenty of particles of the opposite charge around to attract. And they will hit with an energy in eV equal to the voltage on your spacecraft. In our example of defending against a 1 GeV proton beam with a gigavolt field, the electrons from the solar wind will come in and hit with an energy of 1 GeV each. This will generate dangerous radiation.

The second problem is a consequence of the first. Each of those GeV electrons from our example need to get their energy from somewhere, which turns out to be from the electric field we are using as shielding. You will need to make up for this energy loss with power from your electrical system, which gives a constant drain on your power. Some estimates put the electrical effort to maintain a several hundred megavolt shield in space at many tens of gigawatts<ref name="Mechmann2019">Claire Mechmann, "Analysis of Proposed Active Radiation Shielding Design Concept for Spacecraft" (2019) Thesis, College of Engineering and Science of Florida Institute of Technology</ref>.

And the third problem is similar to one for magnetic fields. Your electrodes will experience high self forces and forces between separate electrodes that will try to tear your shield generator apart. Consequently you will need heavy structural reinforcement in order for your shield to work.

There are potential ways to mitigate some of these problems. Interested readers are directed to [[Space_Radiation#Electrostatic_Shielding|the active shielding section in our page on space radiation]] for more of the juicy details; there's no need to repeat the same stuff here.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Elctrostatic_active_shielding.png|200 px|frameless]]
<td>
[[File:Electrostatic_active_shielding_2.png|200 px|frameless]]
<tr><td colspan=2>
Proposed designs for electrostatic shields to protect from positively charged particles<ref name="Tripathi2006">Ram K. Tripathi, John W. Wilson, and Robert C. Youngquist, "Electrostatic Active Radiation Shielding - Revisited", 2006 IEEE Aerospace Conference, Big Sky, MT, USA, 2006, pp. 9 pp.-, doi: 10.1109/AERO.2006.1655760.</ref><ref name="Chowdhury2023">Rajarshi Pal Chowdhury, Luke A. Stegeman, Matthew L. Lund, Dan Fry, Stojan Madzunkov, and Amir A. Bahadori, "Hybrid methods of radiation shielding against deep-space radiation", Life Sciences in Space Research, Volume 38, 2023, Pages 67-78, ISSN 2214-5524, https://doi.org/10.1016/j.lssr.2023.04.004.</ref>.
</table>
</div>

== Damage ==

Particle beams are, first and foremost, devices that produce beams of [[Nuclear_radiation|ionizing radiation]]. The radiation of weaponizable particle beams is extremely penetrating; thus of the [[Particle_Accelerators#Energy_loss_in_matter|interaction mechanisms listed for radiation from particle beams]], we mostly need to worry about bremsstrahlung and nuclear collisions and these produce extensive secondary showers of radiation that increase the penetration of the radiation exposure. Because of this, any target hit by a particle beam weapon will be irradiated, and radiation dose will always be one of the primary damage mechanisms of a particle beam weapon strike. Further, this radiation will be distributed along a track passing deeply through the material of the target, allowing it to affect critical systems through heavy armor, structural material, robust machine components, and passive material such as fuel load, coolant, or cargo. As the radiation deposits its energy into the target material, that material will be heated. If the target is heated sufficiently, it may experience [[Beam-Target_Interactions|thermal or mechanical]] damage from melting, cooking, combustion, vapor explosions, and warping or cracking due to thermal expansion.

The discussion of the radiation penetration into armor also applies to the same radiation penetrating through the target. So the formulas and rules of thumb for estimating the depth of radiation penetration into armor can also be used to estimate how far into a tank or spacecraft a particle beam strike delivers effects.

In addition to the radiation and thermal effects, a particle beam can also produce a sharp electromagnetic pulse that can damage semiconductor electronics.

== Hitting Your Target ==

It doesn't do much to have the most horrifically powerful death ray if it can't hit. What are the challenges for directing your beam of radioactive fury onto your enemies?

We have already mentioned the challenges of stray magnetic fields that are part of the ambient environment messing with your targeting. This will be a challenge when you are launching beams of charged particles. And we've discussed some of the means of getting around this, like using neutralized beams.

But another issue that will be a challenge at very long ranges is dealing with [[Jitter|jitter]]. A weapon made out of a linac will be very long, and so it will be rather wibbly. This could make its jitter problems worse. A synchrotron might not be so badly affected; and for any future hypothetical high energy wakefield accelerators, no one really knows what they'll be like. There are certainly tricks you can play to reduce jitter (see the previously linked jitter page), but at ranges approaching a light second it's certainly plausible that it will be difficult to land a hit.

== 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, Gerrit Bruhaug, and Luke Campbell for writing

Gerrit Bruhaug and Luke Campbell for scientific research, math wrangling and advise

ATriffleMind and Luke W. Campbell for proof reading.

==References==

[[Category:Beams]][[Category:Physics & Engineering‏‎]][[Category:Engineering‏‎]][[Category:Military Technology]][[Category:Warfare‏‎]]

Particle Beam Weapons

2026-10-27T04:58:18Z

Lwcamp: /* Plasma magnets */

{{PageConstructionNotice}}

''For a general overview on particle accelerator technology, see [[Particle Accelerators]]

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.

::: Z = 1.32×102 (γ2-1)3/4 r / √I<ref name=USPAS2016>Martin Reiser, "Theory and Design of Charged Particle Beams", 2008, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim</ref>

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

::: Z = β γ c r √[m/T]

Where Z is the doubling distance like before, r is the initial beam radius, T is the beam temperature in eV, m is the ion mass and β, c, and γ 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.

== Defenses ==

Sure, it's great to have a great big particle cannon to blast your enemies. But what if you are the one being shot at? In that case, it's best to figure out ways to keep from being blown to kingdom come.

=== Armor ===

The simplest way to defend against the high energy particles of a particle beam is to pile a whole bunch of stuff up and put it in the way of the beam. Particle beams may produce extremely penetrating radiation, but enough stuff will block most of that radiation and may allow you to survive. The details depend on what kind of particle is being thrown at you, and are explained below. However, a general rule of thumb is that you probably want about 10 tons of material per square meter of surface you want to armor in order to get sufficient protection.

==== Electrons ====

At the very high energy of an electron accelerated by a weapons-grade particle beam, the primary interaction with matter will be the emission of bremsstrahlung photons in the gamma-ray range as the electron slams into particles of matter. These high energy photons themselves primarily interact by the pair production mechanism – the photon disappears, but leaves behind a highly energetic positron and electron. These two particles are themselves energetic enough to produce further bremsstrahlung, and so the process continues. You get the formation of an electron-gamma shower (also called an electromagnetic cascade), which only stops after either the particles have penetrated through the target or the initial energy of the electron has been spread through so many shower particles (including all the electrons, positrons, and photons) that no particle has enough individual energy to pair produce or to be likely to bremsstrahlung again before it is stopped by other mechanisms.

To stop electrons, you'll want the atoms in your armor to be as heavy as you can practically get them. High atomic number both encourages bremsstrahlung and pair production, giving a double whammy for stopping that cascade. However, occasionally one of those electrons, positrons, or gammas will hit a nucleus and knock out neutrons. So you probably want a back-up layer of several tens of centimeters of hydrogen rich material such as plastic or water mixed with neutron absorbing materials like boron, lithium, cadmium, or gadolinium. High density polyethylene loaded with 5% boron by weight is a typical go-to neutron blocker.

The actual methods to calculate the penetration of the electromagnetic cascade is given are the box below. However, for some examples:
<ul>
<li> A 10 GeV beam incident on thick iron (or steel) will stop cascading at about 9 cm, will deposit half of its energy in the first 12 cm, and will have dumped 98% of its energy within 35 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 2.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 100 GeV beam incident on thick iron (or steel) will stop cascading at about 13 cm, will deposit half of its energy in the first 16 cm, and will have dumped 98% of its energy within 47 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 3.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick iron (or steel) will stop cascading at about 17 cm, will deposit half of its energy in the first 20 cm, and will have dumped 98% of its energy within 59 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 4.6 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick tungsten will stop cascading at about 3.5 cm, will deposit half of its energy in the first 4 cm, and will have dumped 98% of its energy within 12 cm. 90% of the cascade will be contained within about 0.93 cm and 95% of the cascade will be contained within 1.86 cm of the beam axis. You will get 98% protection with a mass thickness of 2.3 tons/m2 and an additional 1/e reduction for each further 0.13 tons/m2 beyond that.
<li> A 1 TeV beam incident on water will stop cascading at about 290 cm, will deposit half of its energy in the first 340 cm, and will have dumped 98% of its energy within 1030 cm. 90% of the cascade will be contained within about 11 cm and 95% of the cascade will be contained within 22 cm of the beam axis. You will get 98% protection with a mass thickness of 10.3 tons/m2 and an additional 1/e reduction for each further 0.72 tons/m2 beyond that.
</ul>
Note that for sufficiently intense particle beams, 98% protection may not be enough. You might need to throw on several more tons per square meter to be sure.

<table border=1>
<tr><td> How Far it Goes:
(This section follows methods presented in Leroy and Rancoita<ref>C. Leroy and P-G Rancoita, "Physics of cascading shower generation and propagation in matter: principles of high-energy, ultrahigh-energy and compensating calorimetry", Reports on Progress in Physics 63 (2000) 505–606. DOI: 10.1088/0034-4885/63/4/202</ref>.)

When trying to estimate the extent of the irradiated region of the target, it is helpful to know the radiation length X0 in the material. This is the distance that an electron will travel, on average, before emitting a bremsstrahlung photon. Conveniently, it is also the distance a high energy photon will travel through the material before pair producing. The radiation length can be found from
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   = 4 α  
<td nowrap="nowrap"> NA
<td rowspan="2" nowrap="nowrap" align="right">   Z (Z+ζ) re2 ln
<td rowspan="2" nowrap="nowrap" align="right"> [
<td nowrap="nowrap"> 183
<td rowspan="2" nowrap="nowrap" align="right"> ]
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> A
<td style="border-top:solid 1px black;"> Z1/3
</table>
</div>
where α = 1/137 is the fine structure constant, NA = 6.02214076×1023 is Avogadro's number (or one mole), A is the atomic weight of the atoms in the material (usually expressed in grams per mole), Z is the atomic number (nuclear charge) of the atoms, and re = 2.8179403205×10-13 cm is the classical electron radius. ζ is a constant of approximately 1.3. This gives the thickness of material in terms of the areal density (mass of material per unit area perpendicular to the beam) – if you use A in g/mol and re in cm the result will be in cm2/g and X0 will be in g/cm2. If your target is made up of more than one kind of atom, you can get the total radiation length by finding the radiation length of each atom of type i, X0i and using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   =  
<td rowspan="2" nowrap="nowrap" align="right"> Σi  
<td nowrap="nowrap"> fi
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> X0i
</table>
</div>
where fi is the fraction by weight of the atom type i. Once you have the radiation length in terms of areal density, you can easily convert it to distance by dividing by the density ρ. If X0i is in g/cm2 and density is in g/cm3, then X0i/ρ will be in cm.

We will also need to know the critical energy εc. This is the energy where an electron is losing energy as rapidly to ionization as it is to radiation. That is, it is the energy at which the cascade stops multiplying, even though the individual particles already made can continue traveling for some distance. The critical energy can be estimated using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> εc = B  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> Z X0
<td rowspan="2" nowrap="nowrap" align="right"> )
<td nowrap="nowrap"> h
<tr>
<td style="border-top:solid 1px black;"> A
<td>  
</table>
</div>
where B = 2.66 MeV and h = 1.11.

We can now describe the distribution of radiation in the target material. The depth tmax where the cascade stops multiplying (in g/cm2) for an incident electron of energy E (in MeV) is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tmax = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> - 1
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
Similarly, the depth where half the energy of the beam has been deposited (in g/cm2) is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tcg = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> + 0.37
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
As usual, you can convert to a depth in cm by dividing by the material density. 98% of the beam energy will be deposited within a distance of &cong; 3 tcg. Beyond this, the beam energy will continue to fall off with thickness t approximately as an exponential exp[-t/λ], where the characteristic attenuation length λ &cong; 2 X0.

The width of a cascade, where 90% of the energy is contained, is approximately given by
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> RM =  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> EM
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
where EM &cong; 21.2 MeV. 95% of the radiation will be contained within a distance of 2 RM from the beam axis.
</table>

==== Protons and Ions ====

<table align=right><tr><td>
[[File:Regolith_Shielding.png|350 px|frameless]]
<tr><td width=350>
Relative effect of particle radiation (compared to no shielding) from galactic cosmic rays behind different thicknesses of water, aluminum, and lunar regolith<ref name="Slaba2022">Tony C. Slaba, "Radiation Shielding for Lunar Missions: Regolith Considerations", LSIC Crosstalk 7/18/2022 https://lsic.jhuapl.edu/uploadedDocs/focus-files/1604-E&C%20+%20EE%20Monthly%20Meeting%20-%202022%2007%20July_Presentation%20-%20NASA%20Slaba.pdf</ref>.
</table>
Protons and ions are grouped together because ions are made of protons and neutrons and when you accelerate these things for a weapon it is the protons and neutrons that are going to be doing the heavy lifting for blasting your target. This is because protons and neutrons are both about 2000 times more massive than electrons so any electrons along for the ride will contribute negligibly to the energy. Protons and neutrons together are called nuclear particles, and both behave quite similarly in nuclear collisions.

Nuclear particles are themselves made of quarks and can interact via the strong nuclear force. At the energies appropriate for weapons use, essentially all of the way protons or ions are blocked is through [[Particle_Accelerators#Nuclear_collisions|direct collisions with the atomic nuclei]] of the armor material. One or more quarks in the projectile will slam into quarks in the target nucleus and stretch them away from the rest of the quarks in the nuclear particles. Physics tells us that you can't have isolated quarks, so the "string" attaching the quark to its neighbors stretches and stretches until it snaps. Snapping releases so much energy that more quarks are made that clump together into all sorts of other nuclear particles, both exotic and familiar. This process is called a hadronic jet. The only particles produced in these jets long-lived enough to travel any noticeable distance through the armor material will be protons, neutrons, the charged pi mesons (or pions), and some of the K mesons (or kaons). The particles too short lived to go far enough to matter will decay into other particles, and these particles are often very high energy gamma rays. Much like with electrons, the long-lived secondary particles can be sufficiently energetic to smash into other nuclei and cause more hadronic jets (or, if they are low enough in energy, just knock more nuclear particles out of target nuclei) and this produces a similar shower or cascade. But in addition, the armor nuclei that were hit will be very hot after the impact from all of the energy given to them from the incident particles. These nuclei will cool off by evaporation – but instead of evaporating away molecules, they evaporate off nuclear particles from the nucleus. These evaporated particles are mostly neutrons, so any armor against proton or ion beam strikes will produce quite a bit of neutron radiation.

Eventually, the original particle's energy will be distributed among so many other particles that the charged particles will come to a stop by making ion tracks rather than from colliding with more nuclei, and the cascade halts. But you still have all the neutrons to worry about. And the gamma rays as well, which are high enough energy to kick off an electromagnetic cascade in the same way that electrons do.

Unlike electrons, which have reasonable approximations of their cascade behavior from relatively simple equations, the cascades of protons and ions usually require sophisticated simulations or experiments to get the resulting distribution of radiation. However, we can get a reasonable idea of their general behavior by comparing them to [[Space_Radiation#Shielding_Against_Cosmic_Rays|cosmic ray particles interacting with shielding]]. Cosmic rays are energetic protons and ions, after all. From this we see that a relatively small amount of armor – in the range of 1 to 3 tons per square meter – can actually make the radiation dose worse. The incident particles will knock off so many secondary particles in their shower that the dose is multiplied from so many tracks passing through the target. Hydrogen rich materials such as water and plastics work the best against proton and ion beam particles; heavy elements are the worst and produce the most secondaries. You could probably obtain adequate radiation protection from most incident beams with about 10 tons per square meter of hydrogen rich material as armor. However, you will want to add neutron absorbing elements to your hydrogen rich armor, such as lithium, boron, cadmium, or gadolinium. Further, in order to deal with the gamma rays you will want a layer of at least several centimeters of heavy elements, such as tungsten or lead, behind the hydrogen rich particle armor.

==== Structural Considerations ====

When your armor material absorbs a radiation beam, the energy of that radiation turns into heat inside the armor where the radiation was absorbed. If the beam pulse has enough energy as was sufficiently tightly focused, this heat can damage your armor. Your armor may melt in places, or thermal expansion could cause it to crack or warp. Perhaps the worst thing it could do is vaporize or flash into a plasma, because the pressure from the vapor or plasma can be sufficient to plastically deform the armor. This leaves a hole along the beam path where the expanding gas or plasma blasted out a long, deep crater. This means a beam that fires rapid shots spaced maybe milliseconds apart can have subsequent shots go down the same hole so your armor will not protect against them. Each shot will dig out a deeper hole until either your armor is penetrated or the attacking beam runs out of shots in its pulse train.

A consequence of this is that when designing your armor you may wish to use refractory materials with a high structural strength (like tungsten), or that is otherwise resistant to being blasted apart by its own evaporated remains.

=== Magnetic fields ===

[[Particle_Accelerators#Magnetic_fields|A charged particle crossing a magnetic field will bend so as to spiral around the field lines]]. If you try to shoot your particle beam through a magnetic field, it will bend away from the direction you shot it. This can make it hard to hit your target. Unfortunately for you, there are magnetic fields everywhere – around planets, and even trapped in the solar wind as it crosses the solar system.

As an example, the Earth has a magnetic field strength of about 5×10-5 tesla near its surface, and falling off farther away. If you are shooting a 1 TeV electron beam past the Earth, your beam is likely to bend by several milliradians (with the exact deflection depending on how close you are skimming the Earth). If you are in a medium Earth orbit and shooting at another target also in a medium Earth orbit on a trajectory that goes near the Earth, then without correcting for the magnetic field your beam will likely be off-target by tens to hundreds of kilometers.

For another example, the interplanetary magnetic field is about 10-9 tesla at Earth's distance from the sun. If you shoot that same 1 TeV electron beam at a target 0.1 light seconds away, you are likely to be off by a couple hundred meters.

<table border=1>
<tr><td>
The radius of curvature of the bend of a particle in a magnetic field is called the gyroradius rc. For a particle with a charge q, mass m, and kinetic energy E, find the relativistic momentum p
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
p = √[E2 - m2 c4] / c
</div>
To convert from energy units of eV to joules, multiply by 1.60×10-19. An electron has a mass of 9.11×10-31 kg, a proton of 1.67×10-27 kg, and an ion of 1.66×10-27×A kg, where A is the atomic weight of the ion. The speed of light c = 3.00×108 m/s.

In a magnetic field of strength B with an angle θ between the field direction and the direction of motion
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rc = |p sin(θ) / (q B)|
</div>
For electrons, q = -e, where the fundamental charge e = 1.60×10-19 coulomb. For a proton, q = +e. For an ion, multiply e by the charge state of the ion.

If the field acts over a distance d that is much smaller than the gyroradius, the deflection angle φ is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = d / rc
</div>
The deflection x at a target at a range R for small φ (φ &Lt; 1) will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
x = φ R.
</div>

Note that d will always be less than or equal to x. Also note that if d is not smaller than twice the gyroradius the beam will never reach the target because it will completely spiral around the field lines before it reaches the target.
</table>

So, it is clear that as the defender it will be helpful to hang out in areas with high magnetic fields, and keep some distance between you and the attacker. Meanwhile, the attacker will benefit from being closer where small deflections in the beam trajectory don't translate into large excursions from where the beam is aimed (and also because there will be less space filled with fields between the two of you). But the attacker can also try to compensate for known fields. If you happen to know that the magnetic fields are between you and your target, you can adjust your fire so that the deflection of the field bends your beam onto the target. This is not a simple task. It is easy enough to measure the field strength and direction where you are, but predicting how it changes over the path of your shot does not have an obvious solution. You can also try to gain intelligence on likely battle sites before you ever fight on them – for example, getting good data on the structure of the fields of a planet using spies and technical intelligence during peacetime (heck, on Earth this information is published for free in academic journals and by government agencies such as NOAA). Planetary magnetospheres are also affected by the solar wind, so different solar weather will change the fields you have to shoot through. But space weather forecasts and predictive simulations of the solar wind interacting with a planetary magnetosphere can give you a good idea of what the field structure is like. But even with all that, there are changes in the fields that happen on a scale of around a second ... anyone who has ever seen the auroras dancing across the sky in the polar night can see this. Compensating for these changes will be a significant challenge.

What would really help is being able to see the path of your own beam. This would allow you to see how far off you were from your target and in what direction, and then adjust fire to walk your shots onto the target. Unfortunately, particle beams in space are invisible. There might be ways to track them, but if so we don't know what they are.

==== Synchrotron radiation ====

When charged particle trajectories are bent, they emit [[Particle_Accelerators#Synchrotron_radiation|synchrotron radiation]]. This can cause the particle beam to lose energy, but if the bend is too tight it can also generate high energy gamma rays that ignore magnetic fields and directly strike the target. Synchrotron radiation can generally be ignored for proton or ion beams, but can be significant for electron beams.

<table border=1><tr><td>
For a beam particle with energy E and mass m, the synchrotron radiation is concentrated in a forward-directed cone with an opening angle of approximately 1/γ, where γ = 1 + E/(m c2). (Interestingly, the radiation intensity peaks an at angle of 1/(2 γ) and vanishes directly on-axis.) Because the beam is bending, you will get emission along the plane along which the beam is being bent. The energy of the photons produced during synchrotron emission is approximately<ref name="Jackson_Electrodynamics">J. D. Jackson, "Classical Electrodynamics: Second Edition", Wiley (1962)</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Eγ &cong; 3 γ3 c &hbar; / rc
</div>
where &hbar; = 1.05×10-34 J s (if you want the energy in joules) or &hbar; = 6.58×10-16 eV s (if you want the energy in eV). The photon energy will not exceed the energy of the incident particles, if you get a photon energy higher than the energy of the incident particles it just means that the synchrotron radiation spectrum extends up to the incident particle energy.

The fraction of the energy (or power) of the beam turned into synchrotron radiation will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = [1/(4 π ε0)] [4 π / 3] [q² / rc] β2 γ4 [φ / (2 π)] / E
</div>
where β = √[1 - 1/γ2] is the particle's speed measured as a fraction of the speed of light, and 1/(4 π ε0) = 8.99×109 is the Coulomb constant. For highly relativistic electrons (like all practical electron beam weapons emit), β &cong; 1 and we can use
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = 8.85×10-2 [E (GeV)3 [φ / (2 π)] / (rc (m))]
</div>
This fraction cannot be more than 1. If f gets close to one, it means that our original assumption that the beam particles mostly keep their original speed no longer holds, and the beam will lose most or almost all of its energy. Instead, when the beam is being deflected it will only make it through an angle of φ/f and will irradiate a region of length x/f starting at where the beam would have been incident if not for the magnets and extending in the direction the beam was deflected.
</table>

For example, a 1 TeV electron beam will have γ &cong; 2,000,000. So if it's bent by a magnetic field the synchtrotron radiation will spread out in a cone of about 5×10-7 radians. If this beam was shooting at a target 0.1 light seconds away from where it was bent, the radiation would arrive at the target over a region with a radius of about 15 meters with a maximum intensity at about 7.5 meters and no radiation directly along the path the beam was pointing while it was being bent. If the beam was bent by the interplanetary magnetic field from our earlier example, it would produce light with photon energies of around 1.3 eV (near infrared) and the lost energy would be a few parts in a hundred trillion. If it was instead deflected by the Earth's magnetic field you might expect a peak photon energy of tens of keV – in the hard x-ray part of the spectrum – and the fraction of energy in the synchrotron radiation would be somewhere in the low parts in 100,000 or less.

==== Separation of neutral beams ====

Magnetic fields only affect charged objects. So one way to overcome magnetic defenses or environments is to use beams that have no net charge. For example, a proton beam could be neutralized with a co-propagating electron beam with an equal number of protons and electrons. Or, potentially, you could fire an ion beam and then neutralize the ions after acceleration to get a beam of neutral atoms.

But, the magnetic fields are still tugging at the charged particles making up the beam as they go past. This will pull the electrons one way and the protons or ions the other way. As the particles are pulled in opposite directions you get a net positive charge on one side of the beam and a net negative charge on the other side (called polarization). This sets up an electric field inside the beam that acts to pull the particles back together. If the particles are going fast enough and the magnetic field is strong enough, the magnetic force can still separate the charges and will split the beam in two, one with the positive charges and one with the negative charges. But if the particles are not that fast or the magnetic field is not that strong this polarization of the beam will keep it together and let it fly right through the magnetic field.

<table border=1><tr><td>
Let's figure out how to analyze this problem. For definiteness, we will have a beam with an electric current of the positive species of I and an electric current of electrons of -I, all moving in the z direction. The beam radius will be R. The particle speed will be v with the usual relativistic parameters β = v/c and γ = √[1/(1-β2)]. If you know the energy per particle E of the positive species and the mass m of the positive species, γ = 1 + E/(m c2) and then β = √[1 - 1/γ2].

Let's assume that the beam is going through a magnetic field of strength B at an angle θ to the field direction. The quantity B sin(θ) is the component of the magnetic field perpendicular to the direction of the beam.

The charge density ρ of the positive component of the beam will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρL = I / (π v R2).
</div>

It is easiest to analyze the problem in the rest frame of the particles in the beam. Now the particles are at rest. Particles at rest are not affected by a magnetic field, but the change of reference frame Lorentz transforms the initial magnetic field into a combination of electric and magnetic fields in the rest frame. We can ignore the magnetic field, but the electric field is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔A = γ v B sin(θ)
</div>
in a direction perpendicular to both v and B. For our purposes, assume the electric field points in the x direction.

At the same time, relativistic length contraction increases the charge density in the rest frame
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρ = γ ρL.
</div>

Let r be the vector from the beam axis to the point inside the beam. It will be useful to separate this into its Cartesian coordinates. If x and y are the unit vectors in the x and y directions, respectively, then
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
r = x x + y y
</div>

If we consider only the positive component of the beam, from [https://en.wikipedia.org/wiki/Gauss%27s_law Gauss's law] it will have an electric field inside of the beam at a distance r from the center axis of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔+ = ρ r / (2 ε0) = (ρ/(2 ε0)) (x x + y y)
</div>
where ε0 = 8.85×10-12 is the vacuum permittivity. Now let's displace the electrons in the beam by a distance δ in the x direction. By itself, this electron beam will have an electric field of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔- = -(ρ/(2 ε0)) ((x-δ) x + y y)
</div>
By the principle of superposition, the electric field inside the beam with both components is the sum of the electric fields of the separate components including the external field
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔B = [(ρ/(2 ε0)) δ + γ v B sin(θ)]x
</div>
Because the charges are free to move, they will do so as long as a field is applied to them. Thus, the beam becomes stable once the charges separate enough that the field internal to the beam is zero. Therefore
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
δ = -2 ε0 γ v B sin(θ) / ρ = -2 π ε0 v2 R2 B sin(θ) / I
</div>
If |δ| > 2 |R|, the beam will fully separate and each component will go its own way. If δ is small compared to R, the beam's polarization suffices to hold it together in the overlap region. However, where the positive and negative charges do not overlap the electric fields no longer cancel, and the external electric field carries away the outer polarized part of the beam. This gives us a rough threshold for which a magnetic field will quickly separate a neutral beam into its charged components and bend them away from their target
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr><td> B sin(θ) < I / (π ε0 v2 R) <td>       <td> Beam is partially eroded.
<tr><td> B sin(θ) ≥ I / (π ε0 v2 R) <td> <td> Beam promptly separates.
</table>
</div>
For a partially eroding beam, the polarized region that was peeled off by the field still acts to screen the remaining neutral portion, at least to within this approximation, leaving the still-neutral beam to hit the target.
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Neutralized_beam_in_B_field.png|5140 px|frameless]]
<tr><td width=700>
A profile of the beam charge distribution of a neutralized beam in a magnetic field. Negative charge density is shown in yellow, positive charge density in cyan, with neutral charge density in light green. The magnetic field is vertical, the electric field in the beam reference frame is shown in magenta including both the polarization field and the Lorentz transformed electric field resulting from the magnetic field. As positive charges are pulled in the direction of the field and negative charges opposite to the field direction, the polarized boundary will be pulled off the beam by the electric field even though the electric field is zero in nearly all the beam interior. This expanding polarization layer will continue to largely screen the still-neutral portion, however; but the extent to which the screening is only partial would require more sophisticated analysis methods than used here. The material removed by the field will expand to a maximum of the deflection distance of a uniform charge based on the magnetic field by the time it reaches the target.
</table>
</div>
<tr><td>
For a beam of neutral atoms, let a be the radius of the atom and let Eb be the binding energy of the least tightly bound electron to the atom. An external electric field in excess of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
|𝓔A| > Eb / (e a)
</div>
will promptly ionize the atom, ripping away its electron. Here, e = 1.60×10-19 C is the magnitude of the electric charge of the electron. For fields close to but a bit weaker than this, the electron can still rapidly pop off due to quantum tunneling<ref>Ph. Durand and I. Paidarová, “Ionization of the hydrogen atom: from weak to strong static electric fields,” The European Physical Journal D, Vol. 26, pg. 253-259, 2003.</ref>. In general, expect that for fields greater in magnitude than (very approximately) Eb / (10 e a) the electron will be quickly removed, while for fields weaker than this the electrons will stay on their atoms.

It is useful to note that Eb/e is just the binding energy in units of eV. The most weakly bound electrons are usually bound to their atom with about 10 eV of energy. Atoms are typically about 10-10 m in radius. Thus, you can expect fields of around 1010 to 1011 V/m to directly ionize atoms. If we call this threshold value 𝓔T, then a beam will be ionized when passing through a field with strength greater than
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B sin(θ) > 𝓔T / (γ v)
</div>
</table>

Example: Let's consider a 1 GeV neutralized proton beam from a synchrotron, with a beam current of 200 kA in each bunch (incidentally, if a bunch is 5 cm long this means that each bunch packs 38 kJ worth of energy). The speed of the bunch is 87% the speed of light, or 262 million m/s, with a γ of 2.07. Consider when the beam's focus takes it to a radius of 5 cm (or 0.05 m, for our calculation). Plugging this in, we find that the protons in the bunch will separate from the electrons in a field with a transverse component greater than about 2.1 T, and the bunch will pass through the field with only minimal losses if the transverse component is much smaller than this.

Example: We'll take the same 1 GeV beam, but this time recombine the electrons with protons to make hydrogen atoms. An electron is bound to a hydrogen atom with an energy of 13.6 eV (the Ryberg constant) and the radius of the electron orbital in a hydrogen atom is 0.529×10-10 m (the Bohr radius). This gives a threshold electric field for prompt ionization of hydrogen of 2.57×1011 V/m. We will get this with a transverse component of the magnetic field of 474 T. Above this field the atoms promptly ionize and electrons are stripped away from the protons. For fields higher than roughly 1/10 of this – so about 40 to 50 T – you may still get significant ionization by quantum tunneling; lower than this your losses from the beam due to field-induced ionization are likely negligible.

==== Magnetic shields ====

What happens when we can't rely on magnetic fields in the environment to protect us? One idea is to produce powerful artificial magnetic fields around our spacecraft to deflect incoming particle beams. This is very similar in concept to using active magnetic shielding to protect against space radiation – [[Space_Radiation#Magnetic_Shielding|check out our article on space radiation for more]].
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr>
<td width=100>
[[File:Unconfined_FRC_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:racetrack_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:Magnetic_shielding_Halback_Array.png|200 px|frameless]]
<tr><td colspan=3>
Some potential configurations for magnetic shields. The currents generating the fields are in green, the magnetic fields themselves are shown in magenta<ref name="Battiston2012">R. Battiston, W.J. Burger, V. Calvelli, R. Musenich, V. Choutko, V.I. Datskov, A. Della Torre, F. Venditti,
C. Gargiulo, G. Laurenti, S. Lucidi, S. Harrison, and R. Meinke, "ARSSEM Active Radiation Shield for Space Exploration Missions", Final Report ESTEC Contract N° 4200023087/10/NL/AF : “Superconductive Magnet for Radiation Shielding of Human Spacecraft” (2012) https://arxiv.org/abs/1209.1907 https://www.researchgate.net/publication/265945847_Active_Radiation_Shield_for_Space_Exploration_Missions</ref><ref name="Desiati2022">Paolo Desiati and Elena D'Onghia, "CREW HaT: A Magnetic Shielding System for Space Habitats", arXiv:2209.13624 [physics.space-ph] https://doi.org/10.48550/arXiv.2209.13624</ref>.
</table>
</div>

One immediate issue is that you want to keep very high magnetic fields out of the crew area. Fields in excess of 0.5 T can start to be unpleasant, and it is recommended to keep the field below 0.2 T for continuous exposure.

Strong magnets can suffer from high self-forces acting to tear them apart. Most shielding designs will have multiple magnetic coils, which will exert high forces and torques on each other. This will result in the need for strong structural bracing to keep the magnets intact and in place. Such strong structural designs will impose significant extra mass on the spacecraft.

Consider a magnetic field configuration that can keep radiation out of a certain region of space. There is no trajectory of a charged radiation particle up to a certain energy that will allow those particles to enter the protected region. But the dynamics of charged particles in a magnetic field are the same whether going forward or backward in time. If there is no way for a particle to get in, by this time reversal symmetry any particle which is in that forbidden region can never escape. Consequently any field configuration that is a magnetic trap is also a magnetic shield; any field configuration that is a magnetic shield is also a magnetic trap. A spacecraft with a magnetic shield may end up surrounded by strong radiation belts.

Using a magnet to bend a weaponized electron beam very often ends up with the beam being bent so hard that it dumps most of its energy in the form of synchrotron radiation at gamma ray energies. Because gamma rays are not charged, they will not be deflected by the field. Thus, if you would have been hit by the electron beam you will be hit instead by the gamma rays.

As an example, consider a 1 TeV electron beam. We'll surround our spacecraft with a 10 T magnetic field in a 100 meter radius region just outside our spacecraft. This will deflect the beam by 30 meters, which is enough to make it miss. But we are not saved, because the beam dumps nearly all of its energy into synchrotron radiation with an energy spectrum of gamma rays up to 13 GeV. This radiation will hit us in a strip about 2.4 meters long and not much more than the width of the original beam with essentially all of the energy of the original beam.

==== Magnet drones ====

If a magnetic shield around your spacecraft will just end up killing you from synchrotron radiation, what if you put the magnetic shield on a drone and put that drone between yourself and the enemy trying to shoot you? This lets you use a much weaker magnet because you don't need nearly so high of a deflection angle to make the shot miss. This means that less beam energy will turn into synchrotron radiation that could kill you instead and will spread what synchrotron radiation does occur out over a much wider area when it reaches your spacecraft, hopefully so that most of it just misses your spacecraft.

It does mean that your enemy will start shooting your drones so that they can eventually take a shot at you. But hopefully by that time you've already cleared out their drones and cored their spacecraft with your own synchrocannon or UREB.

==== Plasma magnets ====

Usually, to make a really big magnetic field you would run current through a really big conductive loop. But really big things are heavy and expensive. One way around this could be to use the ambient space plasma to carry the electric current rather than any kind of wire. In order to make this current, you use the same mechanism used in a rotary induction electric motor: you generate a rotating magnetic field and induce a current in the conductive loop (in this case, the space plasma). How do you make a rotating magnetic field? The method proposed<ref name="Slough2006">John Slough, "The Plasma Magnet", NASA Institute for Advanced Concepts Phase II Final Report (2006) https://www.niac.usra.edu/files/studies/final_report/917Slough.pdf</ref> is to use an antenna of four much smaller current loops as electromagnets at 90° to each other, and sinusoidally vary the current through the loops with a respective 90° phase difference between them. When this is done, a circular current is induced in the space plasma and the self-force of a magnetic field on its generating current inflates the current loop to much larger radii. You end up with a current sheet in the plane of the rotating field's rotation circulating about the antenna nearly synchronously with the field rotation. This current then creates a normal dipolar magnetic field as with any current loop. The current sheet and the resulting magnetic field expand until the magnetic field's pressure equals that of the ambient space plasma. This gives an enormous region of magnetic field from a relatively small and lightweight system.

The system proposed in the initial study would use an antenna approximately 10 meters in radius and produce a region of field approximately 30 km in radius at Earth's orbit. The field inside would not be constant - at the edges it would be 50 nT, giving the right pressure to keep out the solar wind at Earth's orbit. A circulating current sheet produces a field that falls off as 1/r with distance, meaning that adjacent to the antenna the field would be about 0.15 mT. Using the methods in the box below, we can estimate that this particular design as part of a spacecraft's shields will not deflect a TeV electron beam by more than a few centimeters, which is usually insufficient to force a miss. However, it could potentially be useful for a drone magnet, where the much larger distance between the magnetic field and the target spacecraft would allow a small deflection angle to give perhaps tens or low hundreds of meters change in beam location near the target at near light second ranges. The relatively low field and gradual bend of the incident beam suggests that synchrotron radiation will not be an issue; however this does not exclude more powerful plasma magnet designs causing significant synchrotron radiation effects.

<table border=1><tr><td>
Calculating the deflection from a spatially varying magnetic field affected by the solar wind is no trivial task. However, we can make an approximation that both should be within about an order of magnitude of the true value and also represent an upper limit to the deflection angle; although it does neglect the influence of the driving rotating magnetic field which could in some cases increase the deflection.

We will take R0 as the radius of the antenna. B0 is the field strength at the antenna. Rm is the distance at which the magnetic influence stops because the magnetic pressure equals the space plasma pressure. Bm is the field strength at Rm. According to Slough<ref name="Slough2006"></ref>, the magnetic field strength B(r) at any distance from the antenna center r between R0 and Rm is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B(r) = B0 R0 / r.
</div>
Therefore, once we know Bm and Rm, B0 is completely determined
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B0 = Bm Rm / R0.
</div>

The force on a particle with charge q passing perpendicularly through a magnetic field B(r) at a speed v is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
F(r) = q v B(r)
</div>
For an electron beam, v is so close to c as to make no difference. If the particle is moving down the y direction and the field is in the z direction, the force will be in the x direction. We assume that the deflection will be small, so we can choose a straight-line path in the y direction offset from the center by an amount x. The plasma bubble radius is Rm, so the particle will enter at ymax = √[Rm2-x2] and leave at ymin = -√[Rm2-x2].

The momentum change the particle experiences from this force is the time integral of the force over its path. Because of reflection symmetry about y = 0, we only need to integrate from ymax to 0 and then multiply by 2
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] dt F(r)
<tr> <td> <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] (dy/v) F(r)
<tr> <td> <td> = <td align="left">2 q B(0) R0 [Integral from 0 to √[Rm2-x2]] dy 1/√[y2 + x2]
<tr> <td> <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/x]
</table>
</div>
This holds for |x| > R0. But the expression for the field is not valid for r < R0 and the integral will give nonsense and diverging results if we evaluate for |x| < R0. In this case, we can put the integral lower bound at r = R0, or y = √[R02-x2]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/(√[R02-x2] + R0)]
</table>
</div>

The deflection angle is now estimated as
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = px / p
</div>
where the total momentum of the particle p can be found from the methods in the first box in the Magnetic Fields section.

The deflection for a shot aimed right at the spacecraft at the position of the spacecraft can be estimated by setting x = 0 and only taking the first half of the integral (so ignore the factor of 2)
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
q B(0) R0 ln[Rm/R0] / (p Rm)
</div>
This is an additional overestimate because it fails to account for how the deflection builds up over distance during the integration, but it should still give something of an order-of-magnitude and upper limit estimate of what kind of deflections might be expected.
</table>

=== Electric fields ===

If electric fields can accelerate charged particles, maybe they can keep the charged particles in a weaponized particle beam away from your spacecraft? To completely block a particle beam, you would need a voltage on your spacecraft equal to the particle energy in eV of the particles in the beam. That is, if a 1 GeV proton beam is coming at you, you'd need an electric potential of a billion volts to stop that beam. This leads to three immediate problems.

The first is that particles of the opposite charge will be attracted to your spacecraft. Because space isn't really empty but rather a rarefied plasma (called the solar wind in a solar system or the interstellar medium out between stars), there will be plenty of particles of the opposite charge around to attract. And they will hit with an energy in eV equal to the voltage on your spacecraft. In our example of defending against a 1 GeV proton beam with a gigavolt field, the electrons from the solar wind will come in and hit with an energy of 1 GeV each. This will generate dangerous radiation.

The second problem is a consequence of the first. Each of those GeV electrons from our example need to get their energy from somewhere, which turns out to be from the electric field we are using as shielding. You will need to make up for this energy loss with power from your electrical system, which gives a constant drain on your power. Some estimates put the electrical effort to maintain a several hundred megavolt shield in space at many tens of gigawatts<ref name="Mechmann2019">Claire Mechmann, "Analysis of Proposed Active Radiation Shielding Design Concept for Spacecraft" (2019) Thesis, College of Engineering and Science of Florida Institute of Technology</ref>.

And the third problem is similar to one for magnetic fields. Your electrodes will experience high self forces and forces between separate electrodes that will try to tear your shield generator apart. Consequently you will need heavy structural reinforcement in order for your shield to work.

There are potential ways to mitigate some of these problems. Interested readers are directed to [[Space_Radiation#Electrostatic_Shielding|the active shielding section in our page on space radiation]] for more of the juicy details; there's no need to repeat the same stuff here.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Elctrostatic_active_shielding.png|200 px|frameless]]
<td>
[[File:Electrostatic_active_shielding_2.png|200 px|frameless]]
<tr><td colspan=2>
Proposed designs for electrostatic shields to protect from positively charged particles<ref name="Tripathi2006">Ram K. Tripathi, John W. Wilson, and Robert C. Youngquist, "Electrostatic Active Radiation Shielding - Revisited", 2006 IEEE Aerospace Conference, Big Sky, MT, USA, 2006, pp. 9 pp.-, doi: 10.1109/AERO.2006.1655760.</ref><ref name="Chowdhury2023">Rajarshi Pal Chowdhury, Luke A. Stegeman, Matthew L. Lund, Dan Fry, Stojan Madzunkov, and Amir A. Bahadori, "Hybrid methods of radiation shielding against deep-space radiation", Life Sciences in Space Research, Volume 38, 2023, Pages 67-78, ISSN 2214-5524, https://doi.org/10.1016/j.lssr.2023.04.004.</ref>.
</table>
</div>

== Damage ==

Particle beams are, first and foremost, devices that produce beams of [[Nuclear_radiation|ionizing radiation]]. The radiation of weaponizable particle beams is extremely penetrating; thus of the [[Particle_Accelerators#Energy_loss_in_matter|interaction mechanisms listed for radiation from particle beams]], we mostly need to worry about bremsstrahlung and nuclear collisions and these produce extensive secondary showers of radiation that increase the penetration of the radiation exposure. Because of this, any target hit by a particle beam weapon will be irradiated, and radiation dose will always be one of the primary damage mechanisms of a particle beam weapon strike. Further, this radiation will be distributed along a track passing deeply through the material of the target, allowing it to affect critical systems through heavy armor, structural material, robust machine components, and passive material such as fuel load, coolant, or cargo. As the radiation deposits its energy into the target material, that material will be heated. If the target is heated sufficiently, it may experience [[Beam-Target_Interactions|thermal or mechanical]] damage from melting, cooking, combustion, vapor explosions, and warping or cracking due to thermal expansion.

The discussion of the radiation penetration into armor also applies to the same radiation penetrating through the target. So the formulas and rules of thumb for estimating the depth of radiation penetration into armor can also be used to estimate how far into a tank or spacecraft a particle beam strike delivers effects.

In addition to the radiation and thermal effects, a particle beam can also produce a sharp electromagnetic pulse that can damage semiconductor electronics.

== 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, Gerrit Bruhaug, and Luke Campbell for writing

Gerrit Bruhaug and Luke Campbell for scientific research, math wrangling and advise

ATriffleMind and Luke W. Campbell for proof reading.

==References==

[[Category:Beams]][[Category:Physics & Engineering‏‎]][[Category:Engineering‏‎]][[Category:Military Technology]][[Category:Warfare‏‎]]

Particle Beam Weapons

2026-10-27T04:53:42Z

Lwcamp: /* Separation of neutral beams */

{{PageConstructionNotice}}

''For a general overview on particle accelerator technology, see [[Particle Accelerators]]

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.

::: Z = 1.32×102 (γ2-1)3/4 r / √I<ref name=USPAS2016>Martin Reiser, "Theory and Design of Charged Particle Beams", 2008, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim</ref>

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

::: Z = β γ c r √[m/T]

Where Z is the doubling distance like before, r is the initial beam radius, T is the beam temperature in eV, m is the ion mass and β, c, and γ 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.

== Defenses ==

Sure, it's great to have a great big particle cannon to blast your enemies. But what if you are the one being shot at? In that case, it's best to figure out ways to keep from being blown to kingdom come.

=== Armor ===

The simplest way to defend against the high energy particles of a particle beam is to pile a whole bunch of stuff up and put it in the way of the beam. Particle beams may produce extremely penetrating radiation, but enough stuff will block most of that radiation and may allow you to survive. The details depend on what kind of particle is being thrown at you, and are explained below. However, a general rule of thumb is that you probably want about 10 tons of material per square meter of surface you want to armor in order to get sufficient protection.

==== Electrons ====

At the very high energy of an electron accelerated by a weapons-grade particle beam, the primary interaction with matter will be the emission of bremsstrahlung photons in the gamma-ray range as the electron slams into particles of matter. These high energy photons themselves primarily interact by the pair production mechanism – the photon disappears, but leaves behind a highly energetic positron and electron. These two particles are themselves energetic enough to produce further bremsstrahlung, and so the process continues. You get the formation of an electron-gamma shower (also called an electromagnetic cascade), which only stops after either the particles have penetrated through the target or the initial energy of the electron has been spread through so many shower particles (including all the electrons, positrons, and photons) that no particle has enough individual energy to pair produce or to be likely to bremsstrahlung again before it is stopped by other mechanisms.

To stop electrons, you'll want the atoms in your armor to be as heavy as you can practically get them. High atomic number both encourages bremsstrahlung and pair production, giving a double whammy for stopping that cascade. However, occasionally one of those electrons, positrons, or gammas will hit a nucleus and knock out neutrons. So you probably want a back-up layer of several tens of centimeters of hydrogen rich material such as plastic or water mixed with neutron absorbing materials like boron, lithium, cadmium, or gadolinium. High density polyethylene loaded with 5% boron by weight is a typical go-to neutron blocker.

The actual methods to calculate the penetration of the electromagnetic cascade is given are the box below. However, for some examples:
<ul>
<li> A 10 GeV beam incident on thick iron (or steel) will stop cascading at about 9 cm, will deposit half of its energy in the first 12 cm, and will have dumped 98% of its energy within 35 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 2.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 100 GeV beam incident on thick iron (or steel) will stop cascading at about 13 cm, will deposit half of its energy in the first 16 cm, and will have dumped 98% of its energy within 47 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 3.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick iron (or steel) will stop cascading at about 17 cm, will deposit half of its energy in the first 20 cm, and will have dumped 98% of its energy within 59 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 4.6 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick tungsten will stop cascading at about 3.5 cm, will deposit half of its energy in the first 4 cm, and will have dumped 98% of its energy within 12 cm. 90% of the cascade will be contained within about 0.93 cm and 95% of the cascade will be contained within 1.86 cm of the beam axis. You will get 98% protection with a mass thickness of 2.3 tons/m2 and an additional 1/e reduction for each further 0.13 tons/m2 beyond that.
<li> A 1 TeV beam incident on water will stop cascading at about 290 cm, will deposit half of its energy in the first 340 cm, and will have dumped 98% of its energy within 1030 cm. 90% of the cascade will be contained within about 11 cm and 95% of the cascade will be contained within 22 cm of the beam axis. You will get 98% protection with a mass thickness of 10.3 tons/m2 and an additional 1/e reduction for each further 0.72 tons/m2 beyond that.
</ul>
Note that for sufficiently intense particle beams, 98% protection may not be enough. You might need to throw on several more tons per square meter to be sure.

<table border=1>
<tr><td> How Far it Goes:
(This section follows methods presented in Leroy and Rancoita<ref>C. Leroy and P-G Rancoita, "Physics of cascading shower generation and propagation in matter: principles of high-energy, ultrahigh-energy and compensating calorimetry", Reports on Progress in Physics 63 (2000) 505–606. DOI: 10.1088/0034-4885/63/4/202</ref>.)

When trying to estimate the extent of the irradiated region of the target, it is helpful to know the radiation length X0 in the material. This is the distance that an electron will travel, on average, before emitting a bremsstrahlung photon. Conveniently, it is also the distance a high energy photon will travel through the material before pair producing. The radiation length can be found from
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   = 4 α  
<td nowrap="nowrap"> NA
<td rowspan="2" nowrap="nowrap" align="right">   Z (Z+ζ) re2 ln
<td rowspan="2" nowrap="nowrap" align="right"> [
<td nowrap="nowrap"> 183
<td rowspan="2" nowrap="nowrap" align="right"> ]
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> A
<td style="border-top:solid 1px black;"> Z1/3
</table>
</div>
where α = 1/137 is the fine structure constant, NA = 6.02214076×1023 is Avogadro's number (or one mole), A is the atomic weight of the atoms in the material (usually expressed in grams per mole), Z is the atomic number (nuclear charge) of the atoms, and re = 2.8179403205×10-13 cm is the classical electron radius. ζ is a constant of approximately 1.3. This gives the thickness of material in terms of the areal density (mass of material per unit area perpendicular to the beam) – if you use A in g/mol and re in cm the result will be in cm2/g and X0 will be in g/cm2. If your target is made up of more than one kind of atom, you can get the total radiation length by finding the radiation length of each atom of type i, X0i and using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   =  
<td rowspan="2" nowrap="nowrap" align="right"> Σi  
<td nowrap="nowrap"> fi
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> X0i
</table>
</div>
where fi is the fraction by weight of the atom type i. Once you have the radiation length in terms of areal density, you can easily convert it to distance by dividing by the density ρ. If X0i is in g/cm2 and density is in g/cm3, then X0i/ρ will be in cm.

We will also need to know the critical energy εc. This is the energy where an electron is losing energy as rapidly to ionization as it is to radiation. That is, it is the energy at which the cascade stops multiplying, even though the individual particles already made can continue traveling for some distance. The critical energy can be estimated using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> εc = B  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> Z X0
<td rowspan="2" nowrap="nowrap" align="right"> )
<td nowrap="nowrap"> h
<tr>
<td style="border-top:solid 1px black;"> A
<td>  
</table>
</div>
where B = 2.66 MeV and h = 1.11.

We can now describe the distribution of radiation in the target material. The depth tmax where the cascade stops multiplying (in g/cm2) for an incident electron of energy E (in MeV) is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tmax = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> - 1
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
Similarly, the depth where half the energy of the beam has been deposited (in g/cm2) is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tcg = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> + 0.37
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
As usual, you can convert to a depth in cm by dividing by the material density. 98% of the beam energy will be deposited within a distance of &cong; 3 tcg. Beyond this, the beam energy will continue to fall off with thickness t approximately as an exponential exp[-t/λ], where the characteristic attenuation length λ &cong; 2 X0.

The width of a cascade, where 90% of the energy is contained, is approximately given by
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> RM =  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> EM
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
where EM &cong; 21.2 MeV. 95% of the radiation will be contained within a distance of 2 RM from the beam axis.
</table>

==== Protons and Ions ====

<table align=right><tr><td>
[[File:Regolith_Shielding.png|350 px|frameless]]
<tr><td width=350>
Relative effect of particle radiation (compared to no shielding) from galactic cosmic rays behind different thicknesses of water, aluminum, and lunar regolith<ref name="Slaba2022">Tony C. Slaba, "Radiation Shielding for Lunar Missions: Regolith Considerations", LSIC Crosstalk 7/18/2022 https://lsic.jhuapl.edu/uploadedDocs/focus-files/1604-E&C%20+%20EE%20Monthly%20Meeting%20-%202022%2007%20July_Presentation%20-%20NASA%20Slaba.pdf</ref>.
</table>
Protons and ions are grouped together because ions are made of protons and neutrons and when you accelerate these things for a weapon it is the protons and neutrons that are going to be doing the heavy lifting for blasting your target. This is because protons and neutrons are both about 2000 times more massive than electrons so any electrons along for the ride will contribute negligibly to the energy. Protons and neutrons together are called nuclear particles, and both behave quite similarly in nuclear collisions.

Nuclear particles are themselves made of quarks and can interact via the strong nuclear force. At the energies appropriate for weapons use, essentially all of the way protons or ions are blocked is through [[Particle_Accelerators#Nuclear_collisions|direct collisions with the atomic nuclei]] of the armor material. One or more quarks in the projectile will slam into quarks in the target nucleus and stretch them away from the rest of the quarks in the nuclear particles. Physics tells us that you can't have isolated quarks, so the "string" attaching the quark to its neighbors stretches and stretches until it snaps. Snapping releases so much energy that more quarks are made that clump together into all sorts of other nuclear particles, both exotic and familiar. This process is called a hadronic jet. The only particles produced in these jets long-lived enough to travel any noticeable distance through the armor material will be protons, neutrons, the charged pi mesons (or pions), and some of the K mesons (or kaons). The particles too short lived to go far enough to matter will decay into other particles, and these particles are often very high energy gamma rays. Much like with electrons, the long-lived secondary particles can be sufficiently energetic to smash into other nuclei and cause more hadronic jets (or, if they are low enough in energy, just knock more nuclear particles out of target nuclei) and this produces a similar shower or cascade. But in addition, the armor nuclei that were hit will be very hot after the impact from all of the energy given to them from the incident particles. These nuclei will cool off by evaporation – but instead of evaporating away molecules, they evaporate off nuclear particles from the nucleus. These evaporated particles are mostly neutrons, so any armor against proton or ion beam strikes will produce quite a bit of neutron radiation.

Eventually, the original particle's energy will be distributed among so many other particles that the charged particles will come to a stop by making ion tracks rather than from colliding with more nuclei, and the cascade halts. But you still have all the neutrons to worry about. And the gamma rays as well, which are high enough energy to kick off an electromagnetic cascade in the same way that electrons do.

Unlike electrons, which have reasonable approximations of their cascade behavior from relatively simple equations, the cascades of protons and ions usually require sophisticated simulations or experiments to get the resulting distribution of radiation. However, we can get a reasonable idea of their general behavior by comparing them to [[Space_Radiation#Shielding_Against_Cosmic_Rays|cosmic ray particles interacting with shielding]]. Cosmic rays are energetic protons and ions, after all. From this we see that a relatively small amount of armor – in the range of 1 to 3 tons per square meter – can actually make the radiation dose worse. The incident particles will knock off so many secondary particles in their shower that the dose is multiplied from so many tracks passing through the target. Hydrogen rich materials such as water and plastics work the best against proton and ion beam particles; heavy elements are the worst and produce the most secondaries. You could probably obtain adequate radiation protection from most incident beams with about 10 tons per square meter of hydrogen rich material as armor. However, you will want to add neutron absorbing elements to your hydrogen rich armor, such as lithium, boron, cadmium, or gadolinium. Further, in order to deal with the gamma rays you will want a layer of at least several centimeters of heavy elements, such as tungsten or lead, behind the hydrogen rich particle armor.

==== Structural Considerations ====

When your armor material absorbs a radiation beam, the energy of that radiation turns into heat inside the armor where the radiation was absorbed. If the beam pulse has enough energy as was sufficiently tightly focused, this heat can damage your armor. Your armor may melt in places, or thermal expansion could cause it to crack or warp. Perhaps the worst thing it could do is vaporize or flash into a plasma, because the pressure from the vapor or plasma can be sufficient to plastically deform the armor. This leaves a hole along the beam path where the expanding gas or plasma blasted out a long, deep crater. This means a beam that fires rapid shots spaced maybe milliseconds apart can have subsequent shots go down the same hole so your armor will not protect against them. Each shot will dig out a deeper hole until either your armor is penetrated or the attacking beam runs out of shots in its pulse train.

A consequence of this is that when designing your armor you may wish to use refractory materials with a high structural strength (like tungsten), or that is otherwise resistant to being blasted apart by its own evaporated remains.

=== Magnetic fields ===

[[Particle_Accelerators#Magnetic_fields|A charged particle crossing a magnetic field will bend so as to spiral around the field lines]]. If you try to shoot your particle beam through a magnetic field, it will bend away from the direction you shot it. This can make it hard to hit your target. Unfortunately for you, there are magnetic fields everywhere – around planets, and even trapped in the solar wind as it crosses the solar system.

As an example, the Earth has a magnetic field strength of about 5×10-5 tesla near its surface, and falling off farther away. If you are shooting a 1 TeV electron beam past the Earth, your beam is likely to bend by several milliradians (with the exact deflection depending on how close you are skimming the Earth). If you are in a medium Earth orbit and shooting at another target also in a medium Earth orbit on a trajectory that goes near the Earth, then without correcting for the magnetic field your beam will likely be off-target by tens to hundreds of kilometers.

For another example, the interplanetary magnetic field is about 10-9 tesla at Earth's distance from the sun. If you shoot that same 1 TeV electron beam at a target 0.1 light seconds away, you are likely to be off by a couple hundred meters.

<table border=1>
<tr><td>
The radius of curvature of the bend of a particle in a magnetic field is called the gyroradius rc. For a particle with a charge q, mass m, and kinetic energy E, find the relativistic momentum p
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
p = √[E2 - m2 c4] / c
</div>
To convert from energy units of eV to joules, multiply by 1.60×10-19. An electron has a mass of 9.11×10-31 kg, a proton of 1.67×10-27 kg, and an ion of 1.66×10-27×A kg, where A is the atomic weight of the ion. The speed of light c = 3.00×108 m/s.

In a magnetic field of strength B with an angle θ between the field direction and the direction of motion
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rc = |p sin(θ) / (q B)|
</div>
For electrons, q = -e, where the fundamental charge e = 1.60×10-19 coulomb. For a proton, q = +e. For an ion, multiply e by the charge state of the ion.

If the field acts over a distance d that is much smaller than the gyroradius, the deflection angle φ is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = d / rc
</div>
The deflection x at a target at a range R for small φ (φ &Lt; 1) will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
x = φ R.
</div>

Note that d will always be less than or equal to x. Also note that if d is not smaller than twice the gyroradius the beam will never reach the target because it will completely spiral around the field lines before it reaches the target.
</table>

So, it is clear that as the defender it will be helpful to hang out in areas with high magnetic fields, and keep some distance between you and the attacker. Meanwhile, the attacker will benefit from being closer where small deflections in the beam trajectory don't translate into large excursions from where the beam is aimed (and also because there will be less space filled with fields between the two of you). But the attacker can also try to compensate for known fields. If you happen to know that the magnetic fields are between you and your target, you can adjust your fire so that the deflection of the field bends your beam onto the target. This is not a simple task. It is easy enough to measure the field strength and direction where you are, but predicting how it changes over the path of your shot does not have an obvious solution. You can also try to gain intelligence on likely battle sites before you ever fight on them – for example, getting good data on the structure of the fields of a planet using spies and technical intelligence during peacetime (heck, on Earth this information is published for free in academic journals and by government agencies such as NOAA). Planetary magnetospheres are also affected by the solar wind, so different solar weather will change the fields you have to shoot through. But space weather forecasts and predictive simulations of the solar wind interacting with a planetary magnetosphere can give you a good idea of what the field structure is like. But even with all that, there are changes in the fields that happen on a scale of around a second ... anyone who has ever seen the auroras dancing across the sky in the polar night can see this. Compensating for these changes will be a significant challenge.

What would really help is being able to see the path of your own beam. This would allow you to see how far off you were from your target and in what direction, and then adjust fire to walk your shots onto the target. Unfortunately, particle beams in space are invisible. There might be ways to track them, but if so we don't know what they are.

==== Synchrotron radiation ====

When charged particle trajectories are bent, they emit [[Particle_Accelerators#Synchrotron_radiation|synchrotron radiation]]. This can cause the particle beam to lose energy, but if the bend is too tight it can also generate high energy gamma rays that ignore magnetic fields and directly strike the target. Synchrotron radiation can generally be ignored for proton or ion beams, but can be significant for electron beams.

<table border=1><tr><td>
For a beam particle with energy E and mass m, the synchrotron radiation is concentrated in a forward-directed cone with an opening angle of approximately 1/γ, where γ = 1 + E/(m c2). (Interestingly, the radiation intensity peaks an at angle of 1/(2 γ) and vanishes directly on-axis.) Because the beam is bending, you will get emission along the plane along which the beam is being bent. The energy of the photons produced during synchrotron emission is approximately<ref name="Jackson_Electrodynamics">J. D. Jackson, "Classical Electrodynamics: Second Edition", Wiley (1962)</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Eγ &cong; 3 γ3 c &hbar; / rc
</div>
where &hbar; = 1.05×10-34 J s (if you want the energy in joules) or &hbar; = 6.58×10-16 eV s (if you want the energy in eV). The photon energy will not exceed the energy of the incident particles, if you get a photon energy higher than the energy of the incident particles it just means that the synchrotron radiation spectrum extends up to the incident particle energy.

The fraction of the energy (or power) of the beam turned into synchrotron radiation will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = [1/(4 π ε0)] [4 π / 3] [q² / rc] β2 γ4 [φ / (2 π)] / E
</div>
where β = √[1 - 1/γ2] is the particle's speed measured as a fraction of the speed of light, and 1/(4 π ε0) = 8.99×109 is the Coulomb constant. For highly relativistic electrons (like all practical electron beam weapons emit), β &cong; 1 and we can use
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = 8.85×10-2 [E (GeV)3 [φ / (2 π)] / (rc (m))]
</div>
This fraction cannot be more than 1. If f gets close to one, it means that our original assumption that the beam particles mostly keep their original speed no longer holds, and the beam will lose most or almost all of its energy. Instead, when the beam is being deflected it will only make it through an angle of φ/f and will irradiate a region of length x/f starting at where the beam would have been incident if not for the magnets and extending in the direction the beam was deflected.
</table>

For example, a 1 TeV electron beam will have γ &cong; 2,000,000. So if it's bent by a magnetic field the synchtrotron radiation will spread out in a cone of about 5×10-7 radians. If this beam was shooting at a target 0.1 light seconds away from where it was bent, the radiation would arrive at the target over a region with a radius of about 15 meters with a maximum intensity at about 7.5 meters and no radiation directly along the path the beam was pointing while it was being bent. If the beam was bent by the interplanetary magnetic field from our earlier example, it would produce light with photon energies of around 1.3 eV (near infrared) and the lost energy would be a few parts in a hundred trillion. If it was instead deflected by the Earth's magnetic field you might expect a peak photon energy of tens of keV – in the hard x-ray part of the spectrum – and the fraction of energy in the synchrotron radiation would be somewhere in the low parts in 100,000 or less.

==== Separation of neutral beams ====

Magnetic fields only affect charged objects. So one way to overcome magnetic defenses or environments is to use beams that have no net charge. For example, a proton beam could be neutralized with a co-propagating electron beam with an equal number of protons and electrons. Or, potentially, you could fire an ion beam and then neutralize the ions after acceleration to get a beam of neutral atoms.

But, the magnetic fields are still tugging at the charged particles making up the beam as they go past. This will pull the electrons one way and the protons or ions the other way. As the particles are pulled in opposite directions you get a net positive charge on one side of the beam and a net negative charge on the other side (called polarization). This sets up an electric field inside the beam that acts to pull the particles back together. If the particles are going fast enough and the magnetic field is strong enough, the magnetic force can still separate the charges and will split the beam in two, one with the positive charges and one with the negative charges. But if the particles are not that fast or the magnetic field is not that strong this polarization of the beam will keep it together and let it fly right through the magnetic field.

<table border=1><tr><td>
Let's figure out how to analyze this problem. For definiteness, we will have a beam with an electric current of the positive species of I and an electric current of electrons of -I, all moving in the z direction. The beam radius will be R. The particle speed will be v with the usual relativistic parameters β = v/c and γ = √[1/(1-β2)]. If you know the energy per particle E of the positive species and the mass m of the positive species, γ = 1 + E/(m c2) and then β = √[1 - 1/γ2].

Let's assume that the beam is going through a magnetic field of strength B at an angle θ to the field direction. The quantity B sin(θ) is the component of the magnetic field perpendicular to the direction of the beam.

The charge density ρ of the positive component of the beam will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρL = I / (π v R2).
</div>

It is easiest to analyze the problem in the rest frame of the particles in the beam. Now the particles are at rest. Particles at rest are not affected by a magnetic field, but the change of reference frame Lorentz transforms the initial magnetic field into a combination of electric and magnetic fields in the rest frame. We can ignore the magnetic field, but the electric field is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔A = γ v B sin(θ)
</div>
in a direction perpendicular to both v and B. For our purposes, assume the electric field points in the x direction.

At the same time, relativistic length contraction increases the charge density in the rest frame
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρ = γ ρL.
</div>

Let r be the vector from the beam axis to the point inside the beam. It will be useful to separate this into its Cartesian coordinates. If x and y are the unit vectors in the x and y directions, respectively, then
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
r = x x + y y
</div>

If we consider only the positive component of the beam, from [https://en.wikipedia.org/wiki/Gauss%27s_law Gauss's law] it will have an electric field inside of the beam at a distance r from the center axis of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔+ = ρ r / (2 ε0) = (ρ/(2 ε0)) (x x + y y)
</div>
where ε0 = 8.85×10-12 is the vacuum permittivity. Now let's displace the electrons in the beam by a distance δ in the x direction. By itself, this electron beam will have an electric field of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔- = -(ρ/(2 ε0)) ((x-δ) x + y y)
</div>
By the principle of superposition, the electric field inside the beam with both components is the sum of the electric fields of the separate components including the external field
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔B = [(ρ/(2 ε0)) δ + γ v B sin(θ)]x
</div>
Because the charges are free to move, they will do so as long as a field is applied to them. Thus, the beam becomes stable once the charges separate enough that the field internal to the beam is zero. Therefore
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
δ = -2 ε0 γ v B sin(θ) / ρ = -2 π ε0 v2 R2 B sin(θ) / I
</div>
If |δ| > 2 |R|, the beam will fully separate and each component will go its own way. If δ is small compared to R, the beam's polarization suffices to hold it together in the overlap region. However, where the positive and negative charges do not overlap the electric fields no longer cancel, and the external electric field carries away the outer polarized part of the beam. This gives us a rough threshold for which a magnetic field will quickly separate a neutral beam into its charged components and bend them away from their target
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr><td> B sin(θ) < I / (π ε0 v2 R) <td>       <td> Beam is partially eroded.
<tr><td> B sin(θ) ≥ I / (π ε0 v2 R) <td> <td> Beam promptly separates.
</table>
</div>
For a partially eroding beam, the polarized region that was peeled off by the field still acts to screen the remaining neutral portion, at least to within this approximation, leaving the still-neutral beam to hit the target.
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Neutralized_beam_in_B_field.png|5140 px|frameless]]
<tr><td width=700>
A profile of the beam charge distribution of a neutralized beam in a magnetic field. Negative charge density is shown in yellow, positive charge density in cyan, with neutral charge density in light green. The magnetic field is vertical, the electric field in the beam reference frame is shown in magenta including both the polarization field and the Lorentz transformed electric field resulting from the magnetic field. As positive charges are pulled in the direction of the field and negative charges opposite to the field direction, the polarized boundary will be pulled off the beam by the electric field even though the electric field is zero in nearly all the beam interior. This expanding polarization layer will continue to largely screen the still-neutral portion, however; but the extent to which the screening is only partial would require more sophisticated analysis methods than used here. The material removed by the field will expand to a maximum of the deflection distance of a uniform charge based on the magnetic field by the time it reaches the target.
</table>
</div>
<tr><td>
For a beam of neutral atoms, let a be the radius of the atom and let Eb be the binding energy of the least tightly bound electron to the atom. An external electric field in excess of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
|𝓔A| > Eb / (e a)
</div>
will promptly ionize the atom, ripping away its electron. Here, e = 1.60×10-19 C is the magnitude of the electric charge of the electron. For fields close to but a bit weaker than this, the electron can still rapidly pop off due to quantum tunneling<ref>Ph. Durand and I. Paidarová, “Ionization of the hydrogen atom: from weak to strong static electric fields,” The European Physical Journal D, Vol. 26, pg. 253-259, 2003.</ref>. In general, expect that for fields greater in magnitude than (very approximately) Eb / (10 e a) the electron will be quickly removed, while for fields weaker than this the electrons will stay on their atoms.

It is useful to note that Eb/e is just the binding energy in units of eV. The most weakly bound electrons are usually bound to their atom with about 10 eV of energy. Atoms are typically about 10-10 m in radius. Thus, you can expect fields of around 1010 to 1011 V/m to directly ionize atoms. If we call this threshold value 𝓔T, then a beam will be ionized when passing through a field with strength greater than
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B sin(θ) > 𝓔T / (γ v)
</div>
</table>

Example: Let's consider a 1 GeV neutralized proton beam from a synchrotron, with a beam current of 200 kA in each bunch (incidentally, if a bunch is 5 cm long this means that each bunch packs 38 kJ worth of energy). The speed of the bunch is 87% the speed of light, or 262 million m/s, with a γ of 2.07. Consider when the beam's focus takes it to a radius of 5 cm (or 0.05 m, for our calculation). Plugging this in, we find that the protons in the bunch will separate from the electrons in a field with a transverse component greater than about 2.1 T, and the bunch will pass through the field with only minimal losses if the transverse component is much smaller than this.

Example: We'll take the same 1 GeV beam, but this time recombine the electrons with protons to make hydrogen atoms. An electron is bound to a hydrogen atom with an energy of 13.6 eV (the Ryberg constant) and the radius of the electron orbital in a hydrogen atom is 0.529×10-10 m (the Bohr radius). This gives a threshold electric field for prompt ionization of hydrogen of 2.57×1011 V/m. We will get this with a transverse component of the magnetic field of 474 T. Above this field the atoms promptly ionize and electrons are stripped away from the protons. For fields higher than roughly 1/10 of this – so about 40 to 50 T – you may still get significant ionization by quantum tunneling; lower than this your losses from the beam due to field-induced ionization are likely negligible.

==== Magnetic shields ====

What happens when we can't rely on magnetic fields in the environment to protect us? One idea is to produce powerful artificial magnetic fields around our spacecraft to deflect incoming particle beams. This is very similar in concept to using active magnetic shielding to protect against space radiation – [[Space_Radiation#Magnetic_Shielding|check out our article on space radiation for more]].
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr>
<td width=100>
[[File:Unconfined_FRC_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:racetrack_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:Magnetic_shielding_Halback_Array.png|200 px|frameless]]
<tr><td colspan=3>
Some potential configurations for magnetic shields. The currents generating the fields are in green, the magnetic fields themselves are shown in magenta<ref name="Battiston2012">R. Battiston, W.J. Burger, V. Calvelli, R. Musenich, V. Choutko, V.I. Datskov, A. Della Torre, F. Venditti,
C. Gargiulo, G. Laurenti, S. Lucidi, S. Harrison, and R. Meinke, "ARSSEM Active Radiation Shield for Space Exploration Missions", Final Report ESTEC Contract N° 4200023087/10/NL/AF : “Superconductive Magnet for Radiation Shielding of Human Spacecraft” (2012) https://arxiv.org/abs/1209.1907 https://www.researchgate.net/publication/265945847_Active_Radiation_Shield_for_Space_Exploration_Missions</ref><ref name="Desiati2022">Paolo Desiati and Elena D'Onghia, "CREW HaT: A Magnetic Shielding System for Space Habitats", arXiv:2209.13624 [physics.space-ph] https://doi.org/10.48550/arXiv.2209.13624</ref>.
</table>
</div>

One immediate issue is that you want to keep very high magnetic fields out of the crew area. Fields in excess of 0.5 T can start to be unpleasant, and it is recommended to keep the field below 0.2 T for continuous exposure.

Strong magnets can suffer from high self-forces acting to tear them apart. Most shielding designs will have multiple magnetic coils, which will exert high forces and torques on each other. This will result in the need for strong structural bracing to keep the magnets intact and in place. Such strong structural designs will impose significant extra mass on the spacecraft.

Consider a magnetic field configuration that can keep radiation out of a certain region of space. There is no trajectory of a charged radiation particle up to a certain energy that will allow those particles to enter the protected region. But the dynamics of charged particles in a magnetic field are the same whether going forward or backward in time. If there is no way for a particle to get in, by this time reversal symmetry any particle which is in that forbidden region can never escape. Consequently any field configuration that is a magnetic trap is also a magnetic shield; any field configuration that is a magnetic shield is also a magnetic trap. A spacecraft with a magnetic shield may end up surrounded by strong radiation belts.

Using a magnet to bend a weaponized electron beam very often ends up with the beam being bent so hard that it dumps most of its energy in the form of synchrotron radiation at gamma ray energies. Because gamma rays are not charged, they will not be deflected by the field. Thus, if you would have been hit by the electron beam you will be hit instead by the gamma rays.

As an example, consider a 1 TeV electron beam. We'll surround our spacecraft with a 10 T magnetic field in a 100 meter radius region just outside our spacecraft. This will deflect the beam by 30 meters, which is enough to make it miss. But we are not saved, because the beam dumps nearly all of its energy into synchrotron radiation with an energy spectrum of gamma rays up to 13 GeV. This radiation will hit us in a strip about 2.4 meters long and not much more than the width of the original beam with essentially all of the energy of the original beam.

==== Magnet drones ====

If a magnetic shield around your spacecraft will just end up killing you from synchrotron radiation, what if you put the magnetic shield on a drone and put that drone between yourself and the enemy trying to shoot you? This lets you use a much weaker magnet because you don't need nearly so high of a deflection angle to make the shot miss. This means that less beam energy will turn into synchrotron radiation that could kill you instead and will spread what synchrotron radiation does occur out over a much wider area when it reaches your spacecraft, hopefully so that most of it just misses your spacecraft.

It does mean that your enemy will start shooting your drones so that they can eventually take a shot at you. But hopefully by that time you've already cleared out their drones and cored their spacecraft with your own synchrocannon or UREB.

==== Plasma magnets ====

Usually, to make a really big magnetic field you would run current through a really big conductive loop. But really big things are heavy and expensive. One way around this could be to use the ambient space plasma to carry the electric current rather than any kind of wire. In order to make this current, you use the same mechanism used in a rotary induction electric motor: you generate a rotating magnetic field and induce a current in the conductive loop (in this case, the space plasma). How do you make a rotating magnetic field? The method proposed<ref name="Slough2006">John Slough, "The Plasma Magnet", NASA Institute for Advanced Concepts Phase II Final Report (2006) https://www.niac.usra.edu/files/studies/final_report/917Slough.pdf</ref> is to use an antenna of four much smaller current loops as electromagnets at 90&degree; to each other, and sinusoidally vary the current through the loops with a respective 90&degree; phase difference between them. When this is done, a circular current is induced in the space plasma and the self-force of a magnetic field on its generating current inflates the current loop to much larger radii. You end up with a current sheet in the plane of the rotating field's rotation circulating about the antenna nearly synchronously with the field rotation. This current then creates a normal dipolar magnetic field as with any current loop. The current sheet and the resulting magnetic field expand until the magnetic field's pressure equals that of the ambient space plasma. This gives an enormous region of magnetic field from a relatively small and lightweight system.

The system proposed in the initial study would use an antenna approximately 10 meters in radius and produce a region of field approximately 30 km in radius at Earth's orbit. The field inside would not be constant - at the edges it would be 50 nT, giving the right pressure to keep out the solar wind at Earth's orbit. A circulating current sheet produces a field that falls off as 1/r with distance, meaning that adjacent to the antenna the field would be about 0.15 mT. Using the methods in the box below, we can estimate that this particular design as part of a spacecraft's shields will not deflect a TeV electron beam by more than a few centimeters, which is usually insufficient to force a miss. However, it could potentially be useful for a drone magnet, where the much larger distance between the magnetic field and the target spacecraft would allow a small deflection angle to give perhaps tens or low hundreds of meters change in beam location near the target at near light second ranges. The relatively low field and gradual bend of the incident beam suggests that synchrotron radiation will not be an issue; however this does not exclude more powerful plasma magnet designs causing significant synchrotron radiation effects.

<table border=1><tr><td>
Calculating the deflection from a spatially varying magnetic field affected by the solar wind is no trivial task. However, we can make an approximation that both should be within about an order of magnitude of the true value and also represent an upper limit to the deflection angle; although it does neglect the influence of the driving rotating magnetic field which could in some cases increase the deflection.

We will take R0 as the radius of the antenna. B0 is the field strength at the antenna. Rm is the distance at which the magnetic influence stops because the magnetic pressure equals the space plasma pressure. Bm is the field strength at Rm. According to Slough<ref name="Slough2006"></ref>, the magnetic field strength B(r) at any distance from the antenna center r between R0 and Rm is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B(r) = B0 R0 / r.
</div>
Therefore, once we know Bm and Rm, B0 is completely determined
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B0 = Bm Rm / R0.
</div>

The force on a particle with charge q passing perpendicularly through a magnetic field B(r) at a speed v is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
F(r) = q v B(r)
</div>
For an electron beam, v is so close to c as to make no difference. If the particle is moving down the y direction and the field is in the z direction, the force will be in the x direction. We assume that the deflection will be small, so we can choose a straight-line path in the y direction offset from the center by an amount x. The plasma bubble radius is Rm, so the particle will enter at ymax = √[Rm2-x2] and leave at ymin = -√[Rm2-x2].

The momentum change the particle experiences from this force is the time integral of the force over its path. Because of reflection symmetry about y = 0, we only need to integrate from ymax to 0 and then multiply by 2
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] dt F(r)
<tr> <td> <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] (dy/v) F(r)
<tr> <td> <td> = <td align="left">2 q B(0) R0 [Integral from 0 to √[Rm2-x2]] dy 1/√[y2 + x2]
<tr> <td> <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/x]
</table>
</div>
This holds for |x| > R0. But the expression for the field is not valid for r < R0 and the integral will give nonsense and diverging results if we evaluate for |x| < R0. In this case, we can put the integral lower bound at r = R0, or y = √[R02-x2]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/(√[R02-x2] + R0)]
</table>
</div>

The deflection angle is now estimated as
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = px / p
</div>
where the total momentum of the particle p can be found from the methods in the first box in the Magnetic Fields section.

The deflection for a shot aimed right at the spacecraft at the position of the spacecraft can be estimated by setting x = 0 and only taking the first half of the integral (so ignore the factor of 2)
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
q B(0) R0 ln[Rm/R0] / (p Rm)
</div>
This is an additional overestimate because it fails to account for how the deflection builds up over distance during the integration, but it should still give something of an order-of-magnitude and upper limit estimate of what kind of deflections might be expected.
</table>

=== Electric fields ===

If electric fields can accelerate charged particles, maybe they can keep the charged particles in a weaponized particle beam away from your spacecraft? To completely block a particle beam, you would need a voltage on your spacecraft equal to the particle energy in eV of the particles in the beam. That is, if a 1 GeV proton beam is coming at you, you'd need an electric potential of a billion volts to stop that beam. This leads to three immediate problems.

The first is that particles of the opposite charge will be attracted to your spacecraft. Because space isn't really empty but rather a rarefied plasma (called the solar wind in a solar system or the interstellar medium out between stars), there will be plenty of particles of the opposite charge around to attract. And they will hit with an energy in eV equal to the voltage on your spacecraft. In our example of defending against a 1 GeV proton beam with a gigavolt field, the electrons from the solar wind will come in and hit with an energy of 1 GeV each. This will generate dangerous radiation.

The second problem is a consequence of the first. Each of those GeV electrons from our example need to get their energy from somewhere, which turns out to be from the electric field we are using as shielding. You will need to make up for this energy loss with power from your electrical system, which gives a constant drain on your power. Some estimates put the electrical effort to maintain a several hundred megavolt shield in space at many tens of gigawatts<ref name="Mechmann2019">Claire Mechmann, "Analysis of Proposed Active Radiation Shielding Design Concept for Spacecraft" (2019) Thesis, College of Engineering and Science of Florida Institute of Technology</ref>.

And the third problem is similar to one for magnetic fields. Your electrodes will experience high self forces and forces between separate electrodes that will try to tear your shield generator apart. Consequently you will need heavy structural reinforcement in order for your shield to work.

There are potential ways to mitigate some of these problems. Interested readers are directed to [[Space_Radiation#Electrostatic_Shielding|the active shielding section in our page on space radiation]] for more of the juicy details; there's no need to repeat the same stuff here.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Elctrostatic_active_shielding.png|200 px|frameless]]
<td>
[[File:Electrostatic_active_shielding_2.png|200 px|frameless]]
<tr><td colspan=2>
Proposed designs for electrostatic shields to protect from positively charged particles<ref name="Tripathi2006">Ram K. Tripathi, John W. Wilson, and Robert C. Youngquist, "Electrostatic Active Radiation Shielding - Revisited", 2006 IEEE Aerospace Conference, Big Sky, MT, USA, 2006, pp. 9 pp.-, doi: 10.1109/AERO.2006.1655760.</ref><ref name="Chowdhury2023">Rajarshi Pal Chowdhury, Luke A. Stegeman, Matthew L. Lund, Dan Fry, Stojan Madzunkov, and Amir A. Bahadori, "Hybrid methods of radiation shielding against deep-space radiation", Life Sciences in Space Research, Volume 38, 2023, Pages 67-78, ISSN 2214-5524, https://doi.org/10.1016/j.lssr.2023.04.004.</ref>.
</table>
</div>

== Damage ==

Particle beams are, first and foremost, devices that produce beams of [[Nuclear_radiation|ionizing radiation]]. The radiation of weaponizable particle beams is extremely penetrating; thus of the [[Particle_Accelerators#Energy_loss_in_matter|interaction mechanisms listed for radiation from particle beams]], we mostly need to worry about bremsstrahlung and nuclear collisions and these produce extensive secondary showers of radiation that increase the penetration of the radiation exposure. Because of this, any target hit by a particle beam weapon will be irradiated, and radiation dose will always be one of the primary damage mechanisms of a particle beam weapon strike. Further, this radiation will be distributed along a track passing deeply through the material of the target, allowing it to affect critical systems through heavy armor, structural material, robust machine components, and passive material such as fuel load, coolant, or cargo. As the radiation deposits its energy into the target material, that material will be heated. If the target is heated sufficiently, it may experience [[Beam-Target_Interactions|thermal or mechanical]] damage from melting, cooking, combustion, vapor explosions, and warping or cracking due to thermal expansion.

The discussion of the radiation penetration into armor also applies to the same radiation penetrating through the target. So the formulas and rules of thumb for estimating the depth of radiation penetration into armor can also be used to estimate how far into a tank or spacecraft a particle beam strike delivers effects.

In addition to the radiation and thermal effects, a particle beam can also produce a sharp electromagnetic pulse that can damage semiconductor electronics.

== 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, Gerrit Bruhaug, and Luke Campbell for writing

Gerrit Bruhaug and Luke Campbell for scientific research, math wrangling and advise

ATriffleMind and Luke W. Campbell for proof reading.

==References==

[[Category:Beams]][[Category:Physics & Engineering‏‎]][[Category:Engineering‏‎]][[Category:Military Technology]][[Category:Warfare‏‎]]

Particle Beam Weapons

2026-10-27T04:48:50Z

Lwcamp: /* Separation of neutral beams */

{{PageConstructionNotice}}

''For a general overview on particle accelerator technology, see [[Particle Accelerators]]

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.

::: Z = 1.32×102 (γ2-1)3/4 r / √I<ref name=USPAS2016>Martin Reiser, "Theory and Design of Charged Particle Beams", 2008, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim</ref>

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

::: Z = β γ c r √[m/T]

Where Z is the doubling distance like before, r is the initial beam radius, T is the beam temperature in eV, m is the ion mass and β, c, and γ 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.

== Defenses ==

Sure, it's great to have a great big particle cannon to blast your enemies. But what if you are the one being shot at? In that case, it's best to figure out ways to keep from being blown to kingdom come.

=== Armor ===

The simplest way to defend against the high energy particles of a particle beam is to pile a whole bunch of stuff up and put it in the way of the beam. Particle beams may produce extremely penetrating radiation, but enough stuff will block most of that radiation and may allow you to survive. The details depend on what kind of particle is being thrown at you, and are explained below. However, a general rule of thumb is that you probably want about 10 tons of material per square meter of surface you want to armor in order to get sufficient protection.

==== Electrons ====

At the very high energy of an electron accelerated by a weapons-grade particle beam, the primary interaction with matter will be the emission of bremsstrahlung photons in the gamma-ray range as the electron slams into particles of matter. These high energy photons themselves primarily interact by the pair production mechanism – the photon disappears, but leaves behind a highly energetic positron and electron. These two particles are themselves energetic enough to produce further bremsstrahlung, and so the process continues. You get the formation of an electron-gamma shower (also called an electromagnetic cascade), which only stops after either the particles have penetrated through the target or the initial energy of the electron has been spread through so many shower particles (including all the electrons, positrons, and photons) that no particle has enough individual energy to pair produce or to be likely to bremsstrahlung again before it is stopped by other mechanisms.

To stop electrons, you'll want the atoms in your armor to be as heavy as you can practically get them. High atomic number both encourages bremsstrahlung and pair production, giving a double whammy for stopping that cascade. However, occasionally one of those electrons, positrons, or gammas will hit a nucleus and knock out neutrons. So you probably want a back-up layer of several tens of centimeters of hydrogen rich material such as plastic or water mixed with neutron absorbing materials like boron, lithium, cadmium, or gadolinium. High density polyethylene loaded with 5% boron by weight is a typical go-to neutron blocker.

The actual methods to calculate the penetration of the electromagnetic cascade is given are the box below. However, for some examples:
<ul>
<li> A 10 GeV beam incident on thick iron (or steel) will stop cascading at about 9 cm, will deposit half of its energy in the first 12 cm, and will have dumped 98% of its energy within 35 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 2.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 100 GeV beam incident on thick iron (or steel) will stop cascading at about 13 cm, will deposit half of its energy in the first 16 cm, and will have dumped 98% of its energy within 47 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 3.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick iron (or steel) will stop cascading at about 17 cm, will deposit half of its energy in the first 20 cm, and will have dumped 98% of its energy within 59 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 4.6 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick tungsten will stop cascading at about 3.5 cm, will deposit half of its energy in the first 4 cm, and will have dumped 98% of its energy within 12 cm. 90% of the cascade will be contained within about 0.93 cm and 95% of the cascade will be contained within 1.86 cm of the beam axis. You will get 98% protection with a mass thickness of 2.3 tons/m2 and an additional 1/e reduction for each further 0.13 tons/m2 beyond that.
<li> A 1 TeV beam incident on water will stop cascading at about 290 cm, will deposit half of its energy in the first 340 cm, and will have dumped 98% of its energy within 1030 cm. 90% of the cascade will be contained within about 11 cm and 95% of the cascade will be contained within 22 cm of the beam axis. You will get 98% protection with a mass thickness of 10.3 tons/m2 and an additional 1/e reduction for each further 0.72 tons/m2 beyond that.
</ul>
Note that for sufficiently intense particle beams, 98% protection may not be enough. You might need to throw on several more tons per square meter to be sure.

<table border=1>
<tr><td> How Far it Goes:
(This section follows methods presented in Leroy and Rancoita<ref>C. Leroy and P-G Rancoita, "Physics of cascading shower generation and propagation in matter: principles of high-energy, ultrahigh-energy and compensating calorimetry", Reports on Progress in Physics 63 (2000) 505–606. DOI: 10.1088/0034-4885/63/4/202</ref>.)

When trying to estimate the extent of the irradiated region of the target, it is helpful to know the radiation length X0 in the material. This is the distance that an electron will travel, on average, before emitting a bremsstrahlung photon. Conveniently, it is also the distance a high energy photon will travel through the material before pair producing. The radiation length can be found from
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   = 4 α  
<td nowrap="nowrap"> NA
<td rowspan="2" nowrap="nowrap" align="right">   Z (Z+ζ) re2 ln
<td rowspan="2" nowrap="nowrap" align="right"> [
<td nowrap="nowrap"> 183
<td rowspan="2" nowrap="nowrap" align="right"> ]
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> A
<td style="border-top:solid 1px black;"> Z1/3
</table>
</div>
where α = 1/137 is the fine structure constant, NA = 6.02214076×1023 is Avogadro's number (or one mole), A is the atomic weight of the atoms in the material (usually expressed in grams per mole), Z is the atomic number (nuclear charge) of the atoms, and re = 2.8179403205×10-13 cm is the classical electron radius. ζ is a constant of approximately 1.3. This gives the thickness of material in terms of the areal density (mass of material per unit area perpendicular to the beam) – if you use A in g/mol and re in cm the result will be in cm2/g and X0 will be in g/cm2. If your target is made up of more than one kind of atom, you can get the total radiation length by finding the radiation length of each atom of type i, X0i and using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   =  
<td rowspan="2" nowrap="nowrap" align="right"> Σi  
<td nowrap="nowrap"> fi
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> X0i
</table>
</div>
where fi is the fraction by weight of the atom type i. Once you have the radiation length in terms of areal density, you can easily convert it to distance by dividing by the density ρ. If X0i is in g/cm2 and density is in g/cm3, then X0i/ρ will be in cm.

We will also need to know the critical energy εc. This is the energy where an electron is losing energy as rapidly to ionization as it is to radiation. That is, it is the energy at which the cascade stops multiplying, even though the individual particles already made can continue traveling for some distance. The critical energy can be estimated using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> εc = B  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> Z X0
<td rowspan="2" nowrap="nowrap" align="right"> )
<td nowrap="nowrap"> h
<tr>
<td style="border-top:solid 1px black;"> A
<td>  
</table>
</div>
where B = 2.66 MeV and h = 1.11.

We can now describe the distribution of radiation in the target material. The depth tmax where the cascade stops multiplying (in g/cm2) for an incident electron of energy E (in MeV) is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tmax = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> - 1
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
Similarly, the depth where half the energy of the beam has been deposited (in g/cm2) is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tcg = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> + 0.37
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
As usual, you can convert to a depth in cm by dividing by the material density. 98% of the beam energy will be deposited within a distance of &cong; 3 tcg. Beyond this, the beam energy will continue to fall off with thickness t approximately as an exponential exp[-t/λ], where the characteristic attenuation length λ &cong; 2 X0.

The width of a cascade, where 90% of the energy is contained, is approximately given by
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> RM =  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> EM
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
where EM &cong; 21.2 MeV. 95% of the radiation will be contained within a distance of 2 RM from the beam axis.
</table>

==== Protons and Ions ====

<table align=right><tr><td>
[[File:Regolith_Shielding.png|350 px|frameless]]
<tr><td width=350>
Relative effect of particle radiation (compared to no shielding) from galactic cosmic rays behind different thicknesses of water, aluminum, and lunar regolith<ref name="Slaba2022">Tony C. Slaba, "Radiation Shielding for Lunar Missions: Regolith Considerations", LSIC Crosstalk 7/18/2022 https://lsic.jhuapl.edu/uploadedDocs/focus-files/1604-E&C%20+%20EE%20Monthly%20Meeting%20-%202022%2007%20July_Presentation%20-%20NASA%20Slaba.pdf</ref>.
</table>
Protons and ions are grouped together because ions are made of protons and neutrons and when you accelerate these things for a weapon it is the protons and neutrons that are going to be doing the heavy lifting for blasting your target. This is because protons and neutrons are both about 2000 times more massive than electrons so any electrons along for the ride will contribute negligibly to the energy. Protons and neutrons together are called nuclear particles, and both behave quite similarly in nuclear collisions.

Nuclear particles are themselves made of quarks and can interact via the strong nuclear force. At the energies appropriate for weapons use, essentially all of the way protons or ions are blocked is through [[Particle_Accelerators#Nuclear_collisions|direct collisions with the atomic nuclei]] of the armor material. One or more quarks in the projectile will slam into quarks in the target nucleus and stretch them away from the rest of the quarks in the nuclear particles. Physics tells us that you can't have isolated quarks, so the "string" attaching the quark to its neighbors stretches and stretches until it snaps. Snapping releases so much energy that more quarks are made that clump together into all sorts of other nuclear particles, both exotic and familiar. This process is called a hadronic jet. The only particles produced in these jets long-lived enough to travel any noticeable distance through the armor material will be protons, neutrons, the charged pi mesons (or pions), and some of the K mesons (or kaons). The particles too short lived to go far enough to matter will decay into other particles, and these particles are often very high energy gamma rays. Much like with electrons, the long-lived secondary particles can be sufficiently energetic to smash into other nuclei and cause more hadronic jets (or, if they are low enough in energy, just knock more nuclear particles out of target nuclei) and this produces a similar shower or cascade. But in addition, the armor nuclei that were hit will be very hot after the impact from all of the energy given to them from the incident particles. These nuclei will cool off by evaporation – but instead of evaporating away molecules, they evaporate off nuclear particles from the nucleus. These evaporated particles are mostly neutrons, so any armor against proton or ion beam strikes will produce quite a bit of neutron radiation.

Eventually, the original particle's energy will be distributed among so many other particles that the charged particles will come to a stop by making ion tracks rather than from colliding with more nuclei, and the cascade halts. But you still have all the neutrons to worry about. And the gamma rays as well, which are high enough energy to kick off an electromagnetic cascade in the same way that electrons do.

Unlike electrons, which have reasonable approximations of their cascade behavior from relatively simple equations, the cascades of protons and ions usually require sophisticated simulations or experiments to get the resulting distribution of radiation. However, we can get a reasonable idea of their general behavior by comparing them to [[Space_Radiation#Shielding_Against_Cosmic_Rays|cosmic ray particles interacting with shielding]]. Cosmic rays are energetic protons and ions, after all. From this we see that a relatively small amount of armor – in the range of 1 to 3 tons per square meter – can actually make the radiation dose worse. The incident particles will knock off so many secondary particles in their shower that the dose is multiplied from so many tracks passing through the target. Hydrogen rich materials such as water and plastics work the best against proton and ion beam particles; heavy elements are the worst and produce the most secondaries. You could probably obtain adequate radiation protection from most incident beams with about 10 tons per square meter of hydrogen rich material as armor. However, you will want to add neutron absorbing elements to your hydrogen rich armor, such as lithium, boron, cadmium, or gadolinium. Further, in order to deal with the gamma rays you will want a layer of at least several centimeters of heavy elements, such as tungsten or lead, behind the hydrogen rich particle armor.

==== Structural Considerations ====

When your armor material absorbs a radiation beam, the energy of that radiation turns into heat inside the armor where the radiation was absorbed. If the beam pulse has enough energy as was sufficiently tightly focused, this heat can damage your armor. Your armor may melt in places, or thermal expansion could cause it to crack or warp. Perhaps the worst thing it could do is vaporize or flash into a plasma, because the pressure from the vapor or plasma can be sufficient to plastically deform the armor. This leaves a hole along the beam path where the expanding gas or plasma blasted out a long, deep crater. This means a beam that fires rapid shots spaced maybe milliseconds apart can have subsequent shots go down the same hole so your armor will not protect against them. Each shot will dig out a deeper hole until either your armor is penetrated or the attacking beam runs out of shots in its pulse train.

A consequence of this is that when designing your armor you may wish to use refractory materials with a high structural strength (like tungsten), or that is otherwise resistant to being blasted apart by its own evaporated remains.

=== Magnetic fields ===

[[Particle_Accelerators#Magnetic_fields|A charged particle crossing a magnetic field will bend so as to spiral around the field lines]]. If you try to shoot your particle beam through a magnetic field, it will bend away from the direction you shot it. This can make it hard to hit your target. Unfortunately for you, there are magnetic fields everywhere – around planets, and even trapped in the solar wind as it crosses the solar system.

As an example, the Earth has a magnetic field strength of about 5×10-5 tesla near its surface, and falling off farther away. If you are shooting a 1 TeV electron beam past the Earth, your beam is likely to bend by several milliradians (with the exact deflection depending on how close you are skimming the Earth). If you are in a medium Earth orbit and shooting at another target also in a medium Earth orbit on a trajectory that goes near the Earth, then without correcting for the magnetic field your beam will likely be off-target by tens to hundreds of kilometers.

For another example, the interplanetary magnetic field is about 10-9 tesla at Earth's distance from the sun. If you shoot that same 1 TeV electron beam at a target 0.1 light seconds away, you are likely to be off by a couple hundred meters.

<table border=1>
<tr><td>
The radius of curvature of the bend of a particle in a magnetic field is called the gyroradius rc. For a particle with a charge q, mass m, and kinetic energy E, find the relativistic momentum p
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
p = √[E2 - m2 c4] / c
</div>
To convert from energy units of eV to joules, multiply by 1.60×10-19. An electron has a mass of 9.11×10-31 kg, a proton of 1.67×10-27 kg, and an ion of 1.66×10-27×A kg, where A is the atomic weight of the ion. The speed of light c = 3.00×108 m/s.

In a magnetic field of strength B with an angle θ between the field direction and the direction of motion
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rc = |p sin(θ) / (q B)|
</div>
For electrons, q = -e, where the fundamental charge e = 1.60×10-19 coulomb. For a proton, q = +e. For an ion, multiply e by the charge state of the ion.

If the field acts over a distance d that is much smaller than the gyroradius, the deflection angle φ is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = d / rc
</div>
The deflection x at a target at a range R for small φ (φ &Lt; 1) will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
x = φ R.
</div>

Note that d will always be less than or equal to x. Also note that if d is not smaller than twice the gyroradius the beam will never reach the target because it will completely spiral around the field lines before it reaches the target.
</table>

So, it is clear that as the defender it will be helpful to hang out in areas with high magnetic fields, and keep some distance between you and the attacker. Meanwhile, the attacker will benefit from being closer where small deflections in the beam trajectory don't translate into large excursions from where the beam is aimed (and also because there will be less space filled with fields between the two of you). But the attacker can also try to compensate for known fields. If you happen to know that the magnetic fields are between you and your target, you can adjust your fire so that the deflection of the field bends your beam onto the target. This is not a simple task. It is easy enough to measure the field strength and direction where you are, but predicting how it changes over the path of your shot does not have an obvious solution. You can also try to gain intelligence on likely battle sites before you ever fight on them – for example, getting good data on the structure of the fields of a planet using spies and technical intelligence during peacetime (heck, on Earth this information is published for free in academic journals and by government agencies such as NOAA). Planetary magnetospheres are also affected by the solar wind, so different solar weather will change the fields you have to shoot through. But space weather forecasts and predictive simulations of the solar wind interacting with a planetary magnetosphere can give you a good idea of what the field structure is like. But even with all that, there are changes in the fields that happen on a scale of around a second ... anyone who has ever seen the auroras dancing across the sky in the polar night can see this. Compensating for these changes will be a significant challenge.

What would really help is being able to see the path of your own beam. This would allow you to see how far off you were from your target and in what direction, and then adjust fire to walk your shots onto the target. Unfortunately, particle beams in space are invisible. There might be ways to track them, but if so we don't know what they are.

==== Synchrotron radiation ====

When charged particle trajectories are bent, they emit [[Particle_Accelerators#Synchrotron_radiation|synchrotron radiation]]. This can cause the particle beam to lose energy, but if the bend is too tight it can also generate high energy gamma rays that ignore magnetic fields and directly strike the target. Synchrotron radiation can generally be ignored for proton or ion beams, but can be significant for electron beams.

<table border=1><tr><td>
For a beam particle with energy E and mass m, the synchrotron radiation is concentrated in a forward-directed cone with an opening angle of approximately 1/γ, where γ = 1 + E/(m c2). (Interestingly, the radiation intensity peaks an at angle of 1/(2 γ) and vanishes directly on-axis.) Because the beam is bending, you will get emission along the plane along which the beam is being bent. The energy of the photons produced during synchrotron emission is approximately<ref name="Jackson_Electrodynamics">J. D. Jackson, "Classical Electrodynamics: Second Edition", Wiley (1962)</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Eγ &cong; 3 γ3 c &hbar; / rc
</div>
where &hbar; = 1.05×10-34 J s (if you want the energy in joules) or &hbar; = 6.58×10-16 eV s (if you want the energy in eV). The photon energy will not exceed the energy of the incident particles, if you get a photon energy higher than the energy of the incident particles it just means that the synchrotron radiation spectrum extends up to the incident particle energy.

The fraction of the energy (or power) of the beam turned into synchrotron radiation will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = [1/(4 π ε0)] [4 π / 3] [q² / rc] β2 γ4 [φ / (2 π)] / E
</div>
where β = √[1 - 1/γ2] is the particle's speed measured as a fraction of the speed of light, and 1/(4 π ε0) = 8.99×109 is the Coulomb constant. For highly relativistic electrons (like all practical electron beam weapons emit), β &cong; 1 and we can use
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = 8.85×10-2 [E (GeV)3 [φ / (2 π)] / (rc (m))]
</div>
This fraction cannot be more than 1. If f gets close to one, it means that our original assumption that the beam particles mostly keep their original speed no longer holds, and the beam will lose most or almost all of its energy. Instead, when the beam is being deflected it will only make it through an angle of φ/f and will irradiate a region of length x/f starting at where the beam would have been incident if not for the magnets and extending in the direction the beam was deflected.
</table>

For example, a 1 TeV electron beam will have γ &cong; 2,000,000. So if it's bent by a magnetic field the synchtrotron radiation will spread out in a cone of about 5×10-7 radians. If this beam was shooting at a target 0.1 light seconds away from where it was bent, the radiation would arrive at the target over a region with a radius of about 15 meters with a maximum intensity at about 7.5 meters and no radiation directly along the path the beam was pointing while it was being bent. If the beam was bent by the interplanetary magnetic field from our earlier example, it would produce light with photon energies of around 1.3 eV (near infrared) and the lost energy would be a few parts in a hundred trillion. If it was instead deflected by the Earth's magnetic field you might expect a peak photon energy of tens of keV – in the hard x-ray part of the spectrum – and the fraction of energy in the synchrotron radiation would be somewhere in the low parts in 100,000 or less.

==== Separation of neutral beams ====

Magnetic fields only affect charged objects. So one way to overcome magnetic defenses or environments is to use beams that have no net charge. For example, a proton beam could be neutralized with a co-propagating electron beam with an equal number of protons and electrons. Or, potentially, you could fire an ion beam and then neutralize the ions after acceleration to get a beam of neutral atoms.

But, the magnetic fields are still tugging at the charged particles making up the beam as they go past. This will pull the electrons one way and the protons or ions the other way. As the particles are pulled in opposite directions you get a net positive charge on one side of the beam and a net negative charge on the other side (called polarization). This sets up an electric field inside the beam that acts to pull the particles back together. If the particles are going fast enough and the magnetic field is strong enough, the magnetic force can still separate the charges and will split the beam in two, one with the positive charges and one with the negative charges. But if the particles are not that fast or the magnetic field is not that strong this polarization of the beam will keep it together and let it fly right through the magnetic field.

<table border=1><tr><td>
Let's figure out how to analyze this problem. For definiteness, we will have a beam with an electric current of the positive species of I and an electric current of electrons of -I, all moving in the z direction. The beam radius will be R. The particle speed will be v with the usual relativistic parameters β = v/c and γ = √[1/(1-β2)]. If you know the energy per particle E of the positive species and the mass m of the positive species, γ = 1 + E/(m c2) and then β = √[1 - 1/γ2].

Let's assume that the beam is going through a magnetic field of strength B at an angle θ to the field direction. The quantity B sin(θ) is the component of the magnetic field perpendicular to the direction of the beam.

The charge density ρ of the positive component of the beam will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρL = I / (π v R2).
</div>

It is easiest to analyze the problem in the rest frame of the particles in the beam. Now the particles are at rest. Particles at rest are not affected by a magnetic field, but the change of reference frame Lorentz transforms the initial magnetic field into a combination of electric and magnetic fields in the rest frame. We can ignore the magnetic field, but the electric field is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔A = γ v B sin(θ)
</div>
in a direction perpendicular to both v and B. For our purposes, assume the electric field points in the x direction.

At the same time, relativistic length contraction increases the charge density in the rest frame
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρ = γ ρL.
</div>

Let r be the vector from the beam axis to the point inside the beam. It will be useful to separate this into its Cartesian coordinates. If x and y are the unit vectors in the x and y directions, respectively, then
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
r = x x + y y
</div>

If we consider only the positive component of the beam, from [https://en.wikipedia.org/wiki/Gauss%27s_law Gauss's law] it will have an electric field inside of the beam at a distance r from the center axis of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔+ = ρ r / (2 ε0) = (ρ/(2 ε0)) (x x + y y)
</div>
where ε0 = 8.85×10-12 is the vacuum permittivity. Now let's displace the electrons in the beam by a distance δ in the x direction. By itself, this electron beam will have an electric field of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔- = -(ρ/(2 ε0)) ((x-δ) x + y y)
</div>
By the principle of superposition, the electric field inside the beam with both components is the sum of the electric fields of the separate components including the external field
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔B = [(ρ/(2 ε0)) δ + γ v B sin(θ)]x
</div>
Because the charges are free to move, they will do so as long as a field is applied to them. Thus, the beam becomes stable once the charges separate enough that the field internal to the beam is zero. Therefore
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
δ = -2 ε0 γ v B sin(θ) / ρ = -2 π ε0 v2 R2 B sin(θ) / I
</div>
If |δ| > 2 |R|, the beam will fully separate and each component will go its own way. If δ is small compared to R, the beam's polarization suffices to hold it together in the overlap region. However, where the positive and negative charges do not overlap the electric fields no longer cancel, and the external electric field carries away the outer polarized part of the beam. This gives us a rough threshold for which a magnetic field will quickly separate a neutral beam into its charged components and bend them away from their target
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr><td> B sin(θ) < I / (π ε0 v2 R) <td>       <td> Beam is partially eroded.
<tr><td> B sin(θ) ≥ I / (π ε0 v2 R) <td> <td> Beam promptly separates.
</table>
</div>
For a partially eroding beam, the polarized region that was peeled off by the field still acts to screen the remaining neutral portion, at least to within this approximation, leaving the still-neutral beam to hit the target.
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Neutralized_beam_in_B_field.png|5140 px|frameless]]
<tr><td width=700>
A profile of the beam charge distribution of a neutralized beam in a magnetic field. Negative charge density is shown in yellow, positive charge density in cyan, with neutral charge density in light green. The magnetic field is vertical, the electric field in the beam reference frame is shown in magenta including both the polarization field and the Lorentz transformed electric field resulting from the magnetic field. As positive charges are pulled in the direction of the field and negative charges opposite to the field direction, the polarized boundary will be pulled off the beam by the electric field even though the electric field is zero in nearly all the beam interior. This expanding polarization layer will continue to largely screen the still-neutral portion, however; but the extent to which the screening is only partial would require more sophisticated analysis methods than used here. The material removed by the field will expand to a maximum of the deflection distance of a uniform charge based on the magnetic field by the time it reaches the target.
</table>
</div>
<tr><td>
For a beam of neutral atoms, let a be the radius of the atom and let Eb be the binding energy of the least tightly bound electron to the atom. An external electric field in excess of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
|𝓔A| > Eb / (e a)
</div>
will promptly ionize the atom, ripping away its electron. Here, e = 1.60×10-19 C is the magnitude of the electric charge of the electron. For fields close to but a bit weaker than this, the electron can still rapidly pop off due to quantum tunneling<ref>Ph. Durand and I. Paidarová, “Ionization of the hydrogen atom: from weak to strong static electric fields,” The European Physical Journal D, Vol. 26, pg. 253-259, 2003.</ref>. In general, expect that for fields greater in magnitude than (very approximately) Eb / (10 e a) the electron will be quickly removed, while for fields weaker than this the electrons will stay on their atoms.

It is useful to note that Eb/e is just the binding energy in units of eV. The most weakly bound electrons are usually bound to their atom with about 10 eV of energy. Atoms are typically about 10-10 m in radius. Thus, you can expect fields of around 1010 to 1011 V/m to directly ionize atoms. If we call this threshold value 𝓔T, then a beam will be ionized when passing through a field with strength greater than
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B sin(θ) > 𝓔T / (γ v)
</div>
</table>

Example: Let's consider a 1 GeV neutralized proton beam from a synchrotron, with a beam current of 200 kA in each bunch (incidentally, if a bunch is 5 cm long this means that each bunch packs 38 kJ worth of energy). The speed of the bunch is 87% the speed of light, or 262 million m/s, with a γ of 2.07. Consider when the beam's focus takes it to a radius of 5 cm (or 0.05 m, for our calculation). Plugging this in, we find that the protons in the bunch will separate from the electrons in a field with a transverse component greater than about 2.1 T, and the bunch will pass through the field with only gradual losses if the transverse component is much smaller than this.

Example: We'll take the same 1 GeV beam, but this time recombine the electrons with protons to make hydrogen atoms. An electron is bound to a hydrogen atom with an energy of 13.6 eV (the Ryberg constant) and the radius of the electron orbital in a hydrogen atom is 0.529×10-10 m (the Bohr radius). This gives a threshold electric field for prompt ionization of hydrogen of 2.57×1011 V/m. We will get this with a transverse component of the magnetic field of 474 T. Above this field the atoms promptly ionize and electrons are stripped away from the protons. For fields higher than roughly 1/10 of this – so about 40 to 50 T – you may still get significant ionization by quantum tunneling; lower than this your losses from the beam due to field-induced ionization are likely negligible.

==== Magnetic shields ====

What happens when we can't rely on magnetic fields in the environment to protect us? One idea is to produce powerful artificial magnetic fields around our spacecraft to deflect incoming particle beams. This is very similar in concept to using active magnetic shielding to protect against space radiation – [[Space_Radiation#Magnetic_Shielding|check out our article on space radiation for more]].
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr>
<td width=100>
[[File:Unconfined_FRC_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:racetrack_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:Magnetic_shielding_Halback_Array.png|200 px|frameless]]
<tr><td colspan=3>
Some potential configurations for magnetic shields. The currents generating the fields are in green, the magnetic fields themselves are shown in magenta<ref name="Battiston2012">R. Battiston, W.J. Burger, V. Calvelli, R. Musenich, V. Choutko, V.I. Datskov, A. Della Torre, F. Venditti,
C. Gargiulo, G. Laurenti, S. Lucidi, S. Harrison, and R. Meinke, "ARSSEM Active Radiation Shield for Space Exploration Missions", Final Report ESTEC Contract N° 4200023087/10/NL/AF : “Superconductive Magnet for Radiation Shielding of Human Spacecraft” (2012) https://arxiv.org/abs/1209.1907 https://www.researchgate.net/publication/265945847_Active_Radiation_Shield_for_Space_Exploration_Missions</ref><ref name="Desiati2022">Paolo Desiati and Elena D'Onghia, "CREW HaT: A Magnetic Shielding System for Space Habitats", arXiv:2209.13624 [physics.space-ph] https://doi.org/10.48550/arXiv.2209.13624</ref>.
</table>
</div>

One immediate issue is that you want to keep very high magnetic fields out of the crew area. Fields in excess of 0.5 T can start to be unpleasant, and it is recommended to keep the field below 0.2 T for continuous exposure.

Strong magnets can suffer from high self-forces acting to tear them apart. Most shielding designs will have multiple magnetic coils, which will exert high forces and torques on each other. This will result in the need for strong structural bracing to keep the magnets intact and in place. Such strong structural designs will impose significant extra mass on the spacecraft.

Consider a magnetic field configuration that can keep radiation out of a certain region of space. There is no trajectory of a charged radiation particle up to a certain energy that will allow those particles to enter the protected region. But the dynamics of charged particles in a magnetic field are the same whether going forward or backward in time. If there is no way for a particle to get in, by this time reversal symmetry any particle which is in that forbidden region can never escape. Consequently any field configuration that is a magnetic trap is also a magnetic shield; any field configuration that is a magnetic shield is also a magnetic trap. A spacecraft with a magnetic shield may end up surrounded by strong radiation belts.

Using a magnet to bend a weaponized electron beam very often ends up with the beam being bent so hard that it dumps most of its energy in the form of synchrotron radiation at gamma ray energies. Because gamma rays are not charged, they will not be deflected by the field. Thus, if you would have been hit by the electron beam you will be hit instead by the gamma rays.

As an example, consider a 1 TeV electron beam. We'll surround our spacecraft with a 10 T magnetic field in a 100 meter radius region just outside our spacecraft. This will deflect the beam by 30 meters, which is enough to make it miss. But we are not saved, because the beam dumps nearly all of its energy into synchrotron radiation with an energy spectrum of gamma rays up to 13 GeV. This radiation will hit us in a strip about 2.4 meters long and not much more than the width of the original beam with essentially all of the energy of the original beam.

==== Magnet drones ====

If a magnetic shield around your spacecraft will just end up killing you from synchrotron radiation, what if you put the magnetic shield on a drone and put that drone between yourself and the enemy trying to shoot you? This lets you use a much weaker magnet because you don't need nearly so high of a deflection angle to make the shot miss. This means that less beam energy will turn into synchrotron radiation that could kill you instead and will spread what synchrotron radiation does occur out over a much wider area when it reaches your spacecraft, hopefully so that most of it just misses your spacecraft.

It does mean that your enemy will start shooting your drones so that they can eventually take a shot at you. But hopefully by that time you've already cleared out their drones and cored their spacecraft with your own synchrocannon or UREB.

==== Plasma magnets ====

Usually, to make a really big magnetic field you would run current through a really big conductive loop. But really big things are heavy and expensive. One way around this could be to use the ambient space plasma to carry the electric current rather than any kind of wire. In order to make this current, you use the same mechanism used in a rotary induction electric motor: you generate a rotating magnetic field and induce a current in the conductive loop (in this case, the space plasma). How do you make a rotating magnetic field? The method proposed<ref name="Slough2006">John Slough, "The Plasma Magnet", NASA Institute for Advanced Concepts Phase II Final Report (2006) https://www.niac.usra.edu/files/studies/final_report/917Slough.pdf</ref> is to use an antenna of four much smaller current loops as electromagnets at 90&degree; to each other, and sinusoidally vary the current through the loops with a respective 90&degree; phase difference between them. When this is done, a circular current is induced in the space plasma and the self-force of a magnetic field on its generating current inflates the current loop to much larger radii. You end up with a current sheet in the plane of the rotating field's rotation circulating about the antenna nearly synchronously with the field rotation. This current then creates a normal dipolar magnetic field as with any current loop. The current sheet and the resulting magnetic field expand until the magnetic field's pressure equals that of the ambient space plasma. This gives an enormous region of magnetic field from a relatively small and lightweight system.

The system proposed in the initial study would use an antenna approximately 10 meters in radius and produce a region of field approximately 30 km in radius at Earth's orbit. The field inside would not be constant - at the edges it would be 50 nT, giving the right pressure to keep out the solar wind at Earth's orbit. A circulating current sheet produces a field that falls off as 1/r with distance, meaning that adjacent to the antenna the field would be about 0.15 mT. Using the methods in the box below, we can estimate that this particular design as part of a spacecraft's shields will not deflect a TeV electron beam by more than a few centimeters, which is usually insufficient to force a miss. However, it could potentially be useful for a drone magnet, where the much larger distance between the magnetic field and the target spacecraft would allow a small deflection angle to give perhaps tens or low hundreds of meters change in beam location near the target at near light second ranges. The relatively low field and gradual bend of the incident beam suggests that synchrotron radiation will not be an issue; however this does not exclude more powerful plasma magnet designs causing significant synchrotron radiation effects.

<table border=1><tr><td>
Calculating the deflection from a spatially varying magnetic field affected by the solar wind is no trivial task. However, we can make an approximation that both should be within about an order of magnitude of the true value and also represent an upper limit to the deflection angle; although it does neglect the influence of the driving rotating magnetic field which could in some cases increase the deflection.

We will take R0 as the radius of the antenna. B0 is the field strength at the antenna. Rm is the distance at which the magnetic influence stops because the magnetic pressure equals the space plasma pressure. Bm is the field strength at Rm. According to Slough<ref name="Slough2006"></ref>, the magnetic field strength B(r) at any distance from the antenna center r between R0 and Rm is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B(r) = B0 R0 / r.
</div>
Therefore, once we know Bm and Rm, B0 is completely determined
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B0 = Bm Rm / R0.
</div>

The force on a particle with charge q passing perpendicularly through a magnetic field B(r) at a speed v is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
F(r) = q v B(r)
</div>
For an electron beam, v is so close to c as to make no difference. If the particle is moving down the y direction and the field is in the z direction, the force will be in the x direction. We assume that the deflection will be small, so we can choose a straight-line path in the y direction offset from the center by an amount x. The plasma bubble radius is Rm, so the particle will enter at ymax = √[Rm2-x2] and leave at ymin = -√[Rm2-x2].

The momentum change the particle experiences from this force is the time integral of the force over its path. Because of reflection symmetry about y = 0, we only need to integrate from ymax to 0 and then multiply by 2
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] dt F(r)
<tr> <td> <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] (dy/v) F(r)
<tr> <td> <td> = <td align="left">2 q B(0) R0 [Integral from 0 to √[Rm2-x2]] dy 1/√[y2 + x2]
<tr> <td> <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/x]
</table>
</div>
This holds for |x| > R0. But the expression for the field is not valid for r < R0 and the integral will give nonsense and diverging results if we evaluate for |x| < R0. In this case, we can put the integral lower bound at r = R0, or y = √[R02-x2]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/(√[R02-x2] + R0)]
</table>
</div>

The deflection angle is now estimated as
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = px / p
</div>
where the total momentum of the particle p can be found from the methods in the first box in the Magnetic Fields section.

The deflection for a shot aimed right at the spacecraft at the position of the spacecraft can be estimated by setting x = 0 and only taking the first half of the integral (so ignore the factor of 2)
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
q B(0) R0 ln[Rm/R0] / (p Rm)
</div>
This is an additional overestimate because it fails to account for how the deflection builds up over distance during the integration, but it should still give something of an order-of-magnitude and upper limit estimate of what kind of deflections might be expected.
</table>

=== Electric fields ===

If electric fields can accelerate charged particles, maybe they can keep the charged particles in a weaponized particle beam away from your spacecraft? To completely block a particle beam, you would need a voltage on your spacecraft equal to the particle energy in eV of the particles in the beam. That is, if a 1 GeV proton beam is coming at you, you'd need an electric potential of a billion volts to stop that beam. This leads to three immediate problems.

The first is that particles of the opposite charge will be attracted to your spacecraft. Because space isn't really empty but rather a rarefied plasma (called the solar wind in a solar system or the interstellar medium out between stars), there will be plenty of particles of the opposite charge around to attract. And they will hit with an energy in eV equal to the voltage on your spacecraft. In our example of defending against a 1 GeV proton beam with a gigavolt field, the electrons from the solar wind will come in and hit with an energy of 1 GeV each. This will generate dangerous radiation.

The second problem is a consequence of the first. Each of those GeV electrons from our example need to get their energy from somewhere, which turns out to be from the electric field we are using as shielding. You will need to make up for this energy loss with power from your electrical system, which gives a constant drain on your power. Some estimates put the electrical effort to maintain a several hundred megavolt shield in space at many tens of gigawatts<ref name="Mechmann2019">Claire Mechmann, "Analysis of Proposed Active Radiation Shielding Design Concept for Spacecraft" (2019) Thesis, College of Engineering and Science of Florida Institute of Technology</ref>.

And the third problem is similar to one for magnetic fields. Your electrodes will experience high self forces and forces between separate electrodes that will try to tear your shield generator apart. Consequently you will need heavy structural reinforcement in order for your shield to work.

There are potential ways to mitigate some of these problems. Interested readers are directed to [[Space_Radiation#Electrostatic_Shielding|the active shielding section in our page on space radiation]] for more of the juicy details; there's no need to repeat the same stuff here.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Elctrostatic_active_shielding.png|200 px|frameless]]
<td>
[[File:Electrostatic_active_shielding_2.png|200 px|frameless]]
<tr><td colspan=2>
Proposed designs for electrostatic shields to protect from positively charged particles<ref name="Tripathi2006">Ram K. Tripathi, John W. Wilson, and Robert C. Youngquist, "Electrostatic Active Radiation Shielding - Revisited", 2006 IEEE Aerospace Conference, Big Sky, MT, USA, 2006, pp. 9 pp.-, doi: 10.1109/AERO.2006.1655760.</ref><ref name="Chowdhury2023">Rajarshi Pal Chowdhury, Luke A. Stegeman, Matthew L. Lund, Dan Fry, Stojan Madzunkov, and Amir A. Bahadori, "Hybrid methods of radiation shielding against deep-space radiation", Life Sciences in Space Research, Volume 38, 2023, Pages 67-78, ISSN 2214-5524, https://doi.org/10.1016/j.lssr.2023.04.004.</ref>.
</table>
</div>

== Damage ==

Particle beams are, first and foremost, devices that produce beams of [[Nuclear_radiation|ionizing radiation]]. The radiation of weaponizable particle beams is extremely penetrating; thus of the [[Particle_Accelerators#Energy_loss_in_matter|interaction mechanisms listed for radiation from particle beams]], we mostly need to worry about bremsstrahlung and nuclear collisions and these produce extensive secondary showers of radiation that increase the penetration of the radiation exposure. Because of this, any target hit by a particle beam weapon will be irradiated, and radiation dose will always be one of the primary damage mechanisms of a particle beam weapon strike. Further, this radiation will be distributed along a track passing deeply through the material of the target, allowing it to affect critical systems through heavy armor, structural material, robust machine components, and passive material such as fuel load, coolant, or cargo. As the radiation deposits its energy into the target material, that material will be heated. If the target is heated sufficiently, it may experience [[Beam-Target_Interactions|thermal or mechanical]] damage from melting, cooking, combustion, vapor explosions, and warping or cracking due to thermal expansion.

The discussion of the radiation penetration into armor also applies to the same radiation penetrating through the target. So the formulas and rules of thumb for estimating the depth of radiation penetration into armor can also be used to estimate how far into a tank or spacecraft a particle beam strike delivers effects.

In addition to the radiation and thermal effects, a particle beam can also produce a sharp electromagnetic pulse that can damage semiconductor electronics.

== 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, Gerrit Bruhaug, and Luke Campbell for writing

Gerrit Bruhaug and Luke Campbell for scientific research, math wrangling and advise

ATriffleMind and Luke W. Campbell for proof reading.

==References==

[[Category:Beams]][[Category:Physics & Engineering‏‎]][[Category:Engineering‏‎]][[Category:Military Technology]][[Category:Warfare‏‎]]

Particle Beam Weapons

2026-10-27T02:13:51Z

Lwcamp: /* Magnetic fields */

{{PageConstructionNotice}}

''For a general overview on particle accelerator technology, see [[Particle Accelerators]]

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.

::: Z = 1.32×102 (γ2-1)3/4 r / √I<ref name=USPAS2016>Martin Reiser, "Theory and Design of Charged Particle Beams", 2008, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim</ref>

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

::: Z = β γ c r √[m/T]

Where Z is the doubling distance like before, r is the initial beam radius, T is the beam temperature in eV, m is the ion mass and β, c, and γ 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.

== Defenses ==

Sure, it's great to have a great big particle cannon to blast your enemies. But what if you are the one being shot at? In that case, it's best to figure out ways to keep from being blown to kingdom come.

=== Armor ===

The simplest way to defend against the high energy particles of a particle beam is to pile a whole bunch of stuff up and put it in the way of the beam. Particle beams may produce extremely penetrating radiation, but enough stuff will block most of that radiation and may allow you to survive. The details depend on what kind of particle is being thrown at you, and are explained below. However, a general rule of thumb is that you probably want about 10 tons of material per square meter of surface you want to armor in order to get sufficient protection.

==== Electrons ====

At the very high energy of an electron accelerated by a weapons-grade particle beam, the primary interaction with matter will be the emission of bremsstrahlung photons in the gamma-ray range as the electron slams into particles of matter. These high energy photons themselves primarily interact by the pair production mechanism – the photon disappears, but leaves behind a highly energetic positron and electron. These two particles are themselves energetic enough to produce further bremsstrahlung, and so the process continues. You get the formation of an electron-gamma shower (also called an electromagnetic cascade), which only stops after either the particles have penetrated through the target or the initial energy of the electron has been spread through so many shower particles (including all the electrons, positrons, and photons) that no particle has enough individual energy to pair produce or to be likely to bremsstrahlung again before it is stopped by other mechanisms.

To stop electrons, you'll want the atoms in your armor to be as heavy as you can practically get them. High atomic number both encourages bremsstrahlung and pair production, giving a double whammy for stopping that cascade. However, occasionally one of those electrons, positrons, or gammas will hit a nucleus and knock out neutrons. So you probably want a back-up layer of several tens of centimeters of hydrogen rich material such as plastic or water mixed with neutron absorbing materials like boron, lithium, cadmium, or gadolinium. High density polyethylene loaded with 5% boron by weight is a typical go-to neutron blocker.

The actual methods to calculate the penetration of the electromagnetic cascade is given are the box below. However, for some examples:
<ul>
<li> A 10 GeV beam incident on thick iron (or steel) will stop cascading at about 9 cm, will deposit half of its energy in the first 12 cm, and will have dumped 98% of its energy within 35 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 2.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 100 GeV beam incident on thick iron (or steel) will stop cascading at about 13 cm, will deposit half of its energy in the first 16 cm, and will have dumped 98% of its energy within 47 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 3.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick iron (or steel) will stop cascading at about 17 cm, will deposit half of its energy in the first 20 cm, and will have dumped 98% of its energy within 59 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 4.6 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick tungsten will stop cascading at about 3.5 cm, will deposit half of its energy in the first 4 cm, and will have dumped 98% of its energy within 12 cm. 90% of the cascade will be contained within about 0.93 cm and 95% of the cascade will be contained within 1.86 cm of the beam axis. You will get 98% protection with a mass thickness of 2.3 tons/m2 and an additional 1/e reduction for each further 0.13 tons/m2 beyond that.
<li> A 1 TeV beam incident on water will stop cascading at about 290 cm, will deposit half of its energy in the first 340 cm, and will have dumped 98% of its energy within 1030 cm. 90% of the cascade will be contained within about 11 cm and 95% of the cascade will be contained within 22 cm of the beam axis. You will get 98% protection with a mass thickness of 10.3 tons/m2 and an additional 1/e reduction for each further 0.72 tons/m2 beyond that.
</ul>
Note that for sufficiently intense particle beams, 98% protection may not be enough. You might need to throw on several more tons per square meter to be sure.

<table border=1>
<tr><td> How Far it Goes:
(This section follows methods presented in Leroy and Rancoita<ref>C. Leroy and P-G Rancoita, "Physics of cascading shower generation and propagation in matter: principles of high-energy, ultrahigh-energy and compensating calorimetry", Reports on Progress in Physics 63 (2000) 505–606. DOI: 10.1088/0034-4885/63/4/202</ref>.)

When trying to estimate the extent of the irradiated region of the target, it is helpful to know the radiation length X0 in the material. This is the distance that an electron will travel, on average, before emitting a bremsstrahlung photon. Conveniently, it is also the distance a high energy photon will travel through the material before pair producing. The radiation length can be found from
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   = 4 α  
<td nowrap="nowrap"> NA
<td rowspan="2" nowrap="nowrap" align="right">   Z (Z+ζ) re2 ln
<td rowspan="2" nowrap="nowrap" align="right"> [
<td nowrap="nowrap"> 183
<td rowspan="2" nowrap="nowrap" align="right"> ]
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> A
<td style="border-top:solid 1px black;"> Z1/3
</table>
</div>
where α = 1/137 is the fine structure constant, NA = 6.02214076×1023 is Avogadro's number (or one mole), A is the atomic weight of the atoms in the material (usually expressed in grams per mole), Z is the atomic number (nuclear charge) of the atoms, and re = 2.8179403205×10-13 cm is the classical electron radius. ζ is a constant of approximately 1.3. This gives the thickness of material in terms of the areal density (mass of material per unit area perpendicular to the beam) – if you use A in g/mol and re in cm the result will be in cm2/g and X0 will be in g/cm2. If your target is made up of more than one kind of atom, you can get the total radiation length by finding the radiation length of each atom of type i, X0i and using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   =  
<td rowspan="2" nowrap="nowrap" align="right"> Σi  
<td nowrap="nowrap"> fi
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> X0i
</table>
</div>
where fi is the fraction by weight of the atom type i. Once you have the radiation length in terms of areal density, you can easily convert it to distance by dividing by the density ρ. If X0i is in g/cm2 and density is in g/cm3, then X0i/ρ will be in cm.

We will also need to know the critical energy εc. This is the energy where an electron is losing energy as rapidly to ionization as it is to radiation. That is, it is the energy at which the cascade stops multiplying, even though the individual particles already made can continue traveling for some distance. The critical energy can be estimated using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> εc = B  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> Z X0
<td rowspan="2" nowrap="nowrap" align="right"> )
<td nowrap="nowrap"> h
<tr>
<td style="border-top:solid 1px black;"> A
<td>  
</table>
</div>
where B = 2.66 MeV and h = 1.11.

We can now describe the distribution of radiation in the target material. The depth tmax where the cascade stops multiplying (in g/cm2) for an incident electron of energy E (in MeV) is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tmax = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> - 1
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
Similarly, the depth where half the energy of the beam has been deposited (in g/cm2) is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tcg = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> + 0.37
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
As usual, you can convert to a depth in cm by dividing by the material density. 98% of the beam energy will be deposited within a distance of &cong; 3 tcg. Beyond this, the beam energy will continue to fall off with thickness t approximately as an exponential exp[-t/λ], where the characteristic attenuation length λ &cong; 2 X0.

The width of a cascade, where 90% of the energy is contained, is approximately given by
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> RM =  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> EM
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
where EM &cong; 21.2 MeV. 95% of the radiation will be contained within a distance of 2 RM from the beam axis.
</table>

==== Protons and Ions ====

<table align=right><tr><td>
[[File:Regolith_Shielding.png|350 px|frameless]]
<tr><td width=350>
Relative effect of particle radiation (compared to no shielding) from galactic cosmic rays behind different thicknesses of water, aluminum, and lunar regolith<ref name="Slaba2022">Tony C. Slaba, "Radiation Shielding for Lunar Missions: Regolith Considerations", LSIC Crosstalk 7/18/2022 https://lsic.jhuapl.edu/uploadedDocs/focus-files/1604-E&C%20+%20EE%20Monthly%20Meeting%20-%202022%2007%20July_Presentation%20-%20NASA%20Slaba.pdf</ref>.
</table>
Protons and ions are grouped together because ions are made of protons and neutrons and when you accelerate these things for a weapon it is the protons and neutrons that are going to be doing the heavy lifting for blasting your target. This is because protons and neutrons are both about 2000 times more massive than electrons so any electrons along for the ride will contribute negligibly to the energy. Protons and neutrons together are called nuclear particles, and both behave quite similarly in nuclear collisions.

Nuclear particles are themselves made of quarks and can interact via the strong nuclear force. At the energies appropriate for weapons use, essentially all of the way protons or ions are blocked is through [[Particle_Accelerators#Nuclear_collisions|direct collisions with the atomic nuclei]] of the armor material. One or more quarks in the projectile will slam into quarks in the target nucleus and stretch them away from the rest of the quarks in the nuclear particles. Physics tells us that you can't have isolated quarks, so the "string" attaching the quark to its neighbors stretches and stretches until it snaps. Snapping releases so much energy that more quarks are made that clump together into all sorts of other nuclear particles, both exotic and familiar. This process is called a hadronic jet. The only particles produced in these jets long-lived enough to travel any noticeable distance through the armor material will be protons, neutrons, the charged pi mesons (or pions), and some of the K mesons (or kaons). The particles too short lived to go far enough to matter will decay into other particles, and these particles are often very high energy gamma rays. Much like with electrons, the long-lived secondary particles can be sufficiently energetic to smash into other nuclei and cause more hadronic jets (or, if they are low enough in energy, just knock more nuclear particles out of target nuclei) and this produces a similar shower or cascade. But in addition, the armor nuclei that were hit will be very hot after the impact from all of the energy given to them from the incident particles. These nuclei will cool off by evaporation – but instead of evaporating away molecules, they evaporate off nuclear particles from the nucleus. These evaporated particles are mostly neutrons, so any armor against proton or ion beam strikes will produce quite a bit of neutron radiation.

Eventually, the original particle's energy will be distributed among so many other particles that the charged particles will come to a stop by making ion tracks rather than from colliding with more nuclei, and the cascade halts. But you still have all the neutrons to worry about. And the gamma rays as well, which are high enough energy to kick off an electromagnetic cascade in the same way that electrons do.

Unlike electrons, which have reasonable approximations of their cascade behavior from relatively simple equations, the cascades of protons and ions usually require sophisticated simulations or experiments to get the resulting distribution of radiation. However, we can get a reasonable idea of their general behavior by comparing them to [[Space_Radiation#Shielding_Against_Cosmic_Rays|cosmic ray particles interacting with shielding]]. Cosmic rays are energetic protons and ions, after all. From this we see that a relatively small amount of armor – in the range of 1 to 3 tons per square meter – can actually make the radiation dose worse. The incident particles will knock off so many secondary particles in their shower that the dose is multiplied from so many tracks passing through the target. Hydrogen rich materials such as water and plastics work the best against proton and ion beam particles; heavy elements are the worst and produce the most secondaries. You could probably obtain adequate radiation protection from most incident beams with about 10 tons per square meter of hydrogen rich material as armor. However, you will want to add neutron absorbing elements to your hydrogen rich armor, such as lithium, boron, cadmium, or gadolinium. Further, in order to deal with the gamma rays you will want a layer of at least several centimeters of heavy elements, such as tungsten or lead, behind the hydrogen rich particle armor.

==== Structural Considerations ====

When your armor material absorbs a radiation beam, the energy of that radiation turns into heat inside the armor where the radiation was absorbed. If the beam pulse has enough energy as was sufficiently tightly focused, this heat can damage your armor. Your armor may melt in places, or thermal expansion could cause it to crack or warp. Perhaps the worst thing it could do is vaporize or flash into a plasma, because the pressure from the vapor or plasma can be sufficient to plastically deform the armor. This leaves a hole along the beam path where the expanding gas or plasma blasted out a long, deep crater. This means a beam that fires rapid shots spaced maybe milliseconds apart can have subsequent shots go down the same hole so your armor will not protect against them. Each shot will dig out a deeper hole until either your armor is penetrated or the attacking beam runs out of shots in its pulse train.

A consequence of this is that when designing your armor you may wish to use refractory materials with a high structural strength (like tungsten), or that is otherwise resistant to being blasted apart by its own evaporated remains.

=== Magnetic fields ===

[[Particle_Accelerators#Magnetic_fields|A charged particle crossing a magnetic field will bend so as to spiral around the field lines]]. If you try to shoot your particle beam through a magnetic field, it will bend away from the direction you shot it. This can make it hard to hit your target. Unfortunately for you, there are magnetic fields everywhere – around planets, and even trapped in the solar wind as it crosses the solar system.

As an example, the Earth has a magnetic field strength of about 5×10-5 tesla near its surface, and falling off farther away. If you are shooting a 1 TeV electron beam past the Earth, your beam is likely to bend by several milliradians (with the exact deflection depending on how close you are skimming the Earth). If you are in a medium Earth orbit and shooting at another target also in a medium Earth orbit on a trajectory that goes near the Earth, then without correcting for the magnetic field your beam will likely be off-target by tens to hundreds of kilometers.

For another example, the interplanetary magnetic field is about 10-9 tesla at Earth's distance from the sun. If you shoot that same 1 TeV electron beam at a target 0.1 light seconds away, you are likely to be off by a couple hundred meters.

<table border=1>
<tr><td>
The radius of curvature of the bend of a particle in a magnetic field is called the gyroradius rc. For a particle with a charge q, mass m, and kinetic energy E, find the relativistic momentum p
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
p = √[E2 - m2 c4] / c
</div>
To convert from energy units of eV to joules, multiply by 1.60×10-19. An electron has a mass of 9.11×10-31 kg, a proton of 1.67×10-27 kg, and an ion of 1.66×10-27×A kg, where A is the atomic weight of the ion. The speed of light c = 3.00×108 m/s.

In a magnetic field of strength B with an angle θ between the field direction and the direction of motion
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rc = |p sin(θ) / (q B)|
</div>
For electrons, q = -e, where the fundamental charge e = 1.60×10-19 coulomb. For a proton, q = +e. For an ion, multiply e by the charge state of the ion.

If the field acts over a distance d that is much smaller than the gyroradius, the deflection angle φ is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = d / rc
</div>
The deflection x at a target at a range R for small φ (φ &Lt; 1) will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
x = φ R.
</div>

Note that d will always be less than or equal to x. Also note that if d is not smaller than twice the gyroradius the beam will never reach the target because it will completely spiral around the field lines before it reaches the target.
</table>

So, it is clear that as the defender it will be helpful to hang out in areas with high magnetic fields, and keep some distance between you and the attacker. Meanwhile, the attacker will benefit from being closer where small deflections in the beam trajectory don't translate into large excursions from where the beam is aimed (and also because there will be less space filled with fields between the two of you). But the attacker can also try to compensate for known fields. If you happen to know that the magnetic fields are between you and your target, you can adjust your fire so that the deflection of the field bends your beam onto the target. This is not a simple task. It is easy enough to measure the field strength and direction where you are, but predicting how it changes over the path of your shot does not have an obvious solution. You can also try to gain intelligence on likely battle sites before you ever fight on them – for example, getting good data on the structure of the fields of a planet using spies and technical intelligence during peacetime (heck, on Earth this information is published for free in academic journals and by government agencies such as NOAA). Planetary magnetospheres are also affected by the solar wind, so different solar weather will change the fields you have to shoot through. But space weather forecasts and predictive simulations of the solar wind interacting with a planetary magnetosphere can give you a good idea of what the field structure is like. But even with all that, there are changes in the fields that happen on a scale of around a second ... anyone who has ever seen the auroras dancing across the sky in the polar night can see this. Compensating for these changes will be a significant challenge.

What would really help is being able to see the path of your own beam. This would allow you to see how far off you were from your target and in what direction, and then adjust fire to walk your shots onto the target. Unfortunately, particle beams in space are invisible. There might be ways to track them, but if so we don't know what they are.

==== Synchrotron radiation ====

When charged particle trajectories are bent, they emit [[Particle_Accelerators#Synchrotron_radiation|synchrotron radiation]]. This can cause the particle beam to lose energy, but if the bend is too tight it can also generate high energy gamma rays that ignore magnetic fields and directly strike the target. Synchrotron radiation can generally be ignored for proton or ion beams, but can be significant for electron beams.

<table border=1><tr><td>
For a beam particle with energy E and mass m, the synchrotron radiation is concentrated in a forward-directed cone with an opening angle of approximately 1/γ, where γ = 1 + E/(m c2). (Interestingly, the radiation intensity peaks an at angle of 1/(2 γ) and vanishes directly on-axis.) Because the beam is bending, you will get emission along the plane along which the beam is being bent. The energy of the photons produced during synchrotron emission is approximately<ref name="Jackson_Electrodynamics">J. D. Jackson, "Classical Electrodynamics: Second Edition", Wiley (1962)</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Eγ &cong; 3 γ3 c &hbar; / rc
</div>
where &hbar; = 1.05×10-34 J s (if you want the energy in joules) or &hbar; = 6.58×10-16 eV s (if you want the energy in eV). The photon energy will not exceed the energy of the incident particles, if you get a photon energy higher than the energy of the incident particles it just means that the synchrotron radiation spectrum extends up to the incident particle energy.

The fraction of the energy (or power) of the beam turned into synchrotron radiation will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = [1/(4 π ε0)] [4 π / 3] [q² / rc] β2 γ4 [φ / (2 π)] / E
</div>
where β = √[1 - 1/γ2] is the particle's speed measured as a fraction of the speed of light, and 1/(4 π ε0) = 8.99×109 is the Coulomb constant. For highly relativistic electrons (like all practical electron beam weapons emit), β &cong; 1 and we can use
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = 8.85×10-2 [E (GeV)3 [φ / (2 π)] / (rc (m))]
</div>
This fraction cannot be more than 1. If f gets close to one, it means that our original assumption that the beam particles mostly keep their original speed no longer holds, and the beam will lose most or almost all of its energy. Instead, when the beam is being deflected it will only make it through an angle of φ/f and will irradiate a region of length x/f starting at where the beam would have been incident if not for the magnets and extending in the direction the beam was deflected.
</table>

For example, a 1 TeV electron beam will have γ &cong; 2,000,000. So if it's bent by a magnetic field the synchtrotron radiation will spread out in a cone of about 5×10-7 radians. If this beam was shooting at a target 0.1 light seconds away from where it was bent, the radiation would arrive at the target over a region with a radius of about 15 meters with a maximum intensity at about 7.5 meters and no radiation directly along the path the beam was pointing while it was being bent. If the beam was bent by the interplanetary magnetic field from our earlier example, it would produce light with photon energies of around 1.3 eV (near infrared) and the lost energy would be a few parts in a hundred trillion. If it was instead deflected by the Earth's magnetic field you might expect a peak photon energy of tens of keV – in the hard x-ray part of the spectrum – and the fraction of energy in the synchrotron radiation would be somewhere in the low parts in 100,000 or less.

==== Separation of neutral beams ====

Magnetic fields only affect charged objects. So one way to overcome magnetic defenses or environments is to use beams that have no net charge. For example, a proton beam could be neutralized with a co-propagating electron beam with an equal number of protons and electrons. Or, potentially, you could fire an ion beam and then neutralize the ions after acceleration to get a beam of neutral atoms.

But, the magnetic fields are still tugging at the charged particles making up the beam as they go past. This will pull the electrons one way and the protons or ions the other way. As the particles are pulled in opposite directions you get a net positive charge on one side of the beam and a net negative charge on the other side (called polarization). This sets up an electric field inside the beam that acts to pull the particles back together. If the particles are going fast enough and the magnetic field is strong enough, the magnetic force can still separate the charges and will split the beam in two, one with the positive charges and one with the negative charges. But if the particles are not that fast or the magnetic field is not that strong this polarization of the beam will keep it together and let it fly right through the magnetic field.

<table border=1><tr><td>
Let's figure out how to analyze this problem. For definiteness, we will have a beam with an electric current of the positive species of I and an electric current of electrons of -I, all moving in the z direction. The beam radius will be R. The particle speed will be v with the usual relativistic parameters β = v/c and γ = √[1/(1-β2)]. If you know the energy per particle E of the positive species and the mass m of the positive species, γ = 1 + E/(m c2) and then β = √[1 - 1/γ2].

Let's assume that the beam is going through a magnetic field of strength B at an angle θ to the field direction. The quantity B sin(θ) is the component of the magnetic field perpendicular to the direction of the beam.

The charge density ρ of the positive component of the beam will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρL = I / (π v R2).
</div>

It is easiest to analyze the problem in the rest frame of the particles in the beam. Now the particles are at rest. Particles at rest are not affected by a magnetic field, but the change of reference frame Lorentz transforms the initial magnetic field into a combination of electric and magnetic fields in the rest frame. We can ignore the magnetic field, but the electric field is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔A = γ v B sin(θ)
</div>
in a direction perpendicular to both v and B. For our purposes, assume the electric field points in the x direction.

At the same time, relativistic length contraction increases the charge density in the rest frame
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρ = γ ρL.
</div>

Let r be the vector from the beam axis to the point inside the beam. It will be useful to separate this into its Cartesian coordinates. If x and y are the unit vectors in the x and y directions, respectively, then
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
r = x x + y y
</div>

If we consider only the positive component of the beam, from [https://en.wikipedia.org/wiki/Gauss%27s_law Gauss's law] it will have an electric field inside of the beam at a distance r from the center axis of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔+ = ρ r / (2 ε0) = (ρ/(2 ε0)) (x x + y y)
</div>
where ε0 = 8.85×10-12 is the vacuum permittivity. Now let's displace the electrons in the beam by a distance δ in the x direction. By itself, this electron beam will have an electric field of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔- = -(ρ/(2 ε0)) ((x-δ) x + y y)
</div>
By the principle of superposition, the electric field inside the beam with both components is the sum of the electric fields of the separate components including the external field
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔B = [(ρ/(2 ε0)) δ + γ v B sin(θ)]x
</div>
Because the charges are free to move, they will do so as long as a field is applied to them. Thus, the beam becomes stable once the charges separate enough that the field internal to the beam is zero. Therefore
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
δ = -2 ε0 γ v B sin(θ) / ρ = -2 π ε0 v2 R2 B sin(θ) / I
</div>
If |δ| > 2 |R|, the beam will fully separate and each component will go its own way. If δ is small compared to R, the beam's polarization suffices to hold it together in the overlap region. However, where the positive and negative charges do not overlap the electric fields no longer cancel, and the external electric field carries away the outer polarized part of the beam. This gives us a rough threshold for which a magnetic field will quickly separate a neutral beam into its charged components and bend them away from their target
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr><td> B sin(θ) &Lt; I / (π ε0 v2 R) <td>       <td> Beam is partially eroded.
<tr><td> B sin(θ) > I / (π ε0 v2 R) <td> <td> Beam promptly separates.
</table>
</div>
For a partially eroding beam, the polarized region that was peeled off by the field still acts to screen the remaining neutral portion, at least to within this approximation, leaving the still-neutral beam to hit the target.
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Neutralized_beam_in_B_field.png|5140 px|frameless]]
<tr><td width=700>
A profile of the beam charge distribution of a neutralized beam in a magnetic field. Negative charge density is shown in yellow, positive charge density in cyan, with neutral charge density in light green. The magnetic field is vertical, the electric field in the beam reference frame is shown in magenta including both the polarization field and the Lorentz transformed electric field resulting from the magnetic field. As positive charges are pulled in the direction of the field and negative charges opposite to the field direction, the polarized boundary will be pulled off the beam by the electric field even though the electric field is zero in nearly all the beam interior. This expanding polarization layer will continue to largely screen the still-neutral portion, however; but the extent to which the screening is only partial would require more sophisticated analysis methods than used here. The material removed by the field will expand to a maximum of the deflection distance of a uniform charge based on the magnetic field by the time it reaches the target.
</table>
</div>
<tr><td>
For a beam of neutral atoms, let a be the radius of the atom and let Eb be the binding energy of the least tightly bound electron to the atom. An external electric field in excess of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
|𝓔A| > Eb / (e a)
</div>
will promptly ionize the atom, ripping away its electron. Here, e = 1.60×10-19 C is the magnitude of the electric charge of the electron. For fields close to but a bit weaker than this, the electron can still rapidly pop off due to quantum tunneling<ref>Ph. Durand and I. Paidarová, “Ionization of the hydrogen atom: from weak to strong static electric fields,” The European Physical Journal D, Vol. 26, pg. 253-259, 2003.</ref>. In general, expect that for fields greater in magnitude than (very approximately) Eb / (10 e a) the electron will be quickly removed, while for fields weaker than this the electrons will stay on their atoms.

It is useful to note that Eb/e is just the binding energy in units of eV. The most weakly bound electrons are usually bound to their atom with about 10 eV of energy. Atoms are typically about 10-10 m in radius. Thus, you can expect fields of around 1010 to 1011 V/m to directly ionize atoms. If we call this threshold value 𝓔T, then a beam will be ionized when passing through a field with strength greater than
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B sin(θ) > 𝓔T / (γ v)
</div>
</table>

Example: Let's consider a 1 GeV neutralized proton beam from a synchrotron, with a beam current of 200 kA in each bunch (incidentally, if a bunch is 5 cm long this means that each bunch packs 38 kJ worth of energy). The speed of the bunch is 87% the speed of light, or 262 million m/s, with a γ of 2.07. Consider when the beam's focus takes it to a radius of 5 cm (or 0.05 m, for our calculation). Plugging this in, we find that the protons in the bunch will separate from the electrons in a field with a transverse component greater than about 2.1 T, and the bunch will pass through the field with only gradual losses if the transverse component is much smaller than this.

Example: We'll take the same 1 GeV beam, but this time recombine the electrons with protons to make hydrogen atoms. An electron is bound to a hydrogen atom with an energy of 13.6 eV (the Ryberg constant) and the radius of the electron orbital in a hydrogen atom is 0.529×10-10 m (the Bohr radius). This gives a threshold electric field for prompt ionization of hydrogen of 2.57×1011 V/m. We will get this with a transverse component of the magnetic field of 474 T. Above this field the atoms promptly ionize and electrons are stripped away from the protons. For fields higher than roughly 1/10 of this – so about 40 to 50 T – you may still get significant ionization by quantum tunneling; lower than this your losses from the beam due to field-induced ionization are likely negligible.

==== Magnetic shields ====

What happens when we can't rely on magnetic fields in the environment to protect us? One idea is to produce powerful artificial magnetic fields around our spacecraft to deflect incoming particle beams. This is very similar in concept to using active magnetic shielding to protect against space radiation – [[Space_Radiation#Magnetic_Shielding|check out our article on space radiation for more]].
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr>
<td width=100>
[[File:Unconfined_FRC_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:racetrack_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:Magnetic_shielding_Halback_Array.png|200 px|frameless]]
<tr><td colspan=3>
Some potential configurations for magnetic shields. The currents generating the fields are in green, the magnetic fields themselves are shown in magenta<ref name="Battiston2012">R. Battiston, W.J. Burger, V. Calvelli, R. Musenich, V. Choutko, V.I. Datskov, A. Della Torre, F. Venditti,
C. Gargiulo, G. Laurenti, S. Lucidi, S. Harrison, and R. Meinke, "ARSSEM Active Radiation Shield for Space Exploration Missions", Final Report ESTEC Contract N° 4200023087/10/NL/AF : “Superconductive Magnet for Radiation Shielding of Human Spacecraft” (2012) https://arxiv.org/abs/1209.1907 https://www.researchgate.net/publication/265945847_Active_Radiation_Shield_for_Space_Exploration_Missions</ref><ref name="Desiati2022">Paolo Desiati and Elena D'Onghia, "CREW HaT: A Magnetic Shielding System for Space Habitats", arXiv:2209.13624 [physics.space-ph] https://doi.org/10.48550/arXiv.2209.13624</ref>.
</table>
</div>

One immediate issue is that you want to keep very high magnetic fields out of the crew area. Fields in excess of 0.5 T can start to be unpleasant, and it is recommended to keep the field below 0.2 T for continuous exposure.

Strong magnets can suffer from high self-forces acting to tear them apart. Most shielding designs will have multiple magnetic coils, which will exert high forces and torques on each other. This will result in the need for strong structural bracing to keep the magnets intact and in place. Such strong structural designs will impose significant extra mass on the spacecraft.

Consider a magnetic field configuration that can keep radiation out of a certain region of space. There is no trajectory of a charged radiation particle up to a certain energy that will allow those particles to enter the protected region. But the dynamics of charged particles in a magnetic field are the same whether going forward or backward in time. If there is no way for a particle to get in, by this time reversal symmetry any particle which is in that forbidden region can never escape. Consequently any field configuration that is a magnetic trap is also a magnetic shield; any field configuration that is a magnetic shield is also a magnetic trap. A spacecraft with a magnetic shield may end up surrounded by strong radiation belts.

Using a magnet to bend a weaponized electron beam very often ends up with the beam being bent so hard that it dumps most of its energy in the form of synchrotron radiation at gamma ray energies. Because gamma rays are not charged, they will not be deflected by the field. Thus, if you would have been hit by the electron beam you will be hit instead by the gamma rays.

As an example, consider a 1 TeV electron beam. We'll surround our spacecraft with a 10 T magnetic field in a 100 meter radius region just outside our spacecraft. This will deflect the beam by 30 meters, which is enough to make it miss. But we are not saved, because the beam dumps nearly all of its energy into synchrotron radiation with an energy spectrum of gamma rays up to 13 GeV. This radiation will hit us in a strip about 2.4 meters long and not much more than the width of the original beam with essentially all of the energy of the original beam.

==== Magnet drones ====

If a magnetic shield around your spacecraft will just end up killing you from synchrotron radiation, what if you put the magnetic shield on a drone and put that drone between yourself and the enemy trying to shoot you? This lets you use a much weaker magnet because you don't need nearly so high of a deflection angle to make the shot miss. This means that less beam energy will turn into synchrotron radiation that could kill you instead and will spread what synchrotron radiation does occur out over a much wider area when it reaches your spacecraft, hopefully so that most of it just misses your spacecraft.

It does mean that your enemy will start shooting your drones so that they can eventually take a shot at you. But hopefully by that time you've already cleared out their drones and cored their spacecraft with your own synchrocannon or UREB.

==== Plasma magnets ====

Usually, to make a really big magnetic field you would run current through a really big conductive loop. But really big things are heavy and expensive. One way around this could be to use the ambient space plasma to carry the electric current rather than any kind of wire. In order to make this current, you use the same mechanism used in a rotary induction electric motor: you generate a rotating magnetic field and induce a current in the conductive loop (in this case, the space plasma). How do you make a rotating magnetic field? The method proposed<ref name="Slough2006">John Slough, "The Plasma Magnet", NASA Institute for Advanced Concepts Phase II Final Report (2006) https://www.niac.usra.edu/files/studies/final_report/917Slough.pdf</ref> is to use an antenna of four much smaller current loops as electromagnets at 90&degree; to each other, and sinusoidally vary the current through the loops with a respective 90&degree; phase difference between them. When this is done, a circular current is induced in the space plasma and the self-force of a magnetic field on its generating current inflates the current loop to much larger radii. You end up with a current sheet in the plane of the rotating field's rotation circulating about the antenna nearly synchronously with the field rotation. This current then creates a normal dipolar magnetic field as with any current loop. The current sheet and the resulting magnetic field expand until the magnetic field's pressure equals that of the ambient space plasma. This gives an enormous region of magnetic field from a relatively small and lightweight system.

The system proposed in the initial study would use an antenna approximately 10 meters in radius and produce a region of field approximately 30 km in radius at Earth's orbit. The field inside would not be constant - at the edges it would be 50 nT, giving the right pressure to keep out the solar wind at Earth's orbit. A circulating current sheet produces a field that falls off as 1/r with distance, meaning that adjacent to the antenna the field would be about 0.15 mT. Using the methods in the box below, we can estimate that this particular design as part of a spacecraft's shields will not deflect a TeV electron beam by more than a few centimeters, which is usually insufficient to force a miss. However, it could potentially be useful for a drone magnet, where the much larger distance between the magnetic field and the target spacecraft would allow a small deflection angle to give perhaps tens or low hundreds of meters change in beam location near the target at near light second ranges. The relatively low field and gradual bend of the incident beam suggests that synchrotron radiation will not be an issue; however this does not exclude more powerful plasma magnet designs causing significant synchrotron radiation effects.

<table border=1><tr><td>
Calculating the deflection from a spatially varying magnetic field affected by the solar wind is no trivial task. However, we can make an approximation that both should be within about an order of magnitude of the true value and also represent an upper limit to the deflection angle; although it does neglect the influence of the driving rotating magnetic field which could in some cases increase the deflection.

We will take R0 as the radius of the antenna. B0 is the field strength at the antenna. Rm is the distance at which the magnetic influence stops because the magnetic pressure equals the space plasma pressure. Bm is the field strength at Rm. According to Slough<ref name="Slough2006"></ref>, the magnetic field strength B(r) at any distance from the antenna center r between R0 and Rm is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B(r) = B0 R0 / r.
</div>
Therefore, once we know Bm and Rm, B0 is completely determined
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B0 = Bm Rm / R0.
</div>

The force on a particle with charge q passing perpendicularly through a magnetic field B(r) at a speed v is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
F(r) = q v B(r)
</div>
For an electron beam, v is so close to c as to make no difference. If the particle is moving down the y direction and the field is in the z direction, the force will be in the x direction. We assume that the deflection will be small, so we can choose a straight-line path in the y direction offset from the center by an amount x. The plasma bubble radius is Rm, so the particle will enter at ymax = √[Rm2-x2] and leave at ymin = -√[Rm2-x2].

The momentum change the particle experiences from this force is the time integral of the force over its path. Because of reflection symmetry about y = 0, we only need to integrate from ymax to 0 and then multiply by 2
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] dt F(r)
<tr> <td> <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] (dy/v) F(r)
<tr> <td> <td> = <td align="left">2 q B(0) R0 [Integral from 0 to √[Rm2-x2]] dy 1/√[y2 + x2]
<tr> <td> <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/x]
</table>
</div>
This holds for |x| > R0. But the expression for the field is not valid for r < R0 and the integral will give nonsense and diverging results if we evaluate for |x| < R0. In this case, we can put the integral lower bound at r = R0, or y = √[R02-x2]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/(√[R02-x2] + R0)]
</table>
</div>

The deflection angle is now estimated as
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = px / p
</div>
where the total momentum of the particle p can be found from the methods in the first box in the Magnetic Fields section.

The deflection for a shot aimed right at the spacecraft at the position of the spacecraft can be estimated by setting x = 0 and only taking the first half of the integral (so ignore the factor of 2)
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
q B(0) R0 ln[Rm/R0] / (p Rm)
</div>
This is an additional overestimate because it fails to account for how the deflection builds up over distance during the integration, but it should still give something of an order-of-magnitude and upper limit estimate of what kind of deflections might be expected.
</table>

=== Electric fields ===

If electric fields can accelerate charged particles, maybe they can keep the charged particles in a weaponized particle beam away from your spacecraft? To completely block a particle beam, you would need a voltage on your spacecraft equal to the particle energy in eV of the particles in the beam. That is, if a 1 GeV proton beam is coming at you, you'd need an electric potential of a billion volts to stop that beam. This leads to three immediate problems.

The first is that particles of the opposite charge will be attracted to your spacecraft. Because space isn't really empty but rather a rarefied plasma (called the solar wind in a solar system or the interstellar medium out between stars), there will be plenty of particles of the opposite charge around to attract. And they will hit with an energy in eV equal to the voltage on your spacecraft. In our example of defending against a 1 GeV proton beam with a gigavolt field, the electrons from the solar wind will come in and hit with an energy of 1 GeV each. This will generate dangerous radiation.

The second problem is a consequence of the first. Each of those GeV electrons from our example need to get their energy from somewhere, which turns out to be from the electric field we are using as shielding. You will need to make up for this energy loss with power from your electrical system, which gives a constant drain on your power. Some estimates put the electrical effort to maintain a several hundred megavolt shield in space at many tens of gigawatts<ref name="Mechmann2019">Claire Mechmann, "Analysis of Proposed Active Radiation Shielding Design Concept for Spacecraft" (2019) Thesis, College of Engineering and Science of Florida Institute of Technology</ref>.

And the third problem is similar to one for magnetic fields. Your electrodes will experience high self forces and forces between separate electrodes that will try to tear your shield generator apart. Consequently you will need heavy structural reinforcement in order for your shield to work.

There are potential ways to mitigate some of these problems. Interested readers are directed to [[Space_Radiation#Electrostatic_Shielding|the active shielding section in our page on space radiation]] for more of the juicy details; there's no need to repeat the same stuff here.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Elctrostatic_active_shielding.png|200 px|frameless]]
<td>
[[File:Electrostatic_active_shielding_2.png|200 px|frameless]]
<tr><td colspan=2>
Proposed designs for electrostatic shields to protect from positively charged particles<ref name="Tripathi2006">Ram K. Tripathi, John W. Wilson, and Robert C. Youngquist, "Electrostatic Active Radiation Shielding - Revisited", 2006 IEEE Aerospace Conference, Big Sky, MT, USA, 2006, pp. 9 pp.-, doi: 10.1109/AERO.2006.1655760.</ref><ref name="Chowdhury2023">Rajarshi Pal Chowdhury, Luke A. Stegeman, Matthew L. Lund, Dan Fry, Stojan Madzunkov, and Amir A. Bahadori, "Hybrid methods of radiation shielding against deep-space radiation", Life Sciences in Space Research, Volume 38, 2023, Pages 67-78, ISSN 2214-5524, https://doi.org/10.1016/j.lssr.2023.04.004.</ref>.
</table>
</div>

== Damage ==

Particle beams are, first and foremost, devices that produce beams of [[Nuclear_radiation|ionizing radiation]]. The radiation of weaponizable particle beams is extremely penetrating; thus of the [[Particle_Accelerators#Energy_loss_in_matter|interaction mechanisms listed for radiation from particle beams]], we mostly need to worry about bremsstrahlung and nuclear collisions and these produce extensive secondary showers of radiation that increase the penetration of the radiation exposure. Because of this, any target hit by a particle beam weapon will be irradiated, and radiation dose will always be one of the primary damage mechanisms of a particle beam weapon strike. Further, this radiation will be distributed along a track passing deeply through the material of the target, allowing it to affect critical systems through heavy armor, structural material, robust machine components, and passive material such as fuel load, coolant, or cargo. As the radiation deposits its energy into the target material, that material will be heated. If the target is heated sufficiently, it may experience [[Beam-Target_Interactions|thermal or mechanical]] damage from melting, cooking, combustion, vapor explosions, and warping or cracking due to thermal expansion.

The discussion of the radiation penetration into armor also applies to the same radiation penetrating through the target. So the formulas and rules of thumb for estimating the depth of radiation penetration into armor can also be used to estimate how far into a tank or spacecraft a particle beam strike delivers effects.

In addition to the radiation and thermal effects, a particle beam can also produce a sharp electromagnetic pulse that can damage semiconductor electronics.

== 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, Gerrit Bruhaug, and Luke Campbell for writing

Gerrit Bruhaug and Luke Campbell for scientific research, math wrangling and advise

ATriffleMind and Luke W. Campbell for proof reading.

==References==

[[Category:Beams]][[Category:Physics & Engineering‏‎]][[Category:Engineering‏‎]][[Category:Military Technology]][[Category:Warfare‏‎]]

Particle Beam Weapons

2026-10-27T02:06:28Z

Lwcamp: /* Separation of neutral beams */

{{PageConstructionNotice}}

''For a general overview on particle accelerator technology, see [[Particle Accelerators]]

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.

::: Z = 1.32×102 (γ2-1)3/4 r / √I<ref name=USPAS2016>Martin Reiser, "Theory and Design of Charged Particle Beams", 2008, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim</ref>

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

::: Z = β γ c r √[m/T]

Where Z is the doubling distance like before, r is the initial beam radius, T is the beam temperature in eV, m is the ion mass and β, c, and γ 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.

== Defenses ==

Sure, it's great to have a great big particle cannon to blast your enemies. But what if you are the one being shot at? In that case, it's best to figure out ways to keep from being blown to kingdom come.

=== Armor ===

The simplest way to defend against the high energy particles of a particle beam is to pile a whole bunch of stuff up and put it in the way of the beam. Particle beams may produce extremely penetrating radiation, but enough stuff will block most of that radiation and may allow you to survive. The details depend on what kind of particle is being thrown at you, and are explained below. However, a general rule of thumb is that you probably want about 10 tons of material per square meter of surface you want to armor in order to get sufficient protection.

==== Electrons ====

At the very high energy of an electron accelerated by a weapons-grade particle beam, the primary interaction with matter will be the emission of bremsstrahlung photons in the gamma-ray range as the electron slams into particles of matter. These high energy photons themselves primarily interact by the pair production mechanism – the photon disappears, but leaves behind a highly energetic positron and electron. These two particles are themselves energetic enough to produce further bremsstrahlung, and so the process continues. You get the formation of an electron-gamma shower (also called an electromagnetic cascade), which only stops after either the particles have penetrated through the target or the initial energy of the electron has been spread through so many shower particles (including all the electrons, positrons, and photons) that no particle has enough individual energy to pair produce or to be likely to bremsstrahlung again before it is stopped by other mechanisms.

To stop electrons, you'll want the atoms in your armor to be as heavy as you can practically get them. High atomic number both encourages bremsstrahlung and pair production, giving a double whammy for stopping that cascade. However, occasionally one of those electrons, positrons, or gammas will hit a nucleus and knock out neutrons. So you probably want a back-up layer of several tens of centimeters of hydrogen rich material such as plastic or water mixed with neutron absorbing materials like boron, lithium, cadmium, or gadolinium. High density polyethylene loaded with 5% boron by weight is a typical go-to neutron blocker.

The actual methods to calculate the penetration of the electromagnetic cascade is given are the box below. However, for some examples:
<ul>
<li> A 10 GeV beam incident on thick iron (or steel) will stop cascading at about 9 cm, will deposit half of its energy in the first 12 cm, and will have dumped 98% of its energy within 35 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 2.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 100 GeV beam incident on thick iron (or steel) will stop cascading at about 13 cm, will deposit half of its energy in the first 16 cm, and will have dumped 98% of its energy within 47 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 3.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick iron (or steel) will stop cascading at about 17 cm, will deposit half of its energy in the first 20 cm, and will have dumped 98% of its energy within 59 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 4.6 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick tungsten will stop cascading at about 3.5 cm, will deposit half of its energy in the first 4 cm, and will have dumped 98% of its energy within 12 cm. 90% of the cascade will be contained within about 0.93 cm and 95% of the cascade will be contained within 1.86 cm of the beam axis. You will get 98% protection with a mass thickness of 2.3 tons/m2 and an additional 1/e reduction for each further 0.13 tons/m2 beyond that.
<li> A 1 TeV beam incident on water will stop cascading at about 290 cm, will deposit half of its energy in the first 340 cm, and will have dumped 98% of its energy within 1030 cm. 90% of the cascade will be contained within about 11 cm and 95% of the cascade will be contained within 22 cm of the beam axis. You will get 98% protection with a mass thickness of 10.3 tons/m2 and an additional 1/e reduction for each further 0.72 tons/m2 beyond that.
</ul>
Note that for sufficiently intense particle beams, 98% protection may not be enough. You might need to throw on several more tons per square meter to be sure.

<table border=1>
<tr><td> How Far it Goes:
(This section follows methods presented in Leroy and Rancoita<ref>C. Leroy and P-G Rancoita, "Physics of cascading shower generation and propagation in matter: principles of high-energy, ultrahigh-energy and compensating calorimetry", Reports on Progress in Physics 63 (2000) 505–606. DOI: 10.1088/0034-4885/63/4/202</ref>.)

When trying to estimate the extent of the irradiated region of the target, it is helpful to know the radiation length X0 in the material. This is the distance that an electron will travel, on average, before emitting a bremsstrahlung photon. Conveniently, it is also the distance a high energy photon will travel through the material before pair producing. The radiation length can be found from
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   = 4 α  
<td nowrap="nowrap"> NA
<td rowspan="2" nowrap="nowrap" align="right">   Z (Z+ζ) re2 ln
<td rowspan="2" nowrap="nowrap" align="right"> [
<td nowrap="nowrap"> 183
<td rowspan="2" nowrap="nowrap" align="right"> ]
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> A
<td style="border-top:solid 1px black;"> Z1/3
</table>
</div>
where α = 1/137 is the fine structure constant, NA = 6.02214076×1023 is Avogadro's number (or one mole), A is the atomic weight of the atoms in the material (usually expressed in grams per mole), Z is the atomic number (nuclear charge) of the atoms, and re = 2.8179403205×10-13 cm is the classical electron radius. ζ is a constant of approximately 1.3. This gives the thickness of material in terms of the areal density (mass of material per unit area perpendicular to the beam) – if you use A in g/mol and re in cm the result will be in cm2/g and X0 will be in g/cm2. If your target is made up of more than one kind of atom, you can get the total radiation length by finding the radiation length of each atom of type i, X0i and using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   =  
<td rowspan="2" nowrap="nowrap" align="right"> Σi  
<td nowrap="nowrap"> fi
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> X0i
</table>
</div>
where fi is the fraction by weight of the atom type i. Once you have the radiation length in terms of areal density, you can easily convert it to distance by dividing by the density ρ. If X0i is in g/cm2 and density is in g/cm3, then X0i/ρ will be in cm.

We will also need to know the critical energy εc. This is the energy where an electron is losing energy as rapidly to ionization as it is to radiation. That is, it is the energy at which the cascade stops multiplying, even though the individual particles already made can continue traveling for some distance. The critical energy can be estimated using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> εc = B  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> Z X0
<td rowspan="2" nowrap="nowrap" align="right"> )
<td nowrap="nowrap"> h
<tr>
<td style="border-top:solid 1px black;"> A
<td>  
</table>
</div>
where B = 2.66 MeV and h = 1.11.

We can now describe the distribution of radiation in the target material. The depth tmax where the cascade stops multiplying (in g/cm2) for an incident electron of energy E (in MeV) is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tmax = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> - 1
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
Similarly, the depth where half the energy of the beam has been deposited (in g/cm2) is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tcg = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> + 0.37
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
As usual, you can convert to a depth in cm by dividing by the material density. 98% of the beam energy will be deposited within a distance of &cong; 3 tcg. Beyond this, the beam energy will continue to fall off with thickness t approximately as an exponential exp[-t/λ], where the characteristic attenuation length λ &cong; 2 X0.

The width of a cascade, where 90% of the energy is contained, is approximately given by
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> RM =  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> EM
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
where EM &cong; 21.2 MeV. 95% of the radiation will be contained within a distance of 2 RM from the beam axis.
</table>

==== Protons and Ions ====

<table align=right><tr><td>
[[File:Regolith_Shielding.png|350 px|frameless]]
<tr><td width=350>
Relative effect of particle radiation (compared to no shielding) from galactic cosmic rays behind different thicknesses of water, aluminum, and lunar regolith<ref name="Slaba2022">Tony C. Slaba, "Radiation Shielding for Lunar Missions: Regolith Considerations", LSIC Crosstalk 7/18/2022 https://lsic.jhuapl.edu/uploadedDocs/focus-files/1604-E&C%20+%20EE%20Monthly%20Meeting%20-%202022%2007%20July_Presentation%20-%20NASA%20Slaba.pdf</ref>.
</table>
Protons and ions are grouped together because ions are made of protons and neutrons and when you accelerate these things for a weapon it is the protons and neutrons that are going to be doing the heavy lifting for blasting your target. This is because protons and neutrons are both about 2000 times more massive than electrons so any electrons along for the ride will contribute negligibly to the energy. Protons and neutrons together are called nuclear particles, and both behave quite similarly in nuclear collisions.

Nuclear particles are themselves made of quarks and can interact via the strong nuclear force. At the energies appropriate for weapons use, essentially all of the way protons or ions are blocked is through [[Particle_Accelerators#Nuclear_collisions|direct collisions with the atomic nuclei]] of the armor material. One or more quarks in the projectile will slam into quarks in the target nucleus and stretch them away from the rest of the quarks in the nuclear particles. Physics tells us that you can't have isolated quarks, so the "string" attaching the quark to its neighbors stretches and stretches until it snaps. Snapping releases so much energy that more quarks are made that clump together into all sorts of other nuclear particles, both exotic and familiar. This process is called a hadronic jet. The only particles produced in these jets long-lived enough to travel any noticeable distance through the armor material will be protons, neutrons, the charged pi mesons (or pions), and some of the K mesons (or kaons). The particles too short lived to go far enough to matter will decay into other particles, and these particles are often very high energy gamma rays. Much like with electrons, the long-lived secondary particles can be sufficiently energetic to smash into other nuclei and cause more hadronic jets (or, if they are low enough in energy, just knock more nuclear particles out of target nuclei) and this produces a similar shower or cascade. But in addition, the armor nuclei that were hit will be very hot after the impact from all of the energy given to them from the incident particles. These nuclei will cool off by evaporation – but instead of evaporating away molecules, they evaporate off nuclear particles from the nucleus. These evaporated particles are mostly neutrons, so any armor against proton or ion beam strikes will produce quite a bit of neutron radiation.

Eventually, the original particle's energy will be distributed among so many other particles that the charged particles will come to a stop by making ion tracks rather than from colliding with more nuclei, and the cascade halts. But you still have all the neutrons to worry about. And the gamma rays as well, which are high enough energy to kick off an electromagnetic cascade in the same way that electrons do.

Unlike electrons, which have reasonable approximations of their cascade behavior from relatively simple equations, the cascades of protons and ions usually require sophisticated simulations or experiments to get the resulting distribution of radiation. However, we can get a reasonable idea of their general behavior by comparing them to [[Space_Radiation#Shielding_Against_Cosmic_Rays|cosmic ray particles interacting with shielding]]. Cosmic rays are energetic protons and ions, after all. From this we see that a relatively small amount of armor – in the range of 1 to 3 tons per square meter – can actually make the radiation dose worse. The incident particles will knock off so many secondary particles in their shower that the dose is multiplied from so many tracks passing through the target. Hydrogen rich materials such as water and plastics work the best against proton and ion beam particles; heavy elements are the worst and produce the most secondaries. You could probably obtain adequate radiation protection from most incident beams with about 10 tons per square meter of hydrogen rich material as armor. However, you will want to add neutron absorbing elements to your hydrogen rich armor, such as lithium, boron, cadmium, or gadolinium. Further, in order to deal with the gamma rays you will want a layer of at least several centimeters of heavy elements, such as tungsten or lead, behind the hydrogen rich particle armor.

==== Structural Considerations ====

When your armor material absorbs a radiation beam, the energy of that radiation turns into heat inside the armor where the radiation was absorbed. If the beam pulse has enough energy as was sufficiently tightly focused, this heat can damage your armor. Your armor may melt in places, or thermal expansion could cause it to crack or warp. Perhaps the worst thing it could do is vaporize or flash into a plasma, because the pressure from the vapor or plasma can be sufficient to plastically deform the armor. This leaves a hole along the beam path where the expanding gas or plasma blasted out a long, deep crater. This means a beam that fires rapid shots spaced maybe milliseconds apart can have subsequent shots go down the same hole so your armor will not protect against them. Each shot will dig out a deeper hole until either your armor is penetrated or the attacking beam runs out of shots in its pulse train.

A consequence of this is that when designing your armor you may wish to use refractory materials with a high structural strength (like tungsten), or that is otherwise resistant to being blasted apart by its own evaporated remains.

=== Magnetic fields ===

[[Particle_Accelerators#Magnetic_fields|A charged particle crossing a magnetic field will bend so as to spiral around the field lines]]. If you try to shoot your particle beam through a magnetic field, it will bend away from the direction you shot it. This can make it hard to hit your target. Unfortunately for you, there are magnetic fields everywhere – around planets, and even trapped in the solar wind as it crosses the solar system.

As an example, the Earth has a magnetic field strength of about 5×10-5 tesla near its surface, and falling off farther away. If you are shooting a 1 TeV electron beam past the Earth, your beam is likely to bend by several milliradians (with the exact deflection depending on how close you are skimming the Earth). If you are in a medium Earth orbit and shooting at another target also in a medium Earth orbit on a trajectory that goes near the Earth, then without correcting for the magnetic field your beam will likely be off-target by tens to hundreds of kilometers.

For another example, the interplanetary magnetic field is about 10-9 tesla. If you shoot that same 1 TeV electron beam at a target 0.1 light seconds away, you are likely to be off by a couple hundred meters.

<table border=1>
<tr><td>
The radius of curvature of the bend of a particle in a magnetic field is called the gyroradius rc. For a particle with a charge q, mass m, and kinetic energy E, find the relativistic momentum p
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
p = √[E2 - m2 c4] / c
</div>
To convert from energy units of eV to joules, multiply by 1.60×10-19. An electron has a mass of 9.11×10-31 kg, a proton of 1.67×10-27 kg, and an ion of 1.66×10-27×A kg, where A is the atomic weight of the ion. The speed of light c = 3.00×108 m/s.

In a magnetic field of strength B with an angle θ between the field direction and the direction of motion
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rc = |p sin(θ) / (q B)|
</div>
For electrons, q = -e, where the fundamental charge e = 1.60×10-19 coulomb. For a proton, q = +e. For an ion, multiply e by the charge state of the ion.

If the field acts over a distance d that is much smaller than the gyroradius, the deflection angle φ is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = d / rc
</div>
The deflection x at a target at a range R for small φ (φ &Lt; 1) will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
x = φ R.
</div>

Note that d will always be less than or equal to x. Also note that if d is not smaller than twice the gyroradius the beam will never reach the target because it will completely spiral around the field lines before it reaches the target.
</table>

So, it is clear that as the defender it will be helpful to hang out in areas with high magnetic fields, and keep some distance between you and the attacker. Meanwhile, the attacker will benefit from being closer where small deflections in the beam trajectory don't translate into large excursions from where the beam is aimed (and also because there will be less space filled with fields between the two of you). But the attacker can also try to compensate for known fields. If you happen to know that the magnetic fields are between you and your target, you can adjust your fire so that the deflection of the field bends your beam onto the target. This is not a simple task. It is easy enough to measure the field strength and direction where you are, but predicting how it changes over the path of your shot does not have an obvious solution. You can also try to gain intelligence on likely battle sites before you ever fight on them – for example, getting good data on the structure of the fields of a planet using spies and technical intelligence during peacetime (heck, on Earth this information is published for free in academic journals and by government agencies such as NOAA). Planetary magnetospheres are also affected by the solar wind, so different solar weather will change the fields you have to shoot through. But space weather forecasts and predictive simulations of the solar wind interacting with a planetary magnetosphere can give you a good idea of what the field structure is like. But even with all that, there are changes in the fields that happen on a scale of around a second ... anyone who has ever seen the auroras dancing across the sky in the polar night can see this. Compensating for these changes will be a significant challenge.

What would really help is being able to see the path of your own beam. This would allow you to see how far off you were from your target and in what direction, and then adjust fire to walk your shots onto the target. Unfortunately, particle beams in space are invisible. There might be ways to track them, but if so we don't know what they are.

==== Synchrotron radiation ====

When charged particle trajectories are bent, they emit [[Particle_Accelerators#Synchrotron_radiation|synchrotron radiation]]. This can cause the particle beam to lose energy, but if the bend is too tight it can also generate high energy gamma rays that ignore magnetic fields and directly strike the target. Synchrotron radiation can generally be ignored for proton or ion beams, but can be significant for electron beams.

<table border=1><tr><td>
For a beam particle with energy E and mass m, the synchrotron radiation is concentrated in a forward-directed cone with an opening angle of approximately 1/γ, where γ = 1 + E/(m c2). (Interestingly, the radiation intensity peaks an at angle of 1/(2 γ) and vanishes directly on-axis.) Because the beam is bending, you will get emission along the plane along which the beam is being bent. The energy of the photons produced during synchrotron emission is approximately<ref name="Jackson_Electrodynamics">J. D. Jackson, "Classical Electrodynamics: Second Edition", Wiley (1962)</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Eγ &cong; 3 γ3 c &hbar; / rc
</div>
where &hbar; = 1.05×10-34 J s (if you want the energy in joules) or &hbar; = 6.58×10-16 eV s (if you want the energy in eV). The photon energy will not exceed the energy of the incident particles, if you get a photon energy higher than the energy of the incident particles it just means that the synchrotron radiation spectrum extends up to the incident particle energy.

The fraction of the energy (or power) of the beam turned into synchrotron radiation will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = [1/(4 π ε0)] [4 π / 3] [q² / rc] β2 γ4 [φ / (2 π)] / E
</div>
where β = √[1 - 1/γ2] is the particle's speed measured as a fraction of the speed of light, and 1/(4 π ε0) = 8.99×109 is the Coulomb constant. For highly relativistic electrons (like all practical electron beam weapons emit), β &cong; 1 and we can use
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = 8.85×10-2 [E (GeV)3 [φ / (2 π)] / (rc (m))]
</div>
This fraction cannot be more than 1. If f gets close to one, it means that our original assumption that the beam particles mostly keep their original speed no longer holds, and the beam will lose most or almost all of its energy. Instead, when the beam is being deflected it will only make it through an angle of φ/f and will irradiate a region of length x/f starting at where the beam would have been incident if not for the magnets and extending in the direction the beam was deflected.
</table>

For example, a 1 TeV electron beam will have γ &cong; 2,000,000. So if it's bent by a magnetic field the synchtrotron radiation will spread out in a cone of about 5×10-7 radians. If this beam was shooting at a target 0.1 light seconds away from where it was bent, the radiation would arrive at the target over a region with a radius of about 15 meters with a maximum intensity at about 7.5 meters and no radiation directly along the path the beam was pointing while it was being bent. If the beam was bent by the interplanetary magnetic field from our earlier example, it would produce light with photon energies of around 1.3 eV (near infrared) and the lost energy would be a few parts in a hundred trillion. If it was instead deflected by the Earth's magnetic field you might expect a peak photon energy of tens of keV – in the hard x-ray part of the spectrum – and the fraction of energy in the synchrotron radiation would be somewhere in the low parts in 100,000 or less.

==== Separation of neutral beams ====

Magnetic fields only affect charged objects. So one way to overcome magnetic defenses or environments is to use beams that have no net charge. For example, a proton beam could be neutralized with a co-propagating electron beam with an equal number of protons and electrons. Or, potentially, you could fire an ion beam and then neutralize the ions after acceleration to get a beam of neutral atoms.

But, the magnetic fields are still tugging at the charged particles making up the beam as they go past. This will pull the electrons one way and the protons or ions the other way. As the particles are pulled in opposite directions you get a net positive charge on one side of the beam and a net negative charge on the other side (called polarization). This sets up an electric field inside the beam that acts to pull the particles back together. If the particles are going fast enough and the magnetic field is strong enough, the magnetic force can still separate the charges and will split the beam in two, one with the positive charges and one with the negative charges. But if the particles are not that fast or the magnetic field is not that strong this polarization of the beam will keep it together and let it fly right through the magnetic field.

<table border=1><tr><td>
Let's figure out how to analyze this problem. For definiteness, we will have a beam with an electric current of the positive species of I and an electric current of electrons of -I, all moving in the z direction. The beam radius will be R. The particle speed will be v with the usual relativistic parameters β = v/c and γ = √[1/(1-β2)]. If you know the energy per particle E of the positive species and the mass m of the positive species, γ = 1 + E/(m c2) and then β = √[1 - 1/γ2].

Let's assume that the beam is going through a magnetic field of strength B at an angle θ to the field direction. The quantity B sin(θ) is the component of the magnetic field perpendicular to the direction of the beam.

The charge density ρ of the positive component of the beam will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρL = I / (π v R2).
</div>

It is easiest to analyze the problem in the rest frame of the particles in the beam. Now the particles are at rest. Particles at rest are not affected by a magnetic field, but the change of reference frame Lorentz transforms the initial magnetic field into a combination of electric and magnetic fields in the rest frame. We can ignore the magnetic field, but the electric field is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔A = γ v B sin(θ)
</div>
in a direction perpendicular to both v and B. For our purposes, assume the electric field points in the x direction.

At the same time, relativistic length contraction increases the charge density in the rest frame
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρ = γ ρL.
</div>

Let r be the vector from the beam axis to the point inside the beam. It will be useful to separate this into its Cartesian coordinates. If x and y are the unit vectors in the x and y directions, respectively, then
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
r = x x + y y
</div>

If we consider only the positive component of the beam, from [https://en.wikipedia.org/wiki/Gauss%27s_law Gauss's law] it will have an electric field inside of the beam at a distance r from the center axis of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔+ = ρ r / (2 ε0) = (ρ/(2 ε0)) (x x + y y)
</div>
where ε0 = 8.85×10-12 is the vacuum permittivity. Now let's displace the electrons in the beam by a distance δ in the x direction. By itself, this electron beam will have an electric field of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔- = -(ρ/(2 ε0)) ((x-δ) x + y y)
</div>
By the principle of superposition, the electric field inside the beam with both components is the sum of the electric fields of the separate components including the external field
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔B = [(ρ/(2 ε0)) δ + γ v B sin(θ)]x
</div>
Because the charges are free to move, they will do so as long as a field is applied to them. Thus, the beam becomes stable once the charges separate enough that the field internal to the beam is zero. Therefore
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
δ = -2 ε0 γ v B sin(θ) / ρ = -2 π ε0 v2 R2 B sin(θ) / I
</div>
If |δ| > 2 |R|, the beam will fully separate and each component will go its own way. If δ is small compared to R, the beam's polarization suffices to hold it together in the overlap region. However, where the positive and negative charges do not overlap the electric fields no longer cancel, and the external electric field carries away the outer polarized part of the beam. This gives us a rough threshold for which a magnetic field will quickly separate a neutral beam into its charged components and bend them away from their target
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr><td> B sin(θ) &Lt; I / (π ε0 v2 R) <td>       <td> Beam is partially eroded.
<tr><td> B sin(θ) > I / (π ε0 v2 R) <td> <td> Beam promptly separates.
</table>
</div>
For a partially eroding beam, the polarized region that was peeled off by the field still acts to screen the remaining neutral portion, at least to within this approximation, leaving the still-neutral beam to hit the target.
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Neutralized_beam_in_B_field.png|5140 px|frameless]]
<tr><td width=700>
A profile of the beam charge distribution of a neutralized beam in a magnetic field. Negative charge density is shown in yellow, positive charge density in cyan, with neutral charge density in light green. The magnetic field is vertical, the electric field in the beam reference frame is shown in magenta including both the polarization field and the Lorentz transformed electric field resulting from the magnetic field. As positive charges are pulled in the direction of the field and negative charges opposite to the field direction, the polarized boundary will be pulled off the beam by the electric field even though the electric field is zero in nearly all the beam interior. This expanding polarization layer will continue to largely screen the still-neutral portion, however; but the extent to which the screening is only partial would require more sophisticated analysis methods than used here. The material removed by the field will expand to a maximum of the deflection distance of a uniform charge based on the magnetic field by the time it reaches the target.
</table>
</div>
<tr><td>
For a beam of neutral atoms, let a be the radius of the atom and let Eb be the binding energy of the least tightly bound electron to the atom. An external electric field in excess of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
|𝓔A| > Eb / (e a)
</div>
will promptly ionize the atom, ripping away its electron. Here, e = 1.60×10-19 C is the magnitude of the electric charge of the electron. For fields close to but a bit weaker than this, the electron can still rapidly pop off due to quantum tunneling<ref>Ph. Durand and I. Paidarová, “Ionization of the hydrogen atom: from weak to strong static electric fields,” The European Physical Journal D, Vol. 26, pg. 253-259, 2003.</ref>. In general, expect that for fields greater in magnitude than (very approximately) Eb / (10 e a) the electron will be quickly removed, while for fields weaker than this the electrons will stay on their atoms.

It is useful to note that Eb/e is just the binding energy in units of eV. The most weakly bound electrons are usually bound to their atom with about 10 eV of energy. Atoms are typically about 10-10 m in radius. Thus, you can expect fields of around 1010 to 1011 V/m to directly ionize atoms. If we call this threshold value 𝓔T, then a beam will be ionized when passing through a field with strength greater than
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B sin(θ) > 𝓔T / (γ v)
</div>
</table>

Example: Let's consider a 1 GeV neutralized proton beam from a synchrotron, with a beam current of 200 kA in each bunch (incidentally, if a bunch is 5 cm long this means that each bunch packs 38 kJ worth of energy). The speed of the bunch is 87% the speed of light, or 262 million m/s, with a γ of 2.07. Consider when the beam's focus takes it to a radius of 5 cm (or 0.05 m, for our calculation). Plugging this in, we find that the protons in the bunch will separate from the electrons in a field with a transverse component greater than about 2.1 T, and the bunch will pass through the field with only gradual losses if the transverse component is much smaller than this.

Example: We'll take the same 1 GeV beam, but this time recombine the electrons with protons to make hydrogen atoms. An electron is bound to a hydrogen atom with an energy of 13.6 eV (the Ryberg constant) and the radius of the electron orbital in a hydrogen atom is 0.529×10-10 m (the Bohr radius). This gives a threshold electric field for prompt ionization of hydrogen of 2.57×1011 V/m. We will get this with a transverse component of the magnetic field of 474 T. Above this field the atoms promptly ionize and electrons are stripped away from the protons. For fields higher than roughly 1/10 of this – so about 40 to 50 T – you may still get significant ionization by quantum tunneling; lower than this your losses from the beam due to field-induced ionization are likely negligible.

==== Magnetic shields ====

What happens when we can't rely on magnetic fields in the environment to protect us? One idea is to produce powerful artificial magnetic fields around our spacecraft to deflect incoming particle beams. This is very similar in concept to using active magnetic shielding to protect against space radiation – [[Space_Radiation#Magnetic_Shielding|check out our article on space radiation for more]].
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr>
<td width=100>
[[File:Unconfined_FRC_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:racetrack_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:Magnetic_shielding_Halback_Array.png|200 px|frameless]]
<tr><td colspan=3>
Some potential configurations for magnetic shields. The currents generating the fields are in green, the magnetic fields themselves are shown in magenta<ref name="Battiston2012">R. Battiston, W.J. Burger, V. Calvelli, R. Musenich, V. Choutko, V.I. Datskov, A. Della Torre, F. Venditti,
C. Gargiulo, G. Laurenti, S. Lucidi, S. Harrison, and R. Meinke, "ARSSEM Active Radiation Shield for Space Exploration Missions", Final Report ESTEC Contract N° 4200023087/10/NL/AF : “Superconductive Magnet for Radiation Shielding of Human Spacecraft” (2012) https://arxiv.org/abs/1209.1907 https://www.researchgate.net/publication/265945847_Active_Radiation_Shield_for_Space_Exploration_Missions</ref><ref name="Desiati2022">Paolo Desiati and Elena D'Onghia, "CREW HaT: A Magnetic Shielding System for Space Habitats", arXiv:2209.13624 [physics.space-ph] https://doi.org/10.48550/arXiv.2209.13624</ref>.
</table>
</div>

One immediate issue is that you want to keep very high magnetic fields out of the crew area. Fields in excess of 0.5 T can start to be unpleasant, and it is recommended to keep the field below 0.2 T for continuous exposure.

Strong magnets can suffer from high self-forces acting to tear them apart. Most shielding designs will have multiple magnetic coils, which will exert high forces and torques on each other. This will result in the need for strong structural bracing to keep the magnets intact and in place. Such strong structural designs will impose significant extra mass on the spacecraft.

Consider a magnetic field configuration that can keep radiation out of a certain region of space. There is no trajectory of a charged radiation particle up to a certain energy that will allow those particles to enter the protected region. But the dynamics of charged particles in a magnetic field are the same whether going forward or backward in time. If there is no way for a particle to get in, by this time reversal symmetry any particle which is in that forbidden region can never escape. Consequently any field configuration that is a magnetic trap is also a magnetic shield; any field configuration that is a magnetic shield is also a magnetic trap. A spacecraft with a magnetic shield may end up surrounded by strong radiation belts.

Using a magnet to bend a weaponized electron beam very often ends up with the beam being bent so hard that it dumps most of its energy in the form of synchrotron radiation at gamma ray energies. Because gamma rays are not charged, they will not be deflected by the field. Thus, if you would have been hit by the electron beam you will be hit instead by the gamma rays.

As an example, consider a 1 TeV electron beam. We'll surround our spacecraft with a 10 T magnetic field in a 100 meter radius region just outside our spacecraft. This will deflect the beam by 30 meters, which is enough to make it miss. But we are not saved, because the beam dumps nearly all of its energy into synchrotron radiation with an energy spectrum of gamma rays up to 13 GeV. This radiation will hit us in a strip about 2.4 meters long and not much more than the width of the original beam with essentially all of the energy of the original beam.

==== Magnet drones ====

If a magnetic shield around your spacecraft will just end up killing you from synchrotron radiation, what if you put the magnetic shield on a drone and put that drone between yourself and the enemy trying to shoot you? This lets you use a much weaker magnet because you don't need nearly so high of a deflection angle to make the shot miss. This means that less beam energy will turn into synchrotron radiation that could kill you instead and will spread what synchrotron radiation does occur out over a much wider area when it reaches your spacecraft, hopefully so that most of it just misses your spacecraft.

It does mean that your enemy will start shooting your drones so that they can eventually take a shot at you. But hopefully by that time you've already cleared out their drones and cored their spacecraft with your own synchrocannon or UREB.

==== Plasma magnets ====

Usually, to make a really big magnetic field you would run current through a really big conductive loop. But really big things are heavy and expensive. One way around this could be to use the ambient space plasma to carry the electric current rather than any kind of wire. In order to make this current, you use the same mechanism used in a rotary induction electric motor: you generate a rotating magnetic field and induce a current in the conductive loop (in this case, the space plasma). How do you make a rotating magnetic field? The method proposed<ref name="Slough2006">John Slough, "The Plasma Magnet", NASA Institute for Advanced Concepts Phase II Final Report (2006) https://www.niac.usra.edu/files/studies/final_report/917Slough.pdf</ref> is to use an antenna of four much smaller current loops as electromagnets at 90&degree; to each other, and sinusoidally vary the current through the loops with a respective 90&degree; phase difference between them. When this is done, a circular current is induced in the space plasma and the self-force of a magnetic field on its generating current inflates the current loop to much larger radii. You end up with a current sheet in the plane of the rotating field's rotation circulating about the antenna nearly synchronously with the field rotation. This current then creates a normal dipolar magnetic field as with any current loop. The current sheet and the resulting magnetic field expand until the magnetic field's pressure equals that of the ambient space plasma. This gives an enormous region of magnetic field from a relatively small and lightweight system.

The system proposed in the initial study would use an antenna approximately 10 meters in radius and produce a region of field approximately 30 km in radius at Earth's orbit. The field inside would not be constant - at the edges it would be 50 nT, giving the right pressure to keep out the solar wind at Earth's orbit. A circulating current sheet produces a field that falls off as 1/r with distance, meaning that adjacent to the antenna the field would be about 0.15 mT. Using the methods in the box below, we can estimate that this particular design as part of a spacecraft's shields will not deflect a TeV electron beam by more than a few centimeters, which is usually insufficient to force a miss. However, it could potentially be useful for a drone magnet, where the much larger distance between the magnetic field and the target spacecraft would allow a small deflection angle to give perhaps tens or low hundreds of meters change in beam location near the target at near light second ranges. The relatively low field and gradual bend of the incident beam suggests that synchrotron radiation will not be an issue; however this does not exclude more powerful plasma magnet designs causing significant synchrotron radiation effects.

<table border=1><tr><td>
Calculating the deflection from a spatially varying magnetic field affected by the solar wind is no trivial task. However, we can make an approximation that both should be within about an order of magnitude of the true value and also represent an upper limit to the deflection angle; although it does neglect the influence of the driving rotating magnetic field which could in some cases increase the deflection.

We will take R0 as the radius of the antenna. B0 is the field strength at the antenna. Rm is the distance at which the magnetic influence stops because the magnetic pressure equals the space plasma pressure. Bm is the field strength at Rm. According to Slough<ref name="Slough2006"></ref>, the magnetic field strength B(r) at any distance from the antenna center r between R0 and Rm is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B(r) = B0 R0 / r.
</div>
Therefore, once we know Bm and Rm, B0 is completely determined
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B0 = Bm Rm / R0.
</div>

The force on a particle with charge q passing perpendicularly through a magnetic field B(r) at a speed v is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
F(r) = q v B(r)
</div>
For an electron beam, v is so close to c as to make no difference. If the particle is moving down the y direction and the field is in the z direction, the force will be in the x direction. We assume that the deflection will be small, so we can choose a straight-line path in the y direction offset from the center by an amount x. The plasma bubble radius is Rm, so the particle will enter at ymax = √[Rm2-x2] and leave at ymin = -√[Rm2-x2].

The momentum change the particle experiences from this force is the time integral of the force over its path. Because of reflection symmetry about y = 0, we only need to integrate from ymax to 0 and then multiply by 2
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] dt F(r)
<tr> <td> <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] (dy/v) F(r)
<tr> <td> <td> = <td align="left">2 q B(0) R0 [Integral from 0 to √[Rm2-x2]] dy 1/√[y2 + x2]
<tr> <td> <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/x]
</table>
</div>
This holds for |x| > R0. But the expression for the field is not valid for r < R0 and the integral will give nonsense and diverging results if we evaluate for |x| < R0. In this case, we can put the integral lower bound at r = R0, or y = √[R02-x2]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/(√[R02-x2] + R0)]
</table>
</div>

The deflection angle is now estimated as
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = px / p
</div>
where the total momentum of the particle p can be found from the methods in the first box in the Magnetic Fields section.

The deflection for a shot aimed right at the spacecraft at the position of the spacecraft can be estimated by setting x = 0 and only taking the first half of the integral (so ignore the factor of 2)
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
q B(0) R0 ln[Rm/R0] / (p Rm)
</div>
This is an additional overestimate because it fails to account for how the deflection builds up over distance during the integration, but it should still give something of an order-of-magnitude and upper limit estimate of what kind of deflections might be expected.
</table>

=== Electric fields ===

If electric fields can accelerate charged particles, maybe they can keep the charged particles in a weaponized particle beam away from your spacecraft? To completely block a particle beam, you would need a voltage on your spacecraft equal to the particle energy in eV of the particles in the beam. That is, if a 1 GeV proton beam is coming at you, you'd need an electric potential of a billion volts to stop that beam. This leads to three immediate problems.

The first is that particles of the opposite charge will be attracted to your spacecraft. Because space isn't really empty but rather a rarefied plasma (called the solar wind in a solar system or the interstellar medium out between stars), there will be plenty of particles of the opposite charge around to attract. And they will hit with an energy in eV equal to the voltage on your spacecraft. In our example of defending against a 1 GeV proton beam with a gigavolt field, the electrons from the solar wind will come in and hit with an energy of 1 GeV each. This will generate dangerous radiation.

The second problem is a consequence of the first. Each of those GeV electrons from our example need to get their energy from somewhere, which turns out to be from the electric field we are using as shielding. You will need to make up for this energy loss with power from your electrical system, which gives a constant drain on your power. Some estimates put the electrical effort to maintain a several hundred megavolt shield in space at many tens of gigawatts<ref name="Mechmann2019">Claire Mechmann, "Analysis of Proposed Active Radiation Shielding Design Concept for Spacecraft" (2019) Thesis, College of Engineering and Science of Florida Institute of Technology</ref>.

And the third problem is similar to one for magnetic fields. Your electrodes will experience high self forces and forces between separate electrodes that will try to tear your shield generator apart. Consequently you will need heavy structural reinforcement in order for your shield to work.

There are potential ways to mitigate some of these problems. Interested readers are directed to [[Space_Radiation#Electrostatic_Shielding|the active shielding section in our page on space radiation]] for more of the juicy details; there's no need to repeat the same stuff here.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Elctrostatic_active_shielding.png|200 px|frameless]]
<td>
[[File:Electrostatic_active_shielding_2.png|200 px|frameless]]
<tr><td colspan=2>
Proposed designs for electrostatic shields to protect from positively charged particles<ref name="Tripathi2006">Ram K. Tripathi, John W. Wilson, and Robert C. Youngquist, "Electrostatic Active Radiation Shielding - Revisited", 2006 IEEE Aerospace Conference, Big Sky, MT, USA, 2006, pp. 9 pp.-, doi: 10.1109/AERO.2006.1655760.</ref><ref name="Chowdhury2023">Rajarshi Pal Chowdhury, Luke A. Stegeman, Matthew L. Lund, Dan Fry, Stojan Madzunkov, and Amir A. Bahadori, "Hybrid methods of radiation shielding against deep-space radiation", Life Sciences in Space Research, Volume 38, 2023, Pages 67-78, ISSN 2214-5524, https://doi.org/10.1016/j.lssr.2023.04.004.</ref>.
</table>
</div>

== Damage ==

Particle beams are, first and foremost, devices that produce beams of [[Nuclear_radiation|ionizing radiation]]. The radiation of weaponizable particle beams is extremely penetrating; thus of the [[Particle_Accelerators#Energy_loss_in_matter|interaction mechanisms listed for radiation from particle beams]], we mostly need to worry about bremsstrahlung and nuclear collisions and these produce extensive secondary showers of radiation that increase the penetration of the radiation exposure. Because of this, any target hit by a particle beam weapon will be irradiated, and radiation dose will always be one of the primary damage mechanisms of a particle beam weapon strike. Further, this radiation will be distributed along a track passing deeply through the material of the target, allowing it to affect critical systems through heavy armor, structural material, robust machine components, and passive material such as fuel load, coolant, or cargo. As the radiation deposits its energy into the target material, that material will be heated. If the target is heated sufficiently, it may experience [[Beam-Target_Interactions|thermal or mechanical]] damage from melting, cooking, combustion, vapor explosions, and warping or cracking due to thermal expansion.

The discussion of the radiation penetration into armor also applies to the same radiation penetrating through the target. So the formulas and rules of thumb for estimating the depth of radiation penetration into armor can also be used to estimate how far into a tank or spacecraft a particle beam strike delivers effects.

In addition to the radiation and thermal effects, a particle beam can also produce a sharp electromagnetic pulse that can damage semiconductor electronics.

== 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, Gerrit Bruhaug, and Luke Campbell for writing

Gerrit Bruhaug and Luke Campbell for scientific research, math wrangling and advise

ATriffleMind and Luke W. Campbell for proof reading.

==References==

[[Category:Beams]][[Category:Physics & Engineering‏‎]][[Category:Engineering‏‎]][[Category:Military Technology]][[Category:Warfare‏‎]]

Particle Beam Weapons

2026-10-27T00:22:35Z

Lwcamp: /* Separation of neutral beams */

{{PageConstructionNotice}}

''For a general overview on particle accelerator technology, see [[Particle Accelerators]]

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.

::: Z = 1.32×102 (γ2-1)3/4 r / √I<ref name=USPAS2016>Martin Reiser, "Theory and Design of Charged Particle Beams", 2008, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim</ref>

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

::: Z = β γ c r √[m/T]

Where Z is the doubling distance like before, r is the initial beam radius, T is the beam temperature in eV, m is the ion mass and β, c, and γ 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.

== Defenses ==

Sure, it's great to have a great big particle cannon to blast your enemies. But what if you are the one being shot at? In that case, it's best to figure out ways to keep from being blown to kingdom come.

=== Armor ===

The simplest way to defend against the high energy particles of a particle beam is to pile a whole bunch of stuff up and put it in the way of the beam. Particle beams may produce extremely penetrating radiation, but enough stuff will block most of that radiation and may allow you to survive. The details depend on what kind of particle is being thrown at you, and are explained below. However, a general rule of thumb is that you probably want about 10 tons of material per square meter of surface you want to armor in order to get sufficient protection.

==== Electrons ====

At the very high energy of an electron accelerated by a weapons-grade particle beam, the primary interaction with matter will be the emission of bremsstrahlung photons in the gamma-ray range as the electron slams into particles of matter. These high energy photons themselves primarily interact by the pair production mechanism – the photon disappears, but leaves behind a highly energetic positron and electron. These two particles are themselves energetic enough to produce further bremsstrahlung, and so the process continues. You get the formation of an electron-gamma shower (also called an electromagnetic cascade), which only stops after either the particles have penetrated through the target or the initial energy of the electron has been spread through so many shower particles (including all the electrons, positrons, and photons) that no particle has enough individual energy to pair produce or to be likely to bremsstrahlung again before it is stopped by other mechanisms.

To stop electrons, you'll want the atoms in your armor to be as heavy as you can practically get them. High atomic number both encourages bremsstrahlung and pair production, giving a double whammy for stopping that cascade. However, occasionally one of those electrons, positrons, or gammas will hit a nucleus and knock out neutrons. So you probably want a back-up layer of several tens of centimeters of hydrogen rich material such as plastic or water mixed with neutron absorbing materials like boron, lithium, cadmium, or gadolinium. High density polyethylene loaded with 5% boron by weight is a typical go-to neutron blocker.

The actual methods to calculate the penetration of the electromagnetic cascade is given are the box below. However, for some examples:
<ul>
<li> A 10 GeV beam incident on thick iron (or steel) will stop cascading at about 9 cm, will deposit half of its energy in the first 12 cm, and will have dumped 98% of its energy within 35 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 2.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 100 GeV beam incident on thick iron (or steel) will stop cascading at about 13 cm, will deposit half of its energy in the first 16 cm, and will have dumped 98% of its energy within 47 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 3.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick iron (or steel) will stop cascading at about 17 cm, will deposit half of its energy in the first 20 cm, and will have dumped 98% of its energy within 59 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 4.6 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick tungsten will stop cascading at about 3.5 cm, will deposit half of its energy in the first 4 cm, and will have dumped 98% of its energy within 12 cm. 90% of the cascade will be contained within about 0.93 cm and 95% of the cascade will be contained within 1.86 cm of the beam axis. You will get 98% protection with a mass thickness of 2.3 tons/m2 and an additional 1/e reduction for each further 0.13 tons/m2 beyond that.
<li> A 1 TeV beam incident on water will stop cascading at about 290 cm, will deposit half of its energy in the first 340 cm, and will have dumped 98% of its energy within 1030 cm. 90% of the cascade will be contained within about 11 cm and 95% of the cascade will be contained within 22 cm of the beam axis. You will get 98% protection with a mass thickness of 10.3 tons/m2 and an additional 1/e reduction for each further 0.72 tons/m2 beyond that.
</ul>
Note that for sufficiently intense particle beams, 98% protection may not be enough. You might need to throw on several more tons per square meter to be sure.

<table border=1>
<tr><td> How Far it Goes:
(This section follows methods presented in Leroy and Rancoita<ref>C. Leroy and P-G Rancoita, "Physics of cascading shower generation and propagation in matter: principles of high-energy, ultrahigh-energy and compensating calorimetry", Reports on Progress in Physics 63 (2000) 505–606. DOI: 10.1088/0034-4885/63/4/202</ref>.)

When trying to estimate the extent of the irradiated region of the target, it is helpful to know the radiation length X0 in the material. This is the distance that an electron will travel, on average, before emitting a bremsstrahlung photon. Conveniently, it is also the distance a high energy photon will travel through the material before pair producing. The radiation length can be found from
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   = 4 α  
<td nowrap="nowrap"> NA
<td rowspan="2" nowrap="nowrap" align="right">   Z (Z+ζ) re2 ln
<td rowspan="2" nowrap="nowrap" align="right"> [
<td nowrap="nowrap"> 183
<td rowspan="2" nowrap="nowrap" align="right"> ]
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> A
<td style="border-top:solid 1px black;"> Z1/3
</table>
</div>
where α = 1/137 is the fine structure constant, NA = 6.02214076×1023 is Avogadro's number (or one mole), A is the atomic weight of the atoms in the material (usually expressed in grams per mole), Z is the atomic number (nuclear charge) of the atoms, and re = 2.8179403205×10-13 cm is the classical electron radius. ζ is a constant of approximately 1.3. This gives the thickness of material in terms of the areal density (mass of material per unit area perpendicular to the beam) – if you use A in g/mol and re in cm the result will be in cm2/g and X0 will be in g/cm2. If your target is made up of more than one kind of atom, you can get the total radiation length by finding the radiation length of each atom of type i, X0i and using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   =  
<td rowspan="2" nowrap="nowrap" align="right"> Σi  
<td nowrap="nowrap"> fi
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> X0i
</table>
</div>
where fi is the fraction by weight of the atom type i. Once you have the radiation length in terms of areal density, you can easily convert it to distance by dividing by the density ρ. If X0i is in g/cm2 and density is in g/cm3, then X0i/ρ will be in cm.

We will also need to know the critical energy εc. This is the energy where an electron is losing energy as rapidly to ionization as it is to radiation. That is, it is the energy at which the cascade stops multiplying, even though the individual particles already made can continue traveling for some distance. The critical energy can be estimated using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> εc = B  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> Z X0
<td rowspan="2" nowrap="nowrap" align="right"> )
<td nowrap="nowrap"> h
<tr>
<td style="border-top:solid 1px black;"> A
<td>  
</table>
</div>
where B = 2.66 MeV and h = 1.11.

We can now describe the distribution of radiation in the target material. The depth tmax where the cascade stops multiplying (in g/cm2) for an incident electron of energy E (in MeV) is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tmax = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> - 1
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
Similarly, the depth where half the energy of the beam has been deposited (in g/cm2) is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tcg = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> + 0.37
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
As usual, you can convert to a depth in cm by dividing by the material density. 98% of the beam energy will be deposited within a distance of &cong; 3 tcg. Beyond this, the beam energy will continue to fall off with thickness t approximately as an exponential exp[-t/λ], where the characteristic attenuation length λ &cong; 2 X0.

The width of a cascade, where 90% of the energy is contained, is approximately given by
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> RM =  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> EM
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
where EM &cong; 21.2 MeV. 95% of the radiation will be contained within a distance of 2 RM from the beam axis.
</table>

==== Protons and Ions ====

<table align=right><tr><td>
[[File:Regolith_Shielding.png|350 px|frameless]]
<tr><td width=350>
Relative effect of particle radiation (compared to no shielding) from galactic cosmic rays behind different thicknesses of water, aluminum, and lunar regolith<ref name="Slaba2022">Tony C. Slaba, "Radiation Shielding for Lunar Missions: Regolith Considerations", LSIC Crosstalk 7/18/2022 https://lsic.jhuapl.edu/uploadedDocs/focus-files/1604-E&C%20+%20EE%20Monthly%20Meeting%20-%202022%2007%20July_Presentation%20-%20NASA%20Slaba.pdf</ref>.
</table>
Protons and ions are grouped together because ions are made of protons and neutrons and when you accelerate these things for a weapon it is the protons and neutrons that are going to be doing the heavy lifting for blasting your target. This is because protons and neutrons are both about 2000 times more massive than electrons so any electrons along for the ride will contribute negligibly to the energy. Protons and neutrons together are called nuclear particles, and both behave quite similarly in nuclear collisions.

Nuclear particles are themselves made of quarks and can interact via the strong nuclear force. At the energies appropriate for weapons use, essentially all of the way protons or ions are blocked is through [[Particle_Accelerators#Nuclear_collisions|direct collisions with the atomic nuclei]] of the armor material. One or more quarks in the projectile will slam into quarks in the target nucleus and stretch them away from the rest of the quarks in the nuclear particles. Physics tells us that you can't have isolated quarks, so the "string" attaching the quark to its neighbors stretches and stretches until it snaps. Snapping releases so much energy that more quarks are made that clump together into all sorts of other nuclear particles, both exotic and familiar. This process is called a hadronic jet. The only particles produced in these jets long-lived enough to travel any noticeable distance through the armor material will be protons, neutrons, the charged pi mesons (or pions), and some of the K mesons (or kaons). The particles too short lived to go far enough to matter will decay into other particles, and these particles are often very high energy gamma rays. Much like with electrons, the long-lived secondary particles can be sufficiently energetic to smash into other nuclei and cause more hadronic jets (or, if they are low enough in energy, just knock more nuclear particles out of target nuclei) and this produces a similar shower or cascade. But in addition, the armor nuclei that were hit will be very hot after the impact from all of the energy given to them from the incident particles. These nuclei will cool off by evaporation – but instead of evaporating away molecules, they evaporate off nuclear particles from the nucleus. These evaporated particles are mostly neutrons, so any armor against proton or ion beam strikes will produce quite a bit of neutron radiation.

Eventually, the original particle's energy will be distributed among so many other particles that the charged particles will come to a stop by making ion tracks rather than from colliding with more nuclei, and the cascade halts. But you still have all the neutrons to worry about. And the gamma rays as well, which are high enough energy to kick off an electromagnetic cascade in the same way that electrons do.

Unlike electrons, which have reasonable approximations of their cascade behavior from relatively simple equations, the cascades of protons and ions usually require sophisticated simulations or experiments to get the resulting distribution of radiation. However, we can get a reasonable idea of their general behavior by comparing them to [[Space_Radiation#Shielding_Against_Cosmic_Rays|cosmic ray particles interacting with shielding]]. Cosmic rays are energetic protons and ions, after all. From this we see that a relatively small amount of armor – in the range of 1 to 3 tons per square meter – can actually make the radiation dose worse. The incident particles will knock off so many secondary particles in their shower that the dose is multiplied from so many tracks passing through the target. Hydrogen rich materials such as water and plastics work the best against proton and ion beam particles; heavy elements are the worst and produce the most secondaries. You could probably obtain adequate radiation protection from most incident beams with about 10 tons per square meter of hydrogen rich material as armor. However, you will want to add neutron absorbing elements to your hydrogen rich armor, such as lithium, boron, cadmium, or gadolinium. Further, in order to deal with the gamma rays you will want a layer of at least several centimeters of heavy elements, such as tungsten or lead, behind the hydrogen rich particle armor.

==== Structural Considerations ====

When your armor material absorbs a radiation beam, the energy of that radiation turns into heat inside the armor where the radiation was absorbed. If the beam pulse has enough energy as was sufficiently tightly focused, this heat can damage your armor. Your armor may melt in places, or thermal expansion could cause it to crack or warp. Perhaps the worst thing it could do is vaporize or flash into a plasma, because the pressure from the vapor or plasma can be sufficient to plastically deform the armor. This leaves a hole along the beam path where the expanding gas or plasma blasted out a long, deep crater. This means a beam that fires rapid shots spaced maybe milliseconds apart can have subsequent shots go down the same hole so your armor will not protect against them. Each shot will dig out a deeper hole until either your armor is penetrated or the attacking beam runs out of shots in its pulse train.

A consequence of this is that when designing your armor you may wish to use refractory materials with a high structural strength (like tungsten), or that is otherwise resistant to being blasted apart by its own evaporated remains.

=== Magnetic fields ===

[[Particle_Accelerators#Magnetic_fields|A charged particle crossing a magnetic field will bend so as to spiral around the field lines]]. If you try to shoot your particle beam through a magnetic field, it will bend away from the direction you shot it. This can make it hard to hit your target. Unfortunately for you, there are magnetic fields everywhere – around planets, and even trapped in the solar wind as it crosses the solar system.

As an example, the Earth has a magnetic field strength of about 5×10-5 tesla near its surface, and falling off farther away. If you are shooting a 1 TeV electron beam past the Earth, your beam is likely to bend by several milliradians (with the exact deflection depending on how close you are skimming the Earth). If you are in a medium Earth orbit and shooting at another target also in a medium Earth orbit on a trajectory that goes near the Earth, then without correcting for the magnetic field your beam will likely be off-target by tens to hundreds of kilometers.

For another example, the interplanetary magnetic field is about 10-9 tesla. If you shoot that same 1 TeV electron beam at a target 0.1 light seconds away, you are likely to be off by a couple hundred meters.

<table border=1>
<tr><td>
The radius of curvature of the bend of a particle in a magnetic field is called the gyroradius rc. For a particle with a charge q, mass m, and kinetic energy E, find the relativistic momentum p
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
p = √[E2 - m2 c4] / c
</div>
To convert from energy units of eV to joules, multiply by 1.60×10-19. An electron has a mass of 9.11×10-31 kg, a proton of 1.67×10-27 kg, and an ion of 1.66×10-27×A kg, where A is the atomic weight of the ion. The speed of light c = 3.00×108 m/s.

In a magnetic field of strength B with an angle θ between the field direction and the direction of motion
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rc = |p sin(θ) / (q B)|
</div>
For electrons, q = -e, where the fundamental charge e = 1.60×10-19 coulomb. For a proton, q = +e. For an ion, multiply e by the charge state of the ion.

If the field acts over a distance d that is much smaller than the gyroradius, the deflection angle φ is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = d / rc
</div>
The deflection x at a target at a range R for small φ (φ &Lt; 1) will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
x = φ R.
</div>

Note that d will always be less than or equal to x. Also note that if d is not smaller than twice the gyroradius the beam will never reach the target because it will completely spiral around the field lines before it reaches the target.
</table>

So, it is clear that as the defender it will be helpful to hang out in areas with high magnetic fields, and keep some distance between you and the attacker. Meanwhile, the attacker will benefit from being closer where small deflections in the beam trajectory don't translate into large excursions from where the beam is aimed (and also because there will be less space filled with fields between the two of you). But the attacker can also try to compensate for known fields. If you happen to know that the magnetic fields are between you and your target, you can adjust your fire so that the deflection of the field bends your beam onto the target. This is not a simple task. It is easy enough to measure the field strength and direction where you are, but predicting how it changes over the path of your shot does not have an obvious solution. You can also try to gain intelligence on likely battle sites before you ever fight on them – for example, getting good data on the structure of the fields of a planet using spies and technical intelligence during peacetime (heck, on Earth this information is published for free in academic journals and by government agencies such as NOAA). Planetary magnetospheres are also affected by the solar wind, so different solar weather will change the fields you have to shoot through. But space weather forecasts and predictive simulations of the solar wind interacting with a planetary magnetosphere can give you a good idea of what the field structure is like. But even with all that, there are changes in the fields that happen on a scale of around a second ... anyone who has ever seen the auroras dancing across the sky in the polar night can see this. Compensating for these changes will be a significant challenge.

What would really help is being able to see the path of your own beam. This would allow you to see how far off you were from your target and in what direction, and then adjust fire to walk your shots onto the target. Unfortunately, particle beams in space are invisible. There might be ways to track them, but if so we don't know what they are.

==== Synchrotron radiation ====

When charged particle trajectories are bent, they emit [[Particle_Accelerators#Synchrotron_radiation|synchrotron radiation]]. This can cause the particle beam to lose energy, but if the bend is too tight it can also generate high energy gamma rays that ignore magnetic fields and directly strike the target. Synchrotron radiation can generally be ignored for proton or ion beams, but can be significant for electron beams.

<table border=1><tr><td>
For a beam particle with energy E and mass m, the synchrotron radiation is concentrated in a forward-directed cone with an opening angle of approximately 1/γ, where γ = 1 + E/(m c2). (Interestingly, the radiation intensity peaks an at angle of 1/(2 γ) and vanishes directly on-axis.) Because the beam is bending, you will get emission along the plane along which the beam is being bent. The energy of the photons produced during synchrotron emission is approximately<ref name="Jackson_Electrodynamics">J. D. Jackson, "Classical Electrodynamics: Second Edition", Wiley (1962)</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Eγ &cong; 3 γ3 c &hbar; / rc
</div>
where &hbar; = 1.05×10-34 J s (if you want the energy in joules) or &hbar; = 6.58×10-16 eV s (if you want the energy in eV). The photon energy will not exceed the energy of the incident particles, if you get a photon energy higher than the energy of the incident particles it just means that the synchrotron radiation spectrum extends up to the incident particle energy.

The fraction of the energy (or power) of the beam turned into synchrotron radiation will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = [1/(4 π ε0)] [4 π / 3] [q² / rc] β2 γ4 [φ / (2 π)] / E
</div>
where β = √[1 - 1/γ2] is the particle's speed measured as a fraction of the speed of light, and 1/(4 π ε0) = 8.99×109 is the Coulomb constant. For highly relativistic electrons (like all practical electron beam weapons emit), β &cong; 1 and we can use
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = 8.85×10-2 [E (GeV)3 [φ / (2 π)] / (rc (m))]
</div>
This fraction cannot be more than 1. If f gets close to one, it means that our original assumption that the beam particles mostly keep their original speed no longer holds, and the beam will lose most or almost all of its energy. Instead, when the beam is being deflected it will only make it through an angle of φ/f and will irradiate a region of length x/f starting at where the beam would have been incident if not for the magnets and extending in the direction the beam was deflected.
</table>

For example, a 1 TeV electron beam will have γ &cong; 2,000,000. So if it's bent by a magnetic field the synchtrotron radiation will spread out in a cone of about 5×10-7 radians. If this beam was shooting at a target 0.1 light seconds away from where it was bent, the radiation would arrive at the target over a region with a radius of about 15 meters with a maximum intensity at about 7.5 meters and no radiation directly along the path the beam was pointing while it was being bent. If the beam was bent by the interplanetary magnetic field from our earlier example, it would produce light with photon energies of around 1.3 eV (near infrared) and the lost energy would be a few parts in a hundred trillion. If it was instead deflected by the Earth's magnetic field you might expect a peak photon energy of tens of keV – in the hard x-ray part of the spectrum – and the fraction of energy in the synchrotron radiation would be somewhere in the low parts in 100,000 or less.

==== Separation of neutral beams ====

Magnetic fields only affect charged objects. So one way to overcome magnetic defenses or environments is to use beams that have no net charge. For example, a proton beam could be neutralized with a co-propagating electron beam with an equal number of protons and electrons. Or, potentially, you could fire an ion beam and then neutralize the ions after acceleration to get a beam of neutral atoms.

But, the magnetic fields are still tugging at the charged particles making up the beam as they go past. This will pull the electrons one way and the protons or ions the other way. As the particles are pulled in opposite directions you get a net positive charge on one side of the beam and a net negative charge on the other side (called polarization). This sets up an electric field inside the beam that acts to pull the particles back together. If the particles are going fast enough and the magnetic field is strong enough, the magnetic force can still separate the charges and will split the beam in two, one with the positive charges and one with the negative charges. But if the particles are not that fast or the magnetic field is not that strong this polarization of the beam will keep it together and let it fly right through the magnetic field.

<table border=1><tr><td>
Let's figure out how to analyze this problem. For definiteness, we will have a beam with an electric current of the positive species of I and an electric current of electrons of -I, all moving in the z direction. The beam radius will be R. The particle speed will be v with the usual relativistic parameters β = v/c and γ = √[1/(1-β2)]. If you know the energy per particle E of the positive species and the mass m of the positive species, γ = 1 + E/(m c2) and then β = √[1 - 1/γ2].

Let's assume that the beam is going through a magnetic field of strength B at an angle θ to the field direction. The quantity B sin(θ) is the component of the magnetic field perpendicular to the direction of the beam.

The charge density ρ of the positive component of the beam will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρL = I / (π v R2).
</div>

It is easiest to analyze the problem in the rest frame of the particles in the beam. Now the particles are at rest. Particles at rest are not affected by a magnetic field, but the change of reference frame Lorentz transforms the initial magnetic field into a combination of electric and magnetic fields in the rest frame. We can ignore the magnetic field, but the electric field is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔A = γ v B sin(θ)
</div>
in a direction perpendicular to both v and B. For our purposes, assume the electric field points in the x direction.

At the same time, relativistic length contraction increases the charge density in the rest frame
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρ = γ ρL.
</div>

Let r be the vector from the beam axis to the point inside the beam. It will be useful to separate this into its Cartesian coordinates. If x and y are the unit vectors in the x and y directions, respectively, then
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
r = x x + y y
</div>

If we consider only the positive component of the beam, from [https://en.wikipedia.org/wiki/Gauss%27s_law Gauss's law] it will have an electric field inside of the beam at a distance r from the center axis of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔+ = ρ r / (2 ε0) = (ρ/(2 ε0)) (x x + y y)
</div>
where ε0 = 8.85×10-12 is the vacuum permittivity. Now let's displace the electrons in the beam by a distance δ in the x direction. By itself, this electron beam will have an electric field of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔- = -(ρ/(2 ε0)) ((x-δ) x + y y)
</div>
By the principle of superposition, the electric field inside the beam with both components is the sum of the electric fields of the separate components including the external field
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔B = [(ρ/(2 ε0)) δ + γ v B sin(θ)]x
</div>
Because the charges are free to move, they will do so as long as a field is applied to them. Thus, the beam becomes stable once the charges separate enough that the field internal to the beam is zero. Therefore
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
δ = -2 ε0 γ v B sin(θ) / ρ = -2 π ε0 v2 R2 B sin(θ) / I
</div>
If |δ| > 2 |R|, the beam will fully separate and each component will go its own way. If δ is small compared to R, the beam's polarization suffices to hold it together in the overlap region. However, where the positive and negative charges do not overlap the electric fields no longer cancel, and the external electric field carries away the outer polarized part of the beam. This gives us a rough threshold for which a magnetic field will quickly separate a neutral beam into its charged components and bend them away from their target
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr><td> B sin(θ) &Lt; I / (π ε0 v2 R) <td>       <td> Beam gradually erodes.
<tr><td> B sin(θ) > I / (π ε0 v2 R) <td> <td> Beam promptly separates.
</table>
</div>
For a gradually eroding beam, you can figure that if the magnetic deflection of the positive component is not much larger than the beam radius, most of the beam will still hit its target.
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Neutralized_beam_in_B_field.png|5140 px|frameless]]
<tr><td width=700>
A profile of the beam charge distribution of a neutralized beam in a magnetic field. Negative charge density is shown in yellow, positive charge density in cyan, with neutral charge density in light green. The magnetic field is vertical, the electric field in the beam reference frame is shown in magenta including both the polarization field and the Lorentz transformed electric field resulting from the magnetic field. As positive charges are pulled in the direction of the field and negative charges opposite to the field direction, the polarized boundary will be pulled off the beam by the electric field even though the electric field is zero in nearly all the beam interior. This expanding polarization layer will continue to largely screen the still-neutral portion, however; but the extent to which the screening is only partial would require more sophisticated analysis methods than used here. The material removed by the field will expand to a maximum of the deflection distance of a uniform charge based on the magnetic field by the time it reaches the target.
</table>
</div>
<tr><td>
For a beam of neutral atoms, let a be the radius of the atom and let Eb be the binding energy of the least tightly bound electron to the atom. An external electric field in excess of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
|𝓔A| > Eb / (e a)
</div>
will promptly ionize the atom, ripping away its electron. Here, e = 1.60×10-19 C is the magnitude of the electric charge of the electron. For fields close to but a bit weaker than this, the electron can still rapidly pop off due to quantum tunneling<ref>Ph. Durand and I. Paidarová, “Ionization of the hydrogen atom: from weak to strong static electric fields,” The European Physical Journal D, Vol. 26, pg. 253-259, 2003.</ref>. In general, expect that for fields greater in magnitude than (very approximately) Eb / (10 e a) the electron will be quickly removed, while for fields weaker than this the electrons will stay on their atoms.

It is useful to note that Eb/e is just the binding energy in units of eV. The most weakly bound electrons are usually bound to their atom with about 10 eV of energy. Atoms are typically about 10-10 m in radius. Thus, you can expect fields of around 1010 to 1011 V/m to directly ionize atoms. If we call this threshold value 𝓔T, then a beam will be ionized when passing through a field with strength greater than
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B sin(θ) > 𝓔T / (γ v)
</div>
</table>

Example: Let's consider a 1 GeV neutralized proton beam from a synchrotron, with a beam current of 200 kA in each bunch (incidentally, if a bunch is 5 cm long this means that each bunch packs 38 kJ worth of energy). The speed of the bunch is 87% the speed of light, or 262 million m/s, with a γ of 2.07. Consider when the beam's focus takes it to a radius of 5 cm (or 0.05 m, for our calculation). Plugging this in, we find that the protons in the bunch will separate from the electrons in a field with a transverse component greater than about 2.1 T, and the bunch will pass through the field with only gradual losses if the transverse component is much smaller than this.

Example: We'll take the same 1 GeV beam, but this time recombine the electrons with protons to make hydrogen atoms. An electron is bound to a hydrogen atom with an energy of 13.6 eV (the Ryberg constant) and the radius of the electron orbital in a hydrogen atom is 0.529×10-10 m (the Bohr radius). This gives a threshold electric field for prompt ionization of hydrogen of 2.57×1011 V/m. We will get this with a transverse component of the magnetic field of 474 T. Above this field the atoms promptly ionize and electrons are stripped away from the protons. For fields higher than roughly 1/10 of this – so about 40 to 50 T – you may still get significant ionization by quantum tunneling; lower than this your losses from the beam due to field-induced ionization are likely negligible.

==== Magnetic shields ====

What happens when we can't rely on magnetic fields in the environment to protect us? One idea is to produce powerful artificial magnetic fields around our spacecraft to deflect incoming particle beams. This is very similar in concept to using active magnetic shielding to protect against space radiation – [[Space_Radiation#Magnetic_Shielding|check out our article on space radiation for more]].
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr>
<td width=100>
[[File:Unconfined_FRC_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:racetrack_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:Magnetic_shielding_Halback_Array.png|200 px|frameless]]
<tr><td colspan=3>
Some potential configurations for magnetic shields. The currents generating the fields are in green, the magnetic fields themselves are shown in magenta<ref name="Battiston2012">R. Battiston, W.J. Burger, V. Calvelli, R. Musenich, V. Choutko, V.I. Datskov, A. Della Torre, F. Venditti,
C. Gargiulo, G. Laurenti, S. Lucidi, S. Harrison, and R. Meinke, "ARSSEM Active Radiation Shield for Space Exploration Missions", Final Report ESTEC Contract N° 4200023087/10/NL/AF : “Superconductive Magnet for Radiation Shielding of Human Spacecraft” (2012) https://arxiv.org/abs/1209.1907 https://www.researchgate.net/publication/265945847_Active_Radiation_Shield_for_Space_Exploration_Missions</ref><ref name="Desiati2022">Paolo Desiati and Elena D'Onghia, "CREW HaT: A Magnetic Shielding System for Space Habitats", arXiv:2209.13624 [physics.space-ph] https://doi.org/10.48550/arXiv.2209.13624</ref>.
</table>
</div>

One immediate issue is that you want to keep very high magnetic fields out of the crew area. Fields in excess of 0.5 T can start to be unpleasant, and it is recommended to keep the field below 0.2 T for continuous exposure.

Strong magnets can suffer from high self-forces acting to tear them apart. Most shielding designs will have multiple magnetic coils, which will exert high forces and torques on each other. This will result in the need for strong structural bracing to keep the magnets intact and in place. Such strong structural designs will impose significant extra mass on the spacecraft.

Consider a magnetic field configuration that can keep radiation out of a certain region of space. There is no trajectory of a charged radiation particle up to a certain energy that will allow those particles to enter the protected region. But the dynamics of charged particles in a magnetic field are the same whether going forward or backward in time. If there is no way for a particle to get in, by this time reversal symmetry any particle which is in that forbidden region can never escape. Consequently any field configuration that is a magnetic trap is also a magnetic shield; any field configuration that is a magnetic shield is also a magnetic trap. A spacecraft with a magnetic shield may end up surrounded by strong radiation belts.

Using a magnet to bend a weaponized electron beam very often ends up with the beam being bent so hard that it dumps most of its energy in the form of synchrotron radiation at gamma ray energies. Because gamma rays are not charged, they will not be deflected by the field. Thus, if you would have been hit by the electron beam you will be hit instead by the gamma rays.

As an example, consider a 1 TeV electron beam. We'll surround our spacecraft with a 10 T magnetic field in a 100 meter radius region just outside our spacecraft. This will deflect the beam by 30 meters, which is enough to make it miss. But we are not saved, because the beam dumps nearly all of its energy into synchrotron radiation with an energy spectrum of gamma rays up to 13 GeV. This radiation will hit us in a strip about 2.4 meters long and not much more than the width of the original beam with essentially all of the energy of the original beam.

==== Magnet drones ====

If a magnetic shield around your spacecraft will just end up killing you from synchrotron radiation, what if you put the magnetic shield on a drone and put that drone between yourself and the enemy trying to shoot you? This lets you use a much weaker magnet because you don't need nearly so high of a deflection angle to make the shot miss. This means that less beam energy will turn into synchrotron radiation that could kill you instead and will spread what synchrotron radiation does occur out over a much wider area when it reaches your spacecraft, hopefully so that most of it just misses your spacecraft.

It does mean that your enemy will start shooting your drones so that they can eventually take a shot at you. But hopefully by that time you've already cleared out their drones and cored their spacecraft with your own synchrocannon or UREB.

==== Plasma magnets ====

Usually, to make a really big magnetic field you would run current through a really big conductive loop. But really big things are heavy and expensive. One way around this could be to use the ambient space plasma to carry the electric current rather than any kind of wire. In order to make this current, you use the same mechanism used in a rotary induction electric motor: you generate a rotating magnetic field and induce a current in the conductive loop (in this case, the space plasma). How do you make a rotating magnetic field? The method proposed<ref name="Slough2006">John Slough, "The Plasma Magnet", NASA Institute for Advanced Concepts Phase II Final Report (2006) https://www.niac.usra.edu/files/studies/final_report/917Slough.pdf</ref> is to use an antenna of four much smaller current loops as electromagnets at 90&degree; to each other, and sinusoidally vary the current through the loops with a respective 90&degree; phase difference between them. When this is done, a circular current is induced in the space plasma and the self-force of a magnetic field on its generating current inflates the current loop to much larger radii. You end up with a current sheet in the plane of the rotating field's rotation circulating about the antenna nearly synchronously with the field rotation. This current then creates a normal dipolar magnetic field as with any current loop. The current sheet and the resulting magnetic field expand until the magnetic field's pressure equals that of the ambient space plasma. This gives an enormous region of magnetic field from a relatively small and lightweight system.

The system proposed in the initial study would use an antenna approximately 10 meters in radius and produce a region of field approximately 30 km in radius at Earth's orbit. The field inside would not be constant - at the edges it would be 50 nT, giving the right pressure to keep out the solar wind at Earth's orbit. A circulating current sheet produces a field that falls off as 1/r with distance, meaning that adjacent to the antenna the field would be about 0.15 mT. Using the methods in the box below, we can estimate that this particular design as part of a spacecraft's shields will not deflect a TeV electron beam by more than a few centimeters, which is usually insufficient to force a miss. However, it could potentially be useful for a drone magnet, where the much larger distance between the magnetic field and the target spacecraft would allow a small deflection angle to give perhaps tens or low hundreds of meters change in beam location near the target at near light second ranges. The relatively low field and gradual bend of the incident beam suggests that synchrotron radiation will not be an issue; however this does not exclude more powerful plasma magnet designs causing significant synchrotron radiation effects.

<table border=1><tr><td>
Calculating the deflection from a spatially varying magnetic field affected by the solar wind is no trivial task. However, we can make an approximation that both should be within about an order of magnitude of the true value and also represent an upper limit to the deflection angle; although it does neglect the influence of the driving rotating magnetic field which could in some cases increase the deflection.

We will take R0 as the radius of the antenna. B0 is the field strength at the antenna. Rm is the distance at which the magnetic influence stops because the magnetic pressure equals the space plasma pressure. Bm is the field strength at Rm. According to Slough<ref name="Slough2006"></ref>, the magnetic field strength B(r) at any distance from the antenna center r between R0 and Rm is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B(r) = B0 R0 / r.
</div>
Therefore, once we know Bm and Rm, B0 is completely determined
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B0 = Bm Rm / R0.
</div>

The force on a particle with charge q passing perpendicularly through a magnetic field B(r) at a speed v is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
F(r) = q v B(r)
</div>
For an electron beam, v is so close to c as to make no difference. If the particle is moving down the y direction and the field is in the z direction, the force will be in the x direction. We assume that the deflection will be small, so we can choose a straight-line path in the y direction offset from the center by an amount x. The plasma bubble radius is Rm, so the particle will enter at ymax = √[Rm2-x2] and leave at ymin = -√[Rm2-x2].

The momentum change the particle experiences from this force is the time integral of the force over its path. Because of reflection symmetry about y = 0, we only need to integrate from ymax to 0 and then multiply by 2
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] dt F(r)
<tr> <td> <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] (dy/v) F(r)
<tr> <td> <td> = <td align="left">2 q B(0) R0 [Integral from 0 to √[Rm2-x2]] dy 1/√[y2 + x2]
<tr> <td> <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/x]
</table>
</div>
This holds for |x| > R0. But the expression for the field is not valid for r < R0 and the integral will give nonsense and diverging results if we evaluate for |x| < R0. In this case, we can put the integral lower bound at r = R0, or y = √[R02-x2]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/(√[R02-x2] + R0)]
</table>
</div>

The deflection angle is now estimated as
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = px / p
</div>
where the total momentum of the particle p can be found from the methods in the first box in the Magnetic Fields section.

The deflection for a shot aimed right at the spacecraft at the position of the spacecraft can be estimated by setting x = 0 and only taking the first half of the integral (so ignore the factor of 2)
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
q B(0) R0 ln[Rm/R0] / (p Rm)
</div>
This is an additional overestimate because it fails to account for how the deflection builds up over distance during the integration, but it should still give something of an order-of-magnitude and upper limit estimate of what kind of deflections might be expected.
</table>

=== Electric fields ===

If electric fields can accelerate charged particles, maybe they can keep the charged particles in a weaponized particle beam away from your spacecraft? To completely block a particle beam, you would need a voltage on your spacecraft equal to the particle energy in eV of the particles in the beam. That is, if a 1 GeV proton beam is coming at you, you'd need an electric potential of a billion volts to stop that beam. This leads to three immediate problems.

The first is that particles of the opposite charge will be attracted to your spacecraft. Because space isn't really empty but rather a rarefied plasma (called the solar wind in a solar system or the interstellar medium out between stars), there will be plenty of particles of the opposite charge around to attract. And they will hit with an energy in eV equal to the voltage on your spacecraft. In our example of defending against a 1 GeV proton beam with a gigavolt field, the electrons from the solar wind will come in and hit with an energy of 1 GeV each. This will generate dangerous radiation.

The second problem is a consequence of the first. Each of those GeV electrons from our example need to get their energy from somewhere, which turns out to be from the electric field we are using as shielding. You will need to make up for this energy loss with power from your electrical system, which gives a constant drain on your power. Some estimates put the electrical effort to maintain a several hundred megavolt shield in space at many tens of gigawatts<ref name="Mechmann2019">Claire Mechmann, "Analysis of Proposed Active Radiation Shielding Design Concept for Spacecraft" (2019) Thesis, College of Engineering and Science of Florida Institute of Technology</ref>.

And the third problem is similar to one for magnetic fields. Your electrodes will experience high self forces and forces between separate electrodes that will try to tear your shield generator apart. Consequently you will need heavy structural reinforcement in order for your shield to work.

There are potential ways to mitigate some of these problems. Interested readers are directed to [[Space_Radiation#Electrostatic_Shielding|the active shielding section in our page on space radiation]] for more of the juicy details; there's no need to repeat the same stuff here.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Elctrostatic_active_shielding.png|200 px|frameless]]
<td>
[[File:Electrostatic_active_shielding_2.png|200 px|frameless]]
<tr><td colspan=2>
Proposed designs for electrostatic shields to protect from positively charged particles<ref name="Tripathi2006">Ram K. Tripathi, John W. Wilson, and Robert C. Youngquist, "Electrostatic Active Radiation Shielding - Revisited", 2006 IEEE Aerospace Conference, Big Sky, MT, USA, 2006, pp. 9 pp.-, doi: 10.1109/AERO.2006.1655760.</ref><ref name="Chowdhury2023">Rajarshi Pal Chowdhury, Luke A. Stegeman, Matthew L. Lund, Dan Fry, Stojan Madzunkov, and Amir A. Bahadori, "Hybrid methods of radiation shielding against deep-space radiation", Life Sciences in Space Research, Volume 38, 2023, Pages 67-78, ISSN 2214-5524, https://doi.org/10.1016/j.lssr.2023.04.004.</ref>.
</table>
</div>

== Damage ==

Particle beams are, first and foremost, devices that produce beams of [[Nuclear_radiation|ionizing radiation]]. The radiation of weaponizable particle beams is extremely penetrating; thus of the [[Particle_Accelerators#Energy_loss_in_matter|interaction mechanisms listed for radiation from particle beams]], we mostly need to worry about bremsstrahlung and nuclear collisions and these produce extensive secondary showers of radiation that increase the penetration of the radiation exposure. Because of this, any target hit by a particle beam weapon will be irradiated, and radiation dose will always be one of the primary damage mechanisms of a particle beam weapon strike. Further, this radiation will be distributed along a track passing deeply through the material of the target, allowing it to affect critical systems through heavy armor, structural material, robust machine components, and passive material such as fuel load, coolant, or cargo. As the radiation deposits its energy into the target material, that material will be heated. If the target is heated sufficiently, it may experience [[Beam-Target_Interactions|thermal or mechanical]] damage from melting, cooking, combustion, vapor explosions, and warping or cracking due to thermal expansion.

The discussion of the radiation penetration into armor also applies to the same radiation penetrating through the target. So the formulas and rules of thumb for estimating the depth of radiation penetration into armor can also be used to estimate how far into a tank or spacecraft a particle beam strike delivers effects.

In addition to the radiation and thermal effects, a particle beam can also produce a sharp electromagnetic pulse that can damage semiconductor electronics.

== 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, Gerrit Bruhaug, and Luke Campbell for writing

Gerrit Bruhaug and Luke Campbell for scientific research, math wrangling and advise

ATriffleMind and Luke W. Campbell for proof reading.

==References==

[[Category:Beams]][[Category:Physics & Engineering‏‎]][[Category:Engineering‏‎]][[Category:Military Technology]][[Category:Warfare‏‎]]

File:Neutralized beam in B field.png

2026-10-27T00:21:08Z

Lwcamp: A profile of the beam charge distribution of a neutralized beam in a magnetic field. Negative charge density is shown in yellow, positive charge density in cyan. The magnetic field is vertical, the electric field in the beam reference frame is shown in magenta including both the polarization field and the Lorentz transformed electric field resulting from the magnetic field.

== Summary ==
A profile of the beam charge distribution of a neutralized beam in a magnetic field. Negative charge density is shown in yellow, positive charge density in cyan. The magnetic field is vertical, the electric field in the beam reference frame is shown in magenta including both the polarization field and the Lorentz transformed electric field resulting from the magnetic field.

Dielectric Breakdown

2026-10-26T14:09:27Z

Lwcamp:

An electron is happily minding its own business floating freely around in the air when suddenly a high power laser beam comes by. How rude! The electric field of the laser beam starts tugging on the electric charge of the electron, accelerating it to faster and faster speeds. Soon, the electron has more than enough energy to knock out other electrons that are stuck on to atoms. Now the electric field starts tugging on these newly freed electrons, too, and they start knocking off more free electrons of their own. Soon, you get a runaway process where the air turns into an electron soup. This electron soup is called a plasma, and the air has just experienced dielectric breakdown.

You need to get a couple things right in order to get dielectric breakdown. First, if the electrons bump into air molecules before they gain enough energy to knock out additional electrons, then they just get scattered in random directions and need to be re-accelerated by the field. So too dense of air or not strong enough laser light means free electrons just bounce off air molecules without ever starting the breakdown cascade.

Second, light is a wave, and it has a frequency. This means the electric field periodically turns around and pulls the electrons the other way. All that energy they’d been getting from the field is now being drained away. If the laser light’s frequency is high enough, the electrons just jiggle up and down slightly without ever getting enough energy to start the breakdown process.

As you might have guessed by now, this process is not good if it happens part-way along your beam when you are trying to shoot a bad guy. Ripping off all those electrons takes a lot of energy, energy that comes from your beam! Not only that, but plasma is hot, low density, and has funky electrical properties that turn it into a lens. The light it doesn’t absorb gets scattered out of your beam and away from your target. And finally, if the wavelength of your beam is longer than somewhere between 5 and 2 μm (65 to 150 THz or 0.3 to 0.7 eV, in sea level density air and depending on how much ionization occurs), the plasma just directly absorbs your beam, using your beam’s energy to heat itself up even more so it can absorb and scatter your beam further. Not nice, plasma! Are you working for my target, perhaps?

But will your design for a laser beam death annihilation ray suffer from breakdown? Here’s a way to get a quick estimate.
<ref name=Radziemski_1989>Leon J. Radziemski and David A. Cremers, Ed., "Laser-Induced Plasmas and Applications", Marcel Dekker, New York, 1989.</ref>
<ref name=Ali_1983>A. W. Ali, “On Laser Air Breakdown, Threshold Power and Laser Generated Channel Length”, NRL Memorandum Report 5187 AD-A133 2.11, September 13, 1983.</ref>
Find the threshold intensity <math>I_{Th}</math> in watts per square centimeter
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
\begin{array}{lll}
I_{Th,CW} & = & 0.31 \left[ \dfrac{1 \, \mbox{m}}{\lambda} \, \left( \dfrac{10^5 \, \mbox{Pa}}{p} \right)^{0.6} \right]^2 \\
I_{Th,pulse} & = & 6.35 \times 10^{-5} \left[ \left( \dfrac{1 \, \mbox{m}}{\lambda} \right)^2 \, \left( \dfrac{10^5 \, \mbox{Pa s}}{p \, t} \right)^{0.6} \right]
\end{array}
</math></div>
where <math>P</math> is the beam power, <math>E</math> is the beam energy, <math>t</math> is the beam duration, <math>p</math> is the air pressure, and <math>\lambda</math> is the beam wavelength;
and take <math>I_{Th}</math> as the larger of these two. If
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><math>
I = \frac{4 \, P}{S^2} = \frac{4 \, E}{t S^2}
</math></div>
is close to <math>I_{Th}</math>, there’s a risk you’ll start a cascade breakdown, If it is much larger than <math>I_{Th}</math>, you will almost certainly cause an air breakdown. If it is much less, you’re safe. Go ahead and blaze away!

A curious effect can occur if you start dielectric breakdown. The plasma spark initially grows by absorbing beam energy and conducting that energy to the surrounding air. But once it is big enough, the plasma can shield its back part from the incoming laser while the front region continues to be heated and grow. The result? You launch a wave of plasma back down the beam toward the laser emitter! For focused beams, this plasma wave can only go so far as the beam is still focused enough to sustain the plasma, so you don't risk blowing yourself up when you shoot.

==Credit==
Author: Luke Campbell

==References==

[[Category:Lasers]][[Category:Physics & Engineering]][[Category:Physics & Math & Engineering]]

Particle Beam Weapons

2026-10-26T04:33:48Z

Lwcamp: /* Separation of neutral beams */

{{PageConstructionNotice}}

''For a general overview on particle accelerator technology, see [[Particle Accelerators]]

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.

::: Z = 1.32×102 (γ2-1)3/4 r / √I<ref name=USPAS2016>Martin Reiser, "Theory and Design of Charged Particle Beams", 2008, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim</ref>

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

::: Z = β γ c r √[m/T]

Where Z is the doubling distance like before, r is the initial beam radius, T is the beam temperature in eV, m is the ion mass and β, c, and γ 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.

== Defenses ==

Sure, it's great to have a great big particle cannon to blast your enemies. But what if you are the one being shot at? In that case, it's best to figure out ways to keep from being blown to kingdom come.

=== Armor ===

The simplest way to defend against the high energy particles of a particle beam is to pile a whole bunch of stuff up and put it in the way of the beam. Particle beams may produce extremely penetrating radiation, but enough stuff will block most of that radiation and may allow you to survive. The details depend on what kind of particle is being thrown at you, and are explained below. However, a general rule of thumb is that you probably want about 10 tons of material per square meter of surface you want to armor in order to get sufficient protection.

==== Electrons ====

At the very high energy of an electron accelerated by a weapons-grade particle beam, the primary interaction with matter will be the emission of bremsstrahlung photons in the gamma-ray range as the electron slams into particles of matter. These high energy photons themselves primarily interact by the pair production mechanism – the photon disappears, but leaves behind a highly energetic positron and electron. These two particles are themselves energetic enough to produce further bremsstrahlung, and so the process continues. You get the formation of an electron-gamma shower (also called an electromagnetic cascade), which only stops after either the particles have penetrated through the target or the initial energy of the electron has been spread through so many shower particles (including all the electrons, positrons, and photons) that no particle has enough individual energy to pair produce or to be likely to bremsstrahlung again before it is stopped by other mechanisms.

To stop electrons, you'll want the atoms in your armor to be as heavy as you can practically get them. High atomic number both encourages bremsstrahlung and pair production, giving a double whammy for stopping that cascade. However, occasionally one of those electrons, positrons, or gammas will hit a nucleus and knock out neutrons. So you probably want a back-up layer of several tens of centimeters of hydrogen rich material such as plastic or water mixed with neutron absorbing materials like boron, lithium, cadmium, or gadolinium. High density polyethylene loaded with 5% boron by weight is a typical go-to neutron blocker.

The actual methods to calculate the penetration of the electromagnetic cascade is given are the box below. However, for some examples:
<ul>
<li> A 10 GeV beam incident on thick iron (or steel) will stop cascading at about 9 cm, will deposit half of its energy in the first 12 cm, and will have dumped 98% of its energy within 35 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 2.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 100 GeV beam incident on thick iron (or steel) will stop cascading at about 13 cm, will deposit half of its energy in the first 16 cm, and will have dumped 98% of its energy within 47 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 3.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick iron (or steel) will stop cascading at about 17 cm, will deposit half of its energy in the first 20 cm, and will have dumped 98% of its energy within 59 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 4.6 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick tungsten will stop cascading at about 3.5 cm, will deposit half of its energy in the first 4 cm, and will have dumped 98% of its energy within 12 cm. 90% of the cascade will be contained within about 0.93 cm and 95% of the cascade will be contained within 1.86 cm of the beam axis. You will get 98% protection with a mass thickness of 2.3 tons/m2 and an additional 1/e reduction for each further 0.13 tons/m2 beyond that.
<li> A 1 TeV beam incident on water will stop cascading at about 290 cm, will deposit half of its energy in the first 340 cm, and will have dumped 98% of its energy within 1030 cm. 90% of the cascade will be contained within about 11 cm and 95% of the cascade will be contained within 22 cm of the beam axis. You will get 98% protection with a mass thickness of 10.3 tons/m2 and an additional 1/e reduction for each further 0.72 tons/m2 beyond that.
</ul>
Note that for sufficiently intense particle beams, 98% protection may not be enough. You might need to throw on several more tons per square meter to be sure.

<table border=1>
<tr><td> How Far it Goes:
(This section follows methods presented in Leroy and Rancoita<ref>C. Leroy and P-G Rancoita, "Physics of cascading shower generation and propagation in matter: principles of high-energy, ultrahigh-energy and compensating calorimetry", Reports on Progress in Physics 63 (2000) 505–606. DOI: 10.1088/0034-4885/63/4/202</ref>.)

When trying to estimate the extent of the irradiated region of the target, it is helpful to know the radiation length X0 in the material. This is the distance that an electron will travel, on average, before emitting a bremsstrahlung photon. Conveniently, it is also the distance a high energy photon will travel through the material before pair producing. The radiation length can be found from
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   = 4 α  
<td nowrap="nowrap"> NA
<td rowspan="2" nowrap="nowrap" align="right">   Z (Z+ζ) re2 ln
<td rowspan="2" nowrap="nowrap" align="right"> [
<td nowrap="nowrap"> 183
<td rowspan="2" nowrap="nowrap" align="right"> ]
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> A
<td style="border-top:solid 1px black;"> Z1/3
</table>
</div>
where α = 1/137 is the fine structure constant, NA = 6.02214076×1023 is Avogadro's number (or one mole), A is the atomic weight of the atoms in the material (usually expressed in grams per mole), Z is the atomic number (nuclear charge) of the atoms, and re = 2.8179403205×10-13 cm is the classical electron radius. ζ is a constant of approximately 1.3. This gives the thickness of material in terms of the areal density (mass of material per unit area perpendicular to the beam) – if you use A in g/mol and re in cm the result will be in cm2/g and X0 will be in g/cm2. If your target is made up of more than one kind of atom, you can get the total radiation length by finding the radiation length of each atom of type i, X0i and using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   =  
<td rowspan="2" nowrap="nowrap" align="right"> Σi  
<td nowrap="nowrap"> fi
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> X0i
</table>
</div>
where fi is the fraction by weight of the atom type i. Once you have the radiation length in terms of areal density, you can easily convert it to distance by dividing by the density ρ. If X0i is in g/cm2 and density is in g/cm3, then X0i/ρ will be in cm.

We will also need to know the critical energy εc. This is the energy where an electron is losing energy as rapidly to ionization as it is to radiation. That is, it is the energy at which the cascade stops multiplying, even though the individual particles already made can continue traveling for some distance. The critical energy can be estimated using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> εc = B  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> Z X0
<td rowspan="2" nowrap="nowrap" align="right"> )
<td nowrap="nowrap"> h
<tr>
<td style="border-top:solid 1px black;"> A
<td>  
</table>
</div>
where B = 2.66 MeV and h = 1.11.

We can now describe the distribution of radiation in the target material. The depth tmax where the cascade stops multiplying (in g/cm2) for an incident electron of energy E (in MeV) is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tmax = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> - 1
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
Similarly, the depth where half the energy of the beam has been deposited (in g/cm2) is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tcg = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> + 0.37
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
As usual, you can convert to a depth in cm by dividing by the material density. 98% of the beam energy will be deposited within a distance of &cong; 3 tcg. Beyond this, the beam energy will continue to fall off with thickness t approximately as an exponential exp[-t/λ], where the characteristic attenuation length λ &cong; 2 X0.

The width of a cascade, where 90% of the energy is contained, is approximately given by
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> RM =  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> EM
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
where EM &cong; 21.2 MeV. 95% of the radiation will be contained within a distance of 2 RM from the beam axis.
</table>

==== Protons and Ions ====

<table align=right><tr><td>
[[File:Regolith_Shielding.png|350 px|frameless]]
<tr><td width=350>
Relative effect of particle radiation (compared to no shielding) from galactic cosmic rays behind different thicknesses of water, aluminum, and lunar regolith<ref name="Slaba2022">Tony C. Slaba, "Radiation Shielding for Lunar Missions: Regolith Considerations", LSIC Crosstalk 7/18/2022 https://lsic.jhuapl.edu/uploadedDocs/focus-files/1604-E&C%20+%20EE%20Monthly%20Meeting%20-%202022%2007%20July_Presentation%20-%20NASA%20Slaba.pdf</ref>.
</table>
Protons and ions are grouped together because ions are made of protons and neutrons and when you accelerate these things for a weapon it is the protons and neutrons that are going to be doing the heavy lifting for blasting your target. This is because protons and neutrons are both about 2000 times more massive than electrons so any electrons along for the ride will contribute negligibly to the energy. Protons and neutrons together are called nuclear particles, and both behave quite similarly in nuclear collisions.

Nuclear particles are themselves made of quarks and can interact via the strong nuclear force. At the energies appropriate for weapons use, essentially all of the way protons or ions are blocked is through [[Particle_Accelerators#Nuclear_collisions|direct collisions with the atomic nuclei]] of the armor material. One or more quarks in the projectile will slam into quarks in the target nucleus and stretch them away from the rest of the quarks in the nuclear particles. Physics tells us that you can't have isolated quarks, so the "string" attaching the quark to its neighbors stretches and stretches until it snaps. Snapping releases so much energy that more quarks are made that clump together into all sorts of other nuclear particles, both exotic and familiar. This process is called a hadronic jet. The only particles produced in these jets long-lived enough to travel any noticeable distance through the armor material will be protons, neutrons, the charged pi mesons (or pions), and some of the K mesons (or kaons). The particles too short lived to go far enough to matter will decay into other particles, and these particles are often very high energy gamma rays. Much like with electrons, the long-lived secondary particles can be sufficiently energetic to smash into other nuclei and cause more hadronic jets (or, if they are low enough in energy, just knock more nuclear particles out of target nuclei) and this produces a similar shower or cascade. But in addition, the armor nuclei that were hit will be very hot after the impact from all of the energy given to them from the incident particles. These nuclei will cool off by evaporation – but instead of evaporating away molecules, they evaporate off nuclear particles from the nucleus. These evaporated particles are mostly neutrons, so any armor against proton or ion beam strikes will produce quite a bit of neutron radiation.

Eventually, the original particle's energy will be distributed among so many other particles that the charged particles will come to a stop by making ion tracks rather than from colliding with more nuclei, and the cascade halts. But you still have all the neutrons to worry about. And the gamma rays as well, which are high enough energy to kick off an electromagnetic cascade in the same way that electrons do.

Unlike electrons, which have reasonable approximations of their cascade behavior from relatively simple equations, the cascades of protons and ions usually require sophisticated simulations or experiments to get the resulting distribution of radiation. However, we can get a reasonable idea of their general behavior by comparing them to [[Space_Radiation#Shielding_Against_Cosmic_Rays|cosmic ray particles interacting with shielding]]. Cosmic rays are energetic protons and ions, after all. From this we see that a relatively small amount of armor – in the range of 1 to 3 tons per square meter – can actually make the radiation dose worse. The incident particles will knock off so many secondary particles in their shower that the dose is multiplied from so many tracks passing through the target. Hydrogen rich materials such as water and plastics work the best against proton and ion beam particles; heavy elements are the worst and produce the most secondaries. You could probably obtain adequate radiation protection from most incident beams with about 10 tons per square meter of hydrogen rich material as armor. However, you will want to add neutron absorbing elements to your hydrogen rich armor, such as lithium, boron, cadmium, or gadolinium. Further, in order to deal with the gamma rays you will want a layer of at least several centimeters of heavy elements, such as tungsten or lead, behind the hydrogen rich particle armor.

==== Structural Considerations ====

When your armor material absorbs a radiation beam, the energy of that radiation turns into heat inside the armor where the radiation was absorbed. If the beam pulse has enough energy as was sufficiently tightly focused, this heat can damage your armor. Your armor may melt in places, or thermal expansion could cause it to crack or warp. Perhaps the worst thing it could do is vaporize or flash into a plasma, because the pressure from the vapor or plasma can be sufficient to plastically deform the armor. This leaves a hole along the beam path where the expanding gas or plasma blasted out a long, deep crater. This means a beam that fires rapid shots spaced maybe milliseconds apart can have subsequent shots go down the same hole so your armor will not protect against them. Each shot will dig out a deeper hole until either your armor is penetrated or the attacking beam runs out of shots in its pulse train.

A consequence of this is that when designing your armor you may wish to use refractory materials with a high structural strength (like tungsten), or that is otherwise resistant to being blasted apart by its own evaporated remains.

=== Magnetic fields ===

[[Particle_Accelerators#Magnetic_fields|A charged particle crossing a magnetic field will bend so as to spiral around the field lines]]. If you try to shoot your particle beam through a magnetic field, it will bend away from the direction you shot it. This can make it hard to hit your target. Unfortunately for you, there are magnetic fields everywhere – around planets, and even trapped in the solar wind as it crosses the solar system.

As an example, the Earth has a magnetic field strength of about 5×10-5 tesla near its surface, and falling off farther away. If you are shooting a 1 TeV electron beam past the Earth, your beam is likely to bend by several milliradians (with the exact deflection depending on how close you are skimming the Earth). If you are in a medium Earth orbit and shooting at another target also in a medium Earth orbit on a trajectory that goes near the Earth, then without correcting for the magnetic field your beam will likely be off-target by tens to hundreds of kilometers.

For another example, the interplanetary magnetic field is about 10-9 tesla. If you shoot that same 1 TeV electron beam at a target 0.1 light seconds away, you are likely to be off by a couple hundred meters.

<table border=1>
<tr><td>
The radius of curvature of the bend of a particle in a magnetic field is called the gyroradius rc. For a particle with a charge q, mass m, and kinetic energy E, find the relativistic momentum p
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
p = √[E2 - m2 c4] / c
</div>
To convert from energy units of eV to joules, multiply by 1.60×10-19. An electron has a mass of 9.11×10-31 kg, a proton of 1.67×10-27 kg, and an ion of 1.66×10-27×A kg, where A is the atomic weight of the ion. The speed of light c = 3.00×108 m/s.

In a magnetic field of strength B with an angle θ between the field direction and the direction of motion
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rc = |p sin(θ) / (q B)|
</div>
For electrons, q = -e, where the fundamental charge e = 1.60×10-19 coulomb. For a proton, q = +e. For an ion, multiply e by the charge state of the ion.

If the field acts over a distance d that is much smaller than the gyroradius, the deflection angle φ is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = d / rc
</div>
The deflection x at a target at a range R for small φ (φ &Lt; 1) will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
x = φ R.
</div>

Note that d will always be less than or equal to x. Also note that if d is not smaller than twice the gyroradius the beam will never reach the target because it will completely spiral around the field lines before it reaches the target.
</table>

So, it is clear that as the defender it will be helpful to hang out in areas with high magnetic fields, and keep some distance between you and the attacker. Meanwhile, the attacker will benefit from being closer where small deflections in the beam trajectory don't translate into large excursions from where the beam is aimed (and also because there will be less space filled with fields between the two of you). But the attacker can also try to compensate for known fields. If you happen to know that the magnetic fields are between you and your target, you can adjust your fire so that the deflection of the field bends your beam onto the target. This is not a simple task. It is easy enough to measure the field strength and direction where you are, but predicting how it changes over the path of your shot does not have an obvious solution. You can also try to gain intelligence on likely battle sites before you ever fight on them – for example, getting good data on the structure of the fields of a planet using spies and technical intelligence during peacetime (heck, on Earth this information is published for free in academic journals and by government agencies such as NOAA). Planetary magnetospheres are also affected by the solar wind, so different solar weather will change the fields you have to shoot through. But space weather forecasts and predictive simulations of the solar wind interacting with a planetary magnetosphere can give you a good idea of what the field structure is like. But even with all that, there are changes in the fields that happen on a scale of around a second ... anyone who has ever seen the auroras dancing across the sky in the polar night can see this. Compensating for these changes will be a significant challenge.

What would really help is being able to see the path of your own beam. This would allow you to see how far off you were from your target and in what direction, and then adjust fire to walk your shots onto the target. Unfortunately, particle beams in space are invisible. There might be ways to track them, but if so we don't know what they are.

==== Synchrotron radiation ====

When charged particle trajectories are bent, they emit [[Particle_Accelerators#Synchrotron_radiation|synchrotron radiation]]. This can cause the particle beam to lose energy, but if the bend is too tight it can also generate high energy gamma rays that ignore magnetic fields and directly strike the target. Synchrotron radiation can generally be ignored for proton or ion beams, but can be significant for electron beams.

<table border=1><tr><td>
For a beam particle with energy E and mass m, the synchrotron radiation is concentrated in a forward-directed cone with an opening angle of approximately 1/γ, where γ = 1 + E/(m c2). (Interestingly, the radiation intensity peaks an at angle of 1/(2 γ) and vanishes directly on-axis.) Because the beam is bending, you will get emission along the plane along which the beam is being bent. The energy of the photons produced during synchrotron emission is approximately<ref name="Jackson_Electrodynamics">J. D. Jackson, "Classical Electrodynamics: Second Edition", Wiley (1962)</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Eγ &cong; 3 γ3 c &hbar; / rc
</div>
where &hbar; = 1.05×10-34 J s (if you want the energy in joules) or &hbar; = 6.58×10-16 eV s (if you want the energy in eV). The photon energy will not exceed the energy of the incident particles, if you get a photon energy higher than the energy of the incident particles it just means that the synchrotron radiation spectrum extends up to the incident particle energy.

The fraction of the energy (or power) of the beam turned into synchrotron radiation will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = [1/(4 π ε0)] [4 π / 3] [q² / rc] β2 γ4 [φ / (2 π)] / E
</div>
where β = √[1 - 1/γ2] is the particle's speed measured as a fraction of the speed of light, and 1/(4 π ε0) = 8.99×109 is the Coulomb constant. For highly relativistic electrons (like all practical electron beam weapons emit), β &cong; 1 and we can use
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = 8.85×10-2 [E (GeV)3 [φ / (2 π)] / (rc (m))]
</div>
This fraction cannot be more than 1. If f gets close to one, it means that our original assumption that the beam particles mostly keep their original speed no longer holds, and the beam will lose most or almost all of its energy. Instead, when the beam is being deflected it will only make it through an angle of φ/f and will irradiate a region of length x/f starting at where the beam would have been incident if not for the magnets and extending in the direction the beam was deflected.
</table>

For example, a 1 TeV electron beam will have γ &cong; 2,000,000. So if it's bent by a magnetic field the synchtrotron radiation will spread out in a cone of about 5×10-7 radians. If this beam was shooting at a target 0.1 light seconds away from where it was bent, the radiation would arrive at the target over a region with a radius of about 15 meters with a maximum intensity at about 7.5 meters and no radiation directly along the path the beam was pointing while it was being bent. If the beam was bent by the interplanetary magnetic field from our earlier example, it would produce light with photon energies of around 1.3 eV (near infrared) and the lost energy would be a few parts in a hundred trillion. If it was instead deflected by the Earth's magnetic field you might expect a peak photon energy of tens of keV – in the hard x-ray part of the spectrum – and the fraction of energy in the synchrotron radiation would be somewhere in the low parts in 100,000 or less.

==== Separation of neutral beams ====

Magnetic fields only affect charged objects. So one way to overcome magnetic defenses or environments is to use beams that have no net charge. For example, a proton beam could be neutralized with a co-propagating electron beam with an equal number of protons and electrons. Or, potentially, you could fire an ion beam and then neutralize the ions after acceleration to get a beam of neutral atoms.

But, the magnetic fields are still tugging at the charged particles making up the beam as they go past. This will pull the electrons one way and the protons or ions the other way. As the particles are pulled in opposite directions you get a net positive charge on one side of the beam and a net negative charge on the other side (called polarization). This sets up an electric field inside the beam that acts to pull the particles back together. If the particles are going fast enough and the magnetic field is strong enough, the magnetic force can still separate the charges and will split the beam in two, one with the positive charges and one with the negative charges. But if the particles are not that fast or the magnetic field is not that strong this polarization of the beam will keep it together and let it fly right through the magnetic field.

<table border=1><tr><td>
Let's figure out how to analyze this problem. For definiteness, we will have a beam with an electric current of the positive species of I and an electric current of electrons of -I, all moving in the z direction. The beam radius will be R. The particle speed will be v with the usual relativistic parameters β = v/c and γ = √[1/(1-β2)]. If you know the energy per particle E of the positive species and the mass m of the positive species, γ = 1 + E/(m c2) and then β = √[1 - 1/γ2].

Let's assume that the beam is going through a magnetic field of strength B at an angle θ to the field direction. The quantity B sin(θ) is the component of the magnetic field perpendicular to the direction of the beam.

The charge density ρ of the positive component of the beam will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρL = I / (π v R2).
</div>

It is easiest to analyze the problem in the rest frame of the particles in the beam. Now the particles are at rest. Particles at rest are not affected by a magnetic field, but the change of reference frame Lorentz transforms the initial magnetic field into a combination of electric and magnetic fields in the rest frame. We can ignore the magnetic field, but the electric field is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔A = γ v B sin(θ)
</div>
in a direction perpendicular to both v and B. For our purposes, assume the electric field points in the x direction.

At the same time, relativistic length contraction increases the charge density in the rest frame
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρ = γ ρL.
</div>

Let r be the vector from the beam axis to the point inside the beam. It will be useful to separate this into its Cartesian coordinates. If x and y are the unit vectors in the x and y directions, respectively, then
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
r = x x + y y
</div>

If we consider only the positive component of the beam, from [https://en.wikipedia.org/wiki/Gauss%27s_law Gauss's law] it will have an electric field inside of the beam at a distance r from the center axis of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔+ = ρ r / (2 ε0) = (ρ/(2 ε0)) (x x + y y)
</div>
where ε0 = 8.85×10-12 is the vacuum permittivity. Now let's displace the electrons in the beam by a distance δ in the x direction. By itself, this electron beam will have an electric field of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔- = -(ρ/(2 ε0)) ((x-δ) x + y y)
</div>
By the principle of superposition, the electric field inside the beam with both components is the sum of the electric fields of the separate components including the external field
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔B = [(ρ/(2 ε0)) δ + γ v B sin(θ)]x
</div>
Because the charges are free to move, they will do so as long as a field is applied to them. Thus, the beam becomes stable once the charges separate enough that the field internal to the beam is zero. Therefore
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
δ = -2 ε0 γ v B sin(θ) / ρ = -2 π ε0 v2 R2 B sin(θ) / I
</div>
If |δ| > 2 |R|, the beam will fully separate and each component will go its own way. If δ is small compared to R, the beam's polarization suffices to hold it together in the overlap region. However, where the positive and negative charges do not overlap the electric fields no longer cancel, and the external electric field carries away the outer polarized part of the beam. This gives us a rough threshold for which a magnetic field will quickly separate a neutral beam into its charged components and bend them away from their target
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr><td> B sin(θ) &Lt; I / (π ε0 v2 R) <td>       <td> Beam gradually erodes.
<tr><td> B sin(θ) > I / (π ε0 v2 R) <td> <td> Beam promptly separates.
</table>
</div>
For a gradually eroding beam, you can figure that if the magnetic deflection of the positive component is not much larger than the beam radius, most of the beam will still hit its target.
<tr><td>
For a beam of neutral atoms, let a be the radius of the atom and let Eb be the binding energy of the least tightly bound electron to the atom. An external electric field in excess of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
|𝓔A| > Eb / (e a)
</div>
will promptly ionize the atom, ripping away its electron. Here, e = 1.60×10-19 C is the magnitude of the electric charge of the electron. For fields close to but a bit weaker than this, the electron can still rapidly pop off due to quantum tunneling<ref>Ph. Durand and I. Paidarová, “Ionization of the hydrogen atom: from weak to strong static electric fields,” The European Physical Journal D, Vol. 26, pg. 253-259, 2003.</ref>. In general, expect that for fields greater in magnitude than (very approximately) Eb / (10 e a) the electron will be quickly removed, while for fields weaker than this the electrons will stay on their atoms.

It is useful to note that Eb/e is just the binding energy in units of eV. The most weakly bound electrons are usually bound to their atom with about 10 eV of energy. Atoms are typically about 10-10 m in radius. Thus, you can expect fields of around 1010 to 1011 V/m to directly ionize atoms. If we call this threshold value 𝓔T, then a beam will be ionized when passing through a field with strength greater than
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B sin(θ) > 𝓔T / (γ v)
</div>
</table>

Example: Let's consider a 1 GeV neutralized proton beam from a synchrotron, with a beam current of 200 kA in each bunch (incidentally, if a bunch is 5 cm long this means that each bunch packs 38 kJ worth of energy). The speed of the bunch is 87% the speed of light, or 262 million m/s, with a γ of 2.07. Consider when the beam's focus takes it to a radius of 5 cm (or 0.05 m, for our calculation). Plugging this in, we find that the protons in the bunch will separate from the electrons in a field with a transverse component greater than about 2.1 T, and the bunch will pass through the field with only gradual losses if the transverse component is much smaller than this.

Example: We'll take the same 1 GeV beam, but this time recombine the electrons with protons to make hydrogen atoms. An electron is bound to a hydrogen atom with an energy of 13.6 eV (the Ryberg constant) and the radius of the electron orbital in a hydrogen atom is 0.529×10-10 m (the Bohr radius). This gives a threshold electric field for prompt ionization of hydrogen of 2.57×1011 V/m. We will get this with a transverse component of the magnetic field of 474 T. Above this field the atoms promptly ionize and electrons are stripped away from the protons. For fields higher than roughly 1/10 of this – so about 40 to 50 T – you may still get significant ionization by quantum tunneling; lower than this your losses from the beam due to field-induced ionization are likely negligible.

==== Magnetic shields ====

What happens when we can't rely on magnetic fields in the environment to protect us? One idea is to produce powerful artificial magnetic fields around our spacecraft to deflect incoming particle beams. This is very similar in concept to using active magnetic shielding to protect against space radiation – [[Space_Radiation#Magnetic_Shielding|check out our article on space radiation for more]].
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr>
<td width=100>
[[File:Unconfined_FRC_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:racetrack_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:Magnetic_shielding_Halback_Array.png|200 px|frameless]]
<tr><td colspan=3>
Some potential configurations for magnetic shields. The currents generating the fields are in green, the magnetic fields themselves are shown in magenta<ref name="Battiston2012">R. Battiston, W.J. Burger, V. Calvelli, R. Musenich, V. Choutko, V.I. Datskov, A. Della Torre, F. Venditti,
C. Gargiulo, G. Laurenti, S. Lucidi, S. Harrison, and R. Meinke, "ARSSEM Active Radiation Shield for Space Exploration Missions", Final Report ESTEC Contract N° 4200023087/10/NL/AF : “Superconductive Magnet for Radiation Shielding of Human Spacecraft” (2012) https://arxiv.org/abs/1209.1907 https://www.researchgate.net/publication/265945847_Active_Radiation_Shield_for_Space_Exploration_Missions</ref><ref name="Desiati2022">Paolo Desiati and Elena D'Onghia, "CREW HaT: A Magnetic Shielding System for Space Habitats", arXiv:2209.13624 [physics.space-ph] https://doi.org/10.48550/arXiv.2209.13624</ref>.
</table>
</div>

One immediate issue is that you want to keep very high magnetic fields out of the crew area. Fields in excess of 0.5 T can start to be unpleasant, and it is recommended to keep the field below 0.2 T for continuous exposure.

Strong magnets can suffer from high self-forces acting to tear them apart. Most shielding designs will have multiple magnetic coils, which will exert high forces and torques on each other. This will result in the need for strong structural bracing to keep the magnets intact and in place. Such strong structural designs will impose significant extra mass on the spacecraft.

Consider a magnetic field configuration that can keep radiation out of a certain region of space. There is no trajectory of a charged radiation particle up to a certain energy that will allow those particles to enter the protected region. But the dynamics of charged particles in a magnetic field are the same whether going forward or backward in time. If there is no way for a particle to get in, by this time reversal symmetry any particle which is in that forbidden region can never escape. Consequently any field configuration that is a magnetic trap is also a magnetic shield; any field configuration that is a magnetic shield is also a magnetic trap. A spacecraft with a magnetic shield may end up surrounded by strong radiation belts.

Using a magnet to bend a weaponized electron beam very often ends up with the beam being bent so hard that it dumps most of its energy in the form of synchrotron radiation at gamma ray energies. Because gamma rays are not charged, they will not be deflected by the field. Thus, if you would have been hit by the electron beam you will be hit instead by the gamma rays.

As an example, consider a 1 TeV electron beam. We'll surround our spacecraft with a 10 T magnetic field in a 100 meter radius region just outside our spacecraft. This will deflect the beam by 30 meters, which is enough to make it miss. But we are not saved, because the beam dumps nearly all of its energy into synchrotron radiation with an energy spectrum of gamma rays up to 13 GeV. This radiation will hit us in a strip about 2.4 meters long and not much more than the width of the original beam with essentially all of the energy of the original beam.

==== Magnet drones ====

If a magnetic shield around your spacecraft will just end up killing you from synchrotron radiation, what if you put the magnetic shield on a drone and put that drone between yourself and the enemy trying to shoot you? This lets you use a much weaker magnet because you don't need nearly so high of a deflection angle to make the shot miss. This means that less beam energy will turn into synchrotron radiation that could kill you instead and will spread what synchrotron radiation does occur out over a much wider area when it reaches your spacecraft, hopefully so that most of it just misses your spacecraft.

It does mean that your enemy will start shooting your drones so that they can eventually take a shot at you. But hopefully by that time you've already cleared out their drones and cored their spacecraft with your own synchrocannon or UREB.

==== Plasma magnets ====

Usually, to make a really big magnetic field you would run current through a really big conductive loop. But really big things are heavy and expensive. One way around this could be to use the ambient space plasma to carry the electric current rather than any kind of wire. In order to make this current, you use the same mechanism used in a rotary induction electric motor: you generate a rotating magnetic field and induce a current in the conductive loop (in this case, the space plasma). How do you make a rotating magnetic field? The method proposed<ref name="Slough2006">John Slough, "The Plasma Magnet", NASA Institute for Advanced Concepts Phase II Final Report (2006) https://www.niac.usra.edu/files/studies/final_report/917Slough.pdf</ref> is to use an antenna of four much smaller current loops as electromagnets at 90&degree; to each other, and sinusoidally vary the current through the loops with a respective 90&degree; phase difference between them. When this is done, a circular current is induced in the space plasma and the self-force of a magnetic field on its generating current inflates the current loop to much larger radii. You end up with a current sheet in the plane of the rotating field's rotation circulating about the antenna nearly synchronously with the field rotation. This current then creates a normal dipolar magnetic field as with any current loop. The current sheet and the resulting magnetic field expand until the magnetic field's pressure equals that of the ambient space plasma. This gives an enormous region of magnetic field from a relatively small and lightweight system.

The system proposed in the initial study would use an antenna approximately 10 meters in radius and produce a region of field approximately 30 km in radius at Earth's orbit. The field inside would not be constant - at the edges it would be 50 nT, giving the right pressure to keep out the solar wind at Earth's orbit. A circulating current sheet produces a field that falls off as 1/r with distance, meaning that adjacent to the antenna the field would be about 0.15 mT. Using the methods in the box below, we can estimate that this particular design as part of a spacecraft's shields will not deflect a TeV electron beam by more than a few centimeters, which is usually insufficient to force a miss. However, it could potentially be useful for a drone magnet, where the much larger distance between the magnetic field and the target spacecraft would allow a small deflection angle to give perhaps tens or low hundreds of meters change in beam location near the target at near light second ranges. The relatively low field and gradual bend of the incident beam suggests that synchrotron radiation will not be an issue; however this does not exclude more powerful plasma magnet designs causing significant synchrotron radiation effects.

<table border=1><tr><td>
Calculating the deflection from a spatially varying magnetic field affected by the solar wind is no trivial task. However, we can make an approximation that both should be within about an order of magnitude of the true value and also represent an upper limit to the deflection angle; although it does neglect the influence of the driving rotating magnetic field which could in some cases increase the deflection.

We will take R0 as the radius of the antenna. B0 is the field strength at the antenna. Rm is the distance at which the magnetic influence stops because the magnetic pressure equals the space plasma pressure. Bm is the field strength at Rm. According to Slough<ref name="Slough2006"></ref>, the magnetic field strength B(r) at any distance from the antenna center r between R0 and Rm is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B(r) = B0 R0 / r.
</div>
Therefore, once we know Bm and Rm, B0 is completely determined
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B0 = Bm Rm / R0.
</div>

The force on a particle with charge q passing perpendicularly through a magnetic field B(r) at a speed v is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
F(r) = q v B(r)
</div>
For an electron beam, v is so close to c as to make no difference. If the particle is moving down the y direction and the field is in the z direction, the force will be in the x direction. We assume that the deflection will be small, so we can choose a straight-line path in the y direction offset from the center by an amount x. The plasma bubble radius is Rm, so the particle will enter at ymax = √[Rm2-x2] and leave at ymin = -√[Rm2-x2].

The momentum change the particle experiences from this force is the time integral of the force over its path. Because of reflection symmetry about y = 0, we only need to integrate from ymax to 0 and then multiply by 2
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] dt F(r)
<tr> <td> <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] (dy/v) F(r)
<tr> <td> <td> = <td align="left">2 q B(0) R0 [Integral from 0 to √[Rm2-x2]] dy 1/√[y2 + x2]
<tr> <td> <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/x]
</table>
</div>
This holds for |x| > R0. But the expression for the field is not valid for r < R0 and the integral will give nonsense and diverging results if we evaluate for |x| < R0. In this case, we can put the integral lower bound at r = R0, or y = √[R02-x2]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/(√[R02-x2] + R0)]
</table>
</div>

The deflection angle is now estimated as
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = px / p
</div>
where the total momentum of the particle p can be found from the methods in the first box in the Magnetic Fields section.

The deflection for a shot aimed right at the spacecraft at the position of the spacecraft can be estimated by setting x = 0 and only taking the first half of the integral (so ignore the factor of 2)
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
q B(0) R0 ln[Rm/R0] / (p Rm)
</div>
This is an additional overestimate because it fails to account for how the deflection builds up over distance during the integration, but it should still give something of an order-of-magnitude and upper limit estimate of what kind of deflections might be expected.
</table>

=== Electric fields ===

If electric fields can accelerate charged particles, maybe they can keep the charged particles in a weaponized particle beam away from your spacecraft? To completely block a particle beam, you would need a voltage on your spacecraft equal to the particle energy in eV of the particles in the beam. That is, if a 1 GeV proton beam is coming at you, you'd need an electric potential of a billion volts to stop that beam. This leads to three immediate problems.

The first is that particles of the opposite charge will be attracted to your spacecraft. Because space isn't really empty but rather a rarefied plasma (called the solar wind in a solar system or the interstellar medium out between stars), there will be plenty of particles of the opposite charge around to attract. And they will hit with an energy in eV equal to the voltage on your spacecraft. In our example of defending against a 1 GeV proton beam with a gigavolt field, the electrons from the solar wind will come in and hit with an energy of 1 GeV each. This will generate dangerous radiation.

The second problem is a consequence of the first. Each of those GeV electrons from our example need to get their energy from somewhere, which turns out to be from the electric field we are using as shielding. You will need to make up for this energy loss with power from your electrical system, which gives a constant drain on your power. Some estimates put the electrical effort to maintain a several hundred megavolt shield in space at many tens of gigawatts<ref name="Mechmann2019">Claire Mechmann, "Analysis of Proposed Active Radiation Shielding Design Concept for Spacecraft" (2019) Thesis, College of Engineering and Science of Florida Institute of Technology</ref>.

And the third problem is similar to one for magnetic fields. Your electrodes will experience high self forces and forces between separate electrodes that will try to tear your shield generator apart. Consequently you will need heavy structural reinforcement in order for your shield to work.

There are potential ways to mitigate some of these problems. Interested readers are directed to [[Space_Radiation#Electrostatic_Shielding|the active shielding section in our page on space radiation]] for more of the juicy details; there's no need to repeat the same stuff here.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Elctrostatic_active_shielding.png|200 px|frameless]]
<td>
[[File:Electrostatic_active_shielding_2.png|200 px|frameless]]
<tr><td colspan=2>
Proposed designs for electrostatic shields to protect from positively charged particles<ref name="Tripathi2006">Ram K. Tripathi, John W. Wilson, and Robert C. Youngquist, "Electrostatic Active Radiation Shielding - Revisited", 2006 IEEE Aerospace Conference, Big Sky, MT, USA, 2006, pp. 9 pp.-, doi: 10.1109/AERO.2006.1655760.</ref><ref name="Chowdhury2023">Rajarshi Pal Chowdhury, Luke A. Stegeman, Matthew L. Lund, Dan Fry, Stojan Madzunkov, and Amir A. Bahadori, "Hybrid methods of radiation shielding against deep-space radiation", Life Sciences in Space Research, Volume 38, 2023, Pages 67-78, ISSN 2214-5524, https://doi.org/10.1016/j.lssr.2023.04.004.</ref>.
</table>
</div>

== Damage ==

Particle beams are, first and foremost, devices that produce beams of [[Nuclear_radiation|ionizing radiation]]. The radiation of weaponizable particle beams is extremely penetrating; thus of the [[Particle_Accelerators#Energy_loss_in_matter|interaction mechanisms listed for radiation from particle beams]], we mostly need to worry about bremsstrahlung and nuclear collisions and these produce extensive secondary showers of radiation that increase the penetration of the radiation exposure. Because of this, any target hit by a particle beam weapon will be irradiated, and radiation dose will always be one of the primary damage mechanisms of a particle beam weapon strike. Further, this radiation will be distributed along a track passing deeply through the material of the target, allowing it to affect critical systems through heavy armor, structural material, robust machine components, and passive material such as fuel load, coolant, or cargo. As the radiation deposits its energy into the target material, that material will be heated. If the target is heated sufficiently, it may experience [[Beam-Target_Interactions|thermal or mechanical]] damage from melting, cooking, combustion, vapor explosions, and warping or cracking due to thermal expansion.

The discussion of the radiation penetration into armor also applies to the same radiation penetrating through the target. So the formulas and rules of thumb for estimating the depth of radiation penetration into armor can also be used to estimate how far into a tank or spacecraft a particle beam strike delivers effects.

In addition to the radiation and thermal effects, a particle beam can also produce a sharp electromagnetic pulse that can damage semiconductor electronics.

== 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, Gerrit Bruhaug, and Luke Campbell for writing

Gerrit Bruhaug and Luke Campbell for scientific research, math wrangling and advise

ATriffleMind and Luke W. Campbell for proof reading.

==References==

[[Category:Beams]][[Category:Physics & Engineering‏‎]][[Category:Engineering‏‎]][[Category:Military Technology]][[Category:Warfare‏‎]]

Particle Beam Weapons

2026-10-26T04:31:47Z

Lwcamp: /* Separation of neutral beams */

{{PageConstructionNotice}}

''For a general overview on particle accelerator technology, see [[Particle Accelerators]]

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.

::: Z = 1.32×102 (γ2-1)3/4 r / √I<ref name=USPAS2016>Martin Reiser, "Theory and Design of Charged Particle Beams", 2008, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim</ref>

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

::: Z = β γ c r √[m/T]

Where Z is the doubling distance like before, r is the initial beam radius, T is the beam temperature in eV, m is the ion mass and β, c, and γ 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.

== Defenses ==

Sure, it's great to have a great big particle cannon to blast your enemies. But what if you are the one being shot at? In that case, it's best to figure out ways to keep from being blown to kingdom come.

=== Armor ===

The simplest way to defend against the high energy particles of a particle beam is to pile a whole bunch of stuff up and put it in the way of the beam. Particle beams may produce extremely penetrating radiation, but enough stuff will block most of that radiation and may allow you to survive. The details depend on what kind of particle is being thrown at you, and are explained below. However, a general rule of thumb is that you probably want about 10 tons of material per square meter of surface you want to armor in order to get sufficient protection.

==== Electrons ====

At the very high energy of an electron accelerated by a weapons-grade particle beam, the primary interaction with matter will be the emission of bremsstrahlung photons in the gamma-ray range as the electron slams into particles of matter. These high energy photons themselves primarily interact by the pair production mechanism – the photon disappears, but leaves behind a highly energetic positron and electron. These two particles are themselves energetic enough to produce further bremsstrahlung, and so the process continues. You get the formation of an electron-gamma shower (also called an electromagnetic cascade), which only stops after either the particles have penetrated through the target or the initial energy of the electron has been spread through so many shower particles (including all the electrons, positrons, and photons) that no particle has enough individual energy to pair produce or to be likely to bremsstrahlung again before it is stopped by other mechanisms.

To stop electrons, you'll want the atoms in your armor to be as heavy as you can practically get them. High atomic number both encourages bremsstrahlung and pair production, giving a double whammy for stopping that cascade. However, occasionally one of those electrons, positrons, or gammas will hit a nucleus and knock out neutrons. So you probably want a back-up layer of several tens of centimeters of hydrogen rich material such as plastic or water mixed with neutron absorbing materials like boron, lithium, cadmium, or gadolinium. High density polyethylene loaded with 5% boron by weight is a typical go-to neutron blocker.

The actual methods to calculate the penetration of the electromagnetic cascade is given are the box below. However, for some examples:
<ul>
<li> A 10 GeV beam incident on thick iron (or steel) will stop cascading at about 9 cm, will deposit half of its energy in the first 12 cm, and will have dumped 98% of its energy within 35 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 2.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 100 GeV beam incident on thick iron (or steel) will stop cascading at about 13 cm, will deposit half of its energy in the first 16 cm, and will have dumped 98% of its energy within 47 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 3.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick iron (or steel) will stop cascading at about 17 cm, will deposit half of its energy in the first 20 cm, and will have dumped 98% of its energy within 59 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 4.6 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick tungsten will stop cascading at about 3.5 cm, will deposit half of its energy in the first 4 cm, and will have dumped 98% of its energy within 12 cm. 90% of the cascade will be contained within about 0.93 cm and 95% of the cascade will be contained within 1.86 cm of the beam axis. You will get 98% protection with a mass thickness of 2.3 tons/m2 and an additional 1/e reduction for each further 0.13 tons/m2 beyond that.
<li> A 1 TeV beam incident on water will stop cascading at about 290 cm, will deposit half of its energy in the first 340 cm, and will have dumped 98% of its energy within 1030 cm. 90% of the cascade will be contained within about 11 cm and 95% of the cascade will be contained within 22 cm of the beam axis. You will get 98% protection with a mass thickness of 10.3 tons/m2 and an additional 1/e reduction for each further 0.72 tons/m2 beyond that.
</ul>
Note that for sufficiently intense particle beams, 98% protection may not be enough. You might need to throw on several more tons per square meter to be sure.

<table border=1>
<tr><td> How Far it Goes:
(This section follows methods presented in Leroy and Rancoita<ref>C. Leroy and P-G Rancoita, "Physics of cascading shower generation and propagation in matter: principles of high-energy, ultrahigh-energy and compensating calorimetry", Reports on Progress in Physics 63 (2000) 505–606. DOI: 10.1088/0034-4885/63/4/202</ref>.)

When trying to estimate the extent of the irradiated region of the target, it is helpful to know the radiation length X0 in the material. This is the distance that an electron will travel, on average, before emitting a bremsstrahlung photon. Conveniently, it is also the distance a high energy photon will travel through the material before pair producing. The radiation length can be found from
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   = 4 α  
<td nowrap="nowrap"> NA
<td rowspan="2" nowrap="nowrap" align="right">   Z (Z+ζ) re2 ln
<td rowspan="2" nowrap="nowrap" align="right"> [
<td nowrap="nowrap"> 183
<td rowspan="2" nowrap="nowrap" align="right"> ]
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> A
<td style="border-top:solid 1px black;"> Z1/3
</table>
</div>
where α = 1/137 is the fine structure constant, NA = 6.02214076×1023 is Avogadro's number (or one mole), A is the atomic weight of the atoms in the material (usually expressed in grams per mole), Z is the atomic number (nuclear charge) of the atoms, and re = 2.8179403205×10-13 cm is the classical electron radius. ζ is a constant of approximately 1.3. This gives the thickness of material in terms of the areal density (mass of material per unit area perpendicular to the beam) – if you use A in g/mol and re in cm the result will be in cm2/g and X0 will be in g/cm2. If your target is made up of more than one kind of atom, you can get the total radiation length by finding the radiation length of each atom of type i, X0i and using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   =  
<td rowspan="2" nowrap="nowrap" align="right"> Σi  
<td nowrap="nowrap"> fi
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> X0i
</table>
</div>
where fi is the fraction by weight of the atom type i. Once you have the radiation length in terms of areal density, you can easily convert it to distance by dividing by the density ρ. If X0i is in g/cm2 and density is in g/cm3, then X0i/ρ will be in cm.

We will also need to know the critical energy εc. This is the energy where an electron is losing energy as rapidly to ionization as it is to radiation. That is, it is the energy at which the cascade stops multiplying, even though the individual particles already made can continue traveling for some distance. The critical energy can be estimated using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> εc = B  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> Z X0
<td rowspan="2" nowrap="nowrap" align="right"> )
<td nowrap="nowrap"> h
<tr>
<td style="border-top:solid 1px black;"> A
<td>  
</table>
</div>
where B = 2.66 MeV and h = 1.11.

We can now describe the distribution of radiation in the target material. The depth tmax where the cascade stops multiplying (in g/cm2) for an incident electron of energy E (in MeV) is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tmax = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> - 1
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
Similarly, the depth where half the energy of the beam has been deposited (in g/cm2) is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tcg = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> + 0.37
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
As usual, you can convert to a depth in cm by dividing by the material density. 98% of the beam energy will be deposited within a distance of &cong; 3 tcg. Beyond this, the beam energy will continue to fall off with thickness t approximately as an exponential exp[-t/λ], where the characteristic attenuation length λ &cong; 2 X0.

The width of a cascade, where 90% of the energy is contained, is approximately given by
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> RM =  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> EM
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
where EM &cong; 21.2 MeV. 95% of the radiation will be contained within a distance of 2 RM from the beam axis.
</table>

==== Protons and Ions ====

<table align=right><tr><td>
[[File:Regolith_Shielding.png|350 px|frameless]]
<tr><td width=350>
Relative effect of particle radiation (compared to no shielding) from galactic cosmic rays behind different thicknesses of water, aluminum, and lunar regolith<ref name="Slaba2022">Tony C. Slaba, "Radiation Shielding for Lunar Missions: Regolith Considerations", LSIC Crosstalk 7/18/2022 https://lsic.jhuapl.edu/uploadedDocs/focus-files/1604-E&C%20+%20EE%20Monthly%20Meeting%20-%202022%2007%20July_Presentation%20-%20NASA%20Slaba.pdf</ref>.
</table>
Protons and ions are grouped together because ions are made of protons and neutrons and when you accelerate these things for a weapon it is the protons and neutrons that are going to be doing the heavy lifting for blasting your target. This is because protons and neutrons are both about 2000 times more massive than electrons so any electrons along for the ride will contribute negligibly to the energy. Protons and neutrons together are called nuclear particles, and both behave quite similarly in nuclear collisions.

Nuclear particles are themselves made of quarks and can interact via the strong nuclear force. At the energies appropriate for weapons use, essentially all of the way protons or ions are blocked is through [[Particle_Accelerators#Nuclear_collisions|direct collisions with the atomic nuclei]] of the armor material. One or more quarks in the projectile will slam into quarks in the target nucleus and stretch them away from the rest of the quarks in the nuclear particles. Physics tells us that you can't have isolated quarks, so the "string" attaching the quark to its neighbors stretches and stretches until it snaps. Snapping releases so much energy that more quarks are made that clump together into all sorts of other nuclear particles, both exotic and familiar. This process is called a hadronic jet. The only particles produced in these jets long-lived enough to travel any noticeable distance through the armor material will be protons, neutrons, the charged pi mesons (or pions), and some of the K mesons (or kaons). The particles too short lived to go far enough to matter will decay into other particles, and these particles are often very high energy gamma rays. Much like with electrons, the long-lived secondary particles can be sufficiently energetic to smash into other nuclei and cause more hadronic jets (or, if they are low enough in energy, just knock more nuclear particles out of target nuclei) and this produces a similar shower or cascade. But in addition, the armor nuclei that were hit will be very hot after the impact from all of the energy given to them from the incident particles. These nuclei will cool off by evaporation – but instead of evaporating away molecules, they evaporate off nuclear particles from the nucleus. These evaporated particles are mostly neutrons, so any armor against proton or ion beam strikes will produce quite a bit of neutron radiation.

Eventually, the original particle's energy will be distributed among so many other particles that the charged particles will come to a stop by making ion tracks rather than from colliding with more nuclei, and the cascade halts. But you still have all the neutrons to worry about. And the gamma rays as well, which are high enough energy to kick off an electromagnetic cascade in the same way that electrons do.

Unlike electrons, which have reasonable approximations of their cascade behavior from relatively simple equations, the cascades of protons and ions usually require sophisticated simulations or experiments to get the resulting distribution of radiation. However, we can get a reasonable idea of their general behavior by comparing them to [[Space_Radiation#Shielding_Against_Cosmic_Rays|cosmic ray particles interacting with shielding]]. Cosmic rays are energetic protons and ions, after all. From this we see that a relatively small amount of armor – in the range of 1 to 3 tons per square meter – can actually make the radiation dose worse. The incident particles will knock off so many secondary particles in their shower that the dose is multiplied from so many tracks passing through the target. Hydrogen rich materials such as water and plastics work the best against proton and ion beam particles; heavy elements are the worst and produce the most secondaries. You could probably obtain adequate radiation protection from most incident beams with about 10 tons per square meter of hydrogen rich material as armor. However, you will want to add neutron absorbing elements to your hydrogen rich armor, such as lithium, boron, cadmium, or gadolinium. Further, in order to deal with the gamma rays you will want a layer of at least several centimeters of heavy elements, such as tungsten or lead, behind the hydrogen rich particle armor.

==== Structural Considerations ====

When your armor material absorbs a radiation beam, the energy of that radiation turns into heat inside the armor where the radiation was absorbed. If the beam pulse has enough energy as was sufficiently tightly focused, this heat can damage your armor. Your armor may melt in places, or thermal expansion could cause it to crack or warp. Perhaps the worst thing it could do is vaporize or flash into a plasma, because the pressure from the vapor or plasma can be sufficient to plastically deform the armor. This leaves a hole along the beam path where the expanding gas or plasma blasted out a long, deep crater. This means a beam that fires rapid shots spaced maybe milliseconds apart can have subsequent shots go down the same hole so your armor will not protect against them. Each shot will dig out a deeper hole until either your armor is penetrated or the attacking beam runs out of shots in its pulse train.

A consequence of this is that when designing your armor you may wish to use refractory materials with a high structural strength (like tungsten), or that is otherwise resistant to being blasted apart by its own evaporated remains.

=== Magnetic fields ===

[[Particle_Accelerators#Magnetic_fields|A charged particle crossing a magnetic field will bend so as to spiral around the field lines]]. If you try to shoot your particle beam through a magnetic field, it will bend away from the direction you shot it. This can make it hard to hit your target. Unfortunately for you, there are magnetic fields everywhere – around planets, and even trapped in the solar wind as it crosses the solar system.

As an example, the Earth has a magnetic field strength of about 5×10-5 tesla near its surface, and falling off farther away. If you are shooting a 1 TeV electron beam past the Earth, your beam is likely to bend by several milliradians (with the exact deflection depending on how close you are skimming the Earth). If you are in a medium Earth orbit and shooting at another target also in a medium Earth orbit on a trajectory that goes near the Earth, then without correcting for the magnetic field your beam will likely be off-target by tens to hundreds of kilometers.

For another example, the interplanetary magnetic field is about 10-9 tesla. If you shoot that same 1 TeV electron beam at a target 0.1 light seconds away, you are likely to be off by a couple hundred meters.

<table border=1>
<tr><td>
The radius of curvature of the bend of a particle in a magnetic field is called the gyroradius rc. For a particle with a charge q, mass m, and kinetic energy E, find the relativistic momentum p
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
p = √[E2 - m2 c4] / c
</div>
To convert from energy units of eV to joules, multiply by 1.60×10-19. An electron has a mass of 9.11×10-31 kg, a proton of 1.67×10-27 kg, and an ion of 1.66×10-27×A kg, where A is the atomic weight of the ion. The speed of light c = 3.00×108 m/s.

In a magnetic field of strength B with an angle θ between the field direction and the direction of motion
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rc = |p sin(θ) / (q B)|
</div>
For electrons, q = -e, where the fundamental charge e = 1.60×10-19 coulomb. For a proton, q = +e. For an ion, multiply e by the charge state of the ion.

If the field acts over a distance d that is much smaller than the gyroradius, the deflection angle φ is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = d / rc
</div>
The deflection x at a target at a range R for small φ (φ &Lt; 1) will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
x = φ R.
</div>

Note that d will always be less than or equal to x. Also note that if d is not smaller than twice the gyroradius the beam will never reach the target because it will completely spiral around the field lines before it reaches the target.
</table>

So, it is clear that as the defender it will be helpful to hang out in areas with high magnetic fields, and keep some distance between you and the attacker. Meanwhile, the attacker will benefit from being closer where small deflections in the beam trajectory don't translate into large excursions from where the beam is aimed (and also because there will be less space filled with fields between the two of you). But the attacker can also try to compensate for known fields. If you happen to know that the magnetic fields are between you and your target, you can adjust your fire so that the deflection of the field bends your beam onto the target. This is not a simple task. It is easy enough to measure the field strength and direction where you are, but predicting how it changes over the path of your shot does not have an obvious solution. You can also try to gain intelligence on likely battle sites before you ever fight on them – for example, getting good data on the structure of the fields of a planet using spies and technical intelligence during peacetime (heck, on Earth this information is published for free in academic journals and by government agencies such as NOAA). Planetary magnetospheres are also affected by the solar wind, so different solar weather will change the fields you have to shoot through. But space weather forecasts and predictive simulations of the solar wind interacting with a planetary magnetosphere can give you a good idea of what the field structure is like. But even with all that, there are changes in the fields that happen on a scale of around a second ... anyone who has ever seen the auroras dancing across the sky in the polar night can see this. Compensating for these changes will be a significant challenge.

What would really help is being able to see the path of your own beam. This would allow you to see how far off you were from your target and in what direction, and then adjust fire to walk your shots onto the target. Unfortunately, particle beams in space are invisible. There might be ways to track them, but if so we don't know what they are.

==== Synchrotron radiation ====

When charged particle trajectories are bent, they emit [[Particle_Accelerators#Synchrotron_radiation|synchrotron radiation]]. This can cause the particle beam to lose energy, but if the bend is too tight it can also generate high energy gamma rays that ignore magnetic fields and directly strike the target. Synchrotron radiation can generally be ignored for proton or ion beams, but can be significant for electron beams.

<table border=1><tr><td>
For a beam particle with energy E and mass m, the synchrotron radiation is concentrated in a forward-directed cone with an opening angle of approximately 1/γ, where γ = 1 + E/(m c2). (Interestingly, the radiation intensity peaks an at angle of 1/(2 γ) and vanishes directly on-axis.) Because the beam is bending, you will get emission along the plane along which the beam is being bent. The energy of the photons produced during synchrotron emission is approximately<ref name="Jackson_Electrodynamics">J. D. Jackson, "Classical Electrodynamics: Second Edition", Wiley (1962)</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Eγ &cong; 3 γ3 c &hbar; / rc
</div>
where &hbar; = 1.05×10-34 J s (if you want the energy in joules) or &hbar; = 6.58×10-16 eV s (if you want the energy in eV). The photon energy will not exceed the energy of the incident particles, if you get a photon energy higher than the energy of the incident particles it just means that the synchrotron radiation spectrum extends up to the incident particle energy.

The fraction of the energy (or power) of the beam turned into synchrotron radiation will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = [1/(4 π ε0)] [4 π / 3] [q² / rc] β2 γ4 [φ / (2 π)] / E
</div>
where β = √[1 - 1/γ2] is the particle's speed measured as a fraction of the speed of light, and 1/(4 π ε0) = 8.99×109 is the Coulomb constant. For highly relativistic electrons (like all practical electron beam weapons emit), β &cong; 1 and we can use
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = 8.85×10-2 [E (GeV)3 [φ / (2 π)] / (rc (m))]
</div>
This fraction cannot be more than 1. If f gets close to one, it means that our original assumption that the beam particles mostly keep their original speed no longer holds, and the beam will lose most or almost all of its energy. Instead, when the beam is being deflected it will only make it through an angle of φ/f and will irradiate a region of length x/f starting at where the beam would have been incident if not for the magnets and extending in the direction the beam was deflected.
</table>

For example, a 1 TeV electron beam will have γ &cong; 2,000,000. So if it's bent by a magnetic field the synchtrotron radiation will spread out in a cone of about 5×10-7 radians. If this beam was shooting at a target 0.1 light seconds away from where it was bent, the radiation would arrive at the target over a region with a radius of about 15 meters with a maximum intensity at about 7.5 meters and no radiation directly along the path the beam was pointing while it was being bent. If the beam was bent by the interplanetary magnetic field from our earlier example, it would produce light with photon energies of around 1.3 eV (near infrared) and the lost energy would be a few parts in a hundred trillion. If it was instead deflected by the Earth's magnetic field you might expect a peak photon energy of tens of keV – in the hard x-ray part of the spectrum – and the fraction of energy in the synchrotron radiation would be somewhere in the low parts in 100,000 or less.

==== Separation of neutral beams ====

Magnetic fields only affect charged objects. So one way to overcome magnetic defenses or environments is to use beams that have no net charge. For example, a proton beam could be neutralized with a co-propagating electron beam with an equal number of protons and electrons. Or, potentially, you could fire an ion beam and then neutralize the ions after acceleration to get a beam of neutral atoms.

But, the magnetic fields are still tugging at the charged particles making up the beam as they go past. This will pull the electrons one way and the protons or ions the other way. As the particles are pulled in opposite directions you get a net positive charge on one side of the beam and a net negative charge on the other side (called polarization). This sets up an electric field inside the beam that acts to pull the particles back together. If the particles are going fast enough and the magnetic field is strong enough, the magnetic force can still separate the charges and will split the beam in two, one with the positive charges and one with the negative charges. But if the particles are not that fast or the magnetic field is not that strong this polarization of the beam will keep it together and let it fly right through the magnetic field.

<table border=1><tr><td>
Let's figure out how to analyze this problem. For definiteness, we will have a beam with an electric current of the positive species of I and an electric current of electrons of -I, all moving in the z direction. The beam radius will be R. The particle speed will be v with the usual relativistic parameters β = v/c and γ = √[1/(1-β2)]. If you know the energy per particle E of the positive species and the mass m of the positive species, γ = 1 + E/(m c2) and then β = √[1 - 1/γ2].

Let's assume that the beam is going through a magnetic field of strength B at an angle θ to the field direction. The quantity B sin(θ) is the component of the magnetic field perpendicular to the direction of the beam.

The charge density ρ of the positive component of the beam will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρL = I / (π v R2).
</div>

It is easiest to analyze the problem in the rest frame of the particles in the beam. Now the particles are at rest. Particles at rest are not affected by a magnetic field, but the change of reference frame Lorentz transforms the initial magnetic field into a combination of electric and magnetic fields in the rest frame. We can ignore the magnetic field, but the electric field is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔A = γ v B sin(θ)
</div>
in a direction perpendicular to both v and B. For our purposes, assume the electric field points in the x direction.

At the same time, relativistic length contraction increases the charge density in the rest frame
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρ = γ ρL.
</div>

Let r be the vector from the beam axis to the point inside the beam. It will be useful to separate this into its Cartesian coordinates. If x and y are the unit vectors in the x and y directions, respectively, then
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
r = x x + y y
</div>

If we consider only the positive component of the beam, from [https://en.wikipedia.org/wiki/Gauss%27s_law Gauss's law] it will have an electric field inside of the beam at a distance r from the center axis of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔+ = ρ r / (2 ε0) = (ρ/(2 ε0)) (x x + y y)
</div>
where ε0 = 8.85×10-12 is the vacuum permittivity. Now let's displace the electrons in the beam by a distance δ in the x direction. By itself, this electron beam will have an electric field of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔- = -(ρ/(2 ε0)) ((x-δ) x + y y)
</div>
By the principle of superposition, the electric field inside the beam with both components is the sum of the electric fields of the separate components including the external field
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔B = [(ρ/(2 ε0)) δ + γ v B sin(θ)]x
</div>
Because the charges are free to move, they will do so as long as a field is applied to them. Thus, the beam becomes stable once the charges separate enough that the field internal to the beam is zero. Therefore
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
δ = -2 ε0 γ v B sin(θ) / ρ = -2 π ε0 v2 R2 B sin(θ) / I
</div>
If |δ| > 2 |R|, the beam will fully separate and each component will go its own way. If δ is small compared to R, the beam's polarization suffices to hold it together in the overlap region. However, where the positive and negative charges do not overlap the electric fields no longer cancel, and the external electric field carries away the outer polarized part of the beam. This gives us a rough threshold for which a magnetic field will quickly separate a neutral beam into its charged components and bend them away from their target
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr><td> B sin(θ) &Lt; I / (π ε0 v2 R) <td>       <td> Beam gradually erodes.
<tr><td> B sin(θ) > I / (π ε0 v2 R) <td> <td> Beam promptly separates.
</table>
</div>
For a gradually eroding beam, you can figure that if the magnetic deflection of the positive component is not much larger than the beam radius, most of the beam will still hit its target.
<tr><td>
For a beam of neutral atoms, let a be the radius of the atom and let Eb be the binding energy of the least tightly bound electron to the atom. An external electric field in excess of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
|𝓔A| > Eb / (e a)
</div>
will promptly ionize the atom, ripping away its electron. Here, e = 1.60×10-19 C is the magnitude of the electric charge of the electron. For fields close to but a bit weaker than this, the electron can still rapidly pop off due to quantum tunneling<ref>Ph. Durand and I. Paidarová, “Ionization of the hydrogen atom: from weak to strong static electric fields,” The European Physical Journal D, Vol. 26, pg. 253-259, 2003.</ref>. In general, expect that for fields greater in magnitude than (very approximately) Eb / (10 e a) the electron will be quickly removed, while for fields weaker than this the electrons will stay on their atoms.

It is useful to note that Eb/e is just the binding energy in units of eV. The most weakly bound electrons are usually bound to their atom with about 10 eV of energy. Atoms are typically about 10-10 m in radius. Thus, you can expect fields of around 1010 to 1011 V/m to directly ionize atoms. If we call this threshold value 𝓔T, then a beam will be ionized when passing through a field with strength greater than
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B sin(θ) > 𝓔T / (γ v)
</div>
</table>

Example: Let's consider a 1 GeV neutralized proton beam from a synchrotron, with a beam current of 200 kA in each bunch (incidentally, if a bunch is 5 cm long this means that each bunch packs 38 kJ worth of energy). The speed of the bunch is 87% the speed of light, or 262 million m/s, with a γ of 2.07. Consider when the beam's focus takes it to a radius of 5 cm (or 0.05 m, for our calculation). Plugging this in, we find that the protons in the bunch will separate from the electrons in a field with a transverse component greater than about 2.1 T, and the bunch will pass through the field without much effect if the transverse component is much smaller than this.

Example: We'll take the same 1 GeV beam, but this time recombine the electrons with protons to make hydrogen atoms. An electron is bound to a hydrogen atom with an energy of 13.6 eV (the Ryberg constant) and the radius of the electron orbital in a hydrogen atom is 0.529×10-10 m (the Bohr radius). This gives a threshold electric field for prompt ionization of hydrogen of 2.57×1011 V/m. We will get this with a transverse component of the magnetic field of 474 T. Above this field the atoms promptly ionize and electrons are stripped away from the protons. For fields higher than roughly 1/10 of this – so about 40 to 50 T – you may still get significant ionization by quantum tunneling; lower than this your losses from the beam due to field-induced ionization are likely negligible.

==== Magnetic shields ====

What happens when we can't rely on magnetic fields in the environment to protect us? One idea is to produce powerful artificial magnetic fields around our spacecraft to deflect incoming particle beams. This is very similar in concept to using active magnetic shielding to protect against space radiation – [[Space_Radiation#Magnetic_Shielding|check out our article on space radiation for more]].
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr>
<td width=100>
[[File:Unconfined_FRC_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:racetrack_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:Magnetic_shielding_Halback_Array.png|200 px|frameless]]
<tr><td colspan=3>
Some potential configurations for magnetic shields. The currents generating the fields are in green, the magnetic fields themselves are shown in magenta<ref name="Battiston2012">R. Battiston, W.J. Burger, V. Calvelli, R. Musenich, V. Choutko, V.I. Datskov, A. Della Torre, F. Venditti,
C. Gargiulo, G. Laurenti, S. Lucidi, S. Harrison, and R. Meinke, "ARSSEM Active Radiation Shield for Space Exploration Missions", Final Report ESTEC Contract N° 4200023087/10/NL/AF : “Superconductive Magnet for Radiation Shielding of Human Spacecraft” (2012) https://arxiv.org/abs/1209.1907 https://www.researchgate.net/publication/265945847_Active_Radiation_Shield_for_Space_Exploration_Missions</ref><ref name="Desiati2022">Paolo Desiati and Elena D'Onghia, "CREW HaT: A Magnetic Shielding System for Space Habitats", arXiv:2209.13624 [physics.space-ph] https://doi.org/10.48550/arXiv.2209.13624</ref>.
</table>
</div>

One immediate issue is that you want to keep very high magnetic fields out of the crew area. Fields in excess of 0.5 T can start to be unpleasant, and it is recommended to keep the field below 0.2 T for continuous exposure.

Strong magnets can suffer from high self-forces acting to tear them apart. Most shielding designs will have multiple magnetic coils, which will exert high forces and torques on each other. This will result in the need for strong structural bracing to keep the magnets intact and in place. Such strong structural designs will impose significant extra mass on the spacecraft.

Consider a magnetic field configuration that can keep radiation out of a certain region of space. There is no trajectory of a charged radiation particle up to a certain energy that will allow those particles to enter the protected region. But the dynamics of charged particles in a magnetic field are the same whether going forward or backward in time. If there is no way for a particle to get in, by this time reversal symmetry any particle which is in that forbidden region can never escape. Consequently any field configuration that is a magnetic trap is also a magnetic shield; any field configuration that is a magnetic shield is also a magnetic trap. A spacecraft with a magnetic shield may end up surrounded by strong radiation belts.

Using a magnet to bend a weaponized electron beam very often ends up with the beam being bent so hard that it dumps most of its energy in the form of synchrotron radiation at gamma ray energies. Because gamma rays are not charged, they will not be deflected by the field. Thus, if you would have been hit by the electron beam you will be hit instead by the gamma rays.

As an example, consider a 1 TeV electron beam. We'll surround our spacecraft with a 10 T magnetic field in a 100 meter radius region just outside our spacecraft. This will deflect the beam by 30 meters, which is enough to make it miss. But we are not saved, because the beam dumps nearly all of its energy into synchrotron radiation with an energy spectrum of gamma rays up to 13 GeV. This radiation will hit us in a strip about 2.4 meters long and not much more than the width of the original beam with essentially all of the energy of the original beam.

==== Magnet drones ====

If a magnetic shield around your spacecraft will just end up killing you from synchrotron radiation, what if you put the magnetic shield on a drone and put that drone between yourself and the enemy trying to shoot you? This lets you use a much weaker magnet because you don't need nearly so high of a deflection angle to make the shot miss. This means that less beam energy will turn into synchrotron radiation that could kill you instead and will spread what synchrotron radiation does occur out over a much wider area when it reaches your spacecraft, hopefully so that most of it just misses your spacecraft.

It does mean that your enemy will start shooting your drones so that they can eventually take a shot at you. But hopefully by that time you've already cleared out their drones and cored their spacecraft with your own synchrocannon or UREB.

==== Plasma magnets ====

Usually, to make a really big magnetic field you would run current through a really big conductive loop. But really big things are heavy and expensive. One way around this could be to use the ambient space plasma to carry the electric current rather than any kind of wire. In order to make this current, you use the same mechanism used in a rotary induction electric motor: you generate a rotating magnetic field and induce a current in the conductive loop (in this case, the space plasma). How do you make a rotating magnetic field? The method proposed<ref name="Slough2006">John Slough, "The Plasma Magnet", NASA Institute for Advanced Concepts Phase II Final Report (2006) https://www.niac.usra.edu/files/studies/final_report/917Slough.pdf</ref> is to use an antenna of four much smaller current loops as electromagnets at 90&degree; to each other, and sinusoidally vary the current through the loops with a respective 90&degree; phase difference between them. When this is done, a circular current is induced in the space plasma and the self-force of a magnetic field on its generating current inflates the current loop to much larger radii. You end up with a current sheet in the plane of the rotating field's rotation circulating about the antenna nearly synchronously with the field rotation. This current then creates a normal dipolar magnetic field as with any current loop. The current sheet and the resulting magnetic field expand until the magnetic field's pressure equals that of the ambient space plasma. This gives an enormous region of magnetic field from a relatively small and lightweight system.

The system proposed in the initial study would use an antenna approximately 10 meters in radius and produce a region of field approximately 30 km in radius at Earth's orbit. The field inside would not be constant - at the edges it would be 50 nT, giving the right pressure to keep out the solar wind at Earth's orbit. A circulating current sheet produces a field that falls off as 1/r with distance, meaning that adjacent to the antenna the field would be about 0.15 mT. Using the methods in the box below, we can estimate that this particular design as part of a spacecraft's shields will not deflect a TeV electron beam by more than a few centimeters, which is usually insufficient to force a miss. However, it could potentially be useful for a drone magnet, where the much larger distance between the magnetic field and the target spacecraft would allow a small deflection angle to give perhaps tens or low hundreds of meters change in beam location near the target at near light second ranges. The relatively low field and gradual bend of the incident beam suggests that synchrotron radiation will not be an issue; however this does not exclude more powerful plasma magnet designs causing significant synchrotron radiation effects.

<table border=1><tr><td>
Calculating the deflection from a spatially varying magnetic field affected by the solar wind is no trivial task. However, we can make an approximation that both should be within about an order of magnitude of the true value and also represent an upper limit to the deflection angle; although it does neglect the influence of the driving rotating magnetic field which could in some cases increase the deflection.

We will take R0 as the radius of the antenna. B0 is the field strength at the antenna. Rm is the distance at which the magnetic influence stops because the magnetic pressure equals the space plasma pressure. Bm is the field strength at Rm. According to Slough<ref name="Slough2006"></ref>, the magnetic field strength B(r) at any distance from the antenna center r between R0 and Rm is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B(r) = B0 R0 / r.
</div>
Therefore, once we know Bm and Rm, B0 is completely determined
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B0 = Bm Rm / R0.
</div>

The force on a particle with charge q passing perpendicularly through a magnetic field B(r) at a speed v is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
F(r) = q v B(r)
</div>
For an electron beam, v is so close to c as to make no difference. If the particle is moving down the y direction and the field is in the z direction, the force will be in the x direction. We assume that the deflection will be small, so we can choose a straight-line path in the y direction offset from the center by an amount x. The plasma bubble radius is Rm, so the particle will enter at ymax = √[Rm2-x2] and leave at ymin = -√[Rm2-x2].

The momentum change the particle experiences from this force is the time integral of the force over its path. Because of reflection symmetry about y = 0, we only need to integrate from ymax to 0 and then multiply by 2
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] dt F(r)
<tr> <td> <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] (dy/v) F(r)
<tr> <td> <td> = <td align="left">2 q B(0) R0 [Integral from 0 to √[Rm2-x2]] dy 1/√[y2 + x2]
<tr> <td> <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/x]
</table>
</div>
This holds for |x| > R0. But the expression for the field is not valid for r < R0 and the integral will give nonsense and diverging results if we evaluate for |x| < R0. In this case, we can put the integral lower bound at r = R0, or y = √[R02-x2]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/(√[R02-x2] + R0)]
</table>
</div>

The deflection angle is now estimated as
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = px / p
</div>
where the total momentum of the particle p can be found from the methods in the first box in the Magnetic Fields section.

The deflection for a shot aimed right at the spacecraft at the position of the spacecraft can be estimated by setting x = 0 and only taking the first half of the integral (so ignore the factor of 2)
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
q B(0) R0 ln[Rm/R0] / (p Rm)
</div>
This is an additional overestimate because it fails to account for how the deflection builds up over distance during the integration, but it should still give something of an order-of-magnitude and upper limit estimate of what kind of deflections might be expected.
</table>

=== Electric fields ===

If electric fields can accelerate charged particles, maybe they can keep the charged particles in a weaponized particle beam away from your spacecraft? To completely block a particle beam, you would need a voltage on your spacecraft equal to the particle energy in eV of the particles in the beam. That is, if a 1 GeV proton beam is coming at you, you'd need an electric potential of a billion volts to stop that beam. This leads to three immediate problems.

The first is that particles of the opposite charge will be attracted to your spacecraft. Because space isn't really empty but rather a rarefied plasma (called the solar wind in a solar system or the interstellar medium out between stars), there will be plenty of particles of the opposite charge around to attract. And they will hit with an energy in eV equal to the voltage on your spacecraft. In our example of defending against a 1 GeV proton beam with a gigavolt field, the electrons from the solar wind will come in and hit with an energy of 1 GeV each. This will generate dangerous radiation.

The second problem is a consequence of the first. Each of those GeV electrons from our example need to get their energy from somewhere, which turns out to be from the electric field we are using as shielding. You will need to make up for this energy loss with power from your electrical system, which gives a constant drain on your power. Some estimates put the electrical effort to maintain a several hundred megavolt shield in space at many tens of gigawatts<ref name="Mechmann2019">Claire Mechmann, "Analysis of Proposed Active Radiation Shielding Design Concept for Spacecraft" (2019) Thesis, College of Engineering and Science of Florida Institute of Technology</ref>.

And the third problem is similar to one for magnetic fields. Your electrodes will experience high self forces and forces between separate electrodes that will try to tear your shield generator apart. Consequently you will need heavy structural reinforcement in order for your shield to work.

There are potential ways to mitigate some of these problems. Interested readers are directed to [[Space_Radiation#Electrostatic_Shielding|the active shielding section in our page on space radiation]] for more of the juicy details; there's no need to repeat the same stuff here.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Elctrostatic_active_shielding.png|200 px|frameless]]
<td>
[[File:Electrostatic_active_shielding_2.png|200 px|frameless]]
<tr><td colspan=2>
Proposed designs for electrostatic shields to protect from positively charged particles<ref name="Tripathi2006">Ram K. Tripathi, John W. Wilson, and Robert C. Youngquist, "Electrostatic Active Radiation Shielding - Revisited", 2006 IEEE Aerospace Conference, Big Sky, MT, USA, 2006, pp. 9 pp.-, doi: 10.1109/AERO.2006.1655760.</ref><ref name="Chowdhury2023">Rajarshi Pal Chowdhury, Luke A. Stegeman, Matthew L. Lund, Dan Fry, Stojan Madzunkov, and Amir A. Bahadori, "Hybrid methods of radiation shielding against deep-space radiation", Life Sciences in Space Research, Volume 38, 2023, Pages 67-78, ISSN 2214-5524, https://doi.org/10.1016/j.lssr.2023.04.004.</ref>.
</table>
</div>

== Damage ==

Particle beams are, first and foremost, devices that produce beams of [[Nuclear_radiation|ionizing radiation]]. The radiation of weaponizable particle beams is extremely penetrating; thus of the [[Particle_Accelerators#Energy_loss_in_matter|interaction mechanisms listed for radiation from particle beams]], we mostly need to worry about bremsstrahlung and nuclear collisions and these produce extensive secondary showers of radiation that increase the penetration of the radiation exposure. Because of this, any target hit by a particle beam weapon will be irradiated, and radiation dose will always be one of the primary damage mechanisms of a particle beam weapon strike. Further, this radiation will be distributed along a track passing deeply through the material of the target, allowing it to affect critical systems through heavy armor, structural material, robust machine components, and passive material such as fuel load, coolant, or cargo. As the radiation deposits its energy into the target material, that material will be heated. If the target is heated sufficiently, it may experience [[Beam-Target_Interactions|thermal or mechanical]] damage from melting, cooking, combustion, vapor explosions, and warping or cracking due to thermal expansion.

The discussion of the radiation penetration into armor also applies to the same radiation penetrating through the target. So the formulas and rules of thumb for estimating the depth of radiation penetration into armor can also be used to estimate how far into a tank or spacecraft a particle beam strike delivers effects.

In addition to the radiation and thermal effects, a particle beam can also produce a sharp electromagnetic pulse that can damage semiconductor electronics.

== 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, Gerrit Bruhaug, and Luke Campbell for writing

Gerrit Bruhaug and Luke Campbell for scientific research, math wrangling and advise

ATriffleMind and Luke W. Campbell for proof reading.

==References==

[[Category:Beams]][[Category:Physics & Engineering‏‎]][[Category:Engineering‏‎]][[Category:Military Technology]][[Category:Warfare‏‎]]

Particle Beam Weapons

2026-10-26T04:30:55Z

Lwcamp: /* Separation of neutral beams */

{{PageConstructionNotice}}

''For a general overview on particle accelerator technology, see [[Particle Accelerators]]

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.

::: Z = 1.32×102 (γ2-1)3/4 r / √I<ref name=USPAS2016>Martin Reiser, "Theory and Design of Charged Particle Beams", 2008, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim</ref>

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

::: Z = β γ c r √[m/T]

Where Z is the doubling distance like before, r is the initial beam radius, T is the beam temperature in eV, m is the ion mass and β, c, and γ 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.

== Defenses ==

Sure, it's great to have a great big particle cannon to blast your enemies. But what if you are the one being shot at? In that case, it's best to figure out ways to keep from being blown to kingdom come.

=== Armor ===

The simplest way to defend against the high energy particles of a particle beam is to pile a whole bunch of stuff up and put it in the way of the beam. Particle beams may produce extremely penetrating radiation, but enough stuff will block most of that radiation and may allow you to survive. The details depend on what kind of particle is being thrown at you, and are explained below. However, a general rule of thumb is that you probably want about 10 tons of material per square meter of surface you want to armor in order to get sufficient protection.

==== Electrons ====

At the very high energy of an electron accelerated by a weapons-grade particle beam, the primary interaction with matter will be the emission of bremsstrahlung photons in the gamma-ray range as the electron slams into particles of matter. These high energy photons themselves primarily interact by the pair production mechanism – the photon disappears, but leaves behind a highly energetic positron and electron. These two particles are themselves energetic enough to produce further bremsstrahlung, and so the process continues. You get the formation of an electron-gamma shower (also called an electromagnetic cascade), which only stops after either the particles have penetrated through the target or the initial energy of the electron has been spread through so many shower particles (including all the electrons, positrons, and photons) that no particle has enough individual energy to pair produce or to be likely to bremsstrahlung again before it is stopped by other mechanisms.

To stop electrons, you'll want the atoms in your armor to be as heavy as you can practically get them. High atomic number both encourages bremsstrahlung and pair production, giving a double whammy for stopping that cascade. However, occasionally one of those electrons, positrons, or gammas will hit a nucleus and knock out neutrons. So you probably want a back-up layer of several tens of centimeters of hydrogen rich material such as plastic or water mixed with neutron absorbing materials like boron, lithium, cadmium, or gadolinium. High density polyethylene loaded with 5% boron by weight is a typical go-to neutron blocker.

The actual methods to calculate the penetration of the electromagnetic cascade is given are the box below. However, for some examples:
<ul>
<li> A 10 GeV beam incident on thick iron (or steel) will stop cascading at about 9 cm, will deposit half of its energy in the first 12 cm, and will have dumped 98% of its energy within 35 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 2.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 100 GeV beam incident on thick iron (or steel) will stop cascading at about 13 cm, will deposit half of its energy in the first 16 cm, and will have dumped 98% of its energy within 47 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 3.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick iron (or steel) will stop cascading at about 17 cm, will deposit half of its energy in the first 20 cm, and will have dumped 98% of its energy within 59 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 4.6 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick tungsten will stop cascading at about 3.5 cm, will deposit half of its energy in the first 4 cm, and will have dumped 98% of its energy within 12 cm. 90% of the cascade will be contained within about 0.93 cm and 95% of the cascade will be contained within 1.86 cm of the beam axis. You will get 98% protection with a mass thickness of 2.3 tons/m2 and an additional 1/e reduction for each further 0.13 tons/m2 beyond that.
<li> A 1 TeV beam incident on water will stop cascading at about 290 cm, will deposit half of its energy in the first 340 cm, and will have dumped 98% of its energy within 1030 cm. 90% of the cascade will be contained within about 11 cm and 95% of the cascade will be contained within 22 cm of the beam axis. You will get 98% protection with a mass thickness of 10.3 tons/m2 and an additional 1/e reduction for each further 0.72 tons/m2 beyond that.
</ul>
Note that for sufficiently intense particle beams, 98% protection may not be enough. You might need to throw on several more tons per square meter to be sure.

<table border=1>
<tr><td> How Far it Goes:
(This section follows methods presented in Leroy and Rancoita<ref>C. Leroy and P-G Rancoita, "Physics of cascading shower generation and propagation in matter: principles of high-energy, ultrahigh-energy and compensating calorimetry", Reports on Progress in Physics 63 (2000) 505–606. DOI: 10.1088/0034-4885/63/4/202</ref>.)

When trying to estimate the extent of the irradiated region of the target, it is helpful to know the radiation length X0 in the material. This is the distance that an electron will travel, on average, before emitting a bremsstrahlung photon. Conveniently, it is also the distance a high energy photon will travel through the material before pair producing. The radiation length can be found from
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   = 4 α  
<td nowrap="nowrap"> NA
<td rowspan="2" nowrap="nowrap" align="right">   Z (Z+ζ) re2 ln
<td rowspan="2" nowrap="nowrap" align="right"> [
<td nowrap="nowrap"> 183
<td rowspan="2" nowrap="nowrap" align="right"> ]
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> A
<td style="border-top:solid 1px black;"> Z1/3
</table>
</div>
where α = 1/137 is the fine structure constant, NA = 6.02214076×1023 is Avogadro's number (or one mole), A is the atomic weight of the atoms in the material (usually expressed in grams per mole), Z is the atomic number (nuclear charge) of the atoms, and re = 2.8179403205×10-13 cm is the classical electron radius. ζ is a constant of approximately 1.3. This gives the thickness of material in terms of the areal density (mass of material per unit area perpendicular to the beam) – if you use A in g/mol and re in cm the result will be in cm2/g and X0 will be in g/cm2. If your target is made up of more than one kind of atom, you can get the total radiation length by finding the radiation length of each atom of type i, X0i and using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   =  
<td rowspan="2" nowrap="nowrap" align="right"> Σi  
<td nowrap="nowrap"> fi
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> X0i
</table>
</div>
where fi is the fraction by weight of the atom type i. Once you have the radiation length in terms of areal density, you can easily convert it to distance by dividing by the density ρ. If X0i is in g/cm2 and density is in g/cm3, then X0i/ρ will be in cm.

We will also need to know the critical energy εc. This is the energy where an electron is losing energy as rapidly to ionization as it is to radiation. That is, it is the energy at which the cascade stops multiplying, even though the individual particles already made can continue traveling for some distance. The critical energy can be estimated using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> εc = B  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> Z X0
<td rowspan="2" nowrap="nowrap" align="right"> )
<td nowrap="nowrap"> h
<tr>
<td style="border-top:solid 1px black;"> A
<td>  
</table>
</div>
where B = 2.66 MeV and h = 1.11.

We can now describe the distribution of radiation in the target material. The depth tmax where the cascade stops multiplying (in g/cm2) for an incident electron of energy E (in MeV) is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tmax = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> - 1
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
Similarly, the depth where half the energy of the beam has been deposited (in g/cm2) is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tcg = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> + 0.37
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
As usual, you can convert to a depth in cm by dividing by the material density. 98% of the beam energy will be deposited within a distance of &cong; 3 tcg. Beyond this, the beam energy will continue to fall off with thickness t approximately as an exponential exp[-t/λ], where the characteristic attenuation length λ &cong; 2 X0.

The width of a cascade, where 90% of the energy is contained, is approximately given by
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> RM =  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> EM
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
where EM &cong; 21.2 MeV. 95% of the radiation will be contained within a distance of 2 RM from the beam axis.
</table>

==== Protons and Ions ====

<table align=right><tr><td>
[[File:Regolith_Shielding.png|350 px|frameless]]
<tr><td width=350>
Relative effect of particle radiation (compared to no shielding) from galactic cosmic rays behind different thicknesses of water, aluminum, and lunar regolith<ref name="Slaba2022">Tony C. Slaba, "Radiation Shielding for Lunar Missions: Regolith Considerations", LSIC Crosstalk 7/18/2022 https://lsic.jhuapl.edu/uploadedDocs/focus-files/1604-E&C%20+%20EE%20Monthly%20Meeting%20-%202022%2007%20July_Presentation%20-%20NASA%20Slaba.pdf</ref>.
</table>
Protons and ions are grouped together because ions are made of protons and neutrons and when you accelerate these things for a weapon it is the protons and neutrons that are going to be doing the heavy lifting for blasting your target. This is because protons and neutrons are both about 2000 times more massive than electrons so any electrons along for the ride will contribute negligibly to the energy. Protons and neutrons together are called nuclear particles, and both behave quite similarly in nuclear collisions.

Nuclear particles are themselves made of quarks and can interact via the strong nuclear force. At the energies appropriate for weapons use, essentially all of the way protons or ions are blocked is through [[Particle_Accelerators#Nuclear_collisions|direct collisions with the atomic nuclei]] of the armor material. One or more quarks in the projectile will slam into quarks in the target nucleus and stretch them away from the rest of the quarks in the nuclear particles. Physics tells us that you can't have isolated quarks, so the "string" attaching the quark to its neighbors stretches and stretches until it snaps. Snapping releases so much energy that more quarks are made that clump together into all sorts of other nuclear particles, both exotic and familiar. This process is called a hadronic jet. The only particles produced in these jets long-lived enough to travel any noticeable distance through the armor material will be protons, neutrons, the charged pi mesons (or pions), and some of the K mesons (or kaons). The particles too short lived to go far enough to matter will decay into other particles, and these particles are often very high energy gamma rays. Much like with electrons, the long-lived secondary particles can be sufficiently energetic to smash into other nuclei and cause more hadronic jets (or, if they are low enough in energy, just knock more nuclear particles out of target nuclei) and this produces a similar shower or cascade. But in addition, the armor nuclei that were hit will be very hot after the impact from all of the energy given to them from the incident particles. These nuclei will cool off by evaporation – but instead of evaporating away molecules, they evaporate off nuclear particles from the nucleus. These evaporated particles are mostly neutrons, so any armor against proton or ion beam strikes will produce quite a bit of neutron radiation.

Eventually, the original particle's energy will be distributed among so many other particles that the charged particles will come to a stop by making ion tracks rather than from colliding with more nuclei, and the cascade halts. But you still have all the neutrons to worry about. And the gamma rays as well, which are high enough energy to kick off an electromagnetic cascade in the same way that electrons do.

Unlike electrons, which have reasonable approximations of their cascade behavior from relatively simple equations, the cascades of protons and ions usually require sophisticated simulations or experiments to get the resulting distribution of radiation. However, we can get a reasonable idea of their general behavior by comparing them to [[Space_Radiation#Shielding_Against_Cosmic_Rays|cosmic ray particles interacting with shielding]]. Cosmic rays are energetic protons and ions, after all. From this we see that a relatively small amount of armor – in the range of 1 to 3 tons per square meter – can actually make the radiation dose worse. The incident particles will knock off so many secondary particles in their shower that the dose is multiplied from so many tracks passing through the target. Hydrogen rich materials such as water and plastics work the best against proton and ion beam particles; heavy elements are the worst and produce the most secondaries. You could probably obtain adequate radiation protection from most incident beams with about 10 tons per square meter of hydrogen rich material as armor. However, you will want to add neutron absorbing elements to your hydrogen rich armor, such as lithium, boron, cadmium, or gadolinium. Further, in order to deal with the gamma rays you will want a layer of at least several centimeters of heavy elements, such as tungsten or lead, behind the hydrogen rich particle armor.

==== Structural Considerations ====

When your armor material absorbs a radiation beam, the energy of that radiation turns into heat inside the armor where the radiation was absorbed. If the beam pulse has enough energy as was sufficiently tightly focused, this heat can damage your armor. Your armor may melt in places, or thermal expansion could cause it to crack or warp. Perhaps the worst thing it could do is vaporize or flash into a plasma, because the pressure from the vapor or plasma can be sufficient to plastically deform the armor. This leaves a hole along the beam path where the expanding gas or plasma blasted out a long, deep crater. This means a beam that fires rapid shots spaced maybe milliseconds apart can have subsequent shots go down the same hole so your armor will not protect against them. Each shot will dig out a deeper hole until either your armor is penetrated or the attacking beam runs out of shots in its pulse train.

A consequence of this is that when designing your armor you may wish to use refractory materials with a high structural strength (like tungsten), or that is otherwise resistant to being blasted apart by its own evaporated remains.

=== Magnetic fields ===

[[Particle_Accelerators#Magnetic_fields|A charged particle crossing a magnetic field will bend so as to spiral around the field lines]]. If you try to shoot your particle beam through a magnetic field, it will bend away from the direction you shot it. This can make it hard to hit your target. Unfortunately for you, there are magnetic fields everywhere – around planets, and even trapped in the solar wind as it crosses the solar system.

As an example, the Earth has a magnetic field strength of about 5×10-5 tesla near its surface, and falling off farther away. If you are shooting a 1 TeV electron beam past the Earth, your beam is likely to bend by several milliradians (with the exact deflection depending on how close you are skimming the Earth). If you are in a medium Earth orbit and shooting at another target also in a medium Earth orbit on a trajectory that goes near the Earth, then without correcting for the magnetic field your beam will likely be off-target by tens to hundreds of kilometers.

For another example, the interplanetary magnetic field is about 10-9 tesla. If you shoot that same 1 TeV electron beam at a target 0.1 light seconds away, you are likely to be off by a couple hundred meters.

<table border=1>
<tr><td>
The radius of curvature of the bend of a particle in a magnetic field is called the gyroradius rc. For a particle with a charge q, mass m, and kinetic energy E, find the relativistic momentum p
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
p = √[E2 - m2 c4] / c
</div>
To convert from energy units of eV to joules, multiply by 1.60×10-19. An electron has a mass of 9.11×10-31 kg, a proton of 1.67×10-27 kg, and an ion of 1.66×10-27×A kg, where A is the atomic weight of the ion. The speed of light c = 3.00×108 m/s.

In a magnetic field of strength B with an angle θ between the field direction and the direction of motion
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rc = |p sin(θ) / (q B)|
</div>
For electrons, q = -e, where the fundamental charge e = 1.60×10-19 coulomb. For a proton, q = +e. For an ion, multiply e by the charge state of the ion.

If the field acts over a distance d that is much smaller than the gyroradius, the deflection angle φ is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = d / rc
</div>
The deflection x at a target at a range R for small φ (φ &Lt; 1) will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
x = φ R.
</div>

Note that d will always be less than or equal to x. Also note that if d is not smaller than twice the gyroradius the beam will never reach the target because it will completely spiral around the field lines before it reaches the target.
</table>

So, it is clear that as the defender it will be helpful to hang out in areas with high magnetic fields, and keep some distance between you and the attacker. Meanwhile, the attacker will benefit from being closer where small deflections in the beam trajectory don't translate into large excursions from where the beam is aimed (and also because there will be less space filled with fields between the two of you). But the attacker can also try to compensate for known fields. If you happen to know that the magnetic fields are between you and your target, you can adjust your fire so that the deflection of the field bends your beam onto the target. This is not a simple task. It is easy enough to measure the field strength and direction where you are, but predicting how it changes over the path of your shot does not have an obvious solution. You can also try to gain intelligence on likely battle sites before you ever fight on them – for example, getting good data on the structure of the fields of a planet using spies and technical intelligence during peacetime (heck, on Earth this information is published for free in academic journals and by government agencies such as NOAA). Planetary magnetospheres are also affected by the solar wind, so different solar weather will change the fields you have to shoot through. But space weather forecasts and predictive simulations of the solar wind interacting with a planetary magnetosphere can give you a good idea of what the field structure is like. But even with all that, there are changes in the fields that happen on a scale of around a second ... anyone who has ever seen the auroras dancing across the sky in the polar night can see this. Compensating for these changes will be a significant challenge.

What would really help is being able to see the path of your own beam. This would allow you to see how far off you were from your target and in what direction, and then adjust fire to walk your shots onto the target. Unfortunately, particle beams in space are invisible. There might be ways to track them, but if so we don't know what they are.

==== Synchrotron radiation ====

When charged particle trajectories are bent, they emit [[Particle_Accelerators#Synchrotron_radiation|synchrotron radiation]]. This can cause the particle beam to lose energy, but if the bend is too tight it can also generate high energy gamma rays that ignore magnetic fields and directly strike the target. Synchrotron radiation can generally be ignored for proton or ion beams, but can be significant for electron beams.

<table border=1><tr><td>
For a beam particle with energy E and mass m, the synchrotron radiation is concentrated in a forward-directed cone with an opening angle of approximately 1/γ, where γ = 1 + E/(m c2). (Interestingly, the radiation intensity peaks an at angle of 1/(2 γ) and vanishes directly on-axis.) Because the beam is bending, you will get emission along the plane along which the beam is being bent. The energy of the photons produced during synchrotron emission is approximately<ref name="Jackson_Electrodynamics">J. D. Jackson, "Classical Electrodynamics: Second Edition", Wiley (1962)</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Eγ &cong; 3 γ3 c &hbar; / rc
</div>
where &hbar; = 1.05×10-34 J s (if you want the energy in joules) or &hbar; = 6.58×10-16 eV s (if you want the energy in eV). The photon energy will not exceed the energy of the incident particles, if you get a photon energy higher than the energy of the incident particles it just means that the synchrotron radiation spectrum extends up to the incident particle energy.

The fraction of the energy (or power) of the beam turned into synchrotron radiation will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = [1/(4 π ε0)] [4 π / 3] [q² / rc] β2 γ4 [φ / (2 π)] / E
</div>
where β = √[1 - 1/γ2] is the particle's speed measured as a fraction of the speed of light, and 1/(4 π ε0) = 8.99×109 is the Coulomb constant. For highly relativistic electrons (like all practical electron beam weapons emit), β &cong; 1 and we can use
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = 8.85×10-2 [E (GeV)3 [φ / (2 π)] / (rc (m))]
</div>
This fraction cannot be more than 1. If f gets close to one, it means that our original assumption that the beam particles mostly keep their original speed no longer holds, and the beam will lose most or almost all of its energy. Instead, when the beam is being deflected it will only make it through an angle of φ/f and will irradiate a region of length x/f starting at where the beam would have been incident if not for the magnets and extending in the direction the beam was deflected.
</table>

For example, a 1 TeV electron beam will have γ &cong; 2,000,000. So if it's bent by a magnetic field the synchtrotron radiation will spread out in a cone of about 5×10-7 radians. If this beam was shooting at a target 0.1 light seconds away from where it was bent, the radiation would arrive at the target over a region with a radius of about 15 meters with a maximum intensity at about 7.5 meters and no radiation directly along the path the beam was pointing while it was being bent. If the beam was bent by the interplanetary magnetic field from our earlier example, it would produce light with photon energies of around 1.3 eV (near infrared) and the lost energy would be a few parts in a hundred trillion. If it was instead deflected by the Earth's magnetic field you might expect a peak photon energy of tens of keV – in the hard x-ray part of the spectrum – and the fraction of energy in the synchrotron radiation would be somewhere in the low parts in 100,000 or less.

==== Separation of neutral beams ====

Magnetic fields only affect charged objects. So one way to overcome magnetic defenses or environments is to use beams that have no net charge. For example, a proton beam could be neutralized with a co-propagating electron beam with an equal number of protons and electrons. Or, potentially, you could fire an ion beam and then neutralize the ions after acceleration to get a beam of neutral atoms.

But, the magnetic fields are still tugging at the charged particles making up the beam as they go past. This will pull the electrons one way and the protons or ions the other way. As the particles are pulled in opposite directions you get a net positive charge on one side of the beam and a net negative charge on the other side (called polarization). This sets up an electric field inside the beam that acts to pull the particles back together. If the particles are going fast enough and the magnetic field is strong enough, the magnetic force can still separate the charges and will split the beam in two, one with the positive charges and one with the negative charges. But if the particles are not that fast or the magnetic field is not that strong this polarization of the beam will keep it together and let it fly right through the magnetic field.

<table border=1><tr><td>
Let's figure out how to analyze this problem. For definiteness, we will have a beam with an electric current of the positive species of I and an electric current of electrons of -I, all moving in the z direction. The beam radius will be R. The particle speed will be v with the usual relativistic parameters β = v/c and γ = √[1/(1-β2)]. If you know the energy per particle E of the positive species and the mass m of the positive species, γ = 1 + E/(m c2) and then β = √[1 - 1/γ2].

Let's assume that the beam is going through a magnetic field of strength B at an angle θ to the field direction. The quantity B sin(θ) is the component of the magnetic field perpendicular to the direction of the beam.

The charge density ρ of the positive component of the beam will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρL = I / (π v R2).
</div>

It is easiest to analyze the problem in the rest frame of the particles in the beam. Now the particles are at rest. Particles at rest are not affected by a magnetic field, but the change of reference frame Lorentz transforms the initial magnetic field into a combination of electric and magnetic fields in the rest frame. We can ignore the magnetic field, but the electric field is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔A = γ v B sin(θ)
</div>
in a direction perpendicular to both v and B. For our purposes, assume the electric field points in the x direction.

At the same time, relativistic length contraction increases the charge density in the rest frame
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρ = γ ρL.
</div>

Let r be the vector from the beam axis to the point inside the beam. It will be useful to separate this into its Cartesian coordinates. If x and y are the unit vectors in the x and y directions, respectively, then
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
r = x x + y y
</div>

If we consider only the positive component of the beam, from [https://en.wikipedia.org/wiki/Gauss%27s_law Gauss's law] it will have an electric field inside of the beam at a distance r from the center axis of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔+ = ρ r / (2 ε0) = (ρ/(2 ε0)) (x x + y y)
</div>
where ε0 = 8.85×10-12 is the vacuum permittivity. Now let's displace the electrons in the beam by a distance δ in the x direction. By itself, this electron beam will have an electric field of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔- = -(ρ/(2 ε0)) ((x-δ) x + y y)
</div>
By the principle of superposition, the electric field inside the beam with both components is the sum of the electric fields of the separate components including the external field
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔B = [(ρ/(2 ε0)) δ + γ v B sin(θ)]x
</div>
Because the charges are free to move, they will do so as long as a field is applied to them. Thus, the beam becomes stable once the charges separate enough that the field internal to the beam is zero. Therefore
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
δ = -2 ε0 γ v B sin(θ) / ρ = -2 π ε0 v2 R2 B sin(θ) / I
</div>
If |δ| > 2 |R|, the beam will fully separate and each component will go its own way. If δ is small compared to R, the beam's polarization suffices to hold it together in the overlap region. However, where the positive and negative charges do not overlap the electric fields no longer cancel, and the external electric field carries away the outer polarized part of the beam. This gives us a rough threshold for which a magnetic field will separate a neutral beam into its charged components and bend them away from their target
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr><td> B sin(θ) &Lt; I / (π ε0 v2 R) <td>       <td> Beam gradually erodes.
<tr><td> B sin(θ) > I / (π ε0 v2 R) <td> <td> Beam promptly separates.
</table>
</div>
For a gradually eroding beam, you can figure that if the magnetic deflection of the positive component is not much larger than the beam radius, most of the beam will still hit its target.
<tr><td>
For a beam of neutral atoms, let a be the radius of the atom and let Eb be the binding energy of the least tightly bound electron to the atom. An external electric field in excess of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
|𝓔A| > Eb / (e a)
</div>
will promptly ionize the atom, ripping away its electron. Here, e = 1.60×10-19 C is the magnitude of the electric charge of the electron. For fields close to but a bit weaker than this, the electron can still rapidly pop off due to quantum tunneling<ref>Ph. Durand and I. Paidarová, “Ionization of the hydrogen atom: from weak to strong static electric fields,” The European Physical Journal D, Vol. 26, pg. 253-259, 2003.</ref>. In general, expect that for fields greater in magnitude than (very approximately) Eb / (10 e a) the electron will be quickly removed, while for fields weaker than this the electrons will stay on their atoms.

It is useful to note that Eb/e is just the binding energy in units of eV. The most weakly bound electrons are usually bound to their atom with about 10 eV of energy. Atoms are typically about 10-10 m in radius. Thus, you can expect fields of around 1010 to 1011 V/m to directly ionize atoms. If we call this threshold value 𝓔T, then a beam will be ionized when passing through a field with strength greater than
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B sin(θ) > 𝓔T / (γ v)
</div>
</table>

Example: Let's consider a 1 GeV neutralized proton beam from a synchrotron, with a beam current of 200 kA in each bunch (incidentally, if a bunch is 5 cm long this means that each bunch packs 38 kJ worth of energy). The speed of the bunch is 87% the speed of light, or 262 million m/s, with a γ of 2.07. Consider when the beam's focus takes it to a radius of 5 cm (or 0.05 m, for our calculation). Plugging this in, we find that the protons in the bunch will separate from the electrons in a field with a transverse component greater than about 2.1 T, and the bunch will pass through the field without much effect if the transverse component is much smaller than this.

Example: We'll take the same 1 GeV beam, but this time recombine the electrons with protons to make hydrogen atoms. An electron is bound to a hydrogen atom with an energy of 13.6 eV (the Ryberg constant) and the radius of the electron orbital in a hydrogen atom is 0.529×10-10 m (the Bohr radius). This gives a threshold electric field for prompt ionization of hydrogen of 2.57×1011 V/m. We will get this with a transverse component of the magnetic field of 474 T. Above this field the atoms promptly ionize and electrons are stripped away from the protons. For fields higher than roughly 1/10 of this – so about 40 to 50 T – you may still get significant ionization by quantum tunneling; lower than this your losses from the beam due to field-induced ionization are likely negligible.

==== Magnetic shields ====

What happens when we can't rely on magnetic fields in the environment to protect us? One idea is to produce powerful artificial magnetic fields around our spacecraft to deflect incoming particle beams. This is very similar in concept to using active magnetic shielding to protect against space radiation – [[Space_Radiation#Magnetic_Shielding|check out our article on space radiation for more]].
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr>
<td width=100>
[[File:Unconfined_FRC_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:racetrack_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:Magnetic_shielding_Halback_Array.png|200 px|frameless]]
<tr><td colspan=3>
Some potential configurations for magnetic shields. The currents generating the fields are in green, the magnetic fields themselves are shown in magenta<ref name="Battiston2012">R. Battiston, W.J. Burger, V. Calvelli, R. Musenich, V. Choutko, V.I. Datskov, A. Della Torre, F. Venditti,
C. Gargiulo, G. Laurenti, S. Lucidi, S. Harrison, and R. Meinke, "ARSSEM Active Radiation Shield for Space Exploration Missions", Final Report ESTEC Contract N° 4200023087/10/NL/AF : “Superconductive Magnet for Radiation Shielding of Human Spacecraft” (2012) https://arxiv.org/abs/1209.1907 https://www.researchgate.net/publication/265945847_Active_Radiation_Shield_for_Space_Exploration_Missions</ref><ref name="Desiati2022">Paolo Desiati and Elena D'Onghia, "CREW HaT: A Magnetic Shielding System for Space Habitats", arXiv:2209.13624 [physics.space-ph] https://doi.org/10.48550/arXiv.2209.13624</ref>.
</table>
</div>

One immediate issue is that you want to keep very high magnetic fields out of the crew area. Fields in excess of 0.5 T can start to be unpleasant, and it is recommended to keep the field below 0.2 T for continuous exposure.

Strong magnets can suffer from high self-forces acting to tear them apart. Most shielding designs will have multiple magnetic coils, which will exert high forces and torques on each other. This will result in the need for strong structural bracing to keep the magnets intact and in place. Such strong structural designs will impose significant extra mass on the spacecraft.

Consider a magnetic field configuration that can keep radiation out of a certain region of space. There is no trajectory of a charged radiation particle up to a certain energy that will allow those particles to enter the protected region. But the dynamics of charged particles in a magnetic field are the same whether going forward or backward in time. If there is no way for a particle to get in, by this time reversal symmetry any particle which is in that forbidden region can never escape. Consequently any field configuration that is a magnetic trap is also a magnetic shield; any field configuration that is a magnetic shield is also a magnetic trap. A spacecraft with a magnetic shield may end up surrounded by strong radiation belts.

Using a magnet to bend a weaponized electron beam very often ends up with the beam being bent so hard that it dumps most of its energy in the form of synchrotron radiation at gamma ray energies. Because gamma rays are not charged, they will not be deflected by the field. Thus, if you would have been hit by the electron beam you will be hit instead by the gamma rays.

As an example, consider a 1 TeV electron beam. We'll surround our spacecraft with a 10 T magnetic field in a 100 meter radius region just outside our spacecraft. This will deflect the beam by 30 meters, which is enough to make it miss. But we are not saved, because the beam dumps nearly all of its energy into synchrotron radiation with an energy spectrum of gamma rays up to 13 GeV. This radiation will hit us in a strip about 2.4 meters long and not much more than the width of the original beam with essentially all of the energy of the original beam.

==== Magnet drones ====

If a magnetic shield around your spacecraft will just end up killing you from synchrotron radiation, what if you put the magnetic shield on a drone and put that drone between yourself and the enemy trying to shoot you? This lets you use a much weaker magnet because you don't need nearly so high of a deflection angle to make the shot miss. This means that less beam energy will turn into synchrotron radiation that could kill you instead and will spread what synchrotron radiation does occur out over a much wider area when it reaches your spacecraft, hopefully so that most of it just misses your spacecraft.

It does mean that your enemy will start shooting your drones so that they can eventually take a shot at you. But hopefully by that time you've already cleared out their drones and cored their spacecraft with your own synchrocannon or UREB.

==== Plasma magnets ====

Usually, to make a really big magnetic field you would run current through a really big conductive loop. But really big things are heavy and expensive. One way around this could be to use the ambient space plasma to carry the electric current rather than any kind of wire. In order to make this current, you use the same mechanism used in a rotary induction electric motor: you generate a rotating magnetic field and induce a current in the conductive loop (in this case, the space plasma). How do you make a rotating magnetic field? The method proposed<ref name="Slough2006">John Slough, "The Plasma Magnet", NASA Institute for Advanced Concepts Phase II Final Report (2006) https://www.niac.usra.edu/files/studies/final_report/917Slough.pdf</ref> is to use an antenna of four much smaller current loops as electromagnets at 90&degree; to each other, and sinusoidally vary the current through the loops with a respective 90&degree; phase difference between them. When this is done, a circular current is induced in the space plasma and the self-force of a magnetic field on its generating current inflates the current loop to much larger radii. You end up with a current sheet in the plane of the rotating field's rotation circulating about the antenna nearly synchronously with the field rotation. This current then creates a normal dipolar magnetic field as with any current loop. The current sheet and the resulting magnetic field expand until the magnetic field's pressure equals that of the ambient space plasma. This gives an enormous region of magnetic field from a relatively small and lightweight system.

The system proposed in the initial study would use an antenna approximately 10 meters in radius and produce a region of field approximately 30 km in radius at Earth's orbit. The field inside would not be constant - at the edges it would be 50 nT, giving the right pressure to keep out the solar wind at Earth's orbit. A circulating current sheet produces a field that falls off as 1/r with distance, meaning that adjacent to the antenna the field would be about 0.15 mT. Using the methods in the box below, we can estimate that this particular design as part of a spacecraft's shields will not deflect a TeV electron beam by more than a few centimeters, which is usually insufficient to force a miss. However, it could potentially be useful for a drone magnet, where the much larger distance between the magnetic field and the target spacecraft would allow a small deflection angle to give perhaps tens or low hundreds of meters change in beam location near the target at near light second ranges. The relatively low field and gradual bend of the incident beam suggests that synchrotron radiation will not be an issue; however this does not exclude more powerful plasma magnet designs causing significant synchrotron radiation effects.

<table border=1><tr><td>
Calculating the deflection from a spatially varying magnetic field affected by the solar wind is no trivial task. However, we can make an approximation that both should be within about an order of magnitude of the true value and also represent an upper limit to the deflection angle; although it does neglect the influence of the driving rotating magnetic field which could in some cases increase the deflection.

We will take R0 as the radius of the antenna. B0 is the field strength at the antenna. Rm is the distance at which the magnetic influence stops because the magnetic pressure equals the space plasma pressure. Bm is the field strength at Rm. According to Slough<ref name="Slough2006"></ref>, the magnetic field strength B(r) at any distance from the antenna center r between R0 and Rm is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B(r) = B0 R0 / r.
</div>
Therefore, once we know Bm and Rm, B0 is completely determined
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B0 = Bm Rm / R0.
</div>

The force on a particle with charge q passing perpendicularly through a magnetic field B(r) at a speed v is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
F(r) = q v B(r)
</div>
For an electron beam, v is so close to c as to make no difference. If the particle is moving down the y direction and the field is in the z direction, the force will be in the x direction. We assume that the deflection will be small, so we can choose a straight-line path in the y direction offset from the center by an amount x. The plasma bubble radius is Rm, so the particle will enter at ymax = √[Rm2-x2] and leave at ymin = -√[Rm2-x2].

The momentum change the particle experiences from this force is the time integral of the force over its path. Because of reflection symmetry about y = 0, we only need to integrate from ymax to 0 and then multiply by 2
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] dt F(r)
<tr> <td> <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] (dy/v) F(r)
<tr> <td> <td> = <td align="left">2 q B(0) R0 [Integral from 0 to √[Rm2-x2]] dy 1/√[y2 + x2]
<tr> <td> <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/x]
</table>
</div>
This holds for |x| > R0. But the expression for the field is not valid for r < R0 and the integral will give nonsense and diverging results if we evaluate for |x| < R0. In this case, we can put the integral lower bound at r = R0, or y = √[R02-x2]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/(√[R02-x2] + R0)]
</table>
</div>

The deflection angle is now estimated as
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = px / p
</div>
where the total momentum of the particle p can be found from the methods in the first box in the Magnetic Fields section.

The deflection for a shot aimed right at the spacecraft at the position of the spacecraft can be estimated by setting x = 0 and only taking the first half of the integral (so ignore the factor of 2)
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
q B(0) R0 ln[Rm/R0] / (p Rm)
</div>
This is an additional overestimate because it fails to account for how the deflection builds up over distance during the integration, but it should still give something of an order-of-magnitude and upper limit estimate of what kind of deflections might be expected.
</table>

=== Electric fields ===

If electric fields can accelerate charged particles, maybe they can keep the charged particles in a weaponized particle beam away from your spacecraft? To completely block a particle beam, you would need a voltage on your spacecraft equal to the particle energy in eV of the particles in the beam. That is, if a 1 GeV proton beam is coming at you, you'd need an electric potential of a billion volts to stop that beam. This leads to three immediate problems.

The first is that particles of the opposite charge will be attracted to your spacecraft. Because space isn't really empty but rather a rarefied plasma (called the solar wind in a solar system or the interstellar medium out between stars), there will be plenty of particles of the opposite charge around to attract. And they will hit with an energy in eV equal to the voltage on your spacecraft. In our example of defending against a 1 GeV proton beam with a gigavolt field, the electrons from the solar wind will come in and hit with an energy of 1 GeV each. This will generate dangerous radiation.

The second problem is a consequence of the first. Each of those GeV electrons from our example need to get their energy from somewhere, which turns out to be from the electric field we are using as shielding. You will need to make up for this energy loss with power from your electrical system, which gives a constant drain on your power. Some estimates put the electrical effort to maintain a several hundred megavolt shield in space at many tens of gigawatts<ref name="Mechmann2019">Claire Mechmann, "Analysis of Proposed Active Radiation Shielding Design Concept for Spacecraft" (2019) Thesis, College of Engineering and Science of Florida Institute of Technology</ref>.

And the third problem is similar to one for magnetic fields. Your electrodes will experience high self forces and forces between separate electrodes that will try to tear your shield generator apart. Consequently you will need heavy structural reinforcement in order for your shield to work.

There are potential ways to mitigate some of these problems. Interested readers are directed to [[Space_Radiation#Electrostatic_Shielding|the active shielding section in our page on space radiation]] for more of the juicy details; there's no need to repeat the same stuff here.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Elctrostatic_active_shielding.png|200 px|frameless]]
<td>
[[File:Electrostatic_active_shielding_2.png|200 px|frameless]]
<tr><td colspan=2>
Proposed designs for electrostatic shields to protect from positively charged particles<ref name="Tripathi2006">Ram K. Tripathi, John W. Wilson, and Robert C. Youngquist, "Electrostatic Active Radiation Shielding - Revisited", 2006 IEEE Aerospace Conference, Big Sky, MT, USA, 2006, pp. 9 pp.-, doi: 10.1109/AERO.2006.1655760.</ref><ref name="Chowdhury2023">Rajarshi Pal Chowdhury, Luke A. Stegeman, Matthew L. Lund, Dan Fry, Stojan Madzunkov, and Amir A. Bahadori, "Hybrid methods of radiation shielding against deep-space radiation", Life Sciences in Space Research, Volume 38, 2023, Pages 67-78, ISSN 2214-5524, https://doi.org/10.1016/j.lssr.2023.04.004.</ref>.
</table>
</div>

== Damage ==

Particle beams are, first and foremost, devices that produce beams of [[Nuclear_radiation|ionizing radiation]]. The radiation of weaponizable particle beams is extremely penetrating; thus of the [[Particle_Accelerators#Energy_loss_in_matter|interaction mechanisms listed for radiation from particle beams]], we mostly need to worry about bremsstrahlung and nuclear collisions and these produce extensive secondary showers of radiation that increase the penetration of the radiation exposure. Because of this, any target hit by a particle beam weapon will be irradiated, and radiation dose will always be one of the primary damage mechanisms of a particle beam weapon strike. Further, this radiation will be distributed along a track passing deeply through the material of the target, allowing it to affect critical systems through heavy armor, structural material, robust machine components, and passive material such as fuel load, coolant, or cargo. As the radiation deposits its energy into the target material, that material will be heated. If the target is heated sufficiently, it may experience [[Beam-Target_Interactions|thermal or mechanical]] damage from melting, cooking, combustion, vapor explosions, and warping or cracking due to thermal expansion.

The discussion of the radiation penetration into armor also applies to the same radiation penetrating through the target. So the formulas and rules of thumb for estimating the depth of radiation penetration into armor can also be used to estimate how far into a tank or spacecraft a particle beam strike delivers effects.

In addition to the radiation and thermal effects, a particle beam can also produce a sharp electromagnetic pulse that can damage semiconductor electronics.

== 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, Gerrit Bruhaug, and Luke Campbell for writing

Gerrit Bruhaug and Luke Campbell for scientific research, math wrangling and advise

ATriffleMind and Luke W. Campbell for proof reading.

==References==

[[Category:Beams]][[Category:Physics & Engineering‏‎]][[Category:Engineering‏‎]][[Category:Military Technology]][[Category:Warfare‏‎]]

Particle Beam Weapons

2026-10-26T04:21:57Z

Lwcamp: /* Separation of neutral beams */

{{PageConstructionNotice}}

''For a general overview on particle accelerator technology, see [[Particle Accelerators]]

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.

::: Z = 1.32×102 (γ2-1)3/4 r / √I<ref name=USPAS2016>Martin Reiser, "Theory and Design of Charged Particle Beams", 2008, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim</ref>

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

::: Z = β γ c r √[m/T]

Where Z is the doubling distance like before, r is the initial beam radius, T is the beam temperature in eV, m is the ion mass and β, c, and γ 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.

== Defenses ==

Sure, it's great to have a great big particle cannon to blast your enemies. But what if you are the one being shot at? In that case, it's best to figure out ways to keep from being blown to kingdom come.

=== Armor ===

The simplest way to defend against the high energy particles of a particle beam is to pile a whole bunch of stuff up and put it in the way of the beam. Particle beams may produce extremely penetrating radiation, but enough stuff will block most of that radiation and may allow you to survive. The details depend on what kind of particle is being thrown at you, and are explained below. However, a general rule of thumb is that you probably want about 10 tons of material per square meter of surface you want to armor in order to get sufficient protection.

==== Electrons ====

At the very high energy of an electron accelerated by a weapons-grade particle beam, the primary interaction with matter will be the emission of bremsstrahlung photons in the gamma-ray range as the electron slams into particles of matter. These high energy photons themselves primarily interact by the pair production mechanism – the photon disappears, but leaves behind a highly energetic positron and electron. These two particles are themselves energetic enough to produce further bremsstrahlung, and so the process continues. You get the formation of an electron-gamma shower (also called an electromagnetic cascade), which only stops after either the particles have penetrated through the target or the initial energy of the electron has been spread through so many shower particles (including all the electrons, positrons, and photons) that no particle has enough individual energy to pair produce or to be likely to bremsstrahlung again before it is stopped by other mechanisms.

To stop electrons, you'll want the atoms in your armor to be as heavy as you can practically get them. High atomic number both encourages bremsstrahlung and pair production, giving a double whammy for stopping that cascade. However, occasionally one of those electrons, positrons, or gammas will hit a nucleus and knock out neutrons. So you probably want a back-up layer of several tens of centimeters of hydrogen rich material such as plastic or water mixed with neutron absorbing materials like boron, lithium, cadmium, or gadolinium. High density polyethylene loaded with 5% boron by weight is a typical go-to neutron blocker.

The actual methods to calculate the penetration of the electromagnetic cascade is given are the box below. However, for some examples:
<ul>
<li> A 10 GeV beam incident on thick iron (or steel) will stop cascading at about 9 cm, will deposit half of its energy in the first 12 cm, and will have dumped 98% of its energy within 35 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 2.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 100 GeV beam incident on thick iron (or steel) will stop cascading at about 13 cm, will deposit half of its energy in the first 16 cm, and will have dumped 98% of its energy within 47 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 3.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick iron (or steel) will stop cascading at about 17 cm, will deposit half of its energy in the first 20 cm, and will have dumped 98% of its energy within 59 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 4.6 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick tungsten will stop cascading at about 3.5 cm, will deposit half of its energy in the first 4 cm, and will have dumped 98% of its energy within 12 cm. 90% of the cascade will be contained within about 0.93 cm and 95% of the cascade will be contained within 1.86 cm of the beam axis. You will get 98% protection with a mass thickness of 2.3 tons/m2 and an additional 1/e reduction for each further 0.13 tons/m2 beyond that.
<li> A 1 TeV beam incident on water will stop cascading at about 290 cm, will deposit half of its energy in the first 340 cm, and will have dumped 98% of its energy within 1030 cm. 90% of the cascade will be contained within about 11 cm and 95% of the cascade will be contained within 22 cm of the beam axis. You will get 98% protection with a mass thickness of 10.3 tons/m2 and an additional 1/e reduction for each further 0.72 tons/m2 beyond that.
</ul>
Note that for sufficiently intense particle beams, 98% protection may not be enough. You might need to throw on several more tons per square meter to be sure.

<table border=1>
<tr><td> How Far it Goes:
(This section follows methods presented in Leroy and Rancoita<ref>C. Leroy and P-G Rancoita, "Physics of cascading shower generation and propagation in matter: principles of high-energy, ultrahigh-energy and compensating calorimetry", Reports on Progress in Physics 63 (2000) 505–606. DOI: 10.1088/0034-4885/63/4/202</ref>.)

When trying to estimate the extent of the irradiated region of the target, it is helpful to know the radiation length X0 in the material. This is the distance that an electron will travel, on average, before emitting a bremsstrahlung photon. Conveniently, it is also the distance a high energy photon will travel through the material before pair producing. The radiation length can be found from
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   = 4 α  
<td nowrap="nowrap"> NA
<td rowspan="2" nowrap="nowrap" align="right">   Z (Z+ζ) re2 ln
<td rowspan="2" nowrap="nowrap" align="right"> [
<td nowrap="nowrap"> 183
<td rowspan="2" nowrap="nowrap" align="right"> ]
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> A
<td style="border-top:solid 1px black;"> Z1/3
</table>
</div>
where α = 1/137 is the fine structure constant, NA = 6.02214076×1023 is Avogadro's number (or one mole), A is the atomic weight of the atoms in the material (usually expressed in grams per mole), Z is the atomic number (nuclear charge) of the atoms, and re = 2.8179403205×10-13 cm is the classical electron radius. ζ is a constant of approximately 1.3. This gives the thickness of material in terms of the areal density (mass of material per unit area perpendicular to the beam) – if you use A in g/mol and re in cm the result will be in cm2/g and X0 will be in g/cm2. If your target is made up of more than one kind of atom, you can get the total radiation length by finding the radiation length of each atom of type i, X0i and using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   =  
<td rowspan="2" nowrap="nowrap" align="right"> Σi  
<td nowrap="nowrap"> fi
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> X0i
</table>
</div>
where fi is the fraction by weight of the atom type i. Once you have the radiation length in terms of areal density, you can easily convert it to distance by dividing by the density ρ. If X0i is in g/cm2 and density is in g/cm3, then X0i/ρ will be in cm.

We will also need to know the critical energy εc. This is the energy where an electron is losing energy as rapidly to ionization as it is to radiation. That is, it is the energy at which the cascade stops multiplying, even though the individual particles already made can continue traveling for some distance. The critical energy can be estimated using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> εc = B  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> Z X0
<td rowspan="2" nowrap="nowrap" align="right"> )
<td nowrap="nowrap"> h
<tr>
<td style="border-top:solid 1px black;"> A
<td>  
</table>
</div>
where B = 2.66 MeV and h = 1.11.

We can now describe the distribution of radiation in the target material. The depth tmax where the cascade stops multiplying (in g/cm2) for an incident electron of energy E (in MeV) is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tmax = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> - 1
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
Similarly, the depth where half the energy of the beam has been deposited (in g/cm2) is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tcg = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> + 0.37
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
As usual, you can convert to a depth in cm by dividing by the material density. 98% of the beam energy will be deposited within a distance of &cong; 3 tcg. Beyond this, the beam energy will continue to fall off with thickness t approximately as an exponential exp[-t/λ], where the characteristic attenuation length λ &cong; 2 X0.

The width of a cascade, where 90% of the energy is contained, is approximately given by
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> RM =  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> EM
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
where EM &cong; 21.2 MeV. 95% of the radiation will be contained within a distance of 2 RM from the beam axis.
</table>

==== Protons and Ions ====

<table align=right><tr><td>
[[File:Regolith_Shielding.png|350 px|frameless]]
<tr><td width=350>
Relative effect of particle radiation (compared to no shielding) from galactic cosmic rays behind different thicknesses of water, aluminum, and lunar regolith<ref name="Slaba2022">Tony C. Slaba, "Radiation Shielding for Lunar Missions: Regolith Considerations", LSIC Crosstalk 7/18/2022 https://lsic.jhuapl.edu/uploadedDocs/focus-files/1604-E&C%20+%20EE%20Monthly%20Meeting%20-%202022%2007%20July_Presentation%20-%20NASA%20Slaba.pdf</ref>.
</table>
Protons and ions are grouped together because ions are made of protons and neutrons and when you accelerate these things for a weapon it is the protons and neutrons that are going to be doing the heavy lifting for blasting your target. This is because protons and neutrons are both about 2000 times more massive than electrons so any electrons along for the ride will contribute negligibly to the energy. Protons and neutrons together are called nuclear particles, and both behave quite similarly in nuclear collisions.

Nuclear particles are themselves made of quarks and can interact via the strong nuclear force. At the energies appropriate for weapons use, essentially all of the way protons or ions are blocked is through [[Particle_Accelerators#Nuclear_collisions|direct collisions with the atomic nuclei]] of the armor material. One or more quarks in the projectile will slam into quarks in the target nucleus and stretch them away from the rest of the quarks in the nuclear particles. Physics tells us that you can't have isolated quarks, so the "string" attaching the quark to its neighbors stretches and stretches until it snaps. Snapping releases so much energy that more quarks are made that clump together into all sorts of other nuclear particles, both exotic and familiar. This process is called a hadronic jet. The only particles produced in these jets long-lived enough to travel any noticeable distance through the armor material will be protons, neutrons, the charged pi mesons (or pions), and some of the K mesons (or kaons). The particles too short lived to go far enough to matter will decay into other particles, and these particles are often very high energy gamma rays. Much like with electrons, the long-lived secondary particles can be sufficiently energetic to smash into other nuclei and cause more hadronic jets (or, if they are low enough in energy, just knock more nuclear particles out of target nuclei) and this produces a similar shower or cascade. But in addition, the armor nuclei that were hit will be very hot after the impact from all of the energy given to them from the incident particles. These nuclei will cool off by evaporation – but instead of evaporating away molecules, they evaporate off nuclear particles from the nucleus. These evaporated particles are mostly neutrons, so any armor against proton or ion beam strikes will produce quite a bit of neutron radiation.

Eventually, the original particle's energy will be distributed among so many other particles that the charged particles will come to a stop by making ion tracks rather than from colliding with more nuclei, and the cascade halts. But you still have all the neutrons to worry about. And the gamma rays as well, which are high enough energy to kick off an electromagnetic cascade in the same way that electrons do.

Unlike electrons, which have reasonable approximations of their cascade behavior from relatively simple equations, the cascades of protons and ions usually require sophisticated simulations or experiments to get the resulting distribution of radiation. However, we can get a reasonable idea of their general behavior by comparing them to [[Space_Radiation#Shielding_Against_Cosmic_Rays|cosmic ray particles interacting with shielding]]. Cosmic rays are energetic protons and ions, after all. From this we see that a relatively small amount of armor – in the range of 1 to 3 tons per square meter – can actually make the radiation dose worse. The incident particles will knock off so many secondary particles in their shower that the dose is multiplied from so many tracks passing through the target. Hydrogen rich materials such as water and plastics work the best against proton and ion beam particles; heavy elements are the worst and produce the most secondaries. You could probably obtain adequate radiation protection from most incident beams with about 10 tons per square meter of hydrogen rich material as armor. However, you will want to add neutron absorbing elements to your hydrogen rich armor, such as lithium, boron, cadmium, or gadolinium. Further, in order to deal with the gamma rays you will want a layer of at least several centimeters of heavy elements, such as tungsten or lead, behind the hydrogen rich particle armor.

==== Structural Considerations ====

When your armor material absorbs a radiation beam, the energy of that radiation turns into heat inside the armor where the radiation was absorbed. If the beam pulse has enough energy as was sufficiently tightly focused, this heat can damage your armor. Your armor may melt in places, or thermal expansion could cause it to crack or warp. Perhaps the worst thing it could do is vaporize or flash into a plasma, because the pressure from the vapor or plasma can be sufficient to plastically deform the armor. This leaves a hole along the beam path where the expanding gas or plasma blasted out a long, deep crater. This means a beam that fires rapid shots spaced maybe milliseconds apart can have subsequent shots go down the same hole so your armor will not protect against them. Each shot will dig out a deeper hole until either your armor is penetrated or the attacking beam runs out of shots in its pulse train.

A consequence of this is that when designing your armor you may wish to use refractory materials with a high structural strength (like tungsten), or that is otherwise resistant to being blasted apart by its own evaporated remains.

=== Magnetic fields ===

[[Particle_Accelerators#Magnetic_fields|A charged particle crossing a magnetic field will bend so as to spiral around the field lines]]. If you try to shoot your particle beam through a magnetic field, it will bend away from the direction you shot it. This can make it hard to hit your target. Unfortunately for you, there are magnetic fields everywhere – around planets, and even trapped in the solar wind as it crosses the solar system.

As an example, the Earth has a magnetic field strength of about 5×10-5 tesla near its surface, and falling off farther away. If you are shooting a 1 TeV electron beam past the Earth, your beam is likely to bend by several milliradians (with the exact deflection depending on how close you are skimming the Earth). If you are in a medium Earth orbit and shooting at another target also in a medium Earth orbit on a trajectory that goes near the Earth, then without correcting for the magnetic field your beam will likely be off-target by tens to hundreds of kilometers.

For another example, the interplanetary magnetic field is about 10-9 tesla. If you shoot that same 1 TeV electron beam at a target 0.1 light seconds away, you are likely to be off by a couple hundred meters.

<table border=1>
<tr><td>
The radius of curvature of the bend of a particle in a magnetic field is called the gyroradius rc. For a particle with a charge q, mass m, and kinetic energy E, find the relativistic momentum p
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
p = √[E2 - m2 c4] / c
</div>
To convert from energy units of eV to joules, multiply by 1.60×10-19. An electron has a mass of 9.11×10-31 kg, a proton of 1.67×10-27 kg, and an ion of 1.66×10-27×A kg, where A is the atomic weight of the ion. The speed of light c = 3.00×108 m/s.

In a magnetic field of strength B with an angle θ between the field direction and the direction of motion
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rc = |p sin(θ) / (q B)|
</div>
For electrons, q = -e, where the fundamental charge e = 1.60×10-19 coulomb. For a proton, q = +e. For an ion, multiply e by the charge state of the ion.

If the field acts over a distance d that is much smaller than the gyroradius, the deflection angle φ is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = d / rc
</div>
The deflection x at a target at a range R for small φ (φ &Lt; 1) will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
x = φ R.
</div>

Note that d will always be less than or equal to x. Also note that if d is not smaller than twice the gyroradius the beam will never reach the target because it will completely spiral around the field lines before it reaches the target.
</table>

So, it is clear that as the defender it will be helpful to hang out in areas with high magnetic fields, and keep some distance between you and the attacker. Meanwhile, the attacker will benefit from being closer where small deflections in the beam trajectory don't translate into large excursions from where the beam is aimed (and also because there will be less space filled with fields between the two of you). But the attacker can also try to compensate for known fields. If you happen to know that the magnetic fields are between you and your target, you can adjust your fire so that the deflection of the field bends your beam onto the target. This is not a simple task. It is easy enough to measure the field strength and direction where you are, but predicting how it changes over the path of your shot does not have an obvious solution. You can also try to gain intelligence on likely battle sites before you ever fight on them – for example, getting good data on the structure of the fields of a planet using spies and technical intelligence during peacetime (heck, on Earth this information is published for free in academic journals and by government agencies such as NOAA). Planetary magnetospheres are also affected by the solar wind, so different solar weather will change the fields you have to shoot through. But space weather forecasts and predictive simulations of the solar wind interacting with a planetary magnetosphere can give you a good idea of what the field structure is like. But even with all that, there are changes in the fields that happen on a scale of around a second ... anyone who has ever seen the auroras dancing across the sky in the polar night can see this. Compensating for these changes will be a significant challenge.

What would really help is being able to see the path of your own beam. This would allow you to see how far off you were from your target and in what direction, and then adjust fire to walk your shots onto the target. Unfortunately, particle beams in space are invisible. There might be ways to track them, but if so we don't know what they are.

==== Synchrotron radiation ====

When charged particle trajectories are bent, they emit [[Particle_Accelerators#Synchrotron_radiation|synchrotron radiation]]. This can cause the particle beam to lose energy, but if the bend is too tight it can also generate high energy gamma rays that ignore magnetic fields and directly strike the target. Synchrotron radiation can generally be ignored for proton or ion beams, but can be significant for electron beams.

<table border=1><tr><td>
For a beam particle with energy E and mass m, the synchrotron radiation is concentrated in a forward-directed cone with an opening angle of approximately 1/γ, where γ = 1 + E/(m c2). (Interestingly, the radiation intensity peaks an at angle of 1/(2 γ) and vanishes directly on-axis.) Because the beam is bending, you will get emission along the plane along which the beam is being bent. The energy of the photons produced during synchrotron emission is approximately<ref name="Jackson_Electrodynamics">J. D. Jackson, "Classical Electrodynamics: Second Edition", Wiley (1962)</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Eγ &cong; 3 γ3 c &hbar; / rc
</div>
where &hbar; = 1.05×10-34 J s (if you want the energy in joules) or &hbar; = 6.58×10-16 eV s (if you want the energy in eV). The photon energy will not exceed the energy of the incident particles, if you get a photon energy higher than the energy of the incident particles it just means that the synchrotron radiation spectrum extends up to the incident particle energy.

The fraction of the energy (or power) of the beam turned into synchrotron radiation will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = [1/(4 π ε0)] [4 π / 3] [q² / rc] β2 γ4 [φ / (2 π)] / E
</div>
where β = √[1 - 1/γ2] is the particle's speed measured as a fraction of the speed of light, and 1/(4 π ε0) = 8.99×109 is the Coulomb constant. For highly relativistic electrons (like all practical electron beam weapons emit), β &cong; 1 and we can use
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = 8.85×10-2 [E (GeV)3 [φ / (2 π)] / (rc (m))]
</div>
This fraction cannot be more than 1. If f gets close to one, it means that our original assumption that the beam particles mostly keep their original speed no longer holds, and the beam will lose most or almost all of its energy. Instead, when the beam is being deflected it will only make it through an angle of φ/f and will irradiate a region of length x/f starting at where the beam would have been incident if not for the magnets and extending in the direction the beam was deflected.
</table>

For example, a 1 TeV electron beam will have γ &cong; 2,000,000. So if it's bent by a magnetic field the synchtrotron radiation will spread out in a cone of about 5×10-7 radians. If this beam was shooting at a target 0.1 light seconds away from where it was bent, the radiation would arrive at the target over a region with a radius of about 15 meters with a maximum intensity at about 7.5 meters and no radiation directly along the path the beam was pointing while it was being bent. If the beam was bent by the interplanetary magnetic field from our earlier example, it would produce light with photon energies of around 1.3 eV (near infrared) and the lost energy would be a few parts in a hundred trillion. If it was instead deflected by the Earth's magnetic field you might expect a peak photon energy of tens of keV – in the hard x-ray part of the spectrum – and the fraction of energy in the synchrotron radiation would be somewhere in the low parts in 100,000 or less.

==== Separation of neutral beams ====

Magnetic fields only affect charged objects. So one way to overcome magnetic defenses or environments is to use beams that have no net charge. For example, a proton beam could be neutralized with a co-propagating electron beam with an equal number of protons and electrons. Or, potentially, you could fire an ion beam and then neutralize the ions after acceleration to get a beam of neutral atoms.

But, the magnetic fields are still tugging at the charged particles making up the beam as they go past. This will pull the electrons one way and the protons or ions the other way. As the particles are pulled in opposite directions you get a net positive charge on one side of the beam and a net negative charge on the other side (called polarization). This sets up an electric field inside the beam that acts to pull the particles back together. If the particles are going fast enough and the magnetic field is strong enough, the magnetic force can still separate the charges and will split the beam in two, one with the positive charges and one with the negative charges. But if the particles are not that fast or the magnetic field is not that strong this polarization of the beam will keep it together and let it fly right through the magnetic field.

<table border=1><tr><td>
Let's figure out how to analyze this problem. For definiteness, we will have a beam with an electric current of the positive species of I and an electric current of electrons of -I, all moving in the z direction. The beam radius will be R. The particle speed will be v with the usual relativistic parameters β = v/c and γ = √[1/(1-β2)]. If you know the energy per particle E of the positive species and the mass m of the positive species, γ = 1 + E/(m c2) and then β = √[1 - 1/γ2].

Let's assume that the beam is going through a magnetic field of strength B at an angle θ to the field direction. The quantity B sin(θ) is the component of the magnetic field perpendicular to the direction of the beam.

The charge density ρ of the positive component of the beam will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρL = I / (π v R2).
</div>

It is easiest to analyze the problem in the rest frame of the particles in the beam. Now the particles are at rest. Particles at rest are not affected by a magnetic field, but the change of reference frame Lorentz transforms the initial magnetic field into a combination of electric and magnetic fields in the rest frame. We can ignore the magnetic field, but the electric field is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔A = γ v B sin(θ)
</div>
in a direction perpendicular to both v and B. For our purposes, assume the electric field points in the x direction.

At the same time, relativistic length contraction increases the charge density in the rest frame
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρ = γ ρL.
</div>

Let r be the vector from the beam axis to the point inside the beam. It will be useful to separate this into its Cartesian coordinates. If x and y are the unit vectors in the x and y directions, respectively, then
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
r = x x + y y
</div>

If we consider only the positive component of the beam, from [https://en.wikipedia.org/wiki/Gauss%27s_law Gauss's law] it will have an electric field inside of the beam at a distance r from the center axis of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔+ = ρ r / (2 ε0) = (ρ/(2 ε0)) (x x + y y)
</div>
where ε0 = 8.85×10-12 is the vacuum permittivity. Now let's displace the electrons in the beam by a distance δ in the x direction. By itself, this electron beam will have an electric field of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔- = -(ρ/(2 ε0)) ((x-δ) x + y y)
</div>
By the principle of superposition, the electric field inside the beam with both components is the sum of the electric fields of the separate components including the external field
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔B = [(ρ/(2 ε0)) δ + γ v B sin(θ)]x
</div>
Because the charges are free to move, they will do so as long as a field is applied to them. Thus, the beam becomes stable once the charges separate enough that the field internal to the beam is zero. Therefore
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
δ = -2 ε0 γ v B sin(θ) / ρ = -2 π ε0 v2 R2 B sin(θ) / I
</div>
If |δ| > 2 |R|, the beam will fully separate and each component will go its own way. If δ is small compared to R, the beam's polarization suffices to hold it together in the overlap region. However, where the positive and negative charges do not overlap the electric fields no longer cancel, and the external electric field carries away the outer polarized part of the beam. This gives us a rough threshold for which a magnetic field will separate a neutral beam into its charged components and bend them away from their target
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr><td> B sin(θ) &Lt; I / (π ε0 v2 R) <td>       <td> Beam gradually erodes.
<tr><td> B sin(θ) > I / (π ε0 v2 R) <td> <td> Beam promptly separates.
</table>
</div>
<tr><td>
For a beam of neutral atoms, let a be the radius of the atom and let Eb be the binding energy of the least tightly bound electron to the atom. An external electric field in excess of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
|𝓔A| > Eb / (e a)
</div>
will promptly ionize the atom, ripping away its electron. Here, e = 1.60×10-19 C is the magnitude of the electric charge of the electron. For fields close to but a bit weaker than this, the electron can still rapidly pop off due to quantum tunneling<ref>Ph. Durand and I. Paidarová, “Ionization of the hydrogen atom: from weak to strong static electric fields,” The European Physical Journal D, Vol. 26, pg. 253-259, 2003.</ref>. In general, expect that for fields greater in magnitude than (very approximately) Eb / (10 e a) the electron will be quickly removed, while for fields weaker than this the electrons will stay on their atoms.

It is useful to note that Eb/e is just the binding energy in units of eV. The most weakly bound electrons are usually bound to their atom with about 10 eV of energy. Atoms are typically about 10-10 m in radius. Thus, you can expect fields of around 1010 to 1011 V/m to directly ionize atoms. If we call this threshold value 𝓔T, then a beam will be ionized when passing through a field with strength greater than
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B sin(θ) > 𝓔T / (γ v)
</div>
</table>

Example: Let's consider a 1 GeV neutralized proton beam from a synchrotron, with a beam current of 200 kA in each bunch (incidentally, if a bunch is 5 cm long this means that each bunch packs 38 kJ worth of energy). The speed of the bunch is 87% the speed of light, or 262 million m/s, with a γ of 2.07. Consider when the beam's focus takes it to a radius of 5 cm (or 0.05 m, for our calculation). Plugging this in, we find that the protons in the bunch will separate from the electrons in a field with a transverse component greater than about 2.1 T, and the bunch will pass through the field without much effect if the transverse component is much smaller than this.

Example: We'll take the same 1 GeV beam, but this time recombine the electrons with protons to make hydrogen atoms. An electron is bound to a hydrogen atom with an energy of 13.6 eV (the Ryberg constant) and the radius of the electron orbital in a hydrogen atom is 0.529×10-10 m (the Bohr radius). This gives a threshold electric field for prompt ionization of hydrogen of 2.57×1011 V/m. We will get this with a transverse component of the magnetic field of 474 T. Above this field the atoms promptly ionize and electrons are stripped away from the protons. For fields higher than roughly 1/10 of this – so about 40 to 50 T – you may still get significant ionization by quantum tunneling; lower than this your losses from the beam due to field-induced ionization are likely negligible.

==== Magnetic shields ====

What happens when we can't rely on magnetic fields in the environment to protect us? One idea is to produce powerful artificial magnetic fields around our spacecraft to deflect incoming particle beams. This is very similar in concept to using active magnetic shielding to protect against space radiation – [[Space_Radiation#Magnetic_Shielding|check out our article on space radiation for more]].
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr>
<td width=100>
[[File:Unconfined_FRC_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:racetrack_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:Magnetic_shielding_Halback_Array.png|200 px|frameless]]
<tr><td colspan=3>
Some potential configurations for magnetic shields. The currents generating the fields are in green, the magnetic fields themselves are shown in magenta<ref name="Battiston2012">R. Battiston, W.J. Burger, V. Calvelli, R. Musenich, V. Choutko, V.I. Datskov, A. Della Torre, F. Venditti,
C. Gargiulo, G. Laurenti, S. Lucidi, S. Harrison, and R. Meinke, "ARSSEM Active Radiation Shield for Space Exploration Missions", Final Report ESTEC Contract N° 4200023087/10/NL/AF : “Superconductive Magnet for Radiation Shielding of Human Spacecraft” (2012) https://arxiv.org/abs/1209.1907 https://www.researchgate.net/publication/265945847_Active_Radiation_Shield_for_Space_Exploration_Missions</ref><ref name="Desiati2022">Paolo Desiati and Elena D'Onghia, "CREW HaT: A Magnetic Shielding System for Space Habitats", arXiv:2209.13624 [physics.space-ph] https://doi.org/10.48550/arXiv.2209.13624</ref>.
</table>
</div>

One immediate issue is that you want to keep very high magnetic fields out of the crew area. Fields in excess of 0.5 T can start to be unpleasant, and it is recommended to keep the field below 0.2 T for continuous exposure.

Strong magnets can suffer from high self-forces acting to tear them apart. Most shielding designs will have multiple magnetic coils, which will exert high forces and torques on each other. This will result in the need for strong structural bracing to keep the magnets intact and in place. Such strong structural designs will impose significant extra mass on the spacecraft.

Consider a magnetic field configuration that can keep radiation out of a certain region of space. There is no trajectory of a charged radiation particle up to a certain energy that will allow those particles to enter the protected region. But the dynamics of charged particles in a magnetic field are the same whether going forward or backward in time. If there is no way for a particle to get in, by this time reversal symmetry any particle which is in that forbidden region can never escape. Consequently any field configuration that is a magnetic trap is also a magnetic shield; any field configuration that is a magnetic shield is also a magnetic trap. A spacecraft with a magnetic shield may end up surrounded by strong radiation belts.

Using a magnet to bend a weaponized electron beam very often ends up with the beam being bent so hard that it dumps most of its energy in the form of synchrotron radiation at gamma ray energies. Because gamma rays are not charged, they will not be deflected by the field. Thus, if you would have been hit by the electron beam you will be hit instead by the gamma rays.

As an example, consider a 1 TeV electron beam. We'll surround our spacecraft with a 10 T magnetic field in a 100 meter radius region just outside our spacecraft. This will deflect the beam by 30 meters, which is enough to make it miss. But we are not saved, because the beam dumps nearly all of its energy into synchrotron radiation with an energy spectrum of gamma rays up to 13 GeV. This radiation will hit us in a strip about 2.4 meters long and not much more than the width of the original beam with essentially all of the energy of the original beam.

==== Magnet drones ====

If a magnetic shield around your spacecraft will just end up killing you from synchrotron radiation, what if you put the magnetic shield on a drone and put that drone between yourself and the enemy trying to shoot you? This lets you use a much weaker magnet because you don't need nearly so high of a deflection angle to make the shot miss. This means that less beam energy will turn into synchrotron radiation that could kill you instead and will spread what synchrotron radiation does occur out over a much wider area when it reaches your spacecraft, hopefully so that most of it just misses your spacecraft.

It does mean that your enemy will start shooting your drones so that they can eventually take a shot at you. But hopefully by that time you've already cleared out their drones and cored their spacecraft with your own synchrocannon or UREB.

==== Plasma magnets ====

Usually, to make a really big magnetic field you would run current through a really big conductive loop. But really big things are heavy and expensive. One way around this could be to use the ambient space plasma to carry the electric current rather than any kind of wire. In order to make this current, you use the same mechanism used in a rotary induction electric motor: you generate a rotating magnetic field and induce a current in the conductive loop (in this case, the space plasma). How do you make a rotating magnetic field? The method proposed<ref name="Slough2006">John Slough, "The Plasma Magnet", NASA Institute for Advanced Concepts Phase II Final Report (2006) https://www.niac.usra.edu/files/studies/final_report/917Slough.pdf</ref> is to use an antenna of four much smaller current loops as electromagnets at 90&degree; to each other, and sinusoidally vary the current through the loops with a respective 90&degree; phase difference between them. When this is done, a circular current is induced in the space plasma and the self-force of a magnetic field on its generating current inflates the current loop to much larger radii. You end up with a current sheet in the plane of the rotating field's rotation circulating about the antenna nearly synchronously with the field rotation. This current then creates a normal dipolar magnetic field as with any current loop. The current sheet and the resulting magnetic field expand until the magnetic field's pressure equals that of the ambient space plasma. This gives an enormous region of magnetic field from a relatively small and lightweight system.

The system proposed in the initial study would use an antenna approximately 10 meters in radius and produce a region of field approximately 30 km in radius at Earth's orbit. The field inside would not be constant - at the edges it would be 50 nT, giving the right pressure to keep out the solar wind at Earth's orbit. A circulating current sheet produces a field that falls off as 1/r with distance, meaning that adjacent to the antenna the field would be about 0.15 mT. Using the methods in the box below, we can estimate that this particular design as part of a spacecraft's shields will not deflect a TeV electron beam by more than a few centimeters, which is usually insufficient to force a miss. However, it could potentially be useful for a drone magnet, where the much larger distance between the magnetic field and the target spacecraft would allow a small deflection angle to give perhaps tens or low hundreds of meters change in beam location near the target at near light second ranges. The relatively low field and gradual bend of the incident beam suggests that synchrotron radiation will not be an issue; however this does not exclude more powerful plasma magnet designs causing significant synchrotron radiation effects.

<table border=1><tr><td>
Calculating the deflection from a spatially varying magnetic field affected by the solar wind is no trivial task. However, we can make an approximation that both should be within about an order of magnitude of the true value and also represent an upper limit to the deflection angle; although it does neglect the influence of the driving rotating magnetic field which could in some cases increase the deflection.

We will take R0 as the radius of the antenna. B0 is the field strength at the antenna. Rm is the distance at which the magnetic influence stops because the magnetic pressure equals the space plasma pressure. Bm is the field strength at Rm. According to Slough<ref name="Slough2006"></ref>, the magnetic field strength B(r) at any distance from the antenna center r between R0 and Rm is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B(r) = B0 R0 / r.
</div>
Therefore, once we know Bm and Rm, B0 is completely determined
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B0 = Bm Rm / R0.
</div>

The force on a particle with charge q passing perpendicularly through a magnetic field B(r) at a speed v is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
F(r) = q v B(r)
</div>
For an electron beam, v is so close to c as to make no difference. If the particle is moving down the y direction and the field is in the z direction, the force will be in the x direction. We assume that the deflection will be small, so we can choose a straight-line path in the y direction offset from the center by an amount x. The plasma bubble radius is Rm, so the particle will enter at ymax = √[Rm2-x2] and leave at ymin = -√[Rm2-x2].

The momentum change the particle experiences from this force is the time integral of the force over its path. Because of reflection symmetry about y = 0, we only need to integrate from ymax to 0 and then multiply by 2
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] dt F(r)
<tr> <td> <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] (dy/v) F(r)
<tr> <td> <td> = <td align="left">2 q B(0) R0 [Integral from 0 to √[Rm2-x2]] dy 1/√[y2 + x2]
<tr> <td> <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/x]
</table>
</div>
This holds for |x| > R0. But the expression for the field is not valid for r < R0 and the integral will give nonsense and diverging results if we evaluate for |x| < R0. In this case, we can put the integral lower bound at r = R0, or y = √[R02-x2]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/(√[R02-x2] + R0)]
</table>
</div>

The deflection angle is now estimated as
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = px / p
</div>
where the total momentum of the particle p can be found from the methods in the first box in the Magnetic Fields section.

The deflection for a shot aimed right at the spacecraft at the position of the spacecraft can be estimated by setting x = 0 and only taking the first half of the integral (so ignore the factor of 2)
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
q B(0) R0 ln[Rm/R0] / (p Rm)
</div>
This is an additional overestimate because it fails to account for how the deflection builds up over distance during the integration, but it should still give something of an order-of-magnitude and upper limit estimate of what kind of deflections might be expected.
</table>

=== Electric fields ===

If electric fields can accelerate charged particles, maybe they can keep the charged particles in a weaponized particle beam away from your spacecraft? To completely block a particle beam, you would need a voltage on your spacecraft equal to the particle energy in eV of the particles in the beam. That is, if a 1 GeV proton beam is coming at you, you'd need an electric potential of a billion volts to stop that beam. This leads to three immediate problems.

The first is that particles of the opposite charge will be attracted to your spacecraft. Because space isn't really empty but rather a rarefied plasma (called the solar wind in a solar system or the interstellar medium out between stars), there will be plenty of particles of the opposite charge around to attract. And they will hit with an energy in eV equal to the voltage on your spacecraft. In our example of defending against a 1 GeV proton beam with a gigavolt field, the electrons from the solar wind will come in and hit with an energy of 1 GeV each. This will generate dangerous radiation.

The second problem is a consequence of the first. Each of those GeV electrons from our example need to get their energy from somewhere, which turns out to be from the electric field we are using as shielding. You will need to make up for this energy loss with power from your electrical system, which gives a constant drain on your power. Some estimates put the electrical effort to maintain a several hundred megavolt shield in space at many tens of gigawatts<ref name="Mechmann2019">Claire Mechmann, "Analysis of Proposed Active Radiation Shielding Design Concept for Spacecraft" (2019) Thesis, College of Engineering and Science of Florida Institute of Technology</ref>.

And the third problem is similar to one for magnetic fields. Your electrodes will experience high self forces and forces between separate electrodes that will try to tear your shield generator apart. Consequently you will need heavy structural reinforcement in order for your shield to work.

There are potential ways to mitigate some of these problems. Interested readers are directed to [[Space_Radiation#Electrostatic_Shielding|the active shielding section in our page on space radiation]] for more of the juicy details; there's no need to repeat the same stuff here.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Elctrostatic_active_shielding.png|200 px|frameless]]
<td>
[[File:Electrostatic_active_shielding_2.png|200 px|frameless]]
<tr><td colspan=2>
Proposed designs for electrostatic shields to protect from positively charged particles<ref name="Tripathi2006">Ram K. Tripathi, John W. Wilson, and Robert C. Youngquist, "Electrostatic Active Radiation Shielding - Revisited", 2006 IEEE Aerospace Conference, Big Sky, MT, USA, 2006, pp. 9 pp.-, doi: 10.1109/AERO.2006.1655760.</ref><ref name="Chowdhury2023">Rajarshi Pal Chowdhury, Luke A. Stegeman, Matthew L. Lund, Dan Fry, Stojan Madzunkov, and Amir A. Bahadori, "Hybrid methods of radiation shielding against deep-space radiation", Life Sciences in Space Research, Volume 38, 2023, Pages 67-78, ISSN 2214-5524, https://doi.org/10.1016/j.lssr.2023.04.004.</ref>.
</table>
</div>

== Damage ==

Particle beams are, first and foremost, devices that produce beams of [[Nuclear_radiation|ionizing radiation]]. The radiation of weaponizable particle beams is extremely penetrating; thus of the [[Particle_Accelerators#Energy_loss_in_matter|interaction mechanisms listed for radiation from particle beams]], we mostly need to worry about bremsstrahlung and nuclear collisions and these produce extensive secondary showers of radiation that increase the penetration of the radiation exposure. Because of this, any target hit by a particle beam weapon will be irradiated, and radiation dose will always be one of the primary damage mechanisms of a particle beam weapon strike. Further, this radiation will be distributed along a track passing deeply through the material of the target, allowing it to affect critical systems through heavy armor, structural material, robust machine components, and passive material such as fuel load, coolant, or cargo. As the radiation deposits its energy into the target material, that material will be heated. If the target is heated sufficiently, it may experience [[Beam-Target_Interactions|thermal or mechanical]] damage from melting, cooking, combustion, vapor explosions, and warping or cracking due to thermal expansion.

The discussion of the radiation penetration into armor also applies to the same radiation penetrating through the target. So the formulas and rules of thumb for estimating the depth of radiation penetration into armor can also be used to estimate how far into a tank or spacecraft a particle beam strike delivers effects.

In addition to the radiation and thermal effects, a particle beam can also produce a sharp electromagnetic pulse that can damage semiconductor electronics.

== 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, Gerrit Bruhaug, and Luke Campbell for writing

Gerrit Bruhaug and Luke Campbell for scientific research, math wrangling and advise

ATriffleMind and Luke W. Campbell for proof reading.

==References==

[[Category:Beams]][[Category:Physics & Engineering‏‎]][[Category:Engineering‏‎]][[Category:Military Technology]][[Category:Warfare‏‎]]

Particle Beam Weapons

2026-10-26T04:18:52Z

Lwcamp: /* Separation of neutral beams */

{{PageConstructionNotice}}

''For a general overview on particle accelerator technology, see [[Particle Accelerators]]

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.

::: Z = 1.32×102 (γ2-1)3/4 r / √I<ref name=USPAS2016>Martin Reiser, "Theory and Design of Charged Particle Beams", 2008, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim</ref>

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

::: Z = β γ c r √[m/T]

Where Z is the doubling distance like before, r is the initial beam radius, T is the beam temperature in eV, m is the ion mass and β, c, and γ 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.

== Defenses ==

Sure, it's great to have a great big particle cannon to blast your enemies. But what if you are the one being shot at? In that case, it's best to figure out ways to keep from being blown to kingdom come.

=== Armor ===

The simplest way to defend against the high energy particles of a particle beam is to pile a whole bunch of stuff up and put it in the way of the beam. Particle beams may produce extremely penetrating radiation, but enough stuff will block most of that radiation and may allow you to survive. The details depend on what kind of particle is being thrown at you, and are explained below. However, a general rule of thumb is that you probably want about 10 tons of material per square meter of surface you want to armor in order to get sufficient protection.

==== Electrons ====

At the very high energy of an electron accelerated by a weapons-grade particle beam, the primary interaction with matter will be the emission of bremsstrahlung photons in the gamma-ray range as the electron slams into particles of matter. These high energy photons themselves primarily interact by the pair production mechanism – the photon disappears, but leaves behind a highly energetic positron and electron. These two particles are themselves energetic enough to produce further bremsstrahlung, and so the process continues. You get the formation of an electron-gamma shower (also called an electromagnetic cascade), which only stops after either the particles have penetrated through the target or the initial energy of the electron has been spread through so many shower particles (including all the electrons, positrons, and photons) that no particle has enough individual energy to pair produce or to be likely to bremsstrahlung again before it is stopped by other mechanisms.

To stop electrons, you'll want the atoms in your armor to be as heavy as you can practically get them. High atomic number both encourages bremsstrahlung and pair production, giving a double whammy for stopping that cascade. However, occasionally one of those electrons, positrons, or gammas will hit a nucleus and knock out neutrons. So you probably want a back-up layer of several tens of centimeters of hydrogen rich material such as plastic or water mixed with neutron absorbing materials like boron, lithium, cadmium, or gadolinium. High density polyethylene loaded with 5% boron by weight is a typical go-to neutron blocker.

The actual methods to calculate the penetration of the electromagnetic cascade is given are the box below. However, for some examples:
<ul>
<li> A 10 GeV beam incident on thick iron (or steel) will stop cascading at about 9 cm, will deposit half of its energy in the first 12 cm, and will have dumped 98% of its energy within 35 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 2.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 100 GeV beam incident on thick iron (or steel) will stop cascading at about 13 cm, will deposit half of its energy in the first 16 cm, and will have dumped 98% of its energy within 47 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 3.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick iron (or steel) will stop cascading at about 17 cm, will deposit half of its energy in the first 20 cm, and will have dumped 98% of its energy within 59 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 4.6 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick tungsten will stop cascading at about 3.5 cm, will deposit half of its energy in the first 4 cm, and will have dumped 98% of its energy within 12 cm. 90% of the cascade will be contained within about 0.93 cm and 95% of the cascade will be contained within 1.86 cm of the beam axis. You will get 98% protection with a mass thickness of 2.3 tons/m2 and an additional 1/e reduction for each further 0.13 tons/m2 beyond that.
<li> A 1 TeV beam incident on water will stop cascading at about 290 cm, will deposit half of its energy in the first 340 cm, and will have dumped 98% of its energy within 1030 cm. 90% of the cascade will be contained within about 11 cm and 95% of the cascade will be contained within 22 cm of the beam axis. You will get 98% protection with a mass thickness of 10.3 tons/m2 and an additional 1/e reduction for each further 0.72 tons/m2 beyond that.
</ul>
Note that for sufficiently intense particle beams, 98% protection may not be enough. You might need to throw on several more tons per square meter to be sure.

<table border=1>
<tr><td> How Far it Goes:
(This section follows methods presented in Leroy and Rancoita<ref>C. Leroy and P-G Rancoita, "Physics of cascading shower generation and propagation in matter: principles of high-energy, ultrahigh-energy and compensating calorimetry", Reports on Progress in Physics 63 (2000) 505–606. DOI: 10.1088/0034-4885/63/4/202</ref>.)

When trying to estimate the extent of the irradiated region of the target, it is helpful to know the radiation length X0 in the material. This is the distance that an electron will travel, on average, before emitting a bremsstrahlung photon. Conveniently, it is also the distance a high energy photon will travel through the material before pair producing. The radiation length can be found from
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   = 4 α  
<td nowrap="nowrap"> NA
<td rowspan="2" nowrap="nowrap" align="right">   Z (Z+ζ) re2 ln
<td rowspan="2" nowrap="nowrap" align="right"> [
<td nowrap="nowrap"> 183
<td rowspan="2" nowrap="nowrap" align="right"> ]
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> A
<td style="border-top:solid 1px black;"> Z1/3
</table>
</div>
where α = 1/137 is the fine structure constant, NA = 6.02214076×1023 is Avogadro's number (or one mole), A is the atomic weight of the atoms in the material (usually expressed in grams per mole), Z is the atomic number (nuclear charge) of the atoms, and re = 2.8179403205×10-13 cm is the classical electron radius. ζ is a constant of approximately 1.3. This gives the thickness of material in terms of the areal density (mass of material per unit area perpendicular to the beam) – if you use A in g/mol and re in cm the result will be in cm2/g and X0 will be in g/cm2. If your target is made up of more than one kind of atom, you can get the total radiation length by finding the radiation length of each atom of type i, X0i and using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   =  
<td rowspan="2" nowrap="nowrap" align="right"> Σi  
<td nowrap="nowrap"> fi
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> X0i
</table>
</div>
where fi is the fraction by weight of the atom type i. Once you have the radiation length in terms of areal density, you can easily convert it to distance by dividing by the density ρ. If X0i is in g/cm2 and density is in g/cm3, then X0i/ρ will be in cm.

We will also need to know the critical energy εc. This is the energy where an electron is losing energy as rapidly to ionization as it is to radiation. That is, it is the energy at which the cascade stops multiplying, even though the individual particles already made can continue traveling for some distance. The critical energy can be estimated using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> εc = B  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> Z X0
<td rowspan="2" nowrap="nowrap" align="right"> )
<td nowrap="nowrap"> h
<tr>
<td style="border-top:solid 1px black;"> A
<td>  
</table>
</div>
where B = 2.66 MeV and h = 1.11.

We can now describe the distribution of radiation in the target material. The depth tmax where the cascade stops multiplying (in g/cm2) for an incident electron of energy E (in MeV) is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tmax = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> - 1
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
Similarly, the depth where half the energy of the beam has been deposited (in g/cm2) is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tcg = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> + 0.37
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
As usual, you can convert to a depth in cm by dividing by the material density. 98% of the beam energy will be deposited within a distance of &cong; 3 tcg. Beyond this, the beam energy will continue to fall off with thickness t approximately as an exponential exp[-t/λ], where the characteristic attenuation length λ &cong; 2 X0.

The width of a cascade, where 90% of the energy is contained, is approximately given by
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> RM =  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> EM
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
where EM &cong; 21.2 MeV. 95% of the radiation will be contained within a distance of 2 RM from the beam axis.
</table>

==== Protons and Ions ====

<table align=right><tr><td>
[[File:Regolith_Shielding.png|350 px|frameless]]
<tr><td width=350>
Relative effect of particle radiation (compared to no shielding) from galactic cosmic rays behind different thicknesses of water, aluminum, and lunar regolith<ref name="Slaba2022">Tony C. Slaba, "Radiation Shielding for Lunar Missions: Regolith Considerations", LSIC Crosstalk 7/18/2022 https://lsic.jhuapl.edu/uploadedDocs/focus-files/1604-E&C%20+%20EE%20Monthly%20Meeting%20-%202022%2007%20July_Presentation%20-%20NASA%20Slaba.pdf</ref>.
</table>
Protons and ions are grouped together because ions are made of protons and neutrons and when you accelerate these things for a weapon it is the protons and neutrons that are going to be doing the heavy lifting for blasting your target. This is because protons and neutrons are both about 2000 times more massive than electrons so any electrons along for the ride will contribute negligibly to the energy. Protons and neutrons together are called nuclear particles, and both behave quite similarly in nuclear collisions.

Nuclear particles are themselves made of quarks and can interact via the strong nuclear force. At the energies appropriate for weapons use, essentially all of the way protons or ions are blocked is through [[Particle_Accelerators#Nuclear_collisions|direct collisions with the atomic nuclei]] of the armor material. One or more quarks in the projectile will slam into quarks in the target nucleus and stretch them away from the rest of the quarks in the nuclear particles. Physics tells us that you can't have isolated quarks, so the "string" attaching the quark to its neighbors stretches and stretches until it snaps. Snapping releases so much energy that more quarks are made that clump together into all sorts of other nuclear particles, both exotic and familiar. This process is called a hadronic jet. The only particles produced in these jets long-lived enough to travel any noticeable distance through the armor material will be protons, neutrons, the charged pi mesons (or pions), and some of the K mesons (or kaons). The particles too short lived to go far enough to matter will decay into other particles, and these particles are often very high energy gamma rays. Much like with electrons, the long-lived secondary particles can be sufficiently energetic to smash into other nuclei and cause more hadronic jets (or, if they are low enough in energy, just knock more nuclear particles out of target nuclei) and this produces a similar shower or cascade. But in addition, the armor nuclei that were hit will be very hot after the impact from all of the energy given to them from the incident particles. These nuclei will cool off by evaporation – but instead of evaporating away molecules, they evaporate off nuclear particles from the nucleus. These evaporated particles are mostly neutrons, so any armor against proton or ion beam strikes will produce quite a bit of neutron radiation.

Eventually, the original particle's energy will be distributed among so many other particles that the charged particles will come to a stop by making ion tracks rather than from colliding with more nuclei, and the cascade halts. But you still have all the neutrons to worry about. And the gamma rays as well, which are high enough energy to kick off an electromagnetic cascade in the same way that electrons do.

Unlike electrons, which have reasonable approximations of their cascade behavior from relatively simple equations, the cascades of protons and ions usually require sophisticated simulations or experiments to get the resulting distribution of radiation. However, we can get a reasonable idea of their general behavior by comparing them to [[Space_Radiation#Shielding_Against_Cosmic_Rays|cosmic ray particles interacting with shielding]]. Cosmic rays are energetic protons and ions, after all. From this we see that a relatively small amount of armor – in the range of 1 to 3 tons per square meter – can actually make the radiation dose worse. The incident particles will knock off so many secondary particles in their shower that the dose is multiplied from so many tracks passing through the target. Hydrogen rich materials such as water and plastics work the best against proton and ion beam particles; heavy elements are the worst and produce the most secondaries. You could probably obtain adequate radiation protection from most incident beams with about 10 tons per square meter of hydrogen rich material as armor. However, you will want to add neutron absorbing elements to your hydrogen rich armor, such as lithium, boron, cadmium, or gadolinium. Further, in order to deal with the gamma rays you will want a layer of at least several centimeters of heavy elements, such as tungsten or lead, behind the hydrogen rich particle armor.

==== Structural Considerations ====

When your armor material absorbs a radiation beam, the energy of that radiation turns into heat inside the armor where the radiation was absorbed. If the beam pulse has enough energy as was sufficiently tightly focused, this heat can damage your armor. Your armor may melt in places, or thermal expansion could cause it to crack or warp. Perhaps the worst thing it could do is vaporize or flash into a plasma, because the pressure from the vapor or plasma can be sufficient to plastically deform the armor. This leaves a hole along the beam path where the expanding gas or plasma blasted out a long, deep crater. This means a beam that fires rapid shots spaced maybe milliseconds apart can have subsequent shots go down the same hole so your armor will not protect against them. Each shot will dig out a deeper hole until either your armor is penetrated or the attacking beam runs out of shots in its pulse train.

A consequence of this is that when designing your armor you may wish to use refractory materials with a high structural strength (like tungsten), or that is otherwise resistant to being blasted apart by its own evaporated remains.

=== Magnetic fields ===

[[Particle_Accelerators#Magnetic_fields|A charged particle crossing a magnetic field will bend so as to spiral around the field lines]]. If you try to shoot your particle beam through a magnetic field, it will bend away from the direction you shot it. This can make it hard to hit your target. Unfortunately for you, there are magnetic fields everywhere – around planets, and even trapped in the solar wind as it crosses the solar system.

As an example, the Earth has a magnetic field strength of about 5×10-5 tesla near its surface, and falling off farther away. If you are shooting a 1 TeV electron beam past the Earth, your beam is likely to bend by several milliradians (with the exact deflection depending on how close you are skimming the Earth). If you are in a medium Earth orbit and shooting at another target also in a medium Earth orbit on a trajectory that goes near the Earth, then without correcting for the magnetic field your beam will likely be off-target by tens to hundreds of kilometers.

For another example, the interplanetary magnetic field is about 10-9 tesla. If you shoot that same 1 TeV electron beam at a target 0.1 light seconds away, you are likely to be off by a couple hundred meters.

<table border=1>
<tr><td>
The radius of curvature of the bend of a particle in a magnetic field is called the gyroradius rc. For a particle with a charge q, mass m, and kinetic energy E, find the relativistic momentum p
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
p = √[E2 - m2 c4] / c
</div>
To convert from energy units of eV to joules, multiply by 1.60×10-19. An electron has a mass of 9.11×10-31 kg, a proton of 1.67×10-27 kg, and an ion of 1.66×10-27×A kg, where A is the atomic weight of the ion. The speed of light c = 3.00×108 m/s.

In a magnetic field of strength B with an angle θ between the field direction and the direction of motion
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rc = |p sin(θ) / (q B)|
</div>
For electrons, q = -e, where the fundamental charge e = 1.60×10-19 coulomb. For a proton, q = +e. For an ion, multiply e by the charge state of the ion.

If the field acts over a distance d that is much smaller than the gyroradius, the deflection angle φ is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = d / rc
</div>
The deflection x at a target at a range R for small φ (φ &Lt; 1) will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
x = φ R.
</div>

Note that d will always be less than or equal to x. Also note that if d is not smaller than twice the gyroradius the beam will never reach the target because it will completely spiral around the field lines before it reaches the target.
</table>

So, it is clear that as the defender it will be helpful to hang out in areas with high magnetic fields, and keep some distance between you and the attacker. Meanwhile, the attacker will benefit from being closer where small deflections in the beam trajectory don't translate into large excursions from where the beam is aimed (and also because there will be less space filled with fields between the two of you). But the attacker can also try to compensate for known fields. If you happen to know that the magnetic fields are between you and your target, you can adjust your fire so that the deflection of the field bends your beam onto the target. This is not a simple task. It is easy enough to measure the field strength and direction where you are, but predicting how it changes over the path of your shot does not have an obvious solution. You can also try to gain intelligence on likely battle sites before you ever fight on them – for example, getting good data on the structure of the fields of a planet using spies and technical intelligence during peacetime (heck, on Earth this information is published for free in academic journals and by government agencies such as NOAA). Planetary magnetospheres are also affected by the solar wind, so different solar weather will change the fields you have to shoot through. But space weather forecasts and predictive simulations of the solar wind interacting with a planetary magnetosphere can give you a good idea of what the field structure is like. But even with all that, there are changes in the fields that happen on a scale of around a second ... anyone who has ever seen the auroras dancing across the sky in the polar night can see this. Compensating for these changes will be a significant challenge.

What would really help is being able to see the path of your own beam. This would allow you to see how far off you were from your target and in what direction, and then adjust fire to walk your shots onto the target. Unfortunately, particle beams in space are invisible. There might be ways to track them, but if so we don't know what they are.

==== Synchrotron radiation ====

When charged particle trajectories are bent, they emit [[Particle_Accelerators#Synchrotron_radiation|synchrotron radiation]]. This can cause the particle beam to lose energy, but if the bend is too tight it can also generate high energy gamma rays that ignore magnetic fields and directly strike the target. Synchrotron radiation can generally be ignored for proton or ion beams, but can be significant for electron beams.

<table border=1><tr><td>
For a beam particle with energy E and mass m, the synchrotron radiation is concentrated in a forward-directed cone with an opening angle of approximately 1/γ, where γ = 1 + E/(m c2). (Interestingly, the radiation intensity peaks an at angle of 1/(2 γ) and vanishes directly on-axis.) Because the beam is bending, you will get emission along the plane along which the beam is being bent. The energy of the photons produced during synchrotron emission is approximately<ref name="Jackson_Electrodynamics">J. D. Jackson, "Classical Electrodynamics: Second Edition", Wiley (1962)</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Eγ &cong; 3 γ3 c &hbar; / rc
</div>
where &hbar; = 1.05×10-34 J s (if you want the energy in joules) or &hbar; = 6.58×10-16 eV s (if you want the energy in eV). The photon energy will not exceed the energy of the incident particles, if you get a photon energy higher than the energy of the incident particles it just means that the synchrotron radiation spectrum extends up to the incident particle energy.

The fraction of the energy (or power) of the beam turned into synchrotron radiation will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = [1/(4 π ε0)] [4 π / 3] [q² / rc] β2 γ4 [φ / (2 π)] / E
</div>
where β = √[1 - 1/γ2] is the particle's speed measured as a fraction of the speed of light, and 1/(4 π ε0) = 8.99×109 is the Coulomb constant. For highly relativistic electrons (like all practical electron beam weapons emit), β &cong; 1 and we can use
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = 8.85×10-2 [E (GeV)3 [φ / (2 π)] / (rc (m))]
</div>
This fraction cannot be more than 1. If f gets close to one, it means that our original assumption that the beam particles mostly keep their original speed no longer holds, and the beam will lose most or almost all of its energy. Instead, when the beam is being deflected it will only make it through an angle of φ/f and will irradiate a region of length x/f starting at where the beam would have been incident if not for the magnets and extending in the direction the beam was deflected.
</table>

For example, a 1 TeV electron beam will have γ &cong; 2,000,000. So if it's bent by a magnetic field the synchtrotron radiation will spread out in a cone of about 5×10-7 radians. If this beam was shooting at a target 0.1 light seconds away from where it was bent, the radiation would arrive at the target over a region with a radius of about 15 meters with a maximum intensity at about 7.5 meters and no radiation directly along the path the beam was pointing while it was being bent. If the beam was bent by the interplanetary magnetic field from our earlier example, it would produce light with photon energies of around 1.3 eV (near infrared) and the lost energy would be a few parts in a hundred trillion. If it was instead deflected by the Earth's magnetic field you might expect a peak photon energy of tens of keV – in the hard x-ray part of the spectrum – and the fraction of energy in the synchrotron radiation would be somewhere in the low parts in 100,000 or less.

==== Separation of neutral beams ====

Magnetic fields only affect charged objects. So one way to overcome magnetic defenses or environments is to use beams that have no net charge. For example, a proton beam could be neutralized with a co-propagating electron beam with an equal number of protons and electrons. Or, potentially, you could fire an ion beam and then neutralize the ions after acceleration to get a beam of neutral atoms.

But, the magnetic fields are still tugging at the charged particles making up the beam as they go past. This will pull the electrons one way and the protons or ions the other way. As the particles are pulled in opposite directions you get a net positive charge on one side of the beam and a net negative charge on the other side (called polarization). This sets up an electric field inside the beam that acts to pull the particles back together. If the particles are going fast enough and the magnetic field is strong enough, the magnetic force can still separate the charges and will split the beam in two, one with the positive charges and one with the negative charges. But if the particles are not that fast or the magnetic field is not that strong this polarization of the beam will keep it together and let it fly right through the magnetic field.

<table border=1><tr><td>
Let's figure out how to analyze this problem. For definiteness, we will have a beam with an electric current of the positive species of I and an electric current of electrons of -I, all moving in the z direction. The beam radius will be R. The particle speed will be v with the usual relativistic parameters β = v/c and γ = √[1/(1-β2)]. If you know the energy per particle E of the positive species and the mass m of the positive species, γ = 1 + E/(m c2) and then β = √[1 - 1/γ2].

Let's assume that the beam is going through a magnetic field of strength B at an angle θ to the field direction. The quantity B sin(θ) is the component of the magnetic field perpendicular to the direction of the beam.

The charge density ρ of the positive component of the beam will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρL = I / (π v R2).
</div>

It is easiest to analyze the problem in the rest frame of the particles in the beam. Now the particles are at rest. Particles at rest are not affected by a magnetic field, but the change of reference frame Lorentz transforms the initial magnetic field into a combination of electric and magnetic fields in the rest frame. We can ignore the magnetic field, but the electric field is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔A = γ v B sin(θ)
</div>
in a direction perpendicular to both v and B. For our purposes, assume the electric field points in the x direction.

At the same time, relativistic length contraction increases the charge density in the rest frame
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρ = γ ρL.
</div>

Let r be the vector from the beam axis to the point inside the beam. It will be useful to separate this into its Cartesian coordinates. If x and y are the unit vectors in the x and y directions, respectively, then
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
r = x x + y y
</div>

If we consider only the positive component of the beam, from [https://en.wikipedia.org/wiki/Gauss%27s_law Gauss's law] it will have an electric field inside of the beam at a distance r from the center axis of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔+ = ρ r / (2 ε0) = (ρ/(2 ε0)) (x x + y y)
</div>
where ε0 = 8.85×10-12 is the vacuum permittivity. Now let's displace the electrons in the beam by a distance δ in the x direction. By itself, this electron beam will have an electric field of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔- = -(ρ/(2 ε0)) ((x-δ) x + y y)
</div>
By the principle of superposition, the electric field inside the beam with both components is the sum of the electric fields of the separate components including the external field
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔B = [(ρ/(2 ε0)) δ + γ v B sin(θ)]x
</div>
Because the charges are free to move, they will do so as long as a field is applied to them. Thus, the beam becomes stable once the charges separate enough that the field internal to the beam is zero. Therefore
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
δ = -2 ε0 γ v B sin(θ) / ρ = -2 π ε0 v2 R2 B sin(θ) / I
</div>
If |δ| > 2 |R|, the beam will fully separate and each component will go its own way. If δ is small compared to R, the beam's polarization suffices to hold it together in the overlap region. However, where the positive and negative charges do not overlap the electric fields no longer cancel, and the external electric field carries away the outer polarized part of the beam. This gives us a rough threshold for which a magnetic field will separate a neutral beam into its charged components and bend them away from their target
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr><td> B sin(θ) &Lt; I / (π ε0 v2 R) <td>       <td> Beam gradually erodes.
<tr><td> B sin(θ) > I / (π ε0 v2 R) <td> <td> Beam promptly separates.
</table>
</div>

For a beam of neutral atoms, let a be the radius of the atom and let Eb be the binding energy of the least tightly bound electron to the atom. An external electric field in excess of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
|𝓔A| > Eb / (e a)
</div>
will promptly ionize the atom, ripping away its electron. Here, e = 1.60×10-19 C is the magnitude of the electric charge of the electron. For fields close to but a bit weaker than this, the electron can still rapidly pop off due to quantum tunneling<ref>Ph. Durand and I. Paidarová, “Ionization of the hydrogen atom: from weak to strong static electric fields,” The European Physical Journal D, Vol. 26, pg. 253-259, 2003.</ref>. In general, expect that for fields greater in magnitude than (very approximately) Eb / (10 e a) the electron will be quickly removed, while for fields weaker than this the electrons will stay on their atoms.

It is useful to note that Eb/e is just the binding energy in units of eV. The most weakly bound electrons are usually bound to their atom with about 10 eV of energy. Atoms are typically about 10-10 m in radius. Thus, you can expect fields of around 1010 to 1011 V/m to directly ionize atoms. If we call this threshold value 𝓔T, then a beam will be ionized when passing through a field with strength greater than
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B sin(θ) > 𝓔T / (γ v)
</div>
</table>

Example: Let's consider a 1 GeV neutralized proton beam from a synchrotron, with a beam current of 200 kA in each bunch (incidentally, if a bunch is 5 cm long this means that each bunch packs 38 kJ worth of energy). The speed of the bunch is 87% the speed of light, or 262 million m/s, with a γ of 2.07. Consider when the beam's focus takes it to a radius of 5 cm (or 0.05 m, for our calculation). Plugging this in, we find that the protons in the bunch will separate from the electrons in a field with a transverse component greater than about 2.1 T, and the bunch will pass through the field without much effect if the transverse component is much smaller than this.

Example: We'll take the same 1 GeV beam, but this time recombine the electrons with protons to make hydrogen atoms. An electron is bound to a hydrogen atom with an energy of 13.6 eV (the Ryberg constant) and the radius of the electron orbital in a hydrogen atom is 0.529×10-10 m (the Bohr radius). This gives a threshold electric field for prompt ionization of hydrogen of 2.57×1011 V/m. We will get this with a transverse component of the magnetic field of 474 T. Above this field the atoms promptly ionize and electrons are stripped away from the protons. For fields higher than roughly 1/10 of this – so about 40 to 50 T – you may still get significant ionization by quantum tunneling; lower than this your losses from the beam due to field-induced ionization are likely negligible.

==== Magnetic shields ====

What happens when we can't rely on magnetic fields in the environment to protect us? One idea is to produce powerful artificial magnetic fields around our spacecraft to deflect incoming particle beams. This is very similar in concept to using active magnetic shielding to protect against space radiation – [[Space_Radiation#Magnetic_Shielding|check out our article on space radiation for more]].
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr>
<td width=100>
[[File:Unconfined_FRC_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:racetrack_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:Magnetic_shielding_Halback_Array.png|200 px|frameless]]
<tr><td colspan=3>
Some potential configurations for magnetic shields. The currents generating the fields are in green, the magnetic fields themselves are shown in magenta<ref name="Battiston2012">R. Battiston, W.J. Burger, V. Calvelli, R. Musenich, V. Choutko, V.I. Datskov, A. Della Torre, F. Venditti,
C. Gargiulo, G. Laurenti, S. Lucidi, S. Harrison, and R. Meinke, "ARSSEM Active Radiation Shield for Space Exploration Missions", Final Report ESTEC Contract N° 4200023087/10/NL/AF : “Superconductive Magnet for Radiation Shielding of Human Spacecraft” (2012) https://arxiv.org/abs/1209.1907 https://www.researchgate.net/publication/265945847_Active_Radiation_Shield_for_Space_Exploration_Missions</ref><ref name="Desiati2022">Paolo Desiati and Elena D'Onghia, "CREW HaT: A Magnetic Shielding System for Space Habitats", arXiv:2209.13624 [physics.space-ph] https://doi.org/10.48550/arXiv.2209.13624</ref>.
</table>
</div>

One immediate issue is that you want to keep very high magnetic fields out of the crew area. Fields in excess of 0.5 T can start to be unpleasant, and it is recommended to keep the field below 0.2 T for continuous exposure.

Strong magnets can suffer from high self-forces acting to tear them apart. Most shielding designs will have multiple magnetic coils, which will exert high forces and torques on each other. This will result in the need for strong structural bracing to keep the magnets intact and in place. Such strong structural designs will impose significant extra mass on the spacecraft.

Consider a magnetic field configuration that can keep radiation out of a certain region of space. There is no trajectory of a charged radiation particle up to a certain energy that will allow those particles to enter the protected region. But the dynamics of charged particles in a magnetic field are the same whether going forward or backward in time. If there is no way for a particle to get in, by this time reversal symmetry any particle which is in that forbidden region can never escape. Consequently any field configuration that is a magnetic trap is also a magnetic shield; any field configuration that is a magnetic shield is also a magnetic trap. A spacecraft with a magnetic shield may end up surrounded by strong radiation belts.

Using a magnet to bend a weaponized electron beam very often ends up with the beam being bent so hard that it dumps most of its energy in the form of synchrotron radiation at gamma ray energies. Because gamma rays are not charged, they will not be deflected by the field. Thus, if you would have been hit by the electron beam you will be hit instead by the gamma rays.

As an example, consider a 1 TeV electron beam. We'll surround our spacecraft with a 10 T magnetic field in a 100 meter radius region just outside our spacecraft. This will deflect the beam by 30 meters, which is enough to make it miss. But we are not saved, because the beam dumps nearly all of its energy into synchrotron radiation with an energy spectrum of gamma rays up to 13 GeV. This radiation will hit us in a strip about 2.4 meters long and not much more than the width of the original beam with essentially all of the energy of the original beam.

==== Magnet drones ====

If a magnetic shield around your spacecraft will just end up killing you from synchrotron radiation, what if you put the magnetic shield on a drone and put that drone between yourself and the enemy trying to shoot you? This lets you use a much weaker magnet because you don't need nearly so high of a deflection angle to make the shot miss. This means that less beam energy will turn into synchrotron radiation that could kill you instead and will spread what synchrotron radiation does occur out over a much wider area when it reaches your spacecraft, hopefully so that most of it just misses your spacecraft.

It does mean that your enemy will start shooting your drones so that they can eventually take a shot at you. But hopefully by that time you've already cleared out their drones and cored their spacecraft with your own synchrocannon or UREB.

==== Plasma magnets ====

Usually, to make a really big magnetic field you would run current through a really big conductive loop. But really big things are heavy and expensive. One way around this could be to use the ambient space plasma to carry the electric current rather than any kind of wire. In order to make this current, you use the same mechanism used in a rotary induction electric motor: you generate a rotating magnetic field and induce a current in the conductive loop (in this case, the space plasma). How do you make a rotating magnetic field? The method proposed<ref name="Slough2006">John Slough, "The Plasma Magnet", NASA Institute for Advanced Concepts Phase II Final Report (2006) https://www.niac.usra.edu/files/studies/final_report/917Slough.pdf</ref> is to use an antenna of four much smaller current loops as electromagnets at 90&degree; to each other, and sinusoidally vary the current through the loops with a respective 90&degree; phase difference between them. When this is done, a circular current is induced in the space plasma and the self-force of a magnetic field on its generating current inflates the current loop to much larger radii. You end up with a current sheet in the plane of the rotating field's rotation circulating about the antenna nearly synchronously with the field rotation. This current then creates a normal dipolar magnetic field as with any current loop. The current sheet and the resulting magnetic field expand until the magnetic field's pressure equals that of the ambient space plasma. This gives an enormous region of magnetic field from a relatively small and lightweight system.

The system proposed in the initial study would use an antenna approximately 10 meters in radius and produce a region of field approximately 30 km in radius at Earth's orbit. The field inside would not be constant - at the edges it would be 50 nT, giving the right pressure to keep out the solar wind at Earth's orbit. A circulating current sheet produces a field that falls off as 1/r with distance, meaning that adjacent to the antenna the field would be about 0.15 mT. Using the methods in the box below, we can estimate that this particular design as part of a spacecraft's shields will not deflect a TeV electron beam by more than a few centimeters, which is usually insufficient to force a miss. However, it could potentially be useful for a drone magnet, where the much larger distance between the magnetic field and the target spacecraft would allow a small deflection angle to give perhaps tens or low hundreds of meters change in beam location near the target at near light second ranges. The relatively low field and gradual bend of the incident beam suggests that synchrotron radiation will not be an issue; however this does not exclude more powerful plasma magnet designs causing significant synchrotron radiation effects.

<table border=1><tr><td>
Calculating the deflection from a spatially varying magnetic field affected by the solar wind is no trivial task. However, we can make an approximation that both should be within about an order of magnitude of the true value and also represent an upper limit to the deflection angle; although it does neglect the influence of the driving rotating magnetic field which could in some cases increase the deflection.

We will take R0 as the radius of the antenna. B0 is the field strength at the antenna. Rm is the distance at which the magnetic influence stops because the magnetic pressure equals the space plasma pressure. Bm is the field strength at Rm. According to Slough<ref name="Slough2006"></ref>, the magnetic field strength B(r) at any distance from the antenna center r between R0 and Rm is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B(r) = B0 R0 / r.
</div>
Therefore, once we know Bm and Rm, B0 is completely determined
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B0 = Bm Rm / R0.
</div>

The force on a particle with charge q passing perpendicularly through a magnetic field B(r) at a speed v is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
F(r) = q v B(r)
</div>
For an electron beam, v is so close to c as to make no difference. If the particle is moving down the y direction and the field is in the z direction, the force will be in the x direction. We assume that the deflection will be small, so we can choose a straight-line path in the y direction offset from the center by an amount x. The plasma bubble radius is Rm, so the particle will enter at ymax = √[Rm2-x2] and leave at ymin = -√[Rm2-x2].

The momentum change the particle experiences from this force is the time integral of the force over its path. Because of reflection symmetry about y = 0, we only need to integrate from ymax to 0 and then multiply by 2
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] dt F(r)
<tr> <td> <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] (dy/v) F(r)
<tr> <td> <td> = <td align="left">2 q B(0) R0 [Integral from 0 to √[Rm2-x2]] dy 1/√[y2 + x2]
<tr> <td> <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/x]
</table>
</div>
This holds for |x| > R0. But the expression for the field is not valid for r < R0 and the integral will give nonsense and diverging results if we evaluate for |x| < R0. In this case, we can put the integral lower bound at r = R0, or y = √[R02-x2]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/(√[R02-x2] + R0)]
</table>
</div>

The deflection angle is now estimated as
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = px / p
</div>
where the total momentum of the particle p can be found from the methods in the first box in the Magnetic Fields section.

The deflection for a shot aimed right at the spacecraft at the position of the spacecraft can be estimated by setting x = 0 and only taking the first half of the integral (so ignore the factor of 2)
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
q B(0) R0 ln[Rm/R0] / (p Rm)
</div>
This is an additional overestimate because it fails to account for how the deflection builds up over distance during the integration, but it should still give something of an order-of-magnitude and upper limit estimate of what kind of deflections might be expected.
</table>

=== Electric fields ===

If electric fields can accelerate charged particles, maybe they can keep the charged particles in a weaponized particle beam away from your spacecraft? To completely block a particle beam, you would need a voltage on your spacecraft equal to the particle energy in eV of the particles in the beam. That is, if a 1 GeV proton beam is coming at you, you'd need an electric potential of a billion volts to stop that beam. This leads to three immediate problems.

The first is that particles of the opposite charge will be attracted to your spacecraft. Because space isn't really empty but rather a rarefied plasma (called the solar wind in a solar system or the interstellar medium out between stars), there will be plenty of particles of the opposite charge around to attract. And they will hit with an energy in eV equal to the voltage on your spacecraft. In our example of defending against a 1 GeV proton beam with a gigavolt field, the electrons from the solar wind will come in and hit with an energy of 1 GeV each. This will generate dangerous radiation.

The second problem is a consequence of the first. Each of those GeV electrons from our example need to get their energy from somewhere, which turns out to be from the electric field we are using as shielding. You will need to make up for this energy loss with power from your electrical system, which gives a constant drain on your power. Some estimates put the electrical effort to maintain a several hundred megavolt shield in space at many tens of gigawatts<ref name="Mechmann2019">Claire Mechmann, "Analysis of Proposed Active Radiation Shielding Design Concept for Spacecraft" (2019) Thesis, College of Engineering and Science of Florida Institute of Technology</ref>.

And the third problem is similar to one for magnetic fields. Your electrodes will experience high self forces and forces between separate electrodes that will try to tear your shield generator apart. Consequently you will need heavy structural reinforcement in order for your shield to work.

There are potential ways to mitigate some of these problems. Interested readers are directed to [[Space_Radiation#Electrostatic_Shielding|the active shielding section in our page on space radiation]] for more of the juicy details; there's no need to repeat the same stuff here.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Elctrostatic_active_shielding.png|200 px|frameless]]
<td>
[[File:Electrostatic_active_shielding_2.png|200 px|frameless]]
<tr><td colspan=2>
Proposed designs for electrostatic shields to protect from positively charged particles<ref name="Tripathi2006">Ram K. Tripathi, John W. Wilson, and Robert C. Youngquist, "Electrostatic Active Radiation Shielding - Revisited", 2006 IEEE Aerospace Conference, Big Sky, MT, USA, 2006, pp. 9 pp.-, doi: 10.1109/AERO.2006.1655760.</ref><ref name="Chowdhury2023">Rajarshi Pal Chowdhury, Luke A. Stegeman, Matthew L. Lund, Dan Fry, Stojan Madzunkov, and Amir A. Bahadori, "Hybrid methods of radiation shielding against deep-space radiation", Life Sciences in Space Research, Volume 38, 2023, Pages 67-78, ISSN 2214-5524, https://doi.org/10.1016/j.lssr.2023.04.004.</ref>.
</table>
</div>

== Damage ==

Particle beams are, first and foremost, devices that produce beams of [[Nuclear_radiation|ionizing radiation]]. The radiation of weaponizable particle beams is extremely penetrating; thus of the [[Particle_Accelerators#Energy_loss_in_matter|interaction mechanisms listed for radiation from particle beams]], we mostly need to worry about bremsstrahlung and nuclear collisions and these produce extensive secondary showers of radiation that increase the penetration of the radiation exposure. Because of this, any target hit by a particle beam weapon will be irradiated, and radiation dose will always be one of the primary damage mechanisms of a particle beam weapon strike. Further, this radiation will be distributed along a track passing deeply through the material of the target, allowing it to affect critical systems through heavy armor, structural material, robust machine components, and passive material such as fuel load, coolant, or cargo. As the radiation deposits its energy into the target material, that material will be heated. If the target is heated sufficiently, it may experience [[Beam-Target_Interactions|thermal or mechanical]] damage from melting, cooking, combustion, vapor explosions, and warping or cracking due to thermal expansion.

The discussion of the radiation penetration into armor also applies to the same radiation penetrating through the target. So the formulas and rules of thumb for estimating the depth of radiation penetration into armor can also be used to estimate how far into a tank or spacecraft a particle beam strike delivers effects.

In addition to the radiation and thermal effects, a particle beam can also produce a sharp electromagnetic pulse that can damage semiconductor electronics.

== 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, Gerrit Bruhaug, and Luke Campbell for writing

Gerrit Bruhaug and Luke Campbell for scientific research, math wrangling and advise

ATriffleMind and Luke W. Campbell for proof reading.

==References==

[[Category:Beams]][[Category:Physics & Engineering‏‎]][[Category:Engineering‏‎]][[Category:Military Technology]][[Category:Warfare‏‎]]

Particle Beam Weapons

2026-10-26T03:10:33Z

Lwcamp: /* Separation of neutral beams */

{{PageConstructionNotice}}

''For a general overview on particle accelerator technology, see [[Particle Accelerators]]

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.

::: Z = 1.32×102 (γ2-1)3/4 r / √I<ref name=USPAS2016>Martin Reiser, "Theory and Design of Charged Particle Beams", 2008, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim</ref>

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

::: Z = β γ c r √[m/T]

Where Z is the doubling distance like before, r is the initial beam radius, T is the beam temperature in eV, m is the ion mass and β, c, and γ 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.

== Defenses ==

Sure, it's great to have a great big particle cannon to blast your enemies. But what if you are the one being shot at? In that case, it's best to figure out ways to keep from being blown to kingdom come.

=== Armor ===

The simplest way to defend against the high energy particles of a particle beam is to pile a whole bunch of stuff up and put it in the way of the beam. Particle beams may produce extremely penetrating radiation, but enough stuff will block most of that radiation and may allow you to survive. The details depend on what kind of particle is being thrown at you, and are explained below. However, a general rule of thumb is that you probably want about 10 tons of material per square meter of surface you want to armor in order to get sufficient protection.

==== Electrons ====

At the very high energy of an electron accelerated by a weapons-grade particle beam, the primary interaction with matter will be the emission of bremsstrahlung photons in the gamma-ray range as the electron slams into particles of matter. These high energy photons themselves primarily interact by the pair production mechanism – the photon disappears, but leaves behind a highly energetic positron and electron. These two particles are themselves energetic enough to produce further bremsstrahlung, and so the process continues. You get the formation of an electron-gamma shower (also called an electromagnetic cascade), which only stops after either the particles have penetrated through the target or the initial energy of the electron has been spread through so many shower particles (including all the electrons, positrons, and photons) that no particle has enough individual energy to pair produce or to be likely to bremsstrahlung again before it is stopped by other mechanisms.

To stop electrons, you'll want the atoms in your armor to be as heavy as you can practically get them. High atomic number both encourages bremsstrahlung and pair production, giving a double whammy for stopping that cascade. However, occasionally one of those electrons, positrons, or gammas will hit a nucleus and knock out neutrons. So you probably want a back-up layer of several tens of centimeters of hydrogen rich material such as plastic or water mixed with neutron absorbing materials like boron, lithium, cadmium, or gadolinium. High density polyethylene loaded with 5% boron by weight is a typical go-to neutron blocker.

The actual methods to calculate the penetration of the electromagnetic cascade is given are the box below. However, for some examples:
<ul>
<li> A 10 GeV beam incident on thick iron (or steel) will stop cascading at about 9 cm, will deposit half of its energy in the first 12 cm, and will have dumped 98% of its energy within 35 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 2.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 100 GeV beam incident on thick iron (or steel) will stop cascading at about 13 cm, will deposit half of its energy in the first 16 cm, and will have dumped 98% of its energy within 47 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 3.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick iron (or steel) will stop cascading at about 17 cm, will deposit half of its energy in the first 20 cm, and will have dumped 98% of its energy within 59 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 4.6 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick tungsten will stop cascading at about 3.5 cm, will deposit half of its energy in the first 4 cm, and will have dumped 98% of its energy within 12 cm. 90% of the cascade will be contained within about 0.93 cm and 95% of the cascade will be contained within 1.86 cm of the beam axis. You will get 98% protection with a mass thickness of 2.3 tons/m2 and an additional 1/e reduction for each further 0.13 tons/m2 beyond that.
<li> A 1 TeV beam incident on water will stop cascading at about 290 cm, will deposit half of its energy in the first 340 cm, and will have dumped 98% of its energy within 1030 cm. 90% of the cascade will be contained within about 11 cm and 95% of the cascade will be contained within 22 cm of the beam axis. You will get 98% protection with a mass thickness of 10.3 tons/m2 and an additional 1/e reduction for each further 0.72 tons/m2 beyond that.
</ul>
Note that for sufficiently intense particle beams, 98% protection may not be enough. You might need to throw on several more tons per square meter to be sure.

<table border=1>
<tr><td> How Far it Goes:
(This section follows methods presented in Leroy and Rancoita<ref>C. Leroy and P-G Rancoita, "Physics of cascading shower generation and propagation in matter: principles of high-energy, ultrahigh-energy and compensating calorimetry", Reports on Progress in Physics 63 (2000) 505–606. DOI: 10.1088/0034-4885/63/4/202</ref>.)

When trying to estimate the extent of the irradiated region of the target, it is helpful to know the radiation length X0 in the material. This is the distance that an electron will travel, on average, before emitting a bremsstrahlung photon. Conveniently, it is also the distance a high energy photon will travel through the material before pair producing. The radiation length can be found from
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   = 4 α  
<td nowrap="nowrap"> NA
<td rowspan="2" nowrap="nowrap" align="right">   Z (Z+ζ) re2 ln
<td rowspan="2" nowrap="nowrap" align="right"> [
<td nowrap="nowrap"> 183
<td rowspan="2" nowrap="nowrap" align="right"> ]
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> A
<td style="border-top:solid 1px black;"> Z1/3
</table>
</div>
where α = 1/137 is the fine structure constant, NA = 6.02214076×1023 is Avogadro's number (or one mole), A is the atomic weight of the atoms in the material (usually expressed in grams per mole), Z is the atomic number (nuclear charge) of the atoms, and re = 2.8179403205×10-13 cm is the classical electron radius. ζ is a constant of approximately 1.3. This gives the thickness of material in terms of the areal density (mass of material per unit area perpendicular to the beam) – if you use A in g/mol and re in cm the result will be in cm2/g and X0 will be in g/cm2. If your target is made up of more than one kind of atom, you can get the total radiation length by finding the radiation length of each atom of type i, X0i and using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   =  
<td rowspan="2" nowrap="nowrap" align="right"> Σi  
<td nowrap="nowrap"> fi
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> X0i
</table>
</div>
where fi is the fraction by weight of the atom type i. Once you have the radiation length in terms of areal density, you can easily convert it to distance by dividing by the density ρ. If X0i is in g/cm2 and density is in g/cm3, then X0i/ρ will be in cm.

We will also need to know the critical energy εc. This is the energy where an electron is losing energy as rapidly to ionization as it is to radiation. That is, it is the energy at which the cascade stops multiplying, even though the individual particles already made can continue traveling for some distance. The critical energy can be estimated using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> εc = B  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> Z X0
<td rowspan="2" nowrap="nowrap" align="right"> )
<td nowrap="nowrap"> h
<tr>
<td style="border-top:solid 1px black;"> A
<td>  
</table>
</div>
where B = 2.66 MeV and h = 1.11.

We can now describe the distribution of radiation in the target material. The depth tmax where the cascade stops multiplying (in g/cm2) for an incident electron of energy E (in MeV) is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tmax = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> - 1
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
Similarly, the depth where half the energy of the beam has been deposited (in g/cm2) is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tcg = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> + 0.37
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
As usual, you can convert to a depth in cm by dividing by the material density. 98% of the beam energy will be deposited within a distance of &cong; 3 tcg. Beyond this, the beam energy will continue to fall off with thickness t approximately as an exponential exp[-t/λ], where the characteristic attenuation length λ &cong; 2 X0.

The width of a cascade, where 90% of the energy is contained, is approximately given by
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> RM =  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> EM
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
where EM &cong; 21.2 MeV. 95% of the radiation will be contained within a distance of 2 RM from the beam axis.
</table>

==== Protons and Ions ====

<table align=right><tr><td>
[[File:Regolith_Shielding.png|350 px|frameless]]
<tr><td width=350>
Relative effect of particle radiation (compared to no shielding) from galactic cosmic rays behind different thicknesses of water, aluminum, and lunar regolith<ref name="Slaba2022">Tony C. Slaba, "Radiation Shielding for Lunar Missions: Regolith Considerations", LSIC Crosstalk 7/18/2022 https://lsic.jhuapl.edu/uploadedDocs/focus-files/1604-E&C%20+%20EE%20Monthly%20Meeting%20-%202022%2007%20July_Presentation%20-%20NASA%20Slaba.pdf</ref>.
</table>
Protons and ions are grouped together because ions are made of protons and neutrons and when you accelerate these things for a weapon it is the protons and neutrons that are going to be doing the heavy lifting for blasting your target. This is because protons and neutrons are both about 2000 times more massive than electrons so any electrons along for the ride will contribute negligibly to the energy. Protons and neutrons together are called nuclear particles, and both behave quite similarly in nuclear collisions.

Nuclear particles are themselves made of quarks and can interact via the strong nuclear force. At the energies appropriate for weapons use, essentially all of the way protons or ions are blocked is through [[Particle_Accelerators#Nuclear_collisions|direct collisions with the atomic nuclei]] of the armor material. One or more quarks in the projectile will slam into quarks in the target nucleus and stretch them away from the rest of the quarks in the nuclear particles. Physics tells us that you can't have isolated quarks, so the "string" attaching the quark to its neighbors stretches and stretches until it snaps. Snapping releases so much energy that more quarks are made that clump together into all sorts of other nuclear particles, both exotic and familiar. This process is called a hadronic jet. The only particles produced in these jets long-lived enough to travel any noticeable distance through the armor material will be protons, neutrons, the charged pi mesons (or pions), and some of the K mesons (or kaons). The particles too short lived to go far enough to matter will decay into other particles, and these particles are often very high energy gamma rays. Much like with electrons, the long-lived secondary particles can be sufficiently energetic to smash into other nuclei and cause more hadronic jets (or, if they are low enough in energy, just knock more nuclear particles out of target nuclei) and this produces a similar shower or cascade. But in addition, the armor nuclei that were hit will be very hot after the impact from all of the energy given to them from the incident particles. These nuclei will cool off by evaporation – but instead of evaporating away molecules, they evaporate off nuclear particles from the nucleus. These evaporated particles are mostly neutrons, so any armor against proton or ion beam strikes will produce quite a bit of neutron radiation.

Eventually, the original particle's energy will be distributed among so many other particles that the charged particles will come to a stop by making ion tracks rather than from colliding with more nuclei, and the cascade halts. But you still have all the neutrons to worry about. And the gamma rays as well, which are high enough energy to kick off an electromagnetic cascade in the same way that electrons do.

Unlike electrons, which have reasonable approximations of their cascade behavior from relatively simple equations, the cascades of protons and ions usually require sophisticated simulations or experiments to get the resulting distribution of radiation. However, we can get a reasonable idea of their general behavior by comparing them to [[Space_Radiation#Shielding_Against_Cosmic_Rays|cosmic ray particles interacting with shielding]]. Cosmic rays are energetic protons and ions, after all. From this we see that a relatively small amount of armor – in the range of 1 to 3 tons per square meter – can actually make the radiation dose worse. The incident particles will knock off so many secondary particles in their shower that the dose is multiplied from so many tracks passing through the target. Hydrogen rich materials such as water and plastics work the best against proton and ion beam particles; heavy elements are the worst and produce the most secondaries. You could probably obtain adequate radiation protection from most incident beams with about 10 tons per square meter of hydrogen rich material as armor. However, you will want to add neutron absorbing elements to your hydrogen rich armor, such as lithium, boron, cadmium, or gadolinium. Further, in order to deal with the gamma rays you will want a layer of at least several centimeters of heavy elements, such as tungsten or lead, behind the hydrogen rich particle armor.

==== Structural Considerations ====

When your armor material absorbs a radiation beam, the energy of that radiation turns into heat inside the armor where the radiation was absorbed. If the beam pulse has enough energy as was sufficiently tightly focused, this heat can damage your armor. Your armor may melt in places, or thermal expansion could cause it to crack or warp. Perhaps the worst thing it could do is vaporize or flash into a plasma, because the pressure from the vapor or plasma can be sufficient to plastically deform the armor. This leaves a hole along the beam path where the expanding gas or plasma blasted out a long, deep crater. This means a beam that fires rapid shots spaced maybe milliseconds apart can have subsequent shots go down the same hole so your armor will not protect against them. Each shot will dig out a deeper hole until either your armor is penetrated or the attacking beam runs out of shots in its pulse train.

A consequence of this is that when designing your armor you may wish to use refractory materials with a high structural strength (like tungsten), or that is otherwise resistant to being blasted apart by its own evaporated remains.

=== Magnetic fields ===

[[Particle_Accelerators#Magnetic_fields|A charged particle crossing a magnetic field will bend so as to spiral around the field lines]]. If you try to shoot your particle beam through a magnetic field, it will bend away from the direction you shot it. This can make it hard to hit your target. Unfortunately for you, there are magnetic fields everywhere – around planets, and even trapped in the solar wind as it crosses the solar system.

As an example, the Earth has a magnetic field strength of about 5×10-5 tesla near its surface, and falling off farther away. If you are shooting a 1 TeV electron beam past the Earth, your beam is likely to bend by several milliradians (with the exact deflection depending on how close you are skimming the Earth). If you are in a medium Earth orbit and shooting at another target also in a medium Earth orbit on a trajectory that goes near the Earth, then without correcting for the magnetic field your beam will likely be off-target by tens to hundreds of kilometers.

For another example, the interplanetary magnetic field is about 10-9 tesla. If you shoot that same 1 TeV electron beam at a target 0.1 light seconds away, you are likely to be off by a couple hundred meters.

<table border=1>
<tr><td>
The radius of curvature of the bend of a particle in a magnetic field is called the gyroradius rc. For a particle with a charge q, mass m, and kinetic energy E, find the relativistic momentum p
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
p = √[E2 - m2 c4] / c
</div>
To convert from energy units of eV to joules, multiply by 1.60×10-19. An electron has a mass of 9.11×10-31 kg, a proton of 1.67×10-27 kg, and an ion of 1.66×10-27×A kg, where A is the atomic weight of the ion. The speed of light c = 3.00×108 m/s.

In a magnetic field of strength B with an angle θ between the field direction and the direction of motion
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rc = |p sin(θ) / (q B)|
</div>
For electrons, q = -e, where the fundamental charge e = 1.60×10-19 coulomb. For a proton, q = +e. For an ion, multiply e by the charge state of the ion.

If the field acts over a distance d that is much smaller than the gyroradius, the deflection angle φ is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = d / rc
</div>
The deflection x at a target at a range R for small φ (φ &Lt; 1) will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
x = φ R.
</div>

Note that d will always be less than or equal to x. Also note that if d is not smaller than twice the gyroradius the beam will never reach the target because it will completely spiral around the field lines before it reaches the target.
</table>

So, it is clear that as the defender it will be helpful to hang out in areas with high magnetic fields, and keep some distance between you and the attacker. Meanwhile, the attacker will benefit from being closer where small deflections in the beam trajectory don't translate into large excursions from where the beam is aimed (and also because there will be less space filled with fields between the two of you). But the attacker can also try to compensate for known fields. If you happen to know that the magnetic fields are between you and your target, you can adjust your fire so that the deflection of the field bends your beam onto the target. This is not a simple task. It is easy enough to measure the field strength and direction where you are, but predicting how it changes over the path of your shot does not have an obvious solution. You can also try to gain intelligence on likely battle sites before you ever fight on them – for example, getting good data on the structure of the fields of a planet using spies and technical intelligence during peacetime (heck, on Earth this information is published for free in academic journals and by government agencies such as NOAA). Planetary magnetospheres are also affected by the solar wind, so different solar weather will change the fields you have to shoot through. But space weather forecasts and predictive simulations of the solar wind interacting with a planetary magnetosphere can give you a good idea of what the field structure is like. But even with all that, there are changes in the fields that happen on a scale of around a second ... anyone who has ever seen the auroras dancing across the sky in the polar night can see this. Compensating for these changes will be a significant challenge.

What would really help is being able to see the path of your own beam. This would allow you to see how far off you were from your target and in what direction, and then adjust fire to walk your shots onto the target. Unfortunately, particle beams in space are invisible. There might be ways to track them, but if so we don't know what they are.

==== Synchrotron radiation ====

When charged particle trajectories are bent, they emit [[Particle_Accelerators#Synchrotron_radiation|synchrotron radiation]]. This can cause the particle beam to lose energy, but if the bend is too tight it can also generate high energy gamma rays that ignore magnetic fields and directly strike the target. Synchrotron radiation can generally be ignored for proton or ion beams, but can be significant for electron beams.

<table border=1><tr><td>
For a beam particle with energy E and mass m, the synchrotron radiation is concentrated in a forward-directed cone with an opening angle of approximately 1/γ, where γ = 1 + E/(m c2). (Interestingly, the radiation intensity peaks an at angle of 1/(2 γ) and vanishes directly on-axis.) Because the beam is bending, you will get emission along the plane along which the beam is being bent. The energy of the photons produced during synchrotron emission is approximately<ref name="Jackson_Electrodynamics">J. D. Jackson, "Classical Electrodynamics: Second Edition", Wiley (1962)</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Eγ &cong; 3 γ3 c &hbar; / rc
</div>
where &hbar; = 1.05×10-34 J s (if you want the energy in joules) or &hbar; = 6.58×10-16 eV s (if you want the energy in eV). The photon energy will not exceed the energy of the incident particles, if you get a photon energy higher than the energy of the incident particles it just means that the synchrotron radiation spectrum extends up to the incident particle energy.

The fraction of the energy (or power) of the beam turned into synchrotron radiation will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = [1/(4 π ε0)] [4 π / 3] [q² / rc] β2 γ4 [φ / (2 π)] / E
</div>
where β = √[1 - 1/γ2] is the particle's speed measured as a fraction of the speed of light, and 1/(4 π ε0) = 8.99×109 is the Coulomb constant. For highly relativistic electrons (like all practical electron beam weapons emit), β &cong; 1 and we can use
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = 8.85×10-2 [E (GeV)3 [φ / (2 π)] / (rc (m))]
</div>
This fraction cannot be more than 1. If f gets close to one, it means that our original assumption that the beam particles mostly keep their original speed no longer holds, and the beam will lose most or almost all of its energy. Instead, when the beam is being deflected it will only make it through an angle of φ/f and will irradiate a region of length x/f starting at where the beam would have been incident if not for the magnets and extending in the direction the beam was deflected.
</table>

For example, a 1 TeV electron beam will have γ &cong; 2,000,000. So if it's bent by a magnetic field the synchtrotron radiation will spread out in a cone of about 5×10-7 radians. If this beam was shooting at a target 0.1 light seconds away from where it was bent, the radiation would arrive at the target over a region with a radius of about 15 meters with a maximum intensity at about 7.5 meters and no radiation directly along the path the beam was pointing while it was being bent. If the beam was bent by the interplanetary magnetic field from our earlier example, it would produce light with photon energies of around 1.3 eV (near infrared) and the lost energy would be a few parts in a hundred trillion. If it was instead deflected by the Earth's magnetic field you might expect a peak photon energy of tens of keV – in the hard x-ray part of the spectrum – and the fraction of energy in the synchrotron radiation would be somewhere in the low parts in 100,000 or less.

==== Separation of neutral beams ====

Magnetic fields only affect charged objects. So one way to overcome magnetic defenses or environments is to use beams that have no net charge. For example, a proton beam could be neutralized with a co-propagating electron beam with an equal number of protons and electrons. Or, potentially, you could fire an ion beam and then neutralize the ions after acceleration to get a beam of neutral atoms.

But, the magnetic fields are still tugging at the charged particles making up the beam as they go past. This will pull the electrons one way and the protons or ions the other way. As the particles are pulled in opposite directions you get a net positive charge on one side of the beam and a net negative charge on the other side (called polarization). This sets up an electric field inside the beam that acts to pull the particles back together. If the particles are going fast enough and the magnetic field is strong enough, the magnetic force can still separate the charges and will split the beam in two, one with the positive charges and one with the negative charges. But if the particles are not that fast or the magnetic field is not that strong this polarization of the beam will keep it together and let it fly right through the magnetic field.

<table border=1><tr><td>
Let's figure out how to analyze this problem. For definiteness, we will have a beam with an electric current of the positive species of I and an electric current of electrons of -I, all moving in the z direction. The beam radius will be R. The particle speed will be v with the usual relativistic parameters β = v/c and γ = √[1/(1-β2)]. If you know the energy per particle E of the positive species and the mass m of the positive species, γ = 1 + E/(m c2) and then β = √[1 - 1/γ2].

Let's assume that the beam is going through a magnetic field of strength B at an angle θ to the field direction. The quantity B sin(θ) is the component of the magnetic field perpendicular to the direction of the beam.

The charge density ρ of the positive component of the beam will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρL = I / (π v R2).
</div>

It is easiest to analyze the problem in the rest frame of the particles in the beam. Now the particles are at rest. Particles at rest are not affected by a magnetic field, but the change of reference frame Lorentz transforms the initial magnetic field into a combination of electric and magnetic fields in the rest frame. We can ignore the magnetic field, but the electric field is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔A = γ v B sin(θ)
</div>
in a direction perpendicular to both v and B. For our purposes, assume the electric field points in the x direction.

At the same time, relativistic length contraction increases the charge density in the rest frame
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρ = γ ρL.
</div>

Let r be the vector from the beam axis to the point inside the beam. It will be useful to separate this into its Cartesian coordinates. If x and y are the unit vectors in the x and y directions, respectively, then
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
r = x x + y y
</div>

If we consider only the positive component of the beam, from [https://en.wikipedia.org/wiki/Gauss%27s_law Gauss's law] it will have an electric field inside of the beam at a distance r from the center axis of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔+ = ρ r / (2 ε0) = (ρ/(2 ε0)) (x x + y y)
</div>
where ε0 = 8.85×10-12 is the vacuum permittivity. Now let's displace the electrons in the beam by a distance δ in the x direction. By itself, this electron beam will have an electric field of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔- = -(ρ/(2 ε0)) ((x-δ) x + y y)
</div>
By the principle of superposition, the electric field inside the beam with both components is the sum of the electric fields of the separate components including the external field
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔B = [(ρ/(2 ε0)) δ + γ v B sin(θ)]x
</div>
Because the charges are free to move, they will do so as long as a field is applied to them. Thus, the beam becomes stable once the charges separate enough that the field internal to the beam is zero. Therefore
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
δ = -2 ε0 γ v B sin(θ) / ρ = -2 π ε0 v2 R2 B sin(θ) / I
</div>
If δ is small compared to R, the beam's polarization suffices to hold it together. If |δ| > 2 |R|, the beam will fully separate and each component will go its own way. This gives us a rough threshold for which a magnetic field will separate a neutral beam into its charged components and bend them away from their target
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr><td> B sin(θ) &Lt; I / (π ε0 v2 R) <td>       <td> Beam is stable
<tr><td> B sin(θ) > I / (π ε0 v2 R) <td> <td> Beam promptly separates.
</table>
</div>

For a beam of neutral atoms, let a be the radius of the atom and let Eb be the binding energy of the least tightly bound electron to the atom. An external electric field in excess of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
|𝓔A| > Eb / (e a)
</div>
will promptly ionize the atom, ripping away its electron. Here, e = 1.60×10-19 C is the magnitude of the electric charge of the electron. For fields close to but a bit weaker than this, the electron can still rapidly pop off due to quantum tunneling<ref>Ph. Durand and I. Paidarová, “Ionization of the hydrogen atom: from weak to strong static electric fields,” The European Physical Journal D, Vol. 26, pg. 253-259, 2003.</ref>. In general, expect that for fields greater in magnitude than (very approximately) Eb / (10 e a) the electron will be quickly removed, while for fields weaker than this the electrons will stay on their atoms.

It is useful to note that Eb/e is just the binding energy in units of eV. The most weakly bound electrons are usually bound to their atom with about 10 eV of energy. Atoms are typically about 10-10 m in radius. Thus, you can expect fields of around 1010 to 1011 V/m to directly ionize atoms. If we call this threshold value 𝓔T, then a beam will be ionized when passing through a field with strength greater than
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B sin(θ) > 𝓔T / (γ v)
</div>
</table>

Example: Let's consider a 1 GeV neutralized proton beam from a synchrotron, with a beam current of 200 kA in each bunch (incidentally, if a bunch is 5 cm long this means that each bunch packs 38 kJ worth of energy). The speed of the bunch is 87% the speed of light, or 262 million m/s, with a γ of 2.07. Consider when the beam's focus takes it to a radius of 5 cm (or 0.05 m, for our calculation). Plugging this in, we find that the protons in the bunch will separate from the electrons in a field with a transverse component greater than about 2.1 T, and the bunch will pass through the field without much effect if the transverse component is much smaller than this.

Example: We'll take the same 1 GeV beam, but this time recombine the electrons with protons to make hydrogen atoms. An electron is bound to a hydrogen atom with an energy of 13.6 eV (the Ryberg constant) and the radius of the electron orbital in a hydrogen atom is 0.529×10-10 m (the Bohr radius). This gives a threshold electric field for prompt ionization of hydrogen of 2.57×1011 V/m. We will get this with a transverse component of the magnetic field of 474 T. Above this field the atoms promptly ionize and electrons are stripped away from the protons. For fields higher than roughly 1/10 of this – so about 40 to 50 T – you may still get significant ionization by quantum tunneling; lower than this your losses from the beam due to field-induced ionization are likely negligible.

==== Magnetic shields ====

What happens when we can't rely on magnetic fields in the environment to protect us? One idea is to produce powerful artificial magnetic fields around our spacecraft to deflect incoming particle beams. This is very similar in concept to using active magnetic shielding to protect against space radiation – [[Space_Radiation#Magnetic_Shielding|check out our article on space radiation for more]].
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr>
<td width=100>
[[File:Unconfined_FRC_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:racetrack_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:Magnetic_shielding_Halback_Array.png|200 px|frameless]]
<tr><td colspan=3>
Some potential configurations for magnetic shields. The currents generating the fields are in green, the magnetic fields themselves are shown in magenta<ref name="Battiston2012">R. Battiston, W.J. Burger, V. Calvelli, R. Musenich, V. Choutko, V.I. Datskov, A. Della Torre, F. Venditti,
C. Gargiulo, G. Laurenti, S. Lucidi, S. Harrison, and R. Meinke, "ARSSEM Active Radiation Shield for Space Exploration Missions", Final Report ESTEC Contract N° 4200023087/10/NL/AF : “Superconductive Magnet for Radiation Shielding of Human Spacecraft” (2012) https://arxiv.org/abs/1209.1907 https://www.researchgate.net/publication/265945847_Active_Radiation_Shield_for_Space_Exploration_Missions</ref><ref name="Desiati2022">Paolo Desiati and Elena D'Onghia, "CREW HaT: A Magnetic Shielding System for Space Habitats", arXiv:2209.13624 [physics.space-ph] https://doi.org/10.48550/arXiv.2209.13624</ref>.
</table>
</div>

One immediate issue is that you want to keep very high magnetic fields out of the crew area. Fields in excess of 0.5 T can start to be unpleasant, and it is recommended to keep the field below 0.2 T for continuous exposure.

Strong magnets can suffer from high self-forces acting to tear them apart. Most shielding designs will have multiple magnetic coils, which will exert high forces and torques on each other. This will result in the need for strong structural bracing to keep the magnets intact and in place. Such strong structural designs will impose significant extra mass on the spacecraft.

Consider a magnetic field configuration that can keep radiation out of a certain region of space. There is no trajectory of a charged radiation particle up to a certain energy that will allow those particles to enter the protected region. But the dynamics of charged particles in a magnetic field are the same whether going forward or backward in time. If there is no way for a particle to get in, by this time reversal symmetry any particle which is in that forbidden region can never escape. Consequently any field configuration that is a magnetic trap is also a magnetic shield; any field configuration that is a magnetic shield is also a magnetic trap. A spacecraft with a magnetic shield may end up surrounded by strong radiation belts.

Using a magnet to bend a weaponized electron beam very often ends up with the beam being bent so hard that it dumps most of its energy in the form of synchrotron radiation at gamma ray energies. Because gamma rays are not charged, they will not be deflected by the field. Thus, if you would have been hit by the electron beam you will be hit instead by the gamma rays.

As an example, consider a 1 TeV electron beam. We'll surround our spacecraft with a 10 T magnetic field in a 100 meter radius region just outside our spacecraft. This will deflect the beam by 30 meters, which is enough to make it miss. But we are not saved, because the beam dumps nearly all of its energy into synchrotron radiation with an energy spectrum of gamma rays up to 13 GeV. This radiation will hit us in a strip about 2.4 meters long and not much more than the width of the original beam with essentially all of the energy of the original beam.

==== Magnet drones ====

If a magnetic shield around your spacecraft will just end up killing you from synchrotron radiation, what if you put the magnetic shield on a drone and put that drone between yourself and the enemy trying to shoot you? This lets you use a much weaker magnet because you don't need nearly so high of a deflection angle to make the shot miss. This means that less beam energy will turn into synchrotron radiation that could kill you instead and will spread what synchrotron radiation does occur out over a much wider area when it reaches your spacecraft, hopefully so that most of it just misses your spacecraft.

It does mean that your enemy will start shooting your drones so that they can eventually take a shot at you. But hopefully by that time you've already cleared out their drones and cored their spacecraft with your own synchrocannon or UREB.

==== Plasma magnets ====

Usually, to make a really big magnetic field you would run current through a really big conductive loop. But really big things are heavy and expensive. One way around this could be to use the ambient space plasma to carry the electric current rather than any kind of wire. In order to make this current, you use the same mechanism used in a rotary induction electric motor: you generate a rotating magnetic field and induce a current in the conductive loop (in this case, the space plasma). How do you make a rotating magnetic field? The method proposed<ref name="Slough2006">John Slough, "The Plasma Magnet", NASA Institute for Advanced Concepts Phase II Final Report (2006) https://www.niac.usra.edu/files/studies/final_report/917Slough.pdf</ref> is to use an antenna of four much smaller current loops as electromagnets at 90&degree; to each other, and sinusoidally vary the current through the loops with a respective 90&degree; phase difference between them. When this is done, a circular current is induced in the space plasma and the self-force of a magnetic field on its generating current inflates the current loop to much larger radii. You end up with a current sheet in the plane of the rotating field's rotation circulating about the antenna nearly synchronously with the field rotation. This current then creates a normal dipolar magnetic field as with any current loop. The current sheet and the resulting magnetic field expand until the magnetic field's pressure equals that of the ambient space plasma. This gives an enormous region of magnetic field from a relatively small and lightweight system.

The system proposed in the initial study would use an antenna approximately 10 meters in radius and produce a region of field approximately 30 km in radius at Earth's orbit. The field inside would not be constant - at the edges it would be 50 nT, giving the right pressure to keep out the solar wind at Earth's orbit. A circulating current sheet produces a field that falls off as 1/r with distance, meaning that adjacent to the antenna the field would be about 0.15 mT. Using the methods in the box below, we can estimate that this particular design as part of a spacecraft's shields will not deflect a TeV electron beam by more than a few centimeters, which is usually insufficient to force a miss. However, it could potentially be useful for a drone magnet, where the much larger distance between the magnetic field and the target spacecraft would allow a small deflection angle to give perhaps tens or low hundreds of meters change in beam location near the target at near light second ranges. The relatively low field and gradual bend of the incident beam suggests that synchrotron radiation will not be an issue; however this does not exclude more powerful plasma magnet designs causing significant synchrotron radiation effects.

<table border=1><tr><td>
Calculating the deflection from a spatially varying magnetic field affected by the solar wind is no trivial task. However, we can make an approximation that both should be within about an order of magnitude of the true value and also represent an upper limit to the deflection angle; although it does neglect the influence of the driving rotating magnetic field which could in some cases increase the deflection.

We will take R0 as the radius of the antenna. B0 is the field strength at the antenna. Rm is the distance at which the magnetic influence stops because the magnetic pressure equals the space plasma pressure. Bm is the field strength at Rm. According to Slough<ref name="Slough2006"></ref>, the magnetic field strength B(r) at any distance from the antenna center r between R0 and Rm is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B(r) = B0 R0 / r.
</div>
Therefore, once we know Bm and Rm, B0 is completely determined
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B0 = Bm Rm / R0.
</div>

The force on a particle with charge q passing perpendicularly through a magnetic field B(r) at a speed v is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
F(r) = q v B(r)
</div>
For an electron beam, v is so close to c as to make no difference. If the particle is moving down the y direction and the field is in the z direction, the force will be in the x direction. We assume that the deflection will be small, so we can choose a straight-line path in the y direction offset from the center by an amount x. The plasma bubble radius is Rm, so the particle will enter at ymax = √[Rm2-x2] and leave at ymin = -√[Rm2-x2].

The momentum change the particle experiences from this force is the time integral of the force over its path. Because of reflection symmetry about y = 0, we only need to integrate from ymax to 0 and then multiply by 2
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] dt F(r)
<tr> <td> <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] (dy/v) F(r)
<tr> <td> <td> = <td align="left">2 q B(0) R0 [Integral from 0 to √[Rm2-x2]] dy 1/√[y2 + x2]
<tr> <td> <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/x]
</table>
</div>
This holds for |x| > R0. But the expression for the field is not valid for r < R0 and the integral will give nonsense and diverging results if we evaluate for |x| < R0. In this case, we can put the integral lower bound at r = R0, or y = √[R02-x2]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/(√[R02-x2] + R0)]
</table>
</div>

The deflection angle is now estimated as
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = px / p
</div>
where the total momentum of the particle p can be found from the methods in the first box in the Magnetic Fields section.

The deflection for a shot aimed right at the spacecraft at the position of the spacecraft can be estimated by setting x = 0 and only taking the first half of the integral (so ignore the factor of 2)
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
q B(0) R0 ln[Rm/R0] / (p Rm)
</div>
This is an additional overestimate because it fails to account for how the deflection builds up over distance during the integration, but it should still give something of an order-of-magnitude and upper limit estimate of what kind of deflections might be expected.
</table>

=== Electric fields ===

If electric fields can accelerate charged particles, maybe they can keep the charged particles in a weaponized particle beam away from your spacecraft? To completely block a particle beam, you would need a voltage on your spacecraft equal to the particle energy in eV of the particles in the beam. That is, if a 1 GeV proton beam is coming at you, you'd need an electric potential of a billion volts to stop that beam. This leads to three immediate problems.

The first is that particles of the opposite charge will be attracted to your spacecraft. Because space isn't really empty but rather a rarefied plasma (called the solar wind in a solar system or the interstellar medium out between stars), there will be plenty of particles of the opposite charge around to attract. And they will hit with an energy in eV equal to the voltage on your spacecraft. In our example of defending against a 1 GeV proton beam with a gigavolt field, the electrons from the solar wind will come in and hit with an energy of 1 GeV each. This will generate dangerous radiation.

The second problem is a consequence of the first. Each of those GeV electrons from our example need to get their energy from somewhere, which turns out to be from the electric field we are using as shielding. You will need to make up for this energy loss with power from your electrical system, which gives a constant drain on your power. Some estimates put the electrical effort to maintain a several hundred megavolt shield in space at many tens of gigawatts<ref name="Mechmann2019">Claire Mechmann, "Analysis of Proposed Active Radiation Shielding Design Concept for Spacecraft" (2019) Thesis, College of Engineering and Science of Florida Institute of Technology</ref>.

And the third problem is similar to one for magnetic fields. Your electrodes will experience high self forces and forces between separate electrodes that will try to tear your shield generator apart. Consequently you will need heavy structural reinforcement in order for your shield to work.

There are potential ways to mitigate some of these problems. Interested readers are directed to [[Space_Radiation#Electrostatic_Shielding|the active shielding section in our page on space radiation]] for more of the juicy details; there's no need to repeat the same stuff here.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Elctrostatic_active_shielding.png|200 px|frameless]]
<td>
[[File:Electrostatic_active_shielding_2.png|200 px|frameless]]
<tr><td colspan=2>
Proposed designs for electrostatic shields to protect from positively charged particles<ref name="Tripathi2006">Ram K. Tripathi, John W. Wilson, and Robert C. Youngquist, "Electrostatic Active Radiation Shielding - Revisited", 2006 IEEE Aerospace Conference, Big Sky, MT, USA, 2006, pp. 9 pp.-, doi: 10.1109/AERO.2006.1655760.</ref><ref name="Chowdhury2023">Rajarshi Pal Chowdhury, Luke A. Stegeman, Matthew L. Lund, Dan Fry, Stojan Madzunkov, and Amir A. Bahadori, "Hybrid methods of radiation shielding against deep-space radiation", Life Sciences in Space Research, Volume 38, 2023, Pages 67-78, ISSN 2214-5524, https://doi.org/10.1016/j.lssr.2023.04.004.</ref>.
</table>
</div>

== Damage ==

Particle beams are, first and foremost, devices that produce beams of [[Nuclear_radiation|ionizing radiation]]. The radiation of weaponizable particle beams is extremely penetrating; thus of the [[Particle_Accelerators#Energy_loss_in_matter|interaction mechanisms listed for radiation from particle beams]], we mostly need to worry about bremsstrahlung and nuclear collisions and these produce extensive secondary showers of radiation that increase the penetration of the radiation exposure. Because of this, any target hit by a particle beam weapon will be irradiated, and radiation dose will always be one of the primary damage mechanisms of a particle beam weapon strike. Further, this radiation will be distributed along a track passing deeply through the material of the target, allowing it to affect critical systems through heavy armor, structural material, robust machine components, and passive material such as fuel load, coolant, or cargo. As the radiation deposits its energy into the target material, that material will be heated. If the target is heated sufficiently, it may experience [[Beam-Target_Interactions|thermal or mechanical]] damage from melting, cooking, combustion, vapor explosions, and warping or cracking due to thermal expansion.

The discussion of the radiation penetration into armor also applies to the same radiation penetrating through the target. So the formulas and rules of thumb for estimating the depth of radiation penetration into armor can also be used to estimate how far into a tank or spacecraft a particle beam strike delivers effects.

In addition to the radiation and thermal effects, a particle beam can also produce a sharp electromagnetic pulse that can damage semiconductor electronics.

== 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, Gerrit Bruhaug, and Luke Campbell for writing

Gerrit Bruhaug and Luke Campbell for scientific research, math wrangling and advise

ATriffleMind and Luke W. Campbell for proof reading.

==References==

[[Category:Beams]][[Category:Physics & Engineering‏‎]][[Category:Engineering‏‎]][[Category:Military Technology]][[Category:Warfare‏‎]]

Particle Beam Weapons

2026-10-26T00:46:32Z

Lwcamp: /* Plasma magnets */

{{PageConstructionNotice}}

''For a general overview on particle accelerator technology, see [[Particle Accelerators]]

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.

::: Z = 1.32×102 (γ2-1)3/4 r / √I<ref name=USPAS2016>Martin Reiser, "Theory and Design of Charged Particle Beams", 2008, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim</ref>

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

::: Z = β γ c r √[m/T]

Where Z is the doubling distance like before, r is the initial beam radius, T is the beam temperature in eV, m is the ion mass and β, c, and γ 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.

== Defenses ==

Sure, it's great to have a great big particle cannon to blast your enemies. But what if you are the one being shot at? In that case, it's best to figure out ways to keep from being blown to kingdom come.

=== Armor ===

The simplest way to defend against the high energy particles of a particle beam is to pile a whole bunch of stuff up and put it in the way of the beam. Particle beams may produce extremely penetrating radiation, but enough stuff will block most of that radiation and may allow you to survive. The details depend on what kind of particle is being thrown at you, and are explained below. However, a general rule of thumb is that you probably want about 10 tons of material per square meter of surface you want to armor in order to get sufficient protection.

==== Electrons ====

At the very high energy of an electron accelerated by a weapons-grade particle beam, the primary interaction with matter will be the emission of bremsstrahlung photons in the gamma-ray range as the electron slams into particles of matter. These high energy photons themselves primarily interact by the pair production mechanism – the photon disappears, but leaves behind a highly energetic positron and electron. These two particles are themselves energetic enough to produce further bremsstrahlung, and so the process continues. You get the formation of an electron-gamma shower (also called an electromagnetic cascade), which only stops after either the particles have penetrated through the target or the initial energy of the electron has been spread through so many shower particles (including all the electrons, positrons, and photons) that no particle has enough individual energy to pair produce or to be likely to bremsstrahlung again before it is stopped by other mechanisms.

To stop electrons, you'll want the atoms in your armor to be as heavy as you can practically get them. High atomic number both encourages bremsstrahlung and pair production, giving a double whammy for stopping that cascade. However, occasionally one of those electrons, positrons, or gammas will hit a nucleus and knock out neutrons. So you probably want a back-up layer of several tens of centimeters of hydrogen rich material such as plastic or water mixed with neutron absorbing materials like boron, lithium, cadmium, or gadolinium. High density polyethylene loaded with 5% boron by weight is a typical go-to neutron blocker.

The actual methods to calculate the penetration of the electromagnetic cascade is given are the box below. However, for some examples:
<ul>
<li> A 10 GeV beam incident on thick iron (or steel) will stop cascading at about 9 cm, will deposit half of its energy in the first 12 cm, and will have dumped 98% of its energy within 35 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 2.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 100 GeV beam incident on thick iron (or steel) will stop cascading at about 13 cm, will deposit half of its energy in the first 16 cm, and will have dumped 98% of its energy within 47 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 3.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick iron (or steel) will stop cascading at about 17 cm, will deposit half of its energy in the first 20 cm, and will have dumped 98% of its energy within 59 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 4.6 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick tungsten will stop cascading at about 3.5 cm, will deposit half of its energy in the first 4 cm, and will have dumped 98% of its energy within 12 cm. 90% of the cascade will be contained within about 0.93 cm and 95% of the cascade will be contained within 1.86 cm of the beam axis. You will get 98% protection with a mass thickness of 2.3 tons/m2 and an additional 1/e reduction for each further 0.13 tons/m2 beyond that.
<li> A 1 TeV beam incident on water will stop cascading at about 290 cm, will deposit half of its energy in the first 340 cm, and will have dumped 98% of its energy within 1030 cm. 90% of the cascade will be contained within about 11 cm and 95% of the cascade will be contained within 22 cm of the beam axis. You will get 98% protection with a mass thickness of 10.3 tons/m2 and an additional 1/e reduction for each further 0.72 tons/m2 beyond that.
</ul>
Note that for sufficiently intense particle beams, 98% protection may not be enough. You might need to throw on several more tons per square meter to be sure.

<table border=1>
<tr><td> How Far it Goes:
(This section follows methods presented in Leroy and Rancoita<ref>C. Leroy and P-G Rancoita, "Physics of cascading shower generation and propagation in matter: principles of high-energy, ultrahigh-energy and compensating calorimetry", Reports on Progress in Physics 63 (2000) 505–606. DOI: 10.1088/0034-4885/63/4/202</ref>.)

When trying to estimate the extent of the irradiated region of the target, it is helpful to know the radiation length X0 in the material. This is the distance that an electron will travel, on average, before emitting a bremsstrahlung photon. Conveniently, it is also the distance a high energy photon will travel through the material before pair producing. The radiation length can be found from
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   = 4 α  
<td nowrap="nowrap"> NA
<td rowspan="2" nowrap="nowrap" align="right">   Z (Z+ζ) re2 ln
<td rowspan="2" nowrap="nowrap" align="right"> [
<td nowrap="nowrap"> 183
<td rowspan="2" nowrap="nowrap" align="right"> ]
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> A
<td style="border-top:solid 1px black;"> Z1/3
</table>
</div>
where α = 1/137 is the fine structure constant, NA = 6.02214076×1023 is Avogadro's number (or one mole), A is the atomic weight of the atoms in the material (usually expressed in grams per mole), Z is the atomic number (nuclear charge) of the atoms, and re = 2.8179403205×10-13 cm is the classical electron radius. ζ is a constant of approximately 1.3. This gives the thickness of material in terms of the areal density (mass of material per unit area perpendicular to the beam) – if you use A in g/mol and re in cm the result will be in cm2/g and X0 will be in g/cm2. If your target is made up of more than one kind of atom, you can get the total radiation length by finding the radiation length of each atom of type i, X0i and using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   =  
<td rowspan="2" nowrap="nowrap" align="right"> Σi  
<td nowrap="nowrap"> fi
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> X0i
</table>
</div>
where fi is the fraction by weight of the atom type i. Once you have the radiation length in terms of areal density, you can easily convert it to distance by dividing by the density ρ. If X0i is in g/cm2 and density is in g/cm3, then X0i/ρ will be in cm.

We will also need to know the critical energy εc. This is the energy where an electron is losing energy as rapidly to ionization as it is to radiation. That is, it is the energy at which the cascade stops multiplying, even though the individual particles already made can continue traveling for some distance. The critical energy can be estimated using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> εc = B  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> Z X0
<td rowspan="2" nowrap="nowrap" align="right"> )
<td nowrap="nowrap"> h
<tr>
<td style="border-top:solid 1px black;"> A
<td>  
</table>
</div>
where B = 2.66 MeV and h = 1.11.

We can now describe the distribution of radiation in the target material. The depth tmax where the cascade stops multiplying (in g/cm2) for an incident electron of energy E (in MeV) is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tmax = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> - 1
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
Similarly, the depth where half the energy of the beam has been deposited (in g/cm2) is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tcg = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> + 0.37
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
As usual, you can convert to a depth in cm by dividing by the material density. 98% of the beam energy will be deposited within a distance of &cong; 3 tcg. Beyond this, the beam energy will continue to fall off with thickness t approximately as an exponential exp[-t/λ], where the characteristic attenuation length λ &cong; 2 X0.

The width of a cascade, where 90% of the energy is contained, is approximately given by
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> RM =  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> EM
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
where EM &cong; 21.2 MeV. 95% of the radiation will be contained within a distance of 2 RM from the beam axis.
</table>

==== Protons and Ions ====

<table align=right><tr><td>
[[File:Regolith_Shielding.png|350 px|frameless]]
<tr><td width=350>
Relative effect of particle radiation (compared to no shielding) from galactic cosmic rays behind different thicknesses of water, aluminum, and lunar regolith<ref name="Slaba2022">Tony C. Slaba, "Radiation Shielding for Lunar Missions: Regolith Considerations", LSIC Crosstalk 7/18/2022 https://lsic.jhuapl.edu/uploadedDocs/focus-files/1604-E&C%20+%20EE%20Monthly%20Meeting%20-%202022%2007%20July_Presentation%20-%20NASA%20Slaba.pdf</ref>.
</table>
Protons and ions are grouped together because ions are made of protons and neutrons and when you accelerate these things for a weapon it is the protons and neutrons that are going to be doing the heavy lifting for blasting your target. This is because protons and neutrons are both about 2000 times more massive than electrons so any electrons along for the ride will contribute negligibly to the energy. Protons and neutrons together are called nuclear particles, and both behave quite similarly in nuclear collisions.

Nuclear particles are themselves made of quarks and can interact via the strong nuclear force. At the energies appropriate for weapons use, essentially all of the way protons or ions are blocked is through [[Particle_Accelerators#Nuclear_collisions|direct collisions with the atomic nuclei]] of the armor material. One or more quarks in the projectile will slam into quarks in the target nucleus and stretch them away from the rest of the quarks in the nuclear particles. Physics tells us that you can't have isolated quarks, so the "string" attaching the quark to its neighbors stretches and stretches until it snaps. Snapping releases so much energy that more quarks are made that clump together into all sorts of other nuclear particles, both exotic and familiar. This process is called a hadronic jet. The only particles produced in these jets long-lived enough to travel any noticeable distance through the armor material will be protons, neutrons, the charged pi mesons (or pions), and some of the K mesons (or kaons). The particles too short lived to go far enough to matter will decay into other particles, and these particles are often very high energy gamma rays. Much like with electrons, the long-lived secondary particles can be sufficiently energetic to smash into other nuclei and cause more hadronic jets (or, if they are low enough in energy, just knock more nuclear particles out of target nuclei) and this produces a similar shower or cascade. But in addition, the armor nuclei that were hit will be very hot after the impact from all of the energy given to them from the incident particles. These nuclei will cool off by evaporation – but instead of evaporating away molecules, they evaporate off nuclear particles from the nucleus. These evaporated particles are mostly neutrons, so any armor against proton or ion beam strikes will produce quite a bit of neutron radiation.

Eventually, the original particle's energy will be distributed among so many other particles that the charged particles will come to a stop by making ion tracks rather than from colliding with more nuclei, and the cascade halts. But you still have all the neutrons to worry about. And the gamma rays as well, which are high enough energy to kick off an electromagnetic cascade in the same way that electrons do.

Unlike electrons, which have reasonable approximations of their cascade behavior from relatively simple equations, the cascades of protons and ions usually require sophisticated simulations or experiments to get the resulting distribution of radiation. However, we can get a reasonable idea of their general behavior by comparing them to [[Space_Radiation#Shielding_Against_Cosmic_Rays|cosmic ray particles interacting with shielding]]. Cosmic rays are energetic protons and ions, after all. From this we see that a relatively small amount of armor – in the range of 1 to 3 tons per square meter – can actually make the radiation dose worse. The incident particles will knock off so many secondary particles in their shower that the dose is multiplied from so many tracks passing through the target. Hydrogen rich materials such as water and plastics work the best against proton and ion beam particles; heavy elements are the worst and produce the most secondaries. You could probably obtain adequate radiation protection from most incident beams with about 10 tons per square meter of hydrogen rich material as armor. However, you will want to add neutron absorbing elements to your hydrogen rich armor, such as lithium, boron, cadmium, or gadolinium. Further, in order to deal with the gamma rays you will want a layer of at least several centimeters of heavy elements, such as tungsten or lead, behind the hydrogen rich particle armor.

==== Structural Considerations ====

When your armor material absorbs a radiation beam, the energy of that radiation turns into heat inside the armor where the radiation was absorbed. If the beam pulse has enough energy as was sufficiently tightly focused, this heat can damage your armor. Your armor may melt in places, or thermal expansion could cause it to crack or warp. Perhaps the worst thing it could do is vaporize or flash into a plasma, because the pressure from the vapor or plasma can be sufficient to plastically deform the armor. This leaves a hole along the beam path where the expanding gas or plasma blasted out a long, deep crater. This means a beam that fires rapid shots spaced maybe milliseconds apart can have subsequent shots go down the same hole so your armor will not protect against them. Each shot will dig out a deeper hole until either your armor is penetrated or the attacking beam runs out of shots in its pulse train.

A consequence of this is that when designing your armor you may wish to use refractory materials with a high structural strength (like tungsten), or that is otherwise resistant to being blasted apart by its own evaporated remains.

=== Magnetic fields ===

[[Particle_Accelerators#Magnetic_fields|A charged particle crossing a magnetic field will bend so as to spiral around the field lines]]. If you try to shoot your particle beam through a magnetic field, it will bend away from the direction you shot it. This can make it hard to hit your target. Unfortunately for you, there are magnetic fields everywhere – around planets, and even trapped in the solar wind as it crosses the solar system.

As an example, the Earth has a magnetic field strength of about 5×10-5 tesla near its surface, and falling off farther away. If you are shooting a 1 TeV electron beam past the Earth, your beam is likely to bend by several milliradians (with the exact deflection depending on how close you are skimming the Earth). If you are in a medium Earth orbit and shooting at another target also in a medium Earth orbit on a trajectory that goes near the Earth, then without correcting for the magnetic field your beam will likely be off-target by tens to hundreds of kilometers.

For another example, the interplanetary magnetic field is about 10-9 tesla. If you shoot that same 1 TeV electron beam at a target 0.1 light seconds away, you are likely to be off by a couple hundred meters.

<table border=1>
<tr><td>
The radius of curvature of the bend of a particle in a magnetic field is called the gyroradius rc. For a particle with a charge q, mass m, and kinetic energy E, find the relativistic momentum p
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
p = √[E2 - m2 c4] / c
</div>
To convert from energy units of eV to joules, multiply by 1.60×10-19. An electron has a mass of 9.11×10-31 kg, a proton of 1.67×10-27 kg, and an ion of 1.66×10-27×A kg, where A is the atomic weight of the ion. The speed of light c = 3.00×108 m/s.

In a magnetic field of strength B with an angle θ between the field direction and the direction of motion
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rc = |p sin(θ) / (q B)|
</div>
For electrons, q = -e, where the fundamental charge e = 1.60×10-19 coulomb. For a proton, q = +e. For an ion, multiply e by the charge state of the ion.

If the field acts over a distance d that is much smaller than the gyroradius, the deflection angle φ is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = d / rc
</div>
The deflection x at a target at a range R for small φ (φ &Lt; 1) will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
x = φ R.
</div>

Note that d will always be less than or equal to x. Also note that if d is not smaller than twice the gyroradius the beam will never reach the target because it will completely spiral around the field lines before it reaches the target.
</table>

So, it is clear that as the defender it will be helpful to hang out in areas with high magnetic fields, and keep some distance between you and the attacker. Meanwhile, the attacker will benefit from being closer where small deflections in the beam trajectory don't translate into large excursions from where the beam is aimed (and also because there will be less space filled with fields between the two of you). But the attacker can also try to compensate for known fields. If you happen to know that the magnetic fields are between you and your target, you can adjust your fire so that the deflection of the field bends your beam onto the target. This is not a simple task. It is easy enough to measure the field strength and direction where you are, but predicting how it changes over the path of your shot does not have an obvious solution. You can also try to gain intelligence on likely battle sites before you ever fight on them – for example, getting good data on the structure of the fields of a planet using spies and technical intelligence during peacetime (heck, on Earth this information is published for free in academic journals and by government agencies such as NOAA). Planetary magnetospheres are also affected by the solar wind, so different solar weather will change the fields you have to shoot through. But space weather forecasts and predictive simulations of the solar wind interacting with a planetary magnetosphere can give you a good idea of what the field structure is like. But even with all that, there are changes in the fields that happen on a scale of around a second ... anyone who has ever seen the auroras dancing across the sky in the polar night can see this. Compensating for these changes will be a significant challenge.

What would really help is being able to see the path of your own beam. This would allow you to see how far off you were from your target and in what direction, and then adjust fire to walk your shots onto the target. Unfortunately, particle beams in space are invisible. There might be ways to track them, but if so we don't know what they are.

==== Synchrotron radiation ====

When charged particle trajectories are bent, they emit [[Particle_Accelerators#Synchrotron_radiation|synchrotron radiation]]. This can cause the particle beam to lose energy, but if the bend is too tight it can also generate high energy gamma rays that ignore magnetic fields and directly strike the target. Synchrotron radiation can generally be ignored for proton or ion beams, but can be significant for electron beams.

<table border=1><tr><td>
For a beam particle with energy E and mass m, the synchrotron radiation is concentrated in a forward-directed cone with an opening angle of approximately 1/γ, where γ = 1 + E/(m c2). (Interestingly, the radiation intensity peaks an at angle of 1/(2 γ) and vanishes directly on-axis.) Because the beam is bending, you will get emission along the plane along which the beam is being bent. The energy of the photons produced during synchrotron emission is approximately<ref name="Jackson_Electrodynamics">J. D. Jackson, "Classical Electrodynamics: Second Edition", Wiley (1962)</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Eγ &cong; 3 γ3 c &hbar; / rc
</div>
where &hbar; = 1.05×10-34 J s (if you want the energy in joules) or &hbar; = 6.58×10-16 eV s (if you want the energy in eV). The photon energy will not exceed the energy of the incident particles, if you get a photon energy higher than the energy of the incident particles it just means that the synchrotron radiation spectrum extends up to the incident particle energy.

The fraction of the energy (or power) of the beam turned into synchrotron radiation will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = [1/(4 π ε0)] [4 π / 3] [q² / rc] β2 γ4 [φ / (2 π)] / E
</div>
where β = √[1 - 1/γ2] is the particle's speed measured as a fraction of the speed of light, and 1/(4 π ε0) = 8.99×109 is the Coulomb constant. For highly relativistic electrons (like all practical electron beam weapons emit), β &cong; 1 and we can use
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = 8.85×10-2 [E (GeV)3 [φ / (2 π)] / (rc (m))]
</div>
This fraction cannot be more than 1. If f gets close to one, it means that our original assumption that the beam particles mostly keep their original speed no longer holds, and the beam will lose most or almost all of its energy. Instead, when the beam is being deflected it will only make it through an angle of φ/f and will irradiate a region of length x/f starting at where the beam would have been incident if not for the magnets and extending in the direction the beam was deflected.
</table>

For example, a 1 TeV electron beam will have γ &cong; 2,000,000. So if it's bent by a magnetic field the synchtrotron radiation will spread out in a cone of about 5×10-7 radians. If this beam was shooting at a target 0.1 light seconds away from where it was bent, the radiation would arrive at the target over a region with a radius of about 15 meters with a maximum intensity at about 7.5 meters and no radiation directly along the path the beam was pointing while it was being bent. If the beam was bent by the interplanetary magnetic field from our earlier example, it would produce light with photon energies of around 1.3 eV (near infrared) and the lost energy would be a few parts in a hundred trillion. If it was instead deflected by the Earth's magnetic field you might expect a peak photon energy of tens of keV – in the hard x-ray part of the spectrum – and the fraction of energy in the synchrotron radiation would be somewhere in the low parts in 100,000 or less.

==== Separation of neutral beams ====

Magnetic fields only affect charged objects. So one way to overcome magnetic defenses or environments is to use beams that have no net charge. For example, a proton beam could be neutralized with a co-propagating electron beam with an equal number of protons and electrons. Or, potentially, you could fire an ion beam and then neutralize the ions after acceleration to get a beam of neutral atoms.

But, the magnetic fields are still tugging at the charged particles making up the beam as they go past. This will pull the electrons one way and the protons or ions the other way. As the particles are pulled in opposite directions you get a net positive charge on one side of the beam and a net negative charge on the other side (called polarization). This sets up an electric field inside the beam that acts to pull the particles back together. If the particles are going fast enough and the magnetic field is strong enough, the magnetic force can still separate the charges and will split the beam in two, one with the positive charges and one with the negative charges. But if the particles are not that fast or the magnetic field is not that strong this polarization of the beam will keep it together and let it fly right through the magnetic field.

<table border=1><tr><td>
Let's figure out how to analyze this problem. For definiteness, we will have a beam with an electric current of the positive species of I and an electric current of electrons of -I, all moving in the z direction. The beam radius will be R. The particle speed will be v with the usual relativistic parameters β = v/c and γ = √[1/(1-β2)]. If you know the energy per particle E of the positive species and the mass m of the positive species, γ = 1 + E/(m c2) and then β = √[1 - 1/γ2].

Let's assume that the beam is going through a magnetic field of strength B at an angle θ to the field direction. The quantity B sin(θ) is the component of the magnetic field perpendicular to the direction of the beam.

The charge density ρ of the positive component of the beam will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρL = I / (π v R2).
</div>

It is easiest to analyze the problem in the rest frame of the particles in the beam. Now the particles are at rest. Particles at rest are not affected by a magnetic field, but the change of reference frame Lorentz transforms the initial magnetic field into a combination of electric and magnetic fields in the rest frame. We can ignore the magnetic field, but the electric field is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔A = γ v B sin(θ)
</div>
in a direction perpendicular to both v and B. For our purposes, assume the electric field points in the x direction.

At the same time, relativistic length contraction increases the charge density in the rest frame
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρ = γ ρL.
</div>

Let r be the vector from the beam axis to the point inside the beam. It will be useful to separate this into its Cartesian coordinates. If x and y are the unit vectors in the x and y directions, respectively, then
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
r = x x + y y
</div>

If we consider only the positive component of the beam, from [https://en.wikipedia.org/wiki/Gauss%27s_law Gauss's law] it will have an electric field inside of the beam at a distance r from the center axis of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔+ = ρ r / (2 ε0) = (ρ/(2 ε0)) (x x + y y)
</div>
where ε0 = 8.85×10-12 is the vacuum permittivity. Now let's displace the electrons in the beam by a distance δ in the x direction. By itself, this electron beam will have an electric field of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔- = -(ρ/(2 ε0)) ((x+δ) x + y y)
</div>
By the principle of superposition, the electric field inside the beam with both components is the sum of the electric fields of the separate components including the external field
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔B = [-(ρ/(2 ε0)) δ + γ v B sin(θ)]x
</div>
Because the charges are free to move, they will do so as long as a field is applied to them. Thus, the beam becomes stable once the charges separate enough that the field internal to the beam is zero. Therefore
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
δ = 2 ε0 γ v B sin(θ) / ρ = 2 π ε0 v2 R2 B sin(θ) / I
</div>
If δ is small compared to R, the beam's polarization suffices to hold it together. If δ > 2 R, the beam will fully separate and each component will go its own way. This gives us a rough threshold for which a magnetic field will separate a neutral beam into its charged components and bend them away from their target
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr><td> B sin(θ) &Lt; I / (π ε0 v2 R) <td>       <td> Beam is stable
<tr><td> B sin(θ) > I / (π ε0 v2 R) <td> <td> Beam promptly separates.
</table>
</div>

For a beam of neutral atoms, let a be the radius of the atom and let Eb be the binding energy of the least tightly bound electron to the atom. An external electric field in excess of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
|𝓔A| > Eb / (e a)
</div>
will promptly ionize the atom, ripping away its electron. Here, e = 1.60×10-19 C is the magnitude of the electric charge of the electron. For fields close to but a bit weaker than this, the electron can still rapidly pop off due to quantum tunneling<ref>Ph. Durand and I. Paidarová, “Ionization of the hydrogen atom: from weak to strong static electric fields,” The European Physical Journal D, Vol. 26, pg. 253-259, 2003.</ref>. In general, expect that for fields greater in magnitude than (very approximately) Eb / (10 e a) the electron will be quickly removed, while for fields weaker than this the electrons will stay on their atoms.

It is useful to note that Eb/e is just the binding energy in units of eV. The most weakly bound electrons are usually bound to their atom with about 10 eV of energy. Atoms are typically about 10-10 m in radius. Thus, you can expect fields of around 1010 to 1011 V/m to directly ionize atoms. If we call this threshold value 𝓔T, then a beam will be ionized when passing through a field with strength greater than
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B sin(θ) > 𝓔T / (γ v)
</div>
</table>

Example: Let's consider a 1 GeV neutralized proton beam from a synchrotron, with a beam current of 200 kA in each bunch (incidentally, if a bunch is 5 cm long this means that each bunch packs 38 kJ worth of energy). The speed of the bunch is 87% the speed of light, or 262 million m/s, with a γ of 2.07. Consider when the beam's focus takes it to a radius of 5 cm (or 0.05 m, for our calculation). Plugging this in, we find that the protons in the bunch will separate from the electrons in a field with a transverse component greater than about 2.1 T, and the bunch will pass through the field without much effect if the transverse component is much smaller than this.

Example: We'll take the same 1 GeV beam, but this time recombine the electrons with protons to make hydrogen atoms. An electron is bound to a hydrogen atom with an energy of 13.6 eV (the Ryberg constant) and the radius of the electron orbital in a hydrogen atom is 0.529×10-10 m (the Bohr radius). This gives a threshold electric field for prompt ionization of hydrogen of 2.57×1011 V/m. We will get this with a transverse component of the magnetic field of 474 T. Above this field the atoms promptly ionize and electrons are stripped away from the protons. For fields higher than roughly 1/10 of this – so about 40 to 50 T – you may still get significant ionization by quantum tunneling; lower than this your losses from the beam due to field-induced ionization are likely negligible.

==== Magnetic shields ====

What happens when we can't rely on magnetic fields in the environment to protect us? One idea is to produce powerful artificial magnetic fields around our spacecraft to deflect incoming particle beams. This is very similar in concept to using active magnetic shielding to protect against space radiation – [[Space_Radiation#Magnetic_Shielding|check out our article on space radiation for more]].
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr>
<td width=100>
[[File:Unconfined_FRC_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:racetrack_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:Magnetic_shielding_Halback_Array.png|200 px|frameless]]
<tr><td colspan=3>
Some potential configurations for magnetic shields. The currents generating the fields are in green, the magnetic fields themselves are shown in magenta<ref name="Battiston2012">R. Battiston, W.J. Burger, V. Calvelli, R. Musenich, V. Choutko, V.I. Datskov, A. Della Torre, F. Venditti,
C. Gargiulo, G. Laurenti, S. Lucidi, S. Harrison, and R. Meinke, "ARSSEM Active Radiation Shield for Space Exploration Missions", Final Report ESTEC Contract N° 4200023087/10/NL/AF : “Superconductive Magnet for Radiation Shielding of Human Spacecraft” (2012) https://arxiv.org/abs/1209.1907 https://www.researchgate.net/publication/265945847_Active_Radiation_Shield_for_Space_Exploration_Missions</ref><ref name="Desiati2022">Paolo Desiati and Elena D'Onghia, "CREW HaT: A Magnetic Shielding System for Space Habitats", arXiv:2209.13624 [physics.space-ph] https://doi.org/10.48550/arXiv.2209.13624</ref>.
</table>
</div>

One immediate issue is that you want to keep very high magnetic fields out of the crew area. Fields in excess of 0.5 T can start to be unpleasant, and it is recommended to keep the field below 0.2 T for continuous exposure.

Strong magnets can suffer from high self-forces acting to tear them apart. Most shielding designs will have multiple magnetic coils, which will exert high forces and torques on each other. This will result in the need for strong structural bracing to keep the magnets intact and in place. Such strong structural designs will impose significant extra mass on the spacecraft.

Consider a magnetic field configuration that can keep radiation out of a certain region of space. There is no trajectory of a charged radiation particle up to a certain energy that will allow those particles to enter the protected region. But the dynamics of charged particles in a magnetic field are the same whether going forward or backward in time. If there is no way for a particle to get in, by this time reversal symmetry any particle which is in that forbidden region can never escape. Consequently any field configuration that is a magnetic trap is also a magnetic shield; any field configuration that is a magnetic shield is also a magnetic trap. A spacecraft with a magnetic shield may end up surrounded by strong radiation belts.

Using a magnet to bend a weaponized electron beam very often ends up with the beam being bent so hard that it dumps most of its energy in the form of synchrotron radiation at gamma ray energies. Because gamma rays are not charged, they will not be deflected by the field. Thus, if you would have been hit by the electron beam you will be hit instead by the gamma rays.

As an example, consider a 1 TeV electron beam. We'll surround our spacecraft with a 10 T magnetic field in a 100 meter radius region just outside our spacecraft. This will deflect the beam by 30 meters, which is enough to make it miss. But we are not saved, because the beam dumps nearly all of its energy into synchrotron radiation with an energy spectrum of gamma rays up to 13 GeV. This radiation will hit us in a strip about 2.4 meters long and not much more than the width of the original beam with essentially all of the energy of the original beam.

==== Magnet drones ====

If a magnetic shield around your spacecraft will just end up killing you from synchrotron radiation, what if you put the magnetic shield on a drone and put that drone between yourself and the enemy trying to shoot you? This lets you use a much weaker magnet because you don't need nearly so high of a deflection angle to make the shot miss. This means that less beam energy will turn into synchrotron radiation that could kill you instead and will spread what synchrotron radiation does occur out over a much wider area when it reaches your spacecraft, hopefully so that most of it just misses your spacecraft.

It does mean that your enemy will start shooting your drones so that they can eventually take a shot at you. But hopefully by that time you've already cleared out their drones and cored their spacecraft with your own synchrocannon or UREB.

==== Plasma magnets ====

Usually, to make a really big magnetic field you would run current through a really big conductive loop. But really big things are heavy and expensive. One way around this could be to use the ambient space plasma to carry the electric current rather than any kind of wire. In order to make this current, you use the same mechanism used in a rotary induction electric motor: you generate a rotating magnetic field and induce a current in the conductive loop (in this case, the space plasma). How do you make a rotating magnetic field? The method proposed<ref name="Slough2006">John Slough, "The Plasma Magnet", NASA Institute for Advanced Concepts Phase II Final Report (2006) https://www.niac.usra.edu/files/studies/final_report/917Slough.pdf</ref> is to use an antenna of four much smaller current loops as electromagnets at 90&degree; to each other, and sinusoidally vary the current through the loops with a respective 90&degree; phase difference between them. When this is done, a circular current is induced in the space plasma and the self-force of a magnetic field on its generating current inflates the current loop to much larger radii. You end up with a current sheet in the plane of the rotating field's rotation circulating about the antenna nearly synchronously with the field rotation. This current then creates a normal dipolar magnetic field as with any current loop. The current sheet and the resulting magnetic field expand until the magnetic field's pressure equals that of the ambient space plasma. This gives an enormous region of magnetic field from a relatively small and lightweight system.

The system proposed in the initial study would use an antenna approximately 10 meters in radius and produce a region of field approximately 30 km in radius at Earth's orbit. The field inside would not be constant - at the edges it would be 50 nT, giving the right pressure to keep out the solar wind at Earth's orbit. A circulating current sheet produces a field that falls off as 1/r with distance, meaning that adjacent to the antenna the field would be about 0.15 mT. Using the methods in the box below, we can estimate that this particular design as part of a spacecraft's shields will not deflect a TeV electron beam by more than a few centimeters, which is usually insufficient to force a miss. However, it could potentially be useful for a drone magnet, where the much larger distance between the magnetic field and the target spacecraft would allow a small deflection angle to give perhaps tens or low hundreds of meters change in beam location near the target at near light second ranges. The relatively low field and gradual bend of the incident beam suggests that synchrotron radiation will not be an issue; however this does not exclude more powerful plasma magnet designs causing significant synchrotron radiation effects.

<table border=1><tr><td>
Calculating the deflection from a spatially varying magnetic field affected by the solar wind is no trivial task. However, we can make an approximation that both should be within about an order of magnitude of the true value and also represent an upper limit to the deflection angle; although it does neglect the influence of the driving rotating magnetic field which could in some cases increase the deflection.

We will take R0 as the radius of the antenna. B0 is the field strength at the antenna. Rm is the distance at which the magnetic influence stops because the magnetic pressure equals the space plasma pressure. Bm is the field strength at Rm. According to Slough<ref name="Slough2006"></ref>, the magnetic field strength B(r) at any distance from the antenna center r between R0 and Rm is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B(r) = B0 R0 / r.
</div>
Therefore, once we know Bm and Rm, B0 is completely determined
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B0 = Bm Rm / R0.
</div>

The force on a particle with charge q passing perpendicularly through a magnetic field B(r) at a speed v is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
F(r) = q v B(r)
</div>
For an electron beam, v is so close to c as to make no difference. If the particle is moving down the y direction and the field is in the z direction, the force will be in the x direction. We assume that the deflection will be small, so we can choose a straight-line path in the y direction offset from the center by an amount x. The plasma bubble radius is Rm, so the particle will enter at ymax = √[Rm2-x2] and leave at ymin = -√[Rm2-x2].

The momentum change the particle experiences from this force is the time integral of the force over its path. Because of reflection symmetry about y = 0, we only need to integrate from ymax to 0 and then multiply by 2
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] dt F(r)
<tr> <td> <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] (dy/v) F(r)
<tr> <td> <td> = <td align="left">2 q B(0) R0 [Integral from 0 to √[Rm2-x2]] dy 1/√[y2 + x2]
<tr> <td> <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/x]
</table>
</div>
This holds for |x| > R0. But the expression for the field is not valid for r < R0 and the integral will give nonsense and diverging results if we evaluate for |x| < R0. In this case, we can put the integral lower bound at r = R0, or y = √[R02-x2]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/(√[R02-x2] + R0)]
</table>
</div>

The deflection angle is now estimated as
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = px / p
</div>
where the total momentum of the particle p can be found from the methods in the first box in the Magnetic Fields section.

The deflection for a shot aimed right at the spacecraft at the position of the spacecraft can be estimated by setting x = 0 and only taking the first half of the integral (so ignore the factor of 2)
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
q B(0) R0 ln[Rm/R0] / (p Rm)
</div>
This is an additional overestimate because it fails to account for how the deflection builds up over distance during the integration, but it should still give something of an order-of-magnitude and upper limit estimate of what kind of deflections might be expected.
</table>

=== Electric fields ===

If electric fields can accelerate charged particles, maybe they can keep the charged particles in a weaponized particle beam away from your spacecraft? To completely block a particle beam, you would need a voltage on your spacecraft equal to the particle energy in eV of the particles in the beam. That is, if a 1 GeV proton beam is coming at you, you'd need an electric potential of a billion volts to stop that beam. This leads to three immediate problems.

The first is that particles of the opposite charge will be attracted to your spacecraft. Because space isn't really empty but rather a rarefied plasma (called the solar wind in a solar system or the interstellar medium out between stars), there will be plenty of particles of the opposite charge around to attract. And they will hit with an energy in eV equal to the voltage on your spacecraft. In our example of defending against a 1 GeV proton beam with a gigavolt field, the electrons from the solar wind will come in and hit with an energy of 1 GeV each. This will generate dangerous radiation.

The second problem is a consequence of the first. Each of those GeV electrons from our example need to get their energy from somewhere, which turns out to be from the electric field we are using as shielding. You will need to make up for this energy loss with power from your electrical system, which gives a constant drain on your power. Some estimates put the electrical effort to maintain a several hundred megavolt shield in space at many tens of gigawatts<ref name="Mechmann2019">Claire Mechmann, "Analysis of Proposed Active Radiation Shielding Design Concept for Spacecraft" (2019) Thesis, College of Engineering and Science of Florida Institute of Technology</ref>.

And the third problem is similar to one for magnetic fields. Your electrodes will experience high self forces and forces between separate electrodes that will try to tear your shield generator apart. Consequently you will need heavy structural reinforcement in order for your shield to work.

There are potential ways to mitigate some of these problems. Interested readers are directed to [[Space_Radiation#Electrostatic_Shielding|the active shielding section in our page on space radiation]] for more of the juicy details; there's no need to repeat the same stuff here.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Elctrostatic_active_shielding.png|200 px|frameless]]
<td>
[[File:Electrostatic_active_shielding_2.png|200 px|frameless]]
<tr><td colspan=2>
Proposed designs for electrostatic shields to protect from positively charged particles<ref name="Tripathi2006">Ram K. Tripathi, John W. Wilson, and Robert C. Youngquist, "Electrostatic Active Radiation Shielding - Revisited", 2006 IEEE Aerospace Conference, Big Sky, MT, USA, 2006, pp. 9 pp.-, doi: 10.1109/AERO.2006.1655760.</ref><ref name="Chowdhury2023">Rajarshi Pal Chowdhury, Luke A. Stegeman, Matthew L. Lund, Dan Fry, Stojan Madzunkov, and Amir A. Bahadori, "Hybrid methods of radiation shielding against deep-space radiation", Life Sciences in Space Research, Volume 38, 2023, Pages 67-78, ISSN 2214-5524, https://doi.org/10.1016/j.lssr.2023.04.004.</ref>.
</table>
</div>

== Damage ==

Particle beams are, first and foremost, devices that produce beams of [[Nuclear_radiation|ionizing radiation]]. The radiation of weaponizable particle beams is extremely penetrating; thus of the [[Particle_Accelerators#Energy_loss_in_matter|interaction mechanisms listed for radiation from particle beams]], we mostly need to worry about bremsstrahlung and nuclear collisions and these produce extensive secondary showers of radiation that increase the penetration of the radiation exposure. Because of this, any target hit by a particle beam weapon will be irradiated, and radiation dose will always be one of the primary damage mechanisms of a particle beam weapon strike. Further, this radiation will be distributed along a track passing deeply through the material of the target, allowing it to affect critical systems through heavy armor, structural material, robust machine components, and passive material such as fuel load, coolant, or cargo. As the radiation deposits its energy into the target material, that material will be heated. If the target is heated sufficiently, it may experience [[Beam-Target_Interactions|thermal or mechanical]] damage from melting, cooking, combustion, vapor explosions, and warping or cracking due to thermal expansion.

The discussion of the radiation penetration into armor also applies to the same radiation penetrating through the target. So the formulas and rules of thumb for estimating the depth of radiation penetration into armor can also be used to estimate how far into a tank or spacecraft a particle beam strike delivers effects.

In addition to the radiation and thermal effects, a particle beam can also produce a sharp electromagnetic pulse that can damage semiconductor electronics.

== 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, Gerrit Bruhaug, and Luke Campbell for writing

Gerrit Bruhaug and Luke Campbell for scientific research, math wrangling and advise

ATriffleMind and Luke W. Campbell for proof reading.

==References==

[[Category:Beams]][[Category:Physics & Engineering‏‎]][[Category:Engineering‏‎]][[Category:Military Technology]][[Category:Warfare‏‎]]

Particle Beam Weapons

2026-10-26T00:44:10Z

Lwcamp: /* Plasma magnets */

{{PageConstructionNotice}}

''For a general overview on particle accelerator technology, see [[Particle Accelerators]]

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.

::: Z = 1.32×102 (γ2-1)3/4 r / √I<ref name=USPAS2016>Martin Reiser, "Theory and Design of Charged Particle Beams", 2008, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim</ref>

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

::: Z = β γ c r √[m/T]

Where Z is the doubling distance like before, r is the initial beam radius, T is the beam temperature in eV, m is the ion mass and β, c, and γ 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.

== Defenses ==

Sure, it's great to have a great big particle cannon to blast your enemies. But what if you are the one being shot at? In that case, it's best to figure out ways to keep from being blown to kingdom come.

=== Armor ===

The simplest way to defend against the high energy particles of a particle beam is to pile a whole bunch of stuff up and put it in the way of the beam. Particle beams may produce extremely penetrating radiation, but enough stuff will block most of that radiation and may allow you to survive. The details depend on what kind of particle is being thrown at you, and are explained below. However, a general rule of thumb is that you probably want about 10 tons of material per square meter of surface you want to armor in order to get sufficient protection.

==== Electrons ====

At the very high energy of an electron accelerated by a weapons-grade particle beam, the primary interaction with matter will be the emission of bremsstrahlung photons in the gamma-ray range as the electron slams into particles of matter. These high energy photons themselves primarily interact by the pair production mechanism – the photon disappears, but leaves behind a highly energetic positron and electron. These two particles are themselves energetic enough to produce further bremsstrahlung, and so the process continues. You get the formation of an electron-gamma shower (also called an electromagnetic cascade), which only stops after either the particles have penetrated through the target or the initial energy of the electron has been spread through so many shower particles (including all the electrons, positrons, and photons) that no particle has enough individual energy to pair produce or to be likely to bremsstrahlung again before it is stopped by other mechanisms.

To stop electrons, you'll want the atoms in your armor to be as heavy as you can practically get them. High atomic number both encourages bremsstrahlung and pair production, giving a double whammy for stopping that cascade. However, occasionally one of those electrons, positrons, or gammas will hit a nucleus and knock out neutrons. So you probably want a back-up layer of several tens of centimeters of hydrogen rich material such as plastic or water mixed with neutron absorbing materials like boron, lithium, cadmium, or gadolinium. High density polyethylene loaded with 5% boron by weight is a typical go-to neutron blocker.

The actual methods to calculate the penetration of the electromagnetic cascade is given are the box below. However, for some examples:
<ul>
<li> A 10 GeV beam incident on thick iron (or steel) will stop cascading at about 9 cm, will deposit half of its energy in the first 12 cm, and will have dumped 98% of its energy within 35 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 2.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 100 GeV beam incident on thick iron (or steel) will stop cascading at about 13 cm, will deposit half of its energy in the first 16 cm, and will have dumped 98% of its energy within 47 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 3.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick iron (or steel) will stop cascading at about 17 cm, will deposit half of its energy in the first 20 cm, and will have dumped 98% of its energy within 59 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 4.6 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick tungsten will stop cascading at about 3.5 cm, will deposit half of its energy in the first 4 cm, and will have dumped 98% of its energy within 12 cm. 90% of the cascade will be contained within about 0.93 cm and 95% of the cascade will be contained within 1.86 cm of the beam axis. You will get 98% protection with a mass thickness of 2.3 tons/m2 and an additional 1/e reduction for each further 0.13 tons/m2 beyond that.
<li> A 1 TeV beam incident on water will stop cascading at about 290 cm, will deposit half of its energy in the first 340 cm, and will have dumped 98% of its energy within 1030 cm. 90% of the cascade will be contained within about 11 cm and 95% of the cascade will be contained within 22 cm of the beam axis. You will get 98% protection with a mass thickness of 10.3 tons/m2 and an additional 1/e reduction for each further 0.72 tons/m2 beyond that.
</ul>
Note that for sufficiently intense particle beams, 98% protection may not be enough. You might need to throw on several more tons per square meter to be sure.

<table border=1>
<tr><td> How Far it Goes:
(This section follows methods presented in Leroy and Rancoita<ref>C. Leroy and P-G Rancoita, "Physics of cascading shower generation and propagation in matter: principles of high-energy, ultrahigh-energy and compensating calorimetry", Reports on Progress in Physics 63 (2000) 505–606. DOI: 10.1088/0034-4885/63/4/202</ref>.)

When trying to estimate the extent of the irradiated region of the target, it is helpful to know the radiation length X0 in the material. This is the distance that an electron will travel, on average, before emitting a bremsstrahlung photon. Conveniently, it is also the distance a high energy photon will travel through the material before pair producing. The radiation length can be found from
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   = 4 α  
<td nowrap="nowrap"> NA
<td rowspan="2" nowrap="nowrap" align="right">   Z (Z+ζ) re2 ln
<td rowspan="2" nowrap="nowrap" align="right"> [
<td nowrap="nowrap"> 183
<td rowspan="2" nowrap="nowrap" align="right"> ]
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> A
<td style="border-top:solid 1px black;"> Z1/3
</table>
</div>
where α = 1/137 is the fine structure constant, NA = 6.02214076×1023 is Avogadro's number (or one mole), A is the atomic weight of the atoms in the material (usually expressed in grams per mole), Z is the atomic number (nuclear charge) of the atoms, and re = 2.8179403205×10-13 cm is the classical electron radius. ζ is a constant of approximately 1.3. This gives the thickness of material in terms of the areal density (mass of material per unit area perpendicular to the beam) – if you use A in g/mol and re in cm the result will be in cm2/g and X0 will be in g/cm2. If your target is made up of more than one kind of atom, you can get the total radiation length by finding the radiation length of each atom of type i, X0i and using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   =  
<td rowspan="2" nowrap="nowrap" align="right"> Σi  
<td nowrap="nowrap"> fi
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> X0i
</table>
</div>
where fi is the fraction by weight of the atom type i. Once you have the radiation length in terms of areal density, you can easily convert it to distance by dividing by the density ρ. If X0i is in g/cm2 and density is in g/cm3, then X0i/ρ will be in cm.

We will also need to know the critical energy εc. This is the energy where an electron is losing energy as rapidly to ionization as it is to radiation. That is, it is the energy at which the cascade stops multiplying, even though the individual particles already made can continue traveling for some distance. The critical energy can be estimated using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> εc = B  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> Z X0
<td rowspan="2" nowrap="nowrap" align="right"> )
<td nowrap="nowrap"> h
<tr>
<td style="border-top:solid 1px black;"> A
<td>  
</table>
</div>
where B = 2.66 MeV and h = 1.11.

We can now describe the distribution of radiation in the target material. The depth tmax where the cascade stops multiplying (in g/cm2) for an incident electron of energy E (in MeV) is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tmax = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> - 1
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
Similarly, the depth where half the energy of the beam has been deposited (in g/cm2) is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tcg = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> + 0.37
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
As usual, you can convert to a depth in cm by dividing by the material density. 98% of the beam energy will be deposited within a distance of &cong; 3 tcg. Beyond this, the beam energy will continue to fall off with thickness t approximately as an exponential exp[-t/λ], where the characteristic attenuation length λ &cong; 2 X0.

The width of a cascade, where 90% of the energy is contained, is approximately given by
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> RM =  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> EM
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
where EM &cong; 21.2 MeV. 95% of the radiation will be contained within a distance of 2 RM from the beam axis.
</table>

==== Protons and Ions ====

<table align=right><tr><td>
[[File:Regolith_Shielding.png|350 px|frameless]]
<tr><td width=350>
Relative effect of particle radiation (compared to no shielding) from galactic cosmic rays behind different thicknesses of water, aluminum, and lunar regolith<ref name="Slaba2022">Tony C. Slaba, "Radiation Shielding for Lunar Missions: Regolith Considerations", LSIC Crosstalk 7/18/2022 https://lsic.jhuapl.edu/uploadedDocs/focus-files/1604-E&C%20+%20EE%20Monthly%20Meeting%20-%202022%2007%20July_Presentation%20-%20NASA%20Slaba.pdf</ref>.
</table>
Protons and ions are grouped together because ions are made of protons and neutrons and when you accelerate these things for a weapon it is the protons and neutrons that are going to be doing the heavy lifting for blasting your target. This is because protons and neutrons are both about 2000 times more massive than electrons so any electrons along for the ride will contribute negligibly to the energy. Protons and neutrons together are called nuclear particles, and both behave quite similarly in nuclear collisions.

Nuclear particles are themselves made of quarks and can interact via the strong nuclear force. At the energies appropriate for weapons use, essentially all of the way protons or ions are blocked is through [[Particle_Accelerators#Nuclear_collisions|direct collisions with the atomic nuclei]] of the armor material. One or more quarks in the projectile will slam into quarks in the target nucleus and stretch them away from the rest of the quarks in the nuclear particles. Physics tells us that you can't have isolated quarks, so the "string" attaching the quark to its neighbors stretches and stretches until it snaps. Snapping releases so much energy that more quarks are made that clump together into all sorts of other nuclear particles, both exotic and familiar. This process is called a hadronic jet. The only particles produced in these jets long-lived enough to travel any noticeable distance through the armor material will be protons, neutrons, the charged pi mesons (or pions), and some of the K mesons (or kaons). The particles too short lived to go far enough to matter will decay into other particles, and these particles are often very high energy gamma rays. Much like with electrons, the long-lived secondary particles can be sufficiently energetic to smash into other nuclei and cause more hadronic jets (or, if they are low enough in energy, just knock more nuclear particles out of target nuclei) and this produces a similar shower or cascade. But in addition, the armor nuclei that were hit will be very hot after the impact from all of the energy given to them from the incident particles. These nuclei will cool off by evaporation – but instead of evaporating away molecules, they evaporate off nuclear particles from the nucleus. These evaporated particles are mostly neutrons, so any armor against proton or ion beam strikes will produce quite a bit of neutron radiation.

Eventually, the original particle's energy will be distributed among so many other particles that the charged particles will come to a stop by making ion tracks rather than from colliding with more nuclei, and the cascade halts. But you still have all the neutrons to worry about. And the gamma rays as well, which are high enough energy to kick off an electromagnetic cascade in the same way that electrons do.

Unlike electrons, which have reasonable approximations of their cascade behavior from relatively simple equations, the cascades of protons and ions usually require sophisticated simulations or experiments to get the resulting distribution of radiation. However, we can get a reasonable idea of their general behavior by comparing them to [[Space_Radiation#Shielding_Against_Cosmic_Rays|cosmic ray particles interacting with shielding]]. Cosmic rays are energetic protons and ions, after all. From this we see that a relatively small amount of armor – in the range of 1 to 3 tons per square meter – can actually make the radiation dose worse. The incident particles will knock off so many secondary particles in their shower that the dose is multiplied from so many tracks passing through the target. Hydrogen rich materials such as water and plastics work the best against proton and ion beam particles; heavy elements are the worst and produce the most secondaries. You could probably obtain adequate radiation protection from most incident beams with about 10 tons per square meter of hydrogen rich material as armor. However, you will want to add neutron absorbing elements to your hydrogen rich armor, such as lithium, boron, cadmium, or gadolinium. Further, in order to deal with the gamma rays you will want a layer of at least several centimeters of heavy elements, such as tungsten or lead, behind the hydrogen rich particle armor.

==== Structural Considerations ====

When your armor material absorbs a radiation beam, the energy of that radiation turns into heat inside the armor where the radiation was absorbed. If the beam pulse has enough energy as was sufficiently tightly focused, this heat can damage your armor. Your armor may melt in places, or thermal expansion could cause it to crack or warp. Perhaps the worst thing it could do is vaporize or flash into a plasma, because the pressure from the vapor or plasma can be sufficient to plastically deform the armor. This leaves a hole along the beam path where the expanding gas or plasma blasted out a long, deep crater. This means a beam that fires rapid shots spaced maybe milliseconds apart can have subsequent shots go down the same hole so your armor will not protect against them. Each shot will dig out a deeper hole until either your armor is penetrated or the attacking beam runs out of shots in its pulse train.

A consequence of this is that when designing your armor you may wish to use refractory materials with a high structural strength (like tungsten), or that is otherwise resistant to being blasted apart by its own evaporated remains.

=== Magnetic fields ===

[[Particle_Accelerators#Magnetic_fields|A charged particle crossing a magnetic field will bend so as to spiral around the field lines]]. If you try to shoot your particle beam through a magnetic field, it will bend away from the direction you shot it. This can make it hard to hit your target. Unfortunately for you, there are magnetic fields everywhere – around planets, and even trapped in the solar wind as it crosses the solar system.

As an example, the Earth has a magnetic field strength of about 5×10-5 tesla near its surface, and falling off farther away. If you are shooting a 1 TeV electron beam past the Earth, your beam is likely to bend by several milliradians (with the exact deflection depending on how close you are skimming the Earth). If you are in a medium Earth orbit and shooting at another target also in a medium Earth orbit on a trajectory that goes near the Earth, then without correcting for the magnetic field your beam will likely be off-target by tens to hundreds of kilometers.

For another example, the interplanetary magnetic field is about 10-9 tesla. If you shoot that same 1 TeV electron beam at a target 0.1 light seconds away, you are likely to be off by a couple hundred meters.

<table border=1>
<tr><td>
The radius of curvature of the bend of a particle in a magnetic field is called the gyroradius rc. For a particle with a charge q, mass m, and kinetic energy E, find the relativistic momentum p
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
p = √[E2 - m2 c4] / c
</div>
To convert from energy units of eV to joules, multiply by 1.60×10-19. An electron has a mass of 9.11×10-31 kg, a proton of 1.67×10-27 kg, and an ion of 1.66×10-27×A kg, where A is the atomic weight of the ion. The speed of light c = 3.00×108 m/s.

In a magnetic field of strength B with an angle θ between the field direction and the direction of motion
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rc = |p sin(θ) / (q B)|
</div>
For electrons, q = -e, where the fundamental charge e = 1.60×10-19 coulomb. For a proton, q = +e. For an ion, multiply e by the charge state of the ion.

If the field acts over a distance d that is much smaller than the gyroradius, the deflection angle φ is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = d / rc
</div>
The deflection x at a target at a range R for small φ (φ &Lt; 1) will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
x = φ R.
</div>

Note that d will always be less than or equal to x. Also note that if d is not smaller than twice the gyroradius the beam will never reach the target because it will completely spiral around the field lines before it reaches the target.
</table>

So, it is clear that as the defender it will be helpful to hang out in areas with high magnetic fields, and keep some distance between you and the attacker. Meanwhile, the attacker will benefit from being closer where small deflections in the beam trajectory don't translate into large excursions from where the beam is aimed (and also because there will be less space filled with fields between the two of you). But the attacker can also try to compensate for known fields. If you happen to know that the magnetic fields are between you and your target, you can adjust your fire so that the deflection of the field bends your beam onto the target. This is not a simple task. It is easy enough to measure the field strength and direction where you are, but predicting how it changes over the path of your shot does not have an obvious solution. You can also try to gain intelligence on likely battle sites before you ever fight on them – for example, getting good data on the structure of the fields of a planet using spies and technical intelligence during peacetime (heck, on Earth this information is published for free in academic journals and by government agencies such as NOAA). Planetary magnetospheres are also affected by the solar wind, so different solar weather will change the fields you have to shoot through. But space weather forecasts and predictive simulations of the solar wind interacting with a planetary magnetosphere can give you a good idea of what the field structure is like. But even with all that, there are changes in the fields that happen on a scale of around a second ... anyone who has ever seen the auroras dancing across the sky in the polar night can see this. Compensating for these changes will be a significant challenge.

What would really help is being able to see the path of your own beam. This would allow you to see how far off you were from your target and in what direction, and then adjust fire to walk your shots onto the target. Unfortunately, particle beams in space are invisible. There might be ways to track them, but if so we don't know what they are.

==== Synchrotron radiation ====

When charged particle trajectories are bent, they emit [[Particle_Accelerators#Synchrotron_radiation|synchrotron radiation]]. This can cause the particle beam to lose energy, but if the bend is too tight it can also generate high energy gamma rays that ignore magnetic fields and directly strike the target. Synchrotron radiation can generally be ignored for proton or ion beams, but can be significant for electron beams.

<table border=1><tr><td>
For a beam particle with energy E and mass m, the synchrotron radiation is concentrated in a forward-directed cone with an opening angle of approximately 1/γ, where γ = 1 + E/(m c2). (Interestingly, the radiation intensity peaks an at angle of 1/(2 γ) and vanishes directly on-axis.) Because the beam is bending, you will get emission along the plane along which the beam is being bent. The energy of the photons produced during synchrotron emission is approximately<ref name="Jackson_Electrodynamics">J. D. Jackson, "Classical Electrodynamics: Second Edition", Wiley (1962)</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Eγ &cong; 3 γ3 c &hbar; / rc
</div>
where &hbar; = 1.05×10-34 J s (if you want the energy in joules) or &hbar; = 6.58×10-16 eV s (if you want the energy in eV). The photon energy will not exceed the energy of the incident particles, if you get a photon energy higher than the energy of the incident particles it just means that the synchrotron radiation spectrum extends up to the incident particle energy.

The fraction of the energy (or power) of the beam turned into synchrotron radiation will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = [1/(4 π ε0)] [4 π / 3] [q² / rc] β2 γ4 [φ / (2 π)] / E
</div>
where β = √[1 - 1/γ2] is the particle's speed measured as a fraction of the speed of light, and 1/(4 π ε0) = 8.99×109 is the Coulomb constant. For highly relativistic electrons (like all practical electron beam weapons emit), β &cong; 1 and we can use
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = 8.85×10-2 [E (GeV)3 [φ / (2 π)] / (rc (m))]
</div>
This fraction cannot be more than 1. If f gets close to one, it means that our original assumption that the beam particles mostly keep their original speed no longer holds, and the beam will lose most or almost all of its energy. Instead, when the beam is being deflected it will only make it through an angle of φ/f and will irradiate a region of length x/f starting at where the beam would have been incident if not for the magnets and extending in the direction the beam was deflected.
</table>

For example, a 1 TeV electron beam will have γ &cong; 2,000,000. So if it's bent by a magnetic field the synchtrotron radiation will spread out in a cone of about 5×10-7 radians. If this beam was shooting at a target 0.1 light seconds away from where it was bent, the radiation would arrive at the target over a region with a radius of about 15 meters with a maximum intensity at about 7.5 meters and no radiation directly along the path the beam was pointing while it was being bent. If the beam was bent by the interplanetary magnetic field from our earlier example, it would produce light with photon energies of around 1.3 eV (near infrared) and the lost energy would be a few parts in a hundred trillion. If it was instead deflected by the Earth's magnetic field you might expect a peak photon energy of tens of keV – in the hard x-ray part of the spectrum – and the fraction of energy in the synchrotron radiation would be somewhere in the low parts in 100,000 or less.

==== Separation of neutral beams ====

Magnetic fields only affect charged objects. So one way to overcome magnetic defenses or environments is to use beams that have no net charge. For example, a proton beam could be neutralized with a co-propagating electron beam with an equal number of protons and electrons. Or, potentially, you could fire an ion beam and then neutralize the ions after acceleration to get a beam of neutral atoms.

But, the magnetic fields are still tugging at the charged particles making up the beam as they go past. This will pull the electrons one way and the protons or ions the other way. As the particles are pulled in opposite directions you get a net positive charge on one side of the beam and a net negative charge on the other side (called polarization). This sets up an electric field inside the beam that acts to pull the particles back together. If the particles are going fast enough and the magnetic field is strong enough, the magnetic force can still separate the charges and will split the beam in two, one with the positive charges and one with the negative charges. But if the particles are not that fast or the magnetic field is not that strong this polarization of the beam will keep it together and let it fly right through the magnetic field.

<table border=1><tr><td>
Let's figure out how to analyze this problem. For definiteness, we will have a beam with an electric current of the positive species of I and an electric current of electrons of -I, all moving in the z direction. The beam radius will be R. The particle speed will be v with the usual relativistic parameters β = v/c and γ = √[1/(1-β2)]. If you know the energy per particle E of the positive species and the mass m of the positive species, γ = 1 + E/(m c2) and then β = √[1 - 1/γ2].

Let's assume that the beam is going through a magnetic field of strength B at an angle θ to the field direction. The quantity B sin(θ) is the component of the magnetic field perpendicular to the direction of the beam.

The charge density ρ of the positive component of the beam will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρL = I / (π v R2).
</div>

It is easiest to analyze the problem in the rest frame of the particles in the beam. Now the particles are at rest. Particles at rest are not affected by a magnetic field, but the change of reference frame Lorentz transforms the initial magnetic field into a combination of electric and magnetic fields in the rest frame. We can ignore the magnetic field, but the electric field is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔A = γ v B sin(θ)
</div>
in a direction perpendicular to both v and B. For our purposes, assume the electric field points in the x direction.

At the same time, relativistic length contraction increases the charge density in the rest frame
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρ = γ ρL.
</div>

Let r be the vector from the beam axis to the point inside the beam. It will be useful to separate this into its Cartesian coordinates. If x and y are the unit vectors in the x and y directions, respectively, then
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
r = x x + y y
</div>

If we consider only the positive component of the beam, from [https://en.wikipedia.org/wiki/Gauss%27s_law Gauss's law] it will have an electric field inside of the beam at a distance r from the center axis of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔+ = ρ r / (2 ε0) = (ρ/(2 ε0)) (x x + y y)
</div>
where ε0 = 8.85×10-12 is the vacuum permittivity. Now let's displace the electrons in the beam by a distance δ in the x direction. By itself, this electron beam will have an electric field of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔- = -(ρ/(2 ε0)) ((x+δ) x + y y)
</div>
By the principle of superposition, the electric field inside the beam with both components is the sum of the electric fields of the separate components including the external field
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔B = [-(ρ/(2 ε0)) δ + γ v B sin(θ)]x
</div>
Because the charges are free to move, they will do so as long as a field is applied to them. Thus, the beam becomes stable once the charges separate enough that the field internal to the beam is zero. Therefore
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
δ = 2 ε0 γ v B sin(θ) / ρ = 2 π ε0 v2 R2 B sin(θ) / I
</div>
If δ is small compared to R, the beam's polarization suffices to hold it together. If δ > 2 R, the beam will fully separate and each component will go its own way. This gives us a rough threshold for which a magnetic field will separate a neutral beam into its charged components and bend them away from their target
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr><td> B sin(θ) &Lt; I / (π ε0 v2 R) <td>       <td> Beam is stable
<tr><td> B sin(θ) > I / (π ε0 v2 R) <td> <td> Beam promptly separates.
</table>
</div>

For a beam of neutral atoms, let a be the radius of the atom and let Eb be the binding energy of the least tightly bound electron to the atom. An external electric field in excess of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
|𝓔A| > Eb / (e a)
</div>
will promptly ionize the atom, ripping away its electron. Here, e = 1.60×10-19 C is the magnitude of the electric charge of the electron. For fields close to but a bit weaker than this, the electron can still rapidly pop off due to quantum tunneling<ref>Ph. Durand and I. Paidarová, “Ionization of the hydrogen atom: from weak to strong static electric fields,” The European Physical Journal D, Vol. 26, pg. 253-259, 2003.</ref>. In general, expect that for fields greater in magnitude than (very approximately) Eb / (10 e a) the electron will be quickly removed, while for fields weaker than this the electrons will stay on their atoms.

It is useful to note that Eb/e is just the binding energy in units of eV. The most weakly bound electrons are usually bound to their atom with about 10 eV of energy. Atoms are typically about 10-10 m in radius. Thus, you can expect fields of around 1010 to 1011 V/m to directly ionize atoms. If we call this threshold value 𝓔T, then a beam will be ionized when passing through a field with strength greater than
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B sin(θ) > 𝓔T / (γ v)
</div>
</table>

Example: Let's consider a 1 GeV neutralized proton beam from a synchrotron, with a beam current of 200 kA in each bunch (incidentally, if a bunch is 5 cm long this means that each bunch packs 38 kJ worth of energy). The speed of the bunch is 87% the speed of light, or 262 million m/s, with a γ of 2.07. Consider when the beam's focus takes it to a radius of 5 cm (or 0.05 m, for our calculation). Plugging this in, we find that the protons in the bunch will separate from the electrons in a field with a transverse component greater than about 2.1 T, and the bunch will pass through the field without much effect if the transverse component is much smaller than this.

Example: We'll take the same 1 GeV beam, but this time recombine the electrons with protons to make hydrogen atoms. An electron is bound to a hydrogen atom with an energy of 13.6 eV (the Ryberg constant) and the radius of the electron orbital in a hydrogen atom is 0.529×10-10 m (the Bohr radius). This gives a threshold electric field for prompt ionization of hydrogen of 2.57×1011 V/m. We will get this with a transverse component of the magnetic field of 474 T. Above this field the atoms promptly ionize and electrons are stripped away from the protons. For fields higher than roughly 1/10 of this – so about 40 to 50 T – you may still get significant ionization by quantum tunneling; lower than this your losses from the beam due to field-induced ionization are likely negligible.

==== Magnetic shields ====

What happens when we can't rely on magnetic fields in the environment to protect us? One idea is to produce powerful artificial magnetic fields around our spacecraft to deflect incoming particle beams. This is very similar in concept to using active magnetic shielding to protect against space radiation – [[Space_Radiation#Magnetic_Shielding|check out our article on space radiation for more]].
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr>
<td width=100>
[[File:Unconfined_FRC_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:racetrack_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:Magnetic_shielding_Halback_Array.png|200 px|frameless]]
<tr><td colspan=3>
Some potential configurations for magnetic shields. The currents generating the fields are in green, the magnetic fields themselves are shown in magenta<ref name="Battiston2012">R. Battiston, W.J. Burger, V. Calvelli, R. Musenich, V. Choutko, V.I. Datskov, A. Della Torre, F. Venditti,
C. Gargiulo, G. Laurenti, S. Lucidi, S. Harrison, and R. Meinke, "ARSSEM Active Radiation Shield for Space Exploration Missions", Final Report ESTEC Contract N° 4200023087/10/NL/AF : “Superconductive Magnet for Radiation Shielding of Human Spacecraft” (2012) https://arxiv.org/abs/1209.1907 https://www.researchgate.net/publication/265945847_Active_Radiation_Shield_for_Space_Exploration_Missions</ref><ref name="Desiati2022">Paolo Desiati and Elena D'Onghia, "CREW HaT: A Magnetic Shielding System for Space Habitats", arXiv:2209.13624 [physics.space-ph] https://doi.org/10.48550/arXiv.2209.13624</ref>.
</table>
</div>

One immediate issue is that you want to keep very high magnetic fields out of the crew area. Fields in excess of 0.5 T can start to be unpleasant, and it is recommended to keep the field below 0.2 T for continuous exposure.

Strong magnets can suffer from high self-forces acting to tear them apart. Most shielding designs will have multiple magnetic coils, which will exert high forces and torques on each other. This will result in the need for strong structural bracing to keep the magnets intact and in place. Such strong structural designs will impose significant extra mass on the spacecraft.

Consider a magnetic field configuration that can keep radiation out of a certain region of space. There is no trajectory of a charged radiation particle up to a certain energy that will allow those particles to enter the protected region. But the dynamics of charged particles in a magnetic field are the same whether going forward or backward in time. If there is no way for a particle to get in, by this time reversal symmetry any particle which is in that forbidden region can never escape. Consequently any field configuration that is a magnetic trap is also a magnetic shield; any field configuration that is a magnetic shield is also a magnetic trap. A spacecraft with a magnetic shield may end up surrounded by strong radiation belts.

Using a magnet to bend a weaponized electron beam very often ends up with the beam being bent so hard that it dumps most of its energy in the form of synchrotron radiation at gamma ray energies. Because gamma rays are not charged, they will not be deflected by the field. Thus, if you would have been hit by the electron beam you will be hit instead by the gamma rays.

As an example, consider a 1 TeV electron beam. We'll surround our spacecraft with a 10 T magnetic field in a 100 meter radius region just outside our spacecraft. This will deflect the beam by 30 meters, which is enough to make it miss. But we are not saved, because the beam dumps nearly all of its energy into synchrotron radiation with an energy spectrum of gamma rays up to 13 GeV. This radiation will hit us in a strip about 2.4 meters long and not much more than the width of the original beam with essentially all of the energy of the original beam.

==== Magnet drones ====

If a magnetic shield around your spacecraft will just end up killing you from synchrotron radiation, what if you put the magnetic shield on a drone and put that drone between yourself and the enemy trying to shoot you? This lets you use a much weaker magnet because you don't need nearly so high of a deflection angle to make the shot miss. This means that less beam energy will turn into synchrotron radiation that could kill you instead and will spread what synchrotron radiation does occur out over a much wider area when it reaches your spacecraft, hopefully so that most of it just misses your spacecraft.

It does mean that your enemy will start shooting your drones so that they can eventually take a shot at you. But hopefully by that time you've already cleared out their drones and cored their spacecraft with your own synchrocannon or UREB.

==== Plasma magnets ====

Usually, to make a really big magnetic field you would run current through a really big conductive loop. But really big things are heavy and expensive. One way around this could be to use the ambient space plasma to carry the electric current rather than any kind of wire. In order to make this current, you use the same mechanism used in a rotary induction electric motor: you generate a rotating magnetic field and induce a current in the conductive loop (in this case, the space plasma). How do you make a rotating magnetic field? The method proposed<ref name="Slough2006">John Slough, "The Plasma Magnet", NASA Institute for Advanced Concepts Phase II Final Report (2006) https://www.niac.usra.edu/files/studies/final_report/917Slough.pdf</ref> is to use an antenna of four much smaller current loops as electromagnets at 90&degree; to each other, and sinusoidally vary the current through the loops with a respective 90&degree; phase difference between them. When this is done, a circular current is induced in the space plasma and the self-force of a magnetic field on its generating current inflates the current loop to much larger radii. You end up with a current sheet in the plane of the rotating field's rotation circulating about the antenna nearly synchronously with the field rotation. This current then creates a normal dipolar magnetic field as with any current loop. The current sheet and the resulting magnetic field expand until the magnetic field's pressure equals that of the ambient space plasma. This gives an enormous region of magnetic field from a relatively small and lightweight system.

The system proposed in the initial study would use an antenna approximately 10 meters in radius and produce a region of field approximately 30 km in radius at Earth's orbit. The field inside would not be constant - at the edges it would be 50 nT, giving the right pressure to keep out the solar wind at Earth's orbit. A circulating current sheet produces a field that falls off as 1/r with distance, meaning that adjacent to the antenna the field would be about 0.15 mT. Using the methods in the box below, we can estimate that this particular design as part of a spacecraft's shields will not deflect a TeV electron beam by more than a few centimeters, which is usually insufficient to force a miss. However, it could potentially be useful for a drone magnet, where the much larger distance between the magnetic field and the target spacecraft would allow a small deflection angle to give perhaps tens or low hundreds of meters change in beam location near the target at near light second ranges.

<table border=1><tr><td>
Calculating the deflection from a spatially varying magnetic field affected by the solar wind is no trivial task. However, we can make an approximation that both should be within about an order of magnitude of the true value and also represent an upper limit to the deflection angle; although it does neglect the influence of the driving rotating magnetic field which could in some cases increase the deflection.

We will take R0 as the radius of the antenna. B0 is the field strength at the antenna. Rm is the distance at which the magnetic influence stops because the magnetic pressure equals the space plasma pressure. Bm is the field strength at Rm. According to Slough<ref name="Slough2006"></ref>, the magnetic field strength B(r) at any distance from the antenna center r between R0 and Rm is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B(r) = B0 R0 / r.
</div>
Therefore, once we know Bm and Rm, B0 is completely determined
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B0 = Bm Rm / R0.
</div>

The force on a particle with charge q passing perpendicularly through a magnetic field B(r) at a speed v is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
F(r) = q v B(r)
</div>
For an electron beam, v is so close to c as to make no difference. If the particle is moving down the y direction and the field is in the z direction, the force will be in the x direction. We assume that the deflection will be small, so we can choose a straight-line path in the y direction offset from the center by an amount x. The plasma bubble radius is Rm, so the particle will enter at ymax = √[Rm2-x2] and leave at ymin = -√[Rm2-x2].

The momentum change the particle experiences from this force is the time integral of the force over its path. Because of reflection symmetry about y = 0, we only need to integrate from ymax to 0 and then multiply by 2
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] dt F(r)
<tr> <td> <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] (dy/v) F(r)
<tr> <td> <td> = <td align="left">2 q B(0) R0 [Integral from 0 to √[Rm2-x2]] dy 1/√[y2 + x2]
<tr> <td> <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/x]
</table>
</div>
This holds for |x| > R0. But the expression for the field is not valid for r < R0 and the integral will give nonsense and diverging results if we evaluate for |x| < R0. In this case, we can put the integral lower bound at r = R0, or y = √[R02-x2]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/(√[R02-x2] + R0)]
</table>
</div>

The deflection angle is now estimated as
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = px / p
</div>
where the total momentum of the particle p can be found from the methods in the first box in the Magnetic Fields section.

The deflection for a shot aimed right at the spacecraft at the position of the spacecraft can be estimated by setting x = 0 and only taking the first half of the integral (so ignore the factor of 2)
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
q B(0) R0 ln[Rm/R0] / (p Rm)
</div>
This is an additional overestimate because it fails to account for how the deflection builds up over distance during the integration, but it should still give something of an order-of-magnitude and upper limit estimate of what kind of deflections might be expected.
</table>

=== Electric fields ===

If electric fields can accelerate charged particles, maybe they can keep the charged particles in a weaponized particle beam away from your spacecraft? To completely block a particle beam, you would need a voltage on your spacecraft equal to the particle energy in eV of the particles in the beam. That is, if a 1 GeV proton beam is coming at you, you'd need an electric potential of a billion volts to stop that beam. This leads to three immediate problems.

The first is that particles of the opposite charge will be attracted to your spacecraft. Because space isn't really empty but rather a rarefied plasma (called the solar wind in a solar system or the interstellar medium out between stars), there will be plenty of particles of the opposite charge around to attract. And they will hit with an energy in eV equal to the voltage on your spacecraft. In our example of defending against a 1 GeV proton beam with a gigavolt field, the electrons from the solar wind will come in and hit with an energy of 1 GeV each. This will generate dangerous radiation.

The second problem is a consequence of the first. Each of those GeV electrons from our example need to get their energy from somewhere, which turns out to be from the electric field we are using as shielding. You will need to make up for this energy loss with power from your electrical system, which gives a constant drain on your power. Some estimates put the electrical effort to maintain a several hundred megavolt shield in space at many tens of gigawatts<ref name="Mechmann2019">Claire Mechmann, "Analysis of Proposed Active Radiation Shielding Design Concept for Spacecraft" (2019) Thesis, College of Engineering and Science of Florida Institute of Technology</ref>.

And the third problem is similar to one for magnetic fields. Your electrodes will experience high self forces and forces between separate electrodes that will try to tear your shield generator apart. Consequently you will need heavy structural reinforcement in order for your shield to work.

There are potential ways to mitigate some of these problems. Interested readers are directed to [[Space_Radiation#Electrostatic_Shielding|the active shielding section in our page on space radiation]] for more of the juicy details; there's no need to repeat the same stuff here.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Elctrostatic_active_shielding.png|200 px|frameless]]
<td>
[[File:Electrostatic_active_shielding_2.png|200 px|frameless]]
<tr><td colspan=2>
Proposed designs for electrostatic shields to protect from positively charged particles<ref name="Tripathi2006">Ram K. Tripathi, John W. Wilson, and Robert C. Youngquist, "Electrostatic Active Radiation Shielding - Revisited", 2006 IEEE Aerospace Conference, Big Sky, MT, USA, 2006, pp. 9 pp.-, doi: 10.1109/AERO.2006.1655760.</ref><ref name="Chowdhury2023">Rajarshi Pal Chowdhury, Luke A. Stegeman, Matthew L. Lund, Dan Fry, Stojan Madzunkov, and Amir A. Bahadori, "Hybrid methods of radiation shielding against deep-space radiation", Life Sciences in Space Research, Volume 38, 2023, Pages 67-78, ISSN 2214-5524, https://doi.org/10.1016/j.lssr.2023.04.004.</ref>.
</table>
</div>

== Damage ==

Particle beams are, first and foremost, devices that produce beams of [[Nuclear_radiation|ionizing radiation]]. The radiation of weaponizable particle beams is extremely penetrating; thus of the [[Particle_Accelerators#Energy_loss_in_matter|interaction mechanisms listed for radiation from particle beams]], we mostly need to worry about bremsstrahlung and nuclear collisions and these produce extensive secondary showers of radiation that increase the penetration of the radiation exposure. Because of this, any target hit by a particle beam weapon will be irradiated, and radiation dose will always be one of the primary damage mechanisms of a particle beam weapon strike. Further, this radiation will be distributed along a track passing deeply through the material of the target, allowing it to affect critical systems through heavy armor, structural material, robust machine components, and passive material such as fuel load, coolant, or cargo. As the radiation deposits its energy into the target material, that material will be heated. If the target is heated sufficiently, it may experience [[Beam-Target_Interactions|thermal or mechanical]] damage from melting, cooking, combustion, vapor explosions, and warping or cracking due to thermal expansion.

The discussion of the radiation penetration into armor also applies to the same radiation penetrating through the target. So the formulas and rules of thumb for estimating the depth of radiation penetration into armor can also be used to estimate how far into a tank or spacecraft a particle beam strike delivers effects.

In addition to the radiation and thermal effects, a particle beam can also produce a sharp electromagnetic pulse that can damage semiconductor electronics.

== 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, Gerrit Bruhaug, and Luke Campbell for writing

Gerrit Bruhaug and Luke Campbell for scientific research, math wrangling and advise

ATriffleMind and Luke W. Campbell for proof reading.

==References==

[[Category:Beams]][[Category:Physics & Engineering‏‎]][[Category:Engineering‏‎]][[Category:Military Technology]][[Category:Warfare‏‎]]

Particle Beam Weapons

2026-10-25T22:28:42Z

Lwcamp: /* Plasma magnets */

{{PageConstructionNotice}}

''For a general overview on particle accelerator technology, see [[Particle Accelerators]]

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.

::: Z = 1.32×102 (γ2-1)3/4 r / √I<ref name=USPAS2016>Martin Reiser, "Theory and Design of Charged Particle Beams", 2008, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim</ref>

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

::: Z = β γ c r √[m/T]

Where Z is the doubling distance like before, r is the initial beam radius, T is the beam temperature in eV, m is the ion mass and β, c, and γ 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.

== Defenses ==

Sure, it's great to have a great big particle cannon to blast your enemies. But what if you are the one being shot at? In that case, it's best to figure out ways to keep from being blown to kingdom come.

=== Armor ===

The simplest way to defend against the high energy particles of a particle beam is to pile a whole bunch of stuff up and put it in the way of the beam. Particle beams may produce extremely penetrating radiation, but enough stuff will block most of that radiation and may allow you to survive. The details depend on what kind of particle is being thrown at you, and are explained below. However, a general rule of thumb is that you probably want about 10 tons of material per square meter of surface you want to armor in order to get sufficient protection.

==== Electrons ====

At the very high energy of an electron accelerated by a weapons-grade particle beam, the primary interaction with matter will be the emission of bremsstrahlung photons in the gamma-ray range as the electron slams into particles of matter. These high energy photons themselves primarily interact by the pair production mechanism – the photon disappears, but leaves behind a highly energetic positron and electron. These two particles are themselves energetic enough to produce further bremsstrahlung, and so the process continues. You get the formation of an electron-gamma shower (also called an electromagnetic cascade), which only stops after either the particles have penetrated through the target or the initial energy of the electron has been spread through so many shower particles (including all the electrons, positrons, and photons) that no particle has enough individual energy to pair produce or to be likely to bremsstrahlung again before it is stopped by other mechanisms.

To stop electrons, you'll want the atoms in your armor to be as heavy as you can practically get them. High atomic number both encourages bremsstrahlung and pair production, giving a double whammy for stopping that cascade. However, occasionally one of those electrons, positrons, or gammas will hit a nucleus and knock out neutrons. So you probably want a back-up layer of several tens of centimeters of hydrogen rich material such as plastic or water mixed with neutron absorbing materials like boron, lithium, cadmium, or gadolinium. High density polyethylene loaded with 5% boron by weight is a typical go-to neutron blocker.

The actual methods to calculate the penetration of the electromagnetic cascade is given are the box below. However, for some examples:
<ul>
<li> A 10 GeV beam incident on thick iron (or steel) will stop cascading at about 9 cm, will deposit half of its energy in the first 12 cm, and will have dumped 98% of its energy within 35 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 2.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 100 GeV beam incident on thick iron (or steel) will stop cascading at about 13 cm, will deposit half of its energy in the first 16 cm, and will have dumped 98% of its energy within 47 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 3.7 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick iron (or steel) will stop cascading at about 17 cm, will deposit half of its energy in the first 20 cm, and will have dumped 98% of its energy within 59 cm. 90% of the cascade will be contained within about 1.8 cm and 95% of the cascade will be contained within 3.6 cm of the beam axis. You will get 98% protection with a mass thickness of 4.6 tons/m2 and an additional 1/e reduction for each further 0.27 tons/m2 beyond that.
<li> A 1 TeV beam incident on thick tungsten will stop cascading at about 3.5 cm, will deposit half of its energy in the first 4 cm, and will have dumped 98% of its energy within 12 cm. 90% of the cascade will be contained within about 0.93 cm and 95% of the cascade will be contained within 1.86 cm of the beam axis. You will get 98% protection with a mass thickness of 2.3 tons/m2 and an additional 1/e reduction for each further 0.13 tons/m2 beyond that.
<li> A 1 TeV beam incident on water will stop cascading at about 290 cm, will deposit half of its energy in the first 340 cm, and will have dumped 98% of its energy within 1030 cm. 90% of the cascade will be contained within about 11 cm and 95% of the cascade will be contained within 22 cm of the beam axis. You will get 98% protection with a mass thickness of 10.3 tons/m2 and an additional 1/e reduction for each further 0.72 tons/m2 beyond that.
</ul>
Note that for sufficiently intense particle beams, 98% protection may not be enough. You might need to throw on several more tons per square meter to be sure.

<table border=1>
<tr><td> How Far it Goes:
(This section follows methods presented in Leroy and Rancoita<ref>C. Leroy and P-G Rancoita, "Physics of cascading shower generation and propagation in matter: principles of high-energy, ultrahigh-energy and compensating calorimetry", Reports on Progress in Physics 63 (2000) 505–606. DOI: 10.1088/0034-4885/63/4/202</ref>.)

When trying to estimate the extent of the irradiated region of the target, it is helpful to know the radiation length X0 in the material. This is the distance that an electron will travel, on average, before emitting a bremsstrahlung photon. Conveniently, it is also the distance a high energy photon will travel through the material before pair producing. The radiation length can be found from
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   = 4 α  
<td nowrap="nowrap"> NA
<td rowspan="2" nowrap="nowrap" align="right">   Z (Z+ζ) re2 ln
<td rowspan="2" nowrap="nowrap" align="right"> [
<td nowrap="nowrap"> 183
<td rowspan="2" nowrap="nowrap" align="right"> ]
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> A
<td style="border-top:solid 1px black;"> Z1/3
</table>
</div>
where α = 1/137 is the fine structure constant, NA = 6.02214076×1023 is Avogadro's number (or one mole), A is the atomic weight of the atoms in the material (usually expressed in grams per mole), Z is the atomic number (nuclear charge) of the atoms, and re = 2.8179403205×10-13 cm is the classical electron radius. ζ is a constant of approximately 1.3. This gives the thickness of material in terms of the areal density (mass of material per unit area perpendicular to the beam) – if you use A in g/mol and re in cm the result will be in cm2/g and X0 will be in g/cm2. If your target is made up of more than one kind of atom, you can get the total radiation length by finding the radiation length of each atom of type i, X0i and using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td nowrap="nowrap"> 1
<td rowspan="2" nowrap="nowrap" align="right">   =  
<td rowspan="2" nowrap="nowrap" align="right"> Σi  
<td nowrap="nowrap"> fi
<tr>
<td style="border-top:solid 1px black;"> X0
<td style="border-top:solid 1px black;"> X0i
</table>
</div>
where fi is the fraction by weight of the atom type i. Once you have the radiation length in terms of areal density, you can easily convert it to distance by dividing by the density ρ. If X0i is in g/cm2 and density is in g/cm3, then X0i/ρ will be in cm.

We will also need to know the critical energy εc. This is the energy where an electron is losing energy as rapidly to ionization as it is to radiation. That is, it is the energy at which the cascade stops multiplying, even though the individual particles already made can continue traveling for some distance. The critical energy can be estimated using
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> εc = B  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> Z X0
<td rowspan="2" nowrap="nowrap" align="right"> )
<td nowrap="nowrap"> h
<tr>
<td style="border-top:solid 1px black;"> A
<td>  
</table>
</div>
where B = 2.66 MeV and h = 1.11.

We can now describe the distribution of radiation in the target material. The depth tmax where the cascade stops multiplying (in g/cm2) for an incident electron of energy E (in MeV) is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tmax = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> - 1
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
Similarly, the depth where half the energy of the beam has been deposited (in g/cm2) is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> tcg = 1.01  
<td rowspan="2" nowrap="nowrap" align="right"> [
<td rowspan="2" nowrap="nowrap" align="right"> ln
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> E
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right"> + 0.37
<td rowspan="2" nowrap="nowrap" align="right"> ]
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
As usual, you can convert to a depth in cm by dividing by the material density. 98% of the beam energy will be deposited within a distance of &cong; 3 tcg. Beyond this, the beam energy will continue to fall off with thickness t approximately as an exponential exp[-t/λ], where the characteristic attenuation length λ &cong; 2 X0.

The width of a cascade, where 90% of the energy is contained, is approximately given by
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table style="margin-top:0.5em; margin-bottom:0.5em; text-align: center;" align="center" cellpadding="0" cellspacing="0" >
<tr>
<td rowspan="2" nowrap="nowrap" align="right"> RM =  
<td rowspan="2" nowrap="nowrap" align="right"> (
<td nowrap="nowrap"> EM
<td rowspan="2" nowrap="nowrap" align="right"> )
<td rowspan="2" nowrap="nowrap" align="right">   X0
<tr>
<td style="border-top:solid 1px black;"> εc
</table>
</div>
where EM &cong; 21.2 MeV. 95% of the radiation will be contained within a distance of 2 RM from the beam axis.
</table>

==== Protons and Ions ====

<table align=right><tr><td>
[[File:Regolith_Shielding.png|350 px|frameless]]
<tr><td width=350>
Relative effect of particle radiation (compared to no shielding) from galactic cosmic rays behind different thicknesses of water, aluminum, and lunar regolith<ref name="Slaba2022">Tony C. Slaba, "Radiation Shielding for Lunar Missions: Regolith Considerations", LSIC Crosstalk 7/18/2022 https://lsic.jhuapl.edu/uploadedDocs/focus-files/1604-E&C%20+%20EE%20Monthly%20Meeting%20-%202022%2007%20July_Presentation%20-%20NASA%20Slaba.pdf</ref>.
</table>
Protons and ions are grouped together because ions are made of protons and neutrons and when you accelerate these things for a weapon it is the protons and neutrons that are going to be doing the heavy lifting for blasting your target. This is because protons and neutrons are both about 2000 times more massive than electrons so any electrons along for the ride will contribute negligibly to the energy. Protons and neutrons together are called nuclear particles, and both behave quite similarly in nuclear collisions.

Nuclear particles are themselves made of quarks and can interact via the strong nuclear force. At the energies appropriate for weapons use, essentially all of the way protons or ions are blocked is through [[Particle_Accelerators#Nuclear_collisions|direct collisions with the atomic nuclei]] of the armor material. One or more quarks in the projectile will slam into quarks in the target nucleus and stretch them away from the rest of the quarks in the nuclear particles. Physics tells us that you can't have isolated quarks, so the "string" attaching the quark to its neighbors stretches and stretches until it snaps. Snapping releases so much energy that more quarks are made that clump together into all sorts of other nuclear particles, both exotic and familiar. This process is called a hadronic jet. The only particles produced in these jets long-lived enough to travel any noticeable distance through the armor material will be protons, neutrons, the charged pi mesons (or pions), and some of the K mesons (or kaons). The particles too short lived to go far enough to matter will decay into other particles, and these particles are often very high energy gamma rays. Much like with electrons, the long-lived secondary particles can be sufficiently energetic to smash into other nuclei and cause more hadronic jets (or, if they are low enough in energy, just knock more nuclear particles out of target nuclei) and this produces a similar shower or cascade. But in addition, the armor nuclei that were hit will be very hot after the impact from all of the energy given to them from the incident particles. These nuclei will cool off by evaporation – but instead of evaporating away molecules, they evaporate off nuclear particles from the nucleus. These evaporated particles are mostly neutrons, so any armor against proton or ion beam strikes will produce quite a bit of neutron radiation.

Eventually, the original particle's energy will be distributed among so many other particles that the charged particles will come to a stop by making ion tracks rather than from colliding with more nuclei, and the cascade halts. But you still have all the neutrons to worry about. And the gamma rays as well, which are high enough energy to kick off an electromagnetic cascade in the same way that electrons do.

Unlike electrons, which have reasonable approximations of their cascade behavior from relatively simple equations, the cascades of protons and ions usually require sophisticated simulations or experiments to get the resulting distribution of radiation. However, we can get a reasonable idea of their general behavior by comparing them to [[Space_Radiation#Shielding_Against_Cosmic_Rays|cosmic ray particles interacting with shielding]]. Cosmic rays are energetic protons and ions, after all. From this we see that a relatively small amount of armor – in the range of 1 to 3 tons per square meter – can actually make the radiation dose worse. The incident particles will knock off so many secondary particles in their shower that the dose is multiplied from so many tracks passing through the target. Hydrogen rich materials such as water and plastics work the best against proton and ion beam particles; heavy elements are the worst and produce the most secondaries. You could probably obtain adequate radiation protection from most incident beams with about 10 tons per square meter of hydrogen rich material as armor. However, you will want to add neutron absorbing elements to your hydrogen rich armor, such as lithium, boron, cadmium, or gadolinium. Further, in order to deal with the gamma rays you will want a layer of at least several centimeters of heavy elements, such as tungsten or lead, behind the hydrogen rich particle armor.

==== Structural Considerations ====

When your armor material absorbs a radiation beam, the energy of that radiation turns into heat inside the armor where the radiation was absorbed. If the beam pulse has enough energy as was sufficiently tightly focused, this heat can damage your armor. Your armor may melt in places, or thermal expansion could cause it to crack or warp. Perhaps the worst thing it could do is vaporize or flash into a plasma, because the pressure from the vapor or plasma can be sufficient to plastically deform the armor. This leaves a hole along the beam path where the expanding gas or plasma blasted out a long, deep crater. This means a beam that fires rapid shots spaced maybe milliseconds apart can have subsequent shots go down the same hole so your armor will not protect against them. Each shot will dig out a deeper hole until either your armor is penetrated or the attacking beam runs out of shots in its pulse train.

A consequence of this is that when designing your armor you may wish to use refractory materials with a high structural strength (like tungsten), or that is otherwise resistant to being blasted apart by its own evaporated remains.

=== Magnetic fields ===

[[Particle_Accelerators#Magnetic_fields|A charged particle crossing a magnetic field will bend so as to spiral around the field lines]]. If you try to shoot your particle beam through a magnetic field, it will bend away from the direction you shot it. This can make it hard to hit your target. Unfortunately for you, there are magnetic fields everywhere – around planets, and even trapped in the solar wind as it crosses the solar system.

As an example, the Earth has a magnetic field strength of about 5×10-5 tesla near its surface, and falling off farther away. If you are shooting a 1 TeV electron beam past the Earth, your beam is likely to bend by several milliradians (with the exact deflection depending on how close you are skimming the Earth). If you are in a medium Earth orbit and shooting at another target also in a medium Earth orbit on a trajectory that goes near the Earth, then without correcting for the magnetic field your beam will likely be off-target by tens to hundreds of kilometers.

For another example, the interplanetary magnetic field is about 10-9 tesla. If you shoot that same 1 TeV electron beam at a target 0.1 light seconds away, you are likely to be off by a couple hundred meters.

<table border=1>
<tr><td>
The radius of curvature of the bend of a particle in a magnetic field is called the gyroradius rc. For a particle with a charge q, mass m, and kinetic energy E, find the relativistic momentum p
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
p = √[E2 - m2 c4] / c
</div>
To convert from energy units of eV to joules, multiply by 1.60×10-19. An electron has a mass of 9.11×10-31 kg, a proton of 1.67×10-27 kg, and an ion of 1.66×10-27×A kg, where A is the atomic weight of the ion. The speed of light c = 3.00×108 m/s.

In a magnetic field of strength B with an angle θ between the field direction and the direction of motion
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
rc = |p sin(θ) / (q B)|
</div>
For electrons, q = -e, where the fundamental charge e = 1.60×10-19 coulomb. For a proton, q = +e. For an ion, multiply e by the charge state of the ion.

If the field acts over a distance d that is much smaller than the gyroradius, the deflection angle φ is approximately
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = d / rc
</div>
The deflection x at a target at a range R for small φ (φ &Lt; 1) will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
x = φ R.
</div>

Note that d will always be less than or equal to x. Also note that if d is not smaller than twice the gyroradius the beam will never reach the target because it will completely spiral around the field lines before it reaches the target.
</table>

So, it is clear that as the defender it will be helpful to hang out in areas with high magnetic fields, and keep some distance between you and the attacker. Meanwhile, the attacker will benefit from being closer where small deflections in the beam trajectory don't translate into large excursions from where the beam is aimed (and also because there will be less space filled with fields between the two of you). But the attacker can also try to compensate for known fields. If you happen to know that the magnetic fields are between you and your target, you can adjust your fire so that the deflection of the field bends your beam onto the target. This is not a simple task. It is easy enough to measure the field strength and direction where you are, but predicting how it changes over the path of your shot does not have an obvious solution. You can also try to gain intelligence on likely battle sites before you ever fight on them – for example, getting good data on the structure of the fields of a planet using spies and technical intelligence during peacetime (heck, on Earth this information is published for free in academic journals and by government agencies such as NOAA). Planetary magnetospheres are also affected by the solar wind, so different solar weather will change the fields you have to shoot through. But space weather forecasts and predictive simulations of the solar wind interacting with a planetary magnetosphere can give you a good idea of what the field structure is like. But even with all that, there are changes in the fields that happen on a scale of around a second ... anyone who has ever seen the auroras dancing across the sky in the polar night can see this. Compensating for these changes will be a significant challenge.

What would really help is being able to see the path of your own beam. This would allow you to see how far off you were from your target and in what direction, and then adjust fire to walk your shots onto the target. Unfortunately, particle beams in space are invisible. There might be ways to track them, but if so we don't know what they are.

==== Synchrotron radiation ====

When charged particle trajectories are bent, they emit [[Particle_Accelerators#Synchrotron_radiation|synchrotron radiation]]. This can cause the particle beam to lose energy, but if the bend is too tight it can also generate high energy gamma rays that ignore magnetic fields and directly strike the target. Synchrotron radiation can generally be ignored for proton or ion beams, but can be significant for electron beams.

<table border=1><tr><td>
For a beam particle with energy E and mass m, the synchrotron radiation is concentrated in a forward-directed cone with an opening angle of approximately 1/γ, where γ = 1 + E/(m c2). (Interestingly, the radiation intensity peaks an at angle of 1/(2 γ) and vanishes directly on-axis.) Because the beam is bending, you will get emission along the plane along which the beam is being bent. The energy of the photons produced during synchrotron emission is approximately<ref name="Jackson_Electrodynamics">J. D. Jackson, "Classical Electrodynamics: Second Edition", Wiley (1962)</ref>
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
Eγ &cong; 3 γ3 c &hbar; / rc
</div>
where &hbar; = 1.05×10-34 J s (if you want the energy in joules) or &hbar; = 6.58×10-16 eV s (if you want the energy in eV). The photon energy will not exceed the energy of the incident particles, if you get a photon energy higher than the energy of the incident particles it just means that the synchrotron radiation spectrum extends up to the incident particle energy.

The fraction of the energy (or power) of the beam turned into synchrotron radiation will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = [1/(4 π ε0)] [4 π / 3] [q² / rc] β2 γ4 [φ / (2 π)] / E
</div>
where β = √[1 - 1/γ2] is the particle's speed measured as a fraction of the speed of light, and 1/(4 π ε0) = 8.99×109 is the Coulomb constant. For highly relativistic electrons (like all practical electron beam weapons emit), β &cong; 1 and we can use
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
f = 8.85×10-2 [E (GeV)3 [φ / (2 π)] / (rc (m))]
</div>
This fraction cannot be more than 1. If f gets close to one, it means that our original assumption that the beam particles mostly keep their original speed no longer holds, and the beam will lose most or almost all of its energy. Instead, when the beam is being deflected it will only make it through an angle of φ/f and will irradiate a region of length x/f starting at where the beam would have been incident if not for the magnets and extending in the direction the beam was deflected.
</table>

For example, a 1 TeV electron beam will have γ &cong; 2,000,000. So if it's bent by a magnetic field the synchtrotron radiation will spread out in a cone of about 5×10-7 radians. If this beam was shooting at a target 0.1 light seconds away from where it was bent, the radiation would arrive at the target over a region with a radius of about 15 meters with a maximum intensity at about 7.5 meters and no radiation directly along the path the beam was pointing while it was being bent. If the beam was bent by the interplanetary magnetic field from our earlier example, it would produce light with photon energies of around 1.3 eV (near infrared) and the lost energy would be a few parts in a hundred trillion. If it was instead deflected by the Earth's magnetic field you might expect a peak photon energy of tens of keV – in the hard x-ray part of the spectrum – and the fraction of energy in the synchrotron radiation would be somewhere in the low parts in 100,000 or less.

==== Separation of neutral beams ====

Magnetic fields only affect charged objects. So one way to overcome magnetic defenses or environments is to use beams that have no net charge. For example, a proton beam could be neutralized with a co-propagating electron beam with an equal number of protons and electrons. Or, potentially, you could fire an ion beam and then neutralize the ions after acceleration to get a beam of neutral atoms.

But, the magnetic fields are still tugging at the charged particles making up the beam as they go past. This will pull the electrons one way and the protons or ions the other way. As the particles are pulled in opposite directions you get a net positive charge on one side of the beam and a net negative charge on the other side (called polarization). This sets up an electric field inside the beam that acts to pull the particles back together. If the particles are going fast enough and the magnetic field is strong enough, the magnetic force can still separate the charges and will split the beam in two, one with the positive charges and one with the negative charges. But if the particles are not that fast or the magnetic field is not that strong this polarization of the beam will keep it together and let it fly right through the magnetic field.

<table border=1><tr><td>
Let's figure out how to analyze this problem. For definiteness, we will have a beam with an electric current of the positive species of I and an electric current of electrons of -I, all moving in the z direction. The beam radius will be R. The particle speed will be v with the usual relativistic parameters β = v/c and γ = √[1/(1-β2)]. If you know the energy per particle E of the positive species and the mass m of the positive species, γ = 1 + E/(m c2) and then β = √[1 - 1/γ2].

Let's assume that the beam is going through a magnetic field of strength B at an angle θ to the field direction. The quantity B sin(θ) is the component of the magnetic field perpendicular to the direction of the beam.

The charge density ρ of the positive component of the beam will be
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρL = I / (π v R2).
</div>

It is easiest to analyze the problem in the rest frame of the particles in the beam. Now the particles are at rest. Particles at rest are not affected by a magnetic field, but the change of reference frame Lorentz transforms the initial magnetic field into a combination of electric and magnetic fields in the rest frame. We can ignore the magnetic field, but the electric field is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔A = γ v B sin(θ)
</div>
in a direction perpendicular to both v and B. For our purposes, assume the electric field points in the x direction.

At the same time, relativistic length contraction increases the charge density in the rest frame
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
ρ = γ ρL.
</div>

Let r be the vector from the beam axis to the point inside the beam. It will be useful to separate this into its Cartesian coordinates. If x and y are the unit vectors in the x and y directions, respectively, then
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
r = x x + y y
</div>

If we consider only the positive component of the beam, from [https://en.wikipedia.org/wiki/Gauss%27s_law Gauss's law] it will have an electric field inside of the beam at a distance r from the center axis of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔+ = ρ r / (2 ε0) = (ρ/(2 ε0)) (x x + y y)
</div>
where ε0 = 8.85×10-12 is the vacuum permittivity. Now let's displace the electrons in the beam by a distance δ in the x direction. By itself, this electron beam will have an electric field of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔- = -(ρ/(2 ε0)) ((x+δ) x + y y)
</div>
By the principle of superposition, the electric field inside the beam with both components is the sum of the electric fields of the separate components including the external field
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
𝓔B = [-(ρ/(2 ε0)) δ + γ v B sin(θ)]x
</div>
Because the charges are free to move, they will do so as long as a field is applied to them. Thus, the beam becomes stable once the charges separate enough that the field internal to the beam is zero. Therefore
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
δ = 2 ε0 γ v B sin(θ) / ρ = 2 π ε0 v2 R2 B sin(θ) / I
</div>
If δ is small compared to R, the beam's polarization suffices to hold it together. If δ > 2 R, the beam will fully separate and each component will go its own way. This gives us a rough threshold for which a magnetic field will separate a neutral beam into its charged components and bend them away from their target
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr><td> B sin(θ) &Lt; I / (π ε0 v2 R) <td>       <td> Beam is stable
<tr><td> B sin(θ) > I / (π ε0 v2 R) <td> <td> Beam promptly separates.
</table>
</div>

For a beam of neutral atoms, let a be the radius of the atom and let Eb be the binding energy of the least tightly bound electron to the atom. An external electric field in excess of
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
|𝓔A| > Eb / (e a)
</div>
will promptly ionize the atom, ripping away its electron. Here, e = 1.60×10-19 C is the magnitude of the electric charge of the electron. For fields close to but a bit weaker than this, the electron can still rapidly pop off due to quantum tunneling<ref>Ph. Durand and I. Paidarová, “Ionization of the hydrogen atom: from weak to strong static electric fields,” The European Physical Journal D, Vol. 26, pg. 253-259, 2003.</ref>. In general, expect that for fields greater in magnitude than (very approximately) Eb / (10 e a) the electron will be quickly removed, while for fields weaker than this the electrons will stay on their atoms.

It is useful to note that Eb/e is just the binding energy in units of eV. The most weakly bound electrons are usually bound to their atom with about 10 eV of energy. Atoms are typically about 10-10 m in radius. Thus, you can expect fields of around 1010 to 1011 V/m to directly ionize atoms. If we call this threshold value 𝓔T, then a beam will be ionized when passing through a field with strength greater than
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B sin(θ) > 𝓔T / (γ v)
</div>
</table>

Example: Let's consider a 1 GeV neutralized proton beam from a synchrotron, with a beam current of 200 kA in each bunch (incidentally, if a bunch is 5 cm long this means that each bunch packs 38 kJ worth of energy). The speed of the bunch is 87% the speed of light, or 262 million m/s, with a γ of 2.07. Consider when the beam's focus takes it to a radius of 5 cm (or 0.05 m, for our calculation). Plugging this in, we find that the protons in the bunch will separate from the electrons in a field with a transverse component greater than about 2.1 T, and the bunch will pass through the field without much effect if the transverse component is much smaller than this.

Example: We'll take the same 1 GeV beam, but this time recombine the electrons with protons to make hydrogen atoms. An electron is bound to a hydrogen atom with an energy of 13.6 eV (the Ryberg constant) and the radius of the electron orbital in a hydrogen atom is 0.529×10-10 m (the Bohr radius). This gives a threshold electric field for prompt ionization of hydrogen of 2.57×1011 V/m. We will get this with a transverse component of the magnetic field of 474 T. Above this field the atoms promptly ionize and electrons are stripped away from the protons. For fields higher than roughly 1/10 of this – so about 40 to 50 T – you may still get significant ionization by quantum tunneling; lower than this your losses from the beam due to field-induced ionization are likely negligible.

==== Magnetic shields ====

What happens when we can't rely on magnetic fields in the environment to protect us? One idea is to produce powerful artificial magnetic fields around our spacecraft to deflect incoming particle beams. This is very similar in concept to using active magnetic shielding to protect against space radiation – [[Space_Radiation#Magnetic_Shielding|check out our article on space radiation for more]].
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr>
<td width=100>
[[File:Unconfined_FRC_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:racetrack_magnetic_active_shielding.png|200 px|frameless]]
<td width=100>
[[File:Magnetic_shielding_Halback_Array.png|200 px|frameless]]
<tr><td colspan=3>
Some potential configurations for magnetic shields. The currents generating the fields are in green, the magnetic fields themselves are shown in magenta<ref name="Battiston2012">R. Battiston, W.J. Burger, V. Calvelli, R. Musenich, V. Choutko, V.I. Datskov, A. Della Torre, F. Venditti,
C. Gargiulo, G. Laurenti, S. Lucidi, S. Harrison, and R. Meinke, "ARSSEM Active Radiation Shield for Space Exploration Missions", Final Report ESTEC Contract N° 4200023087/10/NL/AF : “Superconductive Magnet for Radiation Shielding of Human Spacecraft” (2012) https://arxiv.org/abs/1209.1907 https://www.researchgate.net/publication/265945847_Active_Radiation_Shield_for_Space_Exploration_Missions</ref><ref name="Desiati2022">Paolo Desiati and Elena D'Onghia, "CREW HaT: A Magnetic Shielding System for Space Habitats", arXiv:2209.13624 [physics.space-ph] https://doi.org/10.48550/arXiv.2209.13624</ref>.
</table>
</div>

One immediate issue is that you want to keep very high magnetic fields out of the crew area. Fields in excess of 0.5 T can start to be unpleasant, and it is recommended to keep the field below 0.2 T for continuous exposure.

Strong magnets can suffer from high self-forces acting to tear them apart. Most shielding designs will have multiple magnetic coils, which will exert high forces and torques on each other. This will result in the need for strong structural bracing to keep the magnets intact and in place. Such strong structural designs will impose significant extra mass on the spacecraft.

Consider a magnetic field configuration that can keep radiation out of a certain region of space. There is no trajectory of a charged radiation particle up to a certain energy that will allow those particles to enter the protected region. But the dynamics of charged particles in a magnetic field are the same whether going forward or backward in time. If there is no way for a particle to get in, by this time reversal symmetry any particle which is in that forbidden region can never escape. Consequently any field configuration that is a magnetic trap is also a magnetic shield; any field configuration that is a magnetic shield is also a magnetic trap. A spacecraft with a magnetic shield may end up surrounded by strong radiation belts.

Using a magnet to bend a weaponized electron beam very often ends up with the beam being bent so hard that it dumps most of its energy in the form of synchrotron radiation at gamma ray energies. Because gamma rays are not charged, they will not be deflected by the field. Thus, if you would have been hit by the electron beam you will be hit instead by the gamma rays.

As an example, consider a 1 TeV electron beam. We'll surround our spacecraft with a 10 T magnetic field in a 100 meter radius region just outside our spacecraft. This will deflect the beam by 30 meters, which is enough to make it miss. But we are not saved, because the beam dumps nearly all of its energy into synchrotron radiation with an energy spectrum of gamma rays up to 13 GeV. This radiation will hit us in a strip about 2.4 meters long and not much more than the width of the original beam with essentially all of the energy of the original beam.

==== Magnet drones ====

If a magnetic shield around your spacecraft will just end up killing you from synchrotron radiation, what if you put the magnetic shield on a drone and put that drone between yourself and the enemy trying to shoot you? This lets you use a much weaker magnet because you don't need nearly so high of a deflection angle to make the shot miss. This means that less beam energy will turn into synchrotron radiation that could kill you instead and will spread what synchrotron radiation does occur out over a much wider area when it reaches your spacecraft, hopefully so that most of it just misses your spacecraft.

It does mean that your enemy will start shooting your drones so that they can eventually take a shot at you. But hopefully by that time you've already cleared out their drones and cored their spacecraft with your own synchrocannon or UREB.

==== Plasma magnets ====

Usually, to make a really big magnetic field you would run current through a really big conductive loop. But really big things are heavy and expensive. One way around this could be to use the ambient space plasma to carry the electric current rather than any kind of wire. In order to make this current, you use the same mechanism used in a rotary induction electric motor: you generate a rotating magnetic field and induce a current in the conductive loop (in this case, the space plasma). How do you make a rotating magnetic field? The method proposed<ref name="Slough2006">John Slough, "The Plasma Magnet", NASA Institute for Advanced Concepts Phase II Final Report (2006) https://www.niac.usra.edu/files/studies/final_report/917Slough.pdf</ref> is to use an antenna of four much smaller current loops as electromagnets at 90&degree; to each other, and sinusoidally vary the current through the loops with a respective 90&degree; phase difference between them. When this is done, a circular current is induced in the space plasma and the self-force of a magnetic field on its generating current inflates the current loop to much larger radii. You end up with a current sheet in the plane of the rotating field's rotation circulating about the antenna nearly synchronously with the field rotation. This current then creates a normal dipolar magnetic field as with any current loop. The current sheet and the resulting magnetic field expand until the magnetic field's pressure equals that of the ambient space plasma. This gives an enormous region of magnetic field from a relatively small and lightweight system.

The system proposed in the initial study would use an antenna approximately 10 meters in radius and produce a region of field approximately 30 km in radius at Earth's orbit. The field inside would not be constant - at the edges it would be 50 nT, giving the right pressure to keep out the solar wind at Earth's orbit. A circulating current sheet produces a field that falls off as 1/r with distance, meaning that adjacent to the antenna the field would be about 0.15 mT. Using the methods in the box below, we can estimate that this particular design as part of a spacecraft's shields will not deflect a TeV electron beam by more than a few centimeters, which is usually insufficient to force a miss. However, it could potentially be useful for a drone magnet, where the much larger distance between the magnetic field and the target spacecraft would allow a small deflection angle to give perhaps tens or low hundreds of meters change in beam location near the target at near light second ranges.

<table border=1><tr><td>
Calculating the deflection from a spatially varying magnetic field affected by the solar wind is no trivial task. However, we can make an approximation that both should be within about an order of magnitude of the true value and also represent an upper limit to the deflection angle.

We will take R0 as the radius of the antenna. B0 is the field strength at the antenna. Rm is the distance at which the magnetic influence stops because the magnetic pressure equals the space plasma pressure. Bm is the field strength at Rm. According to Slough<ref name="Slough2006"></ref>, the magnetic field strength B(r) at any distance from the antenna center r between R0 and Rm is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B(r) = B0 R0 / r.
</div>
Therefore, once we know Bm and Rm, B0 is completely determined
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
B0 = Bm Rm / R0.
</div>

The force on a particle with charge q passing perpendicularly through a magnetic field B(r) at a speed v is
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
F(r) = q v B(r)
</div>
For an electron beam, v is so close to c as to make no difference. If the particle is moving down the y direction and the field is in the z direction, the force will be in the x direction. We assume that the deflection will be small, so we can choose a straight-line path in the y direction offset from the center by an amount x. The plasma bubble radius is Rm, so the particle will enter at ymax = √[Rm2-x2] and leave at ymin = -√[Rm2-x2].

The momentum change the particle experiences from this force is the time integral of the force over its path. Because of reflection symmetry about y = 0, we only need to integrate from ymax to 0 and then multiply by 2
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] dt F(r)
<tr> <td> <td> = <td align="left">2 × [Integral from 0 to √[Rm2-x2]] (dy/v) F(r)
<tr> <td> <td> = <td align="left">2 q B(0) R0 [Integral from 0 to √[Rm2-x2]] dy 1/√[y2 + x2]
<tr> <td> <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/x]
</table>
</div>
This holds for |x| > R0. But the expression for the field is not valid for r < R0 and the integral will give nonsense and diverging results if we evaluate for |x| < R0. In this case, we can put the integral lower bound at r = R0, or y = √[R02-x2]
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table>
<tr> <td>px <td> = <td align="left">2 q B(0) R0 ln[(√[Rm2-x2] + Rm)/(√[R02-x2] + R0)]
</table>
</div>

The deflection angle is now estimated as
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
φ = px / p
</div>
where the total momentum of the particle p can be found from the methods in the first box in the Magnetic Fields section.

The deflection for a shot aimed right at the spacecraft at the position of the spacecraft can be estimated by setting x = 0 and only taking the first half of the integral (so ignore the factor of 2)
<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
q B(0) R0 ln[Rm/R0] / (p Rm)
</div>
This is an additional overestimate because it fails to account for how the deflection builds up over distance during the integration, but it should still give something of an order-of-magnitude and upper limit estimate of what kind of deflections might be expected.
</table>

=== Electric fields ===

If electric fields can accelerate charged particles, maybe they can keep the charged particles in a weaponized particle beam away from your spacecraft? To completely block a particle beam, you would need a voltage on your spacecraft equal to the particle energy in eV of the particles in the beam. That is, if a 1 GeV proton beam is coming at you, you'd need an electric potential of a billion volts to stop that beam. This leads to three immediate problems.

The first is that particles of the opposite charge will be attracted to your spacecraft. Because space isn't really empty but rather a rarefied plasma (called the solar wind in a solar system or the interstellar medium out between stars), there will be plenty of particles of the opposite charge around to attract. And they will hit with an energy in eV equal to the voltage on your spacecraft. In our example of defending against a 1 GeV proton beam with a gigavolt field, the electrons from the solar wind will come in and hit with an energy of 1 GeV each. This will generate dangerous radiation.

The second problem is a consequence of the first. Each of those GeV electrons from our example need to get their energy from somewhere, which turns out to be from the electric field we are using as shielding. You will need to make up for this energy loss with power from your electrical system, which gives a constant drain on your power. Some estimates put the electrical effort to maintain a several hundred megavolt shield in space at many tens of gigawatts<ref name="Mechmann2019">Claire Mechmann, "Analysis of Proposed Active Radiation Shielding Design Concept for Spacecraft" (2019) Thesis, College of Engineering and Science of Florida Institute of Technology</ref>.

And the third problem is similar to one for magnetic fields. Your electrodes will experience high self forces and forces between separate electrodes that will try to tear your shield generator apart. Consequently you will need heavy structural reinforcement in order for your shield to work.

There are potential ways to mitigate some of these problems. Interested readers are directed to [[Space_Radiation#Electrostatic_Shielding|the active shielding section in our page on space radiation]] for more of the juicy details; there's no need to repeat the same stuff here.

<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
<table><tr><td>
[[File:Elctrostatic_active_shielding.png|200 px|frameless]]
<td>
[[File:Electrostatic_active_shielding_2.png|200 px|frameless]]
<tr><td colspan=2>
Proposed designs for electrostatic shields to protect from positively charged particles<ref name="Tripathi2006">Ram K. Tripathi, John W. Wilson, and Robert C. Youngquist, "Electrostatic Active Radiation Shielding - Revisited", 2006 IEEE Aerospace Conference, Big Sky, MT, USA, 2006, pp. 9 pp.-, doi: 10.1109/AERO.2006.1655760.</ref><ref name="Chowdhury2023">Rajarshi Pal Chowdhury, Luke A. Stegeman, Matthew L. Lund, Dan Fry, Stojan Madzunkov, and Amir A. Bahadori, "Hybrid methods of radiation shielding against deep-space radiation", Life Sciences in Space Research, Volume 38, 2023, Pages 67-78, ISSN 2214-5524, https://doi.org/10.1016/j.lssr.2023.04.004.</ref>.
</table>
</div>

== Damage ==

Particle beams are, first and foremost, devices that produce beams of [[Nuclear_radiation|ionizing radiation]]. The radiation of weaponizable particle beams is extremely penetrating; thus of the [[Particle_Accelerators#Energy_loss_in_matter|interaction mechanisms listed for radiation from particle beams]], we mostly need to worry about bremsstrahlung and nuclear collisions and these produce extensive secondary showers of radiation that increase the penetration of the radiation exposure. Because of this, any target hit by a particle beam weapon will be irradiated, and radiation dose will always be one of the primary damage mechanisms of a particle beam weapon strike. Further, this radiation will be distributed along a track passing deeply through the material of the target, allowing it to affect critical systems through heavy armor, structural material, robust machine components, and passive material such as fuel load, coolant, or cargo. As the radiation deposits its energy into the target material, that material will be heated. If the target is heated sufficiently, it may experience [[Beam-Target_Interactions|thermal or mechanical]] damage from melting, cooking, combustion, vapor explosions, and warping or cracking due to thermal expansion.

The discussion of the radiation penetration into armor also applies to the same radiation penetrating through the target. So the formulas and rules of thumb for estimating the depth of radiation penetration into armor can also be used to estimate how far into a tank or spacecraft a particle beam strike delivers effects.

In addition to the radiation and thermal effects, a particle beam can also produce a sharp electromagnetic pulse that can damage semiconductor electronics.

== 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, Gerrit Bruhaug, and Luke Campbell for writing

Gerrit Bruhaug and Luke Campbell for scientific research, math wrangling and advise

ATriffleMind and Luke W. Campbell for proof reading.

==References==

[[Category:Beams]][[Category:Physics & Engineering‏‎]][[Category:Engineering‏‎]][[Category:Military Technology]][[Category:Warfare‏‎]]