Particle Accelerators - Revision history

https://www.galacticlibrary.net/mediawiki-1.41.1/index.php?action=history&feed=atom&title=Particle_Accelerators Particle Accelerators - Revision history 2026-05-02T12:31:42Z Revision history for this page on the wiki MediaWiki 1.41.1 https://www.galacticlibrary.net/mediawiki-1.41.1/index.php?title=Particle_Accelerators&diff=3798&oldid=prev Lwcamp at 19:05, 7 March 2026 2026-03-07T19:05:02Z

<p></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 12:05, 7 March 2026</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l739">Line 739:</td> <td colspan="2" class="diff-lineno">Line 739:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Beams]][[Category:Physics & Engineering‏‎]][[Category:Engineering‏‎]]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Beams]][[Category:Physics & Engineering‏‎<ins style="font-weight: bold; text-decoration: none;">]][[Category:Physics & Math & Engineering</ins>]][[Category:Engineering‏‎]]</div></td></tr> </table>

Lwcamp https://www.galacticlibrary.net/mediawiki-1.41.1/index.php?title=Particle_Accelerators&diff=3166&oldid=prev Lwcamp: /* Nuclear collisions */ 2025-04-09T01:15:36Z

<p><span dir="auto"><span class="autocomment">Nuclear collisions</span></span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 18:15, 8 April 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l619">Line 619:</td> <td colspan="2" class="diff-lineno">Line 619:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>But beyond that you can cause any number of nuclear reactions. You can leave the nucleus in an excited nuclear state, from which it usually relaxes back down to its original state by emitting gamma rays. You can knock out nuclear particles, such as neutrons, protons, deuterons (a proton stuck to a neutron), tritons (a proton stuck to two neutrons), helions (a neutron stuck to two protons), alpha particles (two protons stuck to two neutrons), or potentially even larger nuclear fragments. If you hit a heavy nucleus like uranium, you can cause it to split apart by fission. And if you hit the nucleus really hard you might just shatter it into many tiny fragments of those protons, neutrons, deuterons, tritons, helions, alpha particles, and heavier nuclear fragments mentioned earlier. To get any of these <b>inelastic collisions</b>, you need to hit the nucleus hard enough. If your beam particle doesn't have enough energy to excite the nucleus or knock particles, these things simply can't happen. As it gets enough energy to excite a particular interaction channel (as they are called), that process becomes possible and as the energy increases the process becomes more and more likely.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>But beyond that you can cause any number of nuclear reactions. You can leave the nucleus in an excited nuclear state, from which it usually relaxes back down to its original state by emitting gamma rays. You can knock out nuclear particles, such as neutrons, protons, deuterons (a proton stuck to a neutron), tritons (a proton stuck to two neutrons), helions (a neutron stuck to two protons), alpha particles (two protons stuck to two neutrons), or potentially even larger nuclear fragments. If you hit a heavy nucleus like uranium, you can cause it to split apart by fission. And if you hit the nucleus really hard you might just shatter it into many tiny fragments of those protons, neutrons, deuterons, tritons, helions, alpha particles, and heavier nuclear fragments mentioned earlier. To get any of these <b>inelastic collisions</b>, you need to hit the nucleus hard enough. If your beam particle doesn't have enough energy to excite the nucleus or knock particles, these things simply can't happen. As it gets enough energy to excite a particular interaction channel (as they are called), that process becomes possible and as the energy increases the process becomes more and more likely.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>A nucleus is only about a femtometer across (<del style="font-weight: bold; text-decoration: none;"><math></del>1 <del style="font-weight: bold; text-decoration: none;">\</del>times 10<del style="font-weight: bold; text-decoration: none;">^{</del>-15<del style="font-weight: bold; text-decoration: none;">}</del></<del style="font-weight: bold; text-decoration: none;">math</del>> m). Which makes it only about <del style="font-weight: bold; text-decoration: none;"><math></del>1 <del style="font-weight: bold; text-decoration: none;">\</del>times 10<del style="font-weight: bold; text-decoration: none;">^{</del>-30<del style="font-weight: bold; text-decoration: none;">}</del></<del style="font-weight: bold; text-decoration: none;">math</del>> m&sup2; in cross sectional area. In the meantime, chemical bonds between atoms are usually about 1/10th of a nanometer (<del style="font-weight: bold; text-decoration: none;"><math></del>1 <del style="font-weight: bold; text-decoration: none;">\</del>times 10<del style="font-weight: bold; text-decoration: none;">^{</del>-10<del style="font-weight: bold; text-decoration: none;">}</del></<del style="font-weight: bold; text-decoration: none;">math</del>> m) long. In condensed matter (liquids and solids), this means that you have somewhere about <del style="font-weight: bold; text-decoration: none;"><math></del>1 <del style="font-weight: bold; text-decoration: none;">\</del>times 10<del style="font-weight: bold; text-decoration: none;">^{</del>30<del style="font-weight: bold; text-decoration: none;">}</del></<del style="font-weight: bold; text-decoration: none;">math</del>> atoms per cubic meter. A meter thickness of atoms thus presents just about about enough atoms that their nuclei can cover the projected area &ndash; in other words, you can expect a particle to go through (very roughly) about a meter of condensed matter before it whacks into a nucleus.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>A nucleus is only about a femtometer across (1 <ins style="font-weight: bold; text-decoration: none;">&</ins>times<ins style="font-weight: bold; text-decoration: none;">; </ins>10<ins style="font-weight: bold; text-decoration: none;"><sup></ins>-15</<ins style="font-weight: bold; text-decoration: none;">sup</ins>> m). Which makes it only about 1 <ins style="font-weight: bold; text-decoration: none;">&</ins>times<ins style="font-weight: bold; text-decoration: none;">; </ins>10<ins style="font-weight: bold; text-decoration: none;"><sup></ins>-30</<ins style="font-weight: bold; text-decoration: none;">sup</ins>> m&sup2; in cross sectional area. In the meantime, chemical bonds between atoms are usually about 1/10th of a nanometer (1 <ins style="font-weight: bold; text-decoration: none;">&</ins>times<ins style="font-weight: bold; text-decoration: none;">; </ins>10<ins style="font-weight: bold; text-decoration: none;"><sup></ins>-10</<ins style="font-weight: bold; text-decoration: none;">sup</ins>> m) long. In condensed matter (liquids and solids), this means that you have somewhere about 1 <ins style="font-weight: bold; text-decoration: none;">&</ins>times<ins style="font-weight: bold; text-decoration: none;">; </ins>10<ins style="font-weight: bold; text-decoration: none;"><sup></ins>30</<ins style="font-weight: bold; text-decoration: none;">sup</ins>> atoms per cubic meter. A meter thickness of atoms thus presents just about about enough atoms that their nuclei can cover the projected area &ndash; in other words, you can expect a particle to go through (very roughly) about a meter of condensed matter before it whacks into a nucleus<ins style="font-weight: bold; text-decoration: none;">. More detailed estimates usually have the particles penetrating several tens of centimeters into solid or liquid targets between nuclear collisions</ins>.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This assumes that a particle that goes through a nucleus interacts with it. This is usually a good assumption for particles that interact by the strong nuclear force, such as protons, neutrons, mesons, and the atomic nuclei of ions. But many particles cannot interact by the strong nuclear force, such as electrons or muons. These are much less likely to interact with an atomic nucleus even if they pass through it, so they will end up going considerably farther between hitting nuclei than the above assumption. But nuclei can interact electromagnetically (they are made up of charged protons, after all, and at a smaller level of charged quarks), so electrons and muons can produce some nuclear interactions; just far less than actual nuclear particles like protons and ions.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>This assumes that a particle that goes through a nucleus interacts with it. This is usually a good assumption for particles that interact by the strong nuclear force, such as protons, neutrons, mesons, and the atomic nuclei of ions. But many particles cannot interact by the strong nuclear force, such as electrons or muons. These are much less likely to interact with an atomic nucleus even if they pass through it, so they will end up going considerably farther between hitting nuclei than the above assumption. But nuclei can interact electromagnetically (they are made up of charged protons, after all, and at a smaller level of charged quarks), so electrons and muons can produce some nuclear interactions; just far less than actual nuclear particles like protons and ions.</div></td></tr> </table>

Lwcamp https://www.galacticlibrary.net/mediawiki-1.41.1/index.php?title=Particle_Accelerators&diff=3165&oldid=prev Lwcamp: /* Beam propagation in vacuum with self-forces */ 2025-04-09T01:09:16Z

<p><span dir="auto"><span class="autocomment">Beam propagation in vacuum with self-forces</span></span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 18:09, 8 April 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l554">Line 554:</td> <td colspan="2" class="diff-lineno">Line 554:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><blockquote></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><blockquote></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><i>Example</i>: Let's use our 1 MW, 10 MeV electron accelerator from before, with its 2 cm aperture and <del style="font-weight: bold; text-decoration: none;"><math></del>K = 1.37 <del style="font-weight: bold; text-decoration: none;">\</del>times 10<del style="font-weight: bold; text-decoration: none;">^{</del>-9<del style="font-weight: bold; text-decoration: none;">}</del></<del style="font-weight: bold; text-decoration: none;">math</del>>. Suppose we are using this as an electron cannon, and shooting at a target 2 km away. We want to know how small of a spot we can direct on to our target. We use <<del style="font-weight: bold; text-decoration: none;">math</del>><del style="font-weight: bold; text-decoration: none;">R_m </del>= 0.01<del style="font-weight: bold; text-decoration: none;"></math> </del>m (the beam radius at the aperture) and <del style="font-weight: bold; text-decoration: none;"><math></del>z = 2,000<del style="font-weight: bold; text-decoration: none;"></math> </del>m for the distance to the target. This gives us <del style="font-weight: bold; text-decoration: none;"><math></del>F(<del style="font-weight: bold; text-decoration: none;">\chi</del>) = 10.46<del style="font-weight: bold; text-decoration: none;"></math></del>. Looking at the graph, <del style="font-weight: bold; text-decoration: none;"><math></del>F(<del style="font-weight: bold; text-decoration: none;">\chi</del>)<del style="font-weight: bold; text-decoration: none;"></math> </del>is about 10 where <del style="font-weight: bold; text-decoration: none;"><math>\chi</math> </del>is about 12. So we know that at the target the beam spot at the target is about 12 times larger than at the aperture, or about 12 cm across.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><i>Example</i>: Let's use our 1 MW, 10 MeV electron accelerator from before, with its 2 cm aperture and K = 1.37 <ins style="font-weight: bold; text-decoration: none;">&</ins>times<ins style="font-weight: bold; text-decoration: none;">; </ins>10<ins style="font-weight: bold; text-decoration: none;"><sup></ins>-9</<ins style="font-weight: bold; text-decoration: none;">sup</ins>>. Suppose we are using this as an electron cannon, and shooting at a target 2 km away. We want to know how small of a spot we can direct on to our target. We use <ins style="font-weight: bold; text-decoration: none;">R<sub>m</ins><<ins style="font-weight: bold; text-decoration: none;">/sub</ins>> = 0.01 m (the beam radius at the aperture) and z = 2,000 m for the distance to the target. This gives us F(<ins style="font-weight: bold; text-decoration: none;">χ</ins>) = 10.46. Looking at the graph, F(<ins style="font-weight: bold; text-decoration: none;">χ</ins>) is about 10 where <ins style="font-weight: bold; text-decoration: none;">χ </ins>is about 12. So we know that at the target the beam spot at the target is about 12 times larger than at the aperture, or about 12 cm across.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></blockquote></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></blockquote></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table>

Lwcamp https://www.galacticlibrary.net/mediawiki-1.41.1/index.php?title=Particle_Accelerators&diff=3164&oldid=prev Lwcamp: /* Beam propagation in vacuum with self-forces */ 2025-04-09T01:06:56Z

<p><span dir="auto"><span class="autocomment">Beam propagation in vacuum with self-forces</span></span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 18:06, 8 April 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l524">Line 524:</td> <td colspan="2" class="diff-lineno">Line 524:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></blockquote></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></blockquote></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>To get actual numbers, take a look at the picture below. It shows the shape of a beam emitted from an accelerator. The beam reaches a minimum width of <<del style="font-weight: bold; text-decoration: none;">math</del>><del style="font-weight: bold; text-decoration: none;">R_m</del></<del style="font-weight: bold; text-decoration: none;">math</del>> at a position <del style="font-weight: bold; text-decoration: none;"><math></del>z=0<del style="font-weight: bold; text-decoration: none;"></math></del>. At any distance <del style="font-weight: bold; text-decoration: none;"><math></del>z<del style="font-weight: bold; text-decoration: none;"></math> </del>along the beam from this minimum, the beam width will be <del style="font-weight: bold; text-decoration: none;"><math></del>R(z)<del style="font-weight: bold; text-decoration: none;"></math></del>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>To get actual numbers, take a look at the picture below. It shows the shape of a beam emitted from an accelerator. The beam reaches a minimum width of <ins style="font-weight: bold; text-decoration: none;">R</ins><<ins style="font-weight: bold; text-decoration: none;">sub</ins>><ins style="font-weight: bold; text-decoration: none;">m</ins></<ins style="font-weight: bold; text-decoration: none;">sub</ins>> at a position z = 0. At any distance z along the beam from this minimum, the beam width will be R(z).</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><table class="center" style="width: auto; margin-left: auto; margin-right: auto;"></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><table class="center" style="width: auto; margin-left: auto; margin-right: auto;"></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tr></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tr></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><td width=600>[[File:Self-charge_shaped_beam.png|600 px|frameless]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><td width=600>[[File:Self-charge_shaped_beam.png|600 px|frameless]]</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></table></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></table></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>It is useful to define <del style="font-weight: bold; text-decoration: none;"><math>\chi</math> </del>as the fractional amount by which the beam expands</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>It is useful to define <ins style="font-weight: bold; text-decoration: none;">χ </ins>as the fractional amount by which the beam expands</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><div class="center" style="width: auto; margin-left: auto; margin-right: auto;"<del style="font-weight: bold; text-decoration: none;">><math</del>></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><div class="center" style="width: auto; margin-left: auto; margin-right: auto;"></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">\chi </del>= <del style="font-weight: bold; text-decoration: none;">\frac{</del>R(z)<del style="font-weight: bold; text-decoration: none;">}{R_m}.</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">χ </ins>= R(z) <ins style="font-weight: bold; text-decoration: none;">/ R<sub>m</ins></<ins style="font-weight: bold; text-decoration: none;">sub</ins>><ins style="font-weight: bold; text-decoration: none;">.</ins></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div></<del style="font-weight: bold; text-decoration: none;">math</del>></div></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div></div></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>So if the beam doubles in radius, then <del style="font-weight: bold; text-decoration: none;"><math>\chi </del>= 2<del style="font-weight: bold; text-decoration: none;"></math></del>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>So if the beam doubles in radius, then <ins style="font-weight: bold; text-decoration: none;">χ </ins>= 2.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>We also need to introduce a function <del style="font-weight: bold; text-decoration: none;"><math></del>F(<del style="font-weight: bold; text-decoration: none;">\chi</del>)<del style="font-weight: bold; text-decoration: none;"></math></del>, which doesn't have any simple expression (it is defined as an integral over the reciprocal of a logarithm), so it is easiest just to give it in tables and figures.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>We also need to introduce a function F(<ins style="font-weight: bold; text-decoration: none;">χ</ins>), which doesn't have any simple expression (it is defined as an integral over the reciprocal of a logarithm), so it is easiest just to give it in tables and figures.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><table class="center" style="width: auto; margin-left: auto; margin-right: auto;"></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><table class="center" style="width: auto; margin-left: auto; margin-right: auto;"></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tr></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tr></div></td></tr> <tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l545">Line 545:</td> <td colspan="2" class="diff-lineno">Line 545:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></table></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></table></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>To use these to get the beam size at any distance from the beam minimum, we can use<ref>Stanley Humphries, Jr., "Charged Particle Beams", Originally published in 1990 by John Wiley and Sons (QC786.H86 1990, ISBN 0-471-60014-8)</ref></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>To use these to get the beam size at any distance from the beam minimum, we can use<ref>Stanley Humphries, Jr., "Charged Particle Beams", Originally published in 1990 by John Wiley and Sons (QC786.H86 1990, ISBN 0-471-60014-8)</ref></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><div class="center" style="width: auto; margin-left: auto; margin-right: auto;"<del style="font-weight: bold; text-decoration: none;">><math</del>></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><div class="center" style="width: auto; margin-left: auto; margin-right: auto;"></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>F(<del style="font-weight: bold; text-decoration: none;">\chi</del>) = <del style="font-weight: bold; text-decoration: none;">\sqrt{</del>2 K<del style="font-weight: bold; text-decoration: none;">} \, \frac{</del>z<del style="font-weight: bold; text-decoration: none;">}{R_m}</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>F(<ins style="font-weight: bold; text-decoration: none;">χ</ins>) = <ins style="font-weight: bold; text-decoration: none;">&radic;[</ins>2 K<ins style="font-weight: bold; text-decoration: none;">] </ins>z <ins style="font-weight: bold; text-decoration: none;">/ R<sub>m</ins></<ins style="font-weight: bold; text-decoration: none;">sub</ins>></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div></<del style="font-weight: bold; text-decoration: none;">math</del>></div></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div></div></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>and</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>and</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><div class="center" style="width: auto; margin-left: auto; margin-right: auto;"<del style="font-weight: bold; text-decoration: none;">><math</del>></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><div class="center" style="width: auto; margin-left: auto; margin-right: auto;"></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">\frac{</del>F(<del style="font-weight: bold; text-decoration: none;">\chi</del>)<del style="font-weight: bold; text-decoration: none;">}{\chi} </del>= <del style="font-weight: bold; text-decoration: none;">\sqrt{</del>2 K<del style="font-weight: bold; text-decoration: none;">} \, \frac{</del>z<del style="font-weight: bold; text-decoration: none;">}{</del>R(z)<del style="font-weight: bold; text-decoration: none;">}</del>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>F(<ins style="font-weight: bold; text-decoration: none;">χ</ins>) <ins style="font-weight: bold; text-decoration: none;">/ χ </ins>= <ins style="font-weight: bold; text-decoration: none;">&radic;[</ins>2 K<ins style="font-weight: bold; text-decoration: none;">] </ins>z <ins style="font-weight: bold; text-decoration: none;">/ </ins>R(z).</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></math></del></div></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div></div></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><blockquote></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><blockquote></div></td></tr> </table>

Lwcamp https://www.galacticlibrary.net/mediawiki-1.41.1/index.php?title=Particle_Accelerators&diff=3163&oldid=prev Lwcamp: /* Beam propagation in vacuum with self-forces */ 2025-04-09T00:59:56Z

<p><span dir="auto"><span class="autocomment">Beam propagation in vacuum with self-forces</span></span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:59, 8 April 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l513">Line 513:</td> <td colspan="2" class="diff-lineno">Line 513:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><blockquote></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><blockquote></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><i>Example</i>: Let's consider an electron beam with <del style="font-weight: bold; text-decoration: none;"><math></del>P = 1<del style="font-weight: bold; text-decoration: none;"></math> </del>MW of power and <del style="font-weight: bold; text-decoration: none;"><math></del>V = 10<del style="font-weight: bold; text-decoration: none;"></math> </del>GeV energy particles. It needs to have a current of <del style="font-weight: bold; text-decoration: none;"><math></del>I = 0.0001<del style="font-weight: bold; text-decoration: none;"></math> </del>A to get this power with this particle energy. The beam will initially be 2 cm wide (<del style="font-weight: bold; text-decoration: none;"><math></del>R = 0.01<del style="font-weight: bold; text-decoration: none;"></math> </del>m). At this particle energy, <del style="font-weight: bold; text-decoration: none;"><math>\gamma </del>= 19600<del style="font-weight: bold; text-decoration: none;"></math></del>, and <del style="font-weight: bold; text-decoration: none;"><math>\beta</math> </del>is so close to one as to make no difference. For a reasonable normalized emittance of <<del style="font-weight: bold; text-decoration: none;">math</del>><del style="font-weight: bold; text-decoration: none;">\epsilon_n </del>= 1<del style="font-weight: bold; text-decoration: none;">\</del>times 10<del style="font-weight: bold; text-decoration: none;">^{</del>-6<del style="font-weight: bold; text-decoration: none;">}</del></<del style="font-weight: bold; text-decoration: none;">math</del>> m<del style="font-weight: bold; text-decoration: none;"><math>\cdot</math></del>rad, we end up with a geometrical emittance of <del style="font-weight: bold; text-decoration: none;"><math>\epsilon </del>= 5.1 <del style="font-weight: bold; text-decoration: none;">\</del>times 10<del style="font-weight: bold; text-decoration: none;">^{</del>-11<del style="font-weight: bold; text-decoration: none;">}</del></<del style="font-weight: bold; text-decoration: none;">math</del>> m<del style="font-weight: bold; text-decoration: none;"><math>\cdot</math></del>rad.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><i>Example</i>: Let's consider an electron beam with P = 1 MW of power and V = 10 GeV energy particles. It needs to have a current of I = 0.0001 A to get this power with this particle energy. The beam will initially be 2 cm wide (R = 0.01 m). At this particle energy, <ins style="font-weight: bold; text-decoration: none;">γ </ins>= 19600, and <ins style="font-weight: bold; text-decoration: none;">β </ins>is so close to one as to make no difference. For a reasonable normalized emittance of <ins style="font-weight: bold; text-decoration: none;">ε<sub>n</ins><<ins style="font-weight: bold; text-decoration: none;">/sub</ins>> = 1 <ins style="font-weight: bold; text-decoration: none;">&</ins>times<ins style="font-weight: bold; text-decoration: none;">; </ins>10<ins style="font-weight: bold; text-decoration: none;"><sup></ins>-6</<ins style="font-weight: bold; text-decoration: none;">sup</ins>> m<ins style="font-weight: bold; text-decoration: none;">&middot;</ins>rad, we end up with a geometrical emittance of <ins style="font-weight: bold; text-decoration: none;">ε </ins>= 5.1 <ins style="font-weight: bold; text-decoration: none;">&</ins>times<ins style="font-weight: bold; text-decoration: none;">; </ins>10<ins style="font-weight: bold; text-decoration: none;"><sup></ins>-11</<ins style="font-weight: bold; text-decoration: none;">sup</ins>> m<ins style="font-weight: bold; text-decoration: none;">&middot;</ins>rad.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Plugging these numbers in, we find that <del style="font-weight: bold; text-decoration: none;"><math></del>K = 1.56 <del style="font-weight: bold; text-decoration: none;">\</del>times 10<del style="font-weight: bold; text-decoration: none;">^{</del>-21<del style="font-weight: bold; text-decoration: none;">}</del></<del style="font-weight: bold; text-decoration: none;">math</del>>. The quantity <del style="font-weight: bold; text-decoration: none;"><math>(\epsilon</del>/R<del style="font-weight: bold; text-decoration: none;">)^2 </del>= 2.6 <del style="font-weight: bold; text-decoration: none;">\</del>times 10<del style="font-weight: bold; text-decoration: none;">^{</del>-17<del style="font-weight: bold; text-decoration: none;">}</del></<del style="font-weight: bold; text-decoration: none;">math</del>>. So for this particle beam, emittance will dominate over self-forces and we only really need to worry about emittance. Only if you try to focus it down to about a tenth of a millimeter or less will <del style="font-weight: bold; text-decoration: none;"><math>(\epsilon</del>/R<del style="font-weight: bold; text-decoration: none;">)^2</math> </del>be comparable to <del style="font-weight: bold; text-decoration: none;"><math></del>K<del style="font-weight: bold; text-decoration: none;"></math> </del>such that you would need to take self-forces into account.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Plugging these numbers in, we find that K = 1.56 <ins style="font-weight: bold; text-decoration: none;">&</ins>times<ins style="font-weight: bold; text-decoration: none;">; </ins>10<ins style="font-weight: bold; text-decoration: none;"><sup></ins>-21</<ins style="font-weight: bold; text-decoration: none;">sup</ins>>. The quantity <ins style="font-weight: bold; text-decoration: none;">ε&sup2;</ins>/R<ins style="font-weight: bold; text-decoration: none;">&sup2; </ins>= 2.6 <ins style="font-weight: bold; text-decoration: none;">&</ins>times<ins style="font-weight: bold; text-decoration: none;">; </ins>10<ins style="font-weight: bold; text-decoration: none;"><sup></ins>-17</<ins style="font-weight: bold; text-decoration: none;">sup</ins>>. So for this particle beam, emittance will dominate over self-forces and we only really need to worry about emittance. Only if you try to focus it down to about a tenth of a millimeter or less will <ins style="font-weight: bold; text-decoration: none;">ε&sup2;</ins>/R<ins style="font-weight: bold; text-decoration: none;">&sup2; </ins>be comparable to K such that you would need to take self-forces into account.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></blockquote></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></blockquote></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><blockquote></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><blockquote></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><i>Example</i>: Now consider an electron beam with the same <del style="font-weight: bold; text-decoration: none;"><math></del>P = 1<del style="font-weight: bold; text-decoration: none;"></math> </del>MW of power but particles with only <del style="font-weight: bold; text-decoration: none;"><math></del>V = 10<del style="font-weight: bold; text-decoration: none;"></math> </del>MeV energy. This requires a current of <del style="font-weight: bold; text-decoration: none;"><math></del>I = 0.1<del style="font-weight: bold; text-decoration: none;"></math> </del>A. <del style="font-weight: bold; text-decoration: none;">we</del>'ll use the same 2 cm wide (<del style="font-weight: bold; text-decoration: none;"><math></del>R = 0.01<del style="font-weight: bold; text-decoration: none;"></math> </del>m) beam width. At this particle energy, <del style="font-weight: bold; text-decoration: none;"><math>\gamma </del>= 20.57<del style="font-weight: bold; text-decoration: none;"></math></del>, and <del style="font-weight: bold; text-decoration: none;"><math>\beta </del>= 0.999<del style="font-weight: bold; text-decoration: none;"></math></del>. For the same normalized emittance of <<del style="font-weight: bold; text-decoration: none;">math</del>><del style="font-weight: bold; text-decoration: none;">\epsilon_n </del>= 1<del style="font-weight: bold; text-decoration: none;">\</del>times 10<del style="font-weight: bold; text-decoration: none;">^{</del>-6<del style="font-weight: bold; text-decoration: none;">}</del></<del style="font-weight: bold; text-decoration: none;">math</del>> m<del style="font-weight: bold; text-decoration: none;"><math>\cdot</math></del>rad, we end up with a geometrical emittance of <del style="font-weight: bold; text-decoration: none;"><math>\epsilon </del>= 4.86 <del style="font-weight: bold; text-decoration: none;">\</del>times 10<del style="font-weight: bold; text-decoration: none;">^{</del>-8<del style="font-weight: bold; text-decoration: none;">}</del></<del style="font-weight: bold; text-decoration: none;">math</del>> m<del style="font-weight: bold; text-decoration: none;"><math>\cdot</math></del>rad.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><i>Example</i>: Now consider an electron beam with the same P = 1 MW of power but particles with only V = 10 MeV energy. This requires a current of I = 0.1 A. <ins style="font-weight: bold; text-decoration: none;">We</ins>'ll use the same 2 cm wide (R = 0.01 m) beam width. At this particle energy, <ins style="font-weight: bold; text-decoration: none;">γ </ins>= 20.57, and <ins style="font-weight: bold; text-decoration: none;">β </ins>= 0.999. For the same normalized emittance of <ins style="font-weight: bold; text-decoration: none;">ε</ins><<ins style="font-weight: bold; text-decoration: none;">sub>n</sub</ins>> = 1 <ins style="font-weight: bold; text-decoration: none;">&</ins>times<ins style="font-weight: bold; text-decoration: none;">; </ins>10<ins style="font-weight: bold; text-decoration: none;"><sup></ins>-6</<ins style="font-weight: bold; text-decoration: none;">sup</ins>> m<ins style="font-weight: bold; text-decoration: none;">&middot;</ins>rad, we end up with a geometrical emittance of <ins style="font-weight: bold; text-decoration: none;">ε </ins>= 4.86 <ins style="font-weight: bold; text-decoration: none;">&</ins>times<ins style="font-weight: bold; text-decoration: none;">; </ins>10<ins style="font-weight: bold; text-decoration: none;"><sup></ins>-8</<ins style="font-weight: bold; text-decoration: none;">sup</ins>> m<ins style="font-weight: bold; text-decoration: none;">&middot;</ins>rad.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Plugging these numbers in, we find that <del style="font-weight: bold; text-decoration: none;"><math></del>K = 1.37 <del style="font-weight: bold; text-decoration: none;">\</del>times 10<del style="font-weight: bold; text-decoration: none;">^{</del>-9<del style="font-weight: bold; text-decoration: none;">}</del></<del style="font-weight: bold; text-decoration: none;">math</del>>. The quantity <del style="font-weight: bold; text-decoration: none;"><math>(\epsilon</del>/R<del style="font-weight: bold; text-decoration: none;">)^2 </del>= 2.36 <del style="font-weight: bold; text-decoration: none;">\</del>times 10<del style="font-weight: bold; text-decoration: none;">^{</del>-11<del style="font-weight: bold; text-decoration: none;">}</del></<del style="font-weight: bold; text-decoration: none;">math</del>>. For this particle beam, self-forces will have a much greater effect than emittance.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Plugging these numbers in, we find that K = 1.37 <ins style="font-weight: bold; text-decoration: none;">&</ins>times<ins style="font-weight: bold; text-decoration: none;">; </ins>10<ins style="font-weight: bold; text-decoration: none;"><sup></ins>-9</<ins style="font-weight: bold; text-decoration: none;">sup</ins>>. The quantity <ins style="font-weight: bold; text-decoration: none;">ε&sup2;</ins>/R<ins style="font-weight: bold; text-decoration: none;">&sup2; </ins>= 2.36 <ins style="font-weight: bold; text-decoration: none;">&</ins>times<ins style="font-weight: bold; text-decoration: none;">; </ins>10<ins style="font-weight: bold; text-decoration: none;"><sup></ins>-11</<ins style="font-weight: bold; text-decoration: none;">sup</ins>>. For this particle beam, self-forces will have a much greater effect than emittance.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></blockquote></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></blockquote></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table>

Lwcamp https://www.galacticlibrary.net/mediawiki-1.41.1/index.php?title=Particle_Accelerators&diff=3162&oldid=prev Lwcamp: /* Beam propagation in vacuum with self-forces */ 2025-04-09T00:47:28Z

<p><span dir="auto"><span class="autocomment">Beam propagation in vacuum with self-forces</span></span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:47, 8 April 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l510">Line 510:</td> <td colspan="2" class="diff-lineno">Line 510:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where q is the charge per particle, I is the electric current in the beam, 1 / (4 π ε<sub>0</sub>) = 8.9875517923 &times; 10<sup>9</sup> N m&sup2;/C&sup2; is the Coulomb constant, m is the particle mass, β is the speed as a fraction of the speed of light, γ is the Lorentz factor, c is the speed of light, and R is the radius of the beam. If K is much larger than ε&sup2;/R&sup2; the beam will be dominated by self-force expansion; for K much smaller than ε&sup2;/R&sup2; it will be dominated by emittance (and if K &asymp; ε&sup2;/R&sup2; you get a mix of both). </div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>where q is the charge per particle, I is the electric current in the beam, 1 / (4 π ε<sub>0</sub>) = 8.9875517923 &times; 10<sup>9</sup> N m&sup2;/C&sup2; is the Coulomb constant, m is the particle mass, β is the speed as a fraction of the speed of light, γ is the Lorentz factor, c is the speed of light, and R is the radius of the beam. If K is much larger than ε&sup2;/R&sup2; the beam will be dominated by self-force expansion; for K much smaller than ε&sup2;/R&sup2; it will be dominated by emittance (and if K &asymp; ε&sup2;/R&sup2; you get a mix of both). </div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>(Useful values: the charge of an electron is <del style="font-weight: bold; text-decoration: none;"><math></del>q = -1.602176634 <del style="font-weight: bold; text-decoration: none;">\</del>times 10<del style="font-weight: bold; text-decoration: none;">^{</del>-19<del style="font-weight: bold; text-decoration: none;">}</del></<del style="font-weight: bold; text-decoration: none;">math</del>> C and that of the proton is <del style="font-weight: bold; text-decoration: none;"><math></del>q = +1.602176634 <del style="font-weight: bold; text-decoration: none;">\</del>times 10<del style="font-weight: bold; text-decoration: none;">^{</del>-19<del style="font-weight: bold; text-decoration: none;">}</del></<del style="font-weight: bold; text-decoration: none;">math</del>> C. The electron mass <del style="font-weight: bold; text-decoration: none;"><math></del>m = 9.1093837015 <del style="font-weight: bold; text-decoration: none;">\</del>times 10<del style="font-weight: bold; text-decoration: none;">^{</del>-31<del style="font-weight: bold; text-decoration: none;">}</del></<del style="font-weight: bold; text-decoration: none;">math</del>> kg and for a proton <del style="font-weight: bold; text-decoration: none;"><math></del>m = 1.67262192369 <del style="font-weight: bold; text-decoration: none;">\</del>times 10<del style="font-weight: bold; text-decoration: none;">^{</del>-27<del style="font-weight: bold; text-decoration: none;">}</del></<del style="font-weight: bold; text-decoration: none;">math</del>> kg. Ions have a mass of <del style="font-weight: bold; text-decoration: none;"><math></del>m = 1.66053906660 <del style="font-weight: bold; text-decoration: none;">\</del>times 10<del style="font-weight: bold; text-decoration: none;">^{</del>-27<del style="font-weight: bold; text-decoration: none;">}</del></<del style="font-weight: bold; text-decoration: none;">math</del>> kg times their standard atomic weight. The speed of light is <del style="font-weight: bold; text-decoration: none;"><math></del>c = 299792458<del style="font-weight: bold; text-decoration: none;"></math> </del>m/s. If you know the total <del style="font-weight: bold; text-decoration: none;">Power <math></del>P<del style="font-weight: bold; text-decoration: none;"></math> </del>in watts delivered by your beam and you know the energy of each particle <del style="font-weight: bold; text-decoration: none;"><math></del>V<del style="font-weight: bold; text-decoration: none;"></math> </del>in eV, then the beam current in amperes is <del style="font-weight: bold; text-decoration: none;"><math></del>I = P/V<del style="font-weight: bold; text-decoration: none;"></math></del>.)</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>(Useful values: the charge of an electron is q = -1.602176634 <ins style="font-weight: bold; text-decoration: none;">&</ins>times<ins style="font-weight: bold; text-decoration: none;">; </ins>10<ins style="font-weight: bold; text-decoration: none;"><sup></ins>-19</<ins style="font-weight: bold; text-decoration: none;">sup</ins>> C and that of the proton is q = +1.602176634 <ins style="font-weight: bold; text-decoration: none;">&</ins>times<ins style="font-weight: bold; text-decoration: none;">; </ins>10<ins style="font-weight: bold; text-decoration: none;"><sup></ins>-19</<ins style="font-weight: bold; text-decoration: none;">sup</ins>> C. The electron mass m = 9.1093837015 <ins style="font-weight: bold; text-decoration: none;">&</ins>times<ins style="font-weight: bold; text-decoration: none;">; </ins>10<ins style="font-weight: bold; text-decoration: none;"><sup></ins>-31</<ins style="font-weight: bold; text-decoration: none;">sup</ins>> kg and for a proton m = 1.67262192369 <ins style="font-weight: bold; text-decoration: none;">&</ins>times<ins style="font-weight: bold; text-decoration: none;">; </ins>10<ins style="font-weight: bold; text-decoration: none;"><sup></ins>-27</<ins style="font-weight: bold; text-decoration: none;">sup</ins>> kg. Ions have a mass of m = 1.66053906660 <ins style="font-weight: bold; text-decoration: none;">&</ins>times<ins style="font-weight: bold; text-decoration: none;">; </ins>10<ins style="font-weight: bold; text-decoration: none;"><sup></ins>-27</<ins style="font-weight: bold; text-decoration: none;">sup</ins>> kg times their standard atomic weight. The speed of light is c = 299792458 m/s. If you know the total <ins style="font-weight: bold; text-decoration: none;">power </ins>P in watts delivered by your beam and you know the energy of each particle V in eV, then the beam current in amperes is I = P/V.)</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><blockquote></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><blockquote></div></td></tr> </table>

Lwcamp https://www.galacticlibrary.net/mediawiki-1.41.1/index.php?title=Particle_Accelerators&diff=3161&oldid=prev Lwcamp: /* Beam propagation in vacuum with self-forces */ 2025-04-09T00:44:16Z

<p><span dir="auto"><span class="autocomment">Beam propagation in vacuum with self-forces</span></span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:44, 8 April 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l504">Line 504:</td> <td colspan="2" class="diff-lineno">Line 504:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Beam propagation in vacuum with self-forces===</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===Beam propagation in vacuum with self-forces===</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>To quantify the effects of the charge and current self-forces on the beam, you can calculate a number called the <b>perveance</b<del style="font-weight: bold; text-decoration: none;">> <math</del>>K<del style="font-weight: bold; text-decoration: none;"></math></del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>To quantify the effects of the charge and current self-forces on the beam, you can calculate a number called the <b>perveance</b> K</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><div class="center" style="width: auto; margin-left: auto; margin-right: auto;"<del style="font-weight: bold; text-decoration: none;">><math</del>></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><div class="center" style="width: auto; margin-left: auto; margin-right: auto;"></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>K = <del style="font-weight: bold; text-decoration: none;">\frac{</del>2 <del style="font-weight: bold; text-decoration: none;">\,</del>q <del style="font-weight: bold; text-decoration: none;">\, </del>I<del style="font-weight: bold; text-decoration: none;">}{</del>4 <del style="font-weight: bold; text-decoration: none;">\, \pi \, \varepsilon_0 \, </del>m <del style="font-weight: bold; text-decoration: none;">\, </del>(<del style="font-weight: bold; text-decoration: none;">\beta \, \gamma \, </del>c)<del style="font-weight: bold; text-decoration: none;">^</del>3<del style="font-weight: bold; text-decoration: none;">}</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>K = <ins style="font-weight: bold; text-decoration: none;">[</ins>2 q I<ins style="font-weight: bold; text-decoration: none;">] / [</ins>4 <ins style="font-weight: bold; text-decoration: none;">π ε<sub>0</sub> </ins>m (<ins style="font-weight: bold; text-decoration: none;">β γ </ins>c)<ins style="font-weight: bold; text-decoration: none;"><sup></ins>3</<ins style="font-weight: bold; text-decoration: none;">sup</ins>><ins style="font-weight: bold; text-decoration: none;">]</ins></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div></<del style="font-weight: bold; text-decoration: none;">math</del>></div></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div></div></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>where <del style="font-weight: bold; text-decoration: none;"><math></del>q<del style="font-weight: bold; text-decoration: none;"></math> </del>is the charge per particle, <del style="font-weight: bold; text-decoration: none;"><math></del>I<del style="font-weight: bold; text-decoration: none;"></math> </del>is the electric current in the beam, <del style="font-weight: bold; text-decoration: none;"><math></del>1 / (4 <del style="font-weight: bold; text-decoration: none;">\, \pi \, \varepsilon_0</del>) = 8.9875517923 <del style="font-weight: bold; text-decoration: none;">\</del>times 10<del style="font-weight: bold; text-decoration: none;">^{</del>9<del style="font-weight: bold; text-decoration: none;">}</del></<del style="font-weight: bold; text-decoration: none;">math</del>> N m&sup2;/C&sup2; is the Coulomb constant, <del style="font-weight: bold; text-decoration: none;"><math></del>m<del style="font-weight: bold; text-decoration: none;"></math> </del>is the particle mass, <del style="font-weight: bold; text-decoration: none;"><math>\beta</math> </del>is the speed as a fraction of the speed of light, <del style="font-weight: bold; text-decoration: none;"><math>\gamma = 1/\sqrt{1-\beta^2}</math> </del>is the Lorentz factor, <del style="font-weight: bold; text-decoration: none;"><math></del>c<del style="font-weight: bold; text-decoration: none;"></math> </del>is the speed of light, and <del style="font-weight: bold; text-decoration: none;"><math></del>R<del style="font-weight: bold; text-decoration: none;"></math> </del>is the radius of the beam. If <del style="font-weight: bold; text-decoration: none;"><math></del>K<del style="font-weight: bold; text-decoration: none;"></math> </del>is much larger than <del style="font-weight: bold; text-decoration: none;"><math>\epsilon^2</del>/R<del style="font-weight: bold; text-decoration: none;">^2</math> </del>the beam will be dominated by self-force expansion; for <del style="font-weight: bold; text-decoration: none;"><math></del>K<del style="font-weight: bold; text-decoration: none;"></math> </del>much smaller than <del style="font-weight: bold; text-decoration: none;"><math>\epsilon^2</del>/R<del style="font-weight: bold; text-decoration: none;">^2</math> </del>it will be dominated by emittance (and if <del style="font-weight: bold; text-decoration: none;"><math></del>K <del style="font-weight: bold; text-decoration: none;">\approx \epsilon^2</del>/R<del style="font-weight: bold; text-decoration: none;">^2</math> </del>you get a mix of both). </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>where q is the charge per particle, I is the electric current in the beam, 1 / (4 <ins style="font-weight: bold; text-decoration: none;">π ε<sub>0</sub></ins>) = 8.9875517923 <ins style="font-weight: bold; text-decoration: none;">&</ins>times<ins style="font-weight: bold; text-decoration: none;">; </ins>10<ins style="font-weight: bold; text-decoration: none;"><sup></ins>9</<ins style="font-weight: bold; text-decoration: none;">sup</ins>> N m&sup2;/C&sup2; is the Coulomb constant, m is the particle mass, <ins style="font-weight: bold; text-decoration: none;">β </ins>is the speed as a fraction of the speed of light, <ins style="font-weight: bold; text-decoration: none;">γ </ins>is the Lorentz factor, c is the speed of light, and R is the radius of the beam. If K is much larger than <ins style="font-weight: bold; text-decoration: none;">ε&sup2;</ins>/R<ins style="font-weight: bold; text-decoration: none;">&sup2; </ins>the beam will be dominated by self-force expansion; for K much smaller than <ins style="font-weight: bold; text-decoration: none;">ε&sup2;</ins>/R<ins style="font-weight: bold; text-decoration: none;">&sup2; </ins>it will be dominated by emittance (and if K <ins style="font-weight: bold; text-decoration: none;">&asymp; ε&sup2;</ins>/R<ins style="font-weight: bold; text-decoration: none;">&sup2; </ins>you get a mix of both). </div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>(Useful values: the charge of an electron is <math>q = -1.602176634 \times 10^{-19}</math> C and that of the proton is <math>q = +1.602176634 \times 10^{-19}</math> C. The electron mass <math>m = 9.1093837015 \times 10^{-31}</math> kg and for a proton <math>m = 1.67262192369 \times 10^{-27}</math> kg. Ions have a mass of <math>m = 1.66053906660 \times 10^{-27}</math> kg times their standard atomic weight. The speed of light is <math>c = 299792458</math> m/s. If you know the total Power <math>P</math> in watts delivered by your beam and you know the energy of each particle <math>V</math> in eV, then the beam current in amperes is <math>I = P/V</math>.)</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>(Useful values: the charge of an electron is <math>q = -1.602176634 \times 10^{-19}</math> C and that of the proton is <math>q = +1.602176634 \times 10^{-19}</math> C. The electron mass <math>m = 9.1093837015 \times 10^{-31}</math> kg and for a proton <math>m = 1.67262192369 \times 10^{-27}</math> kg. Ions have a mass of <math>m = 1.66053906660 \times 10^{-27}</math> kg times their standard atomic weight. The speed of light is <math>c = 299792458</math> m/s. If you know the total Power <math>P</math> in watts delivered by your beam and you know the energy of each particle <math>V</math> in eV, then the beam current in amperes is <math>I = P/V</math>.)</div></td></tr> </table>

Lwcamp https://www.galacticlibrary.net/mediawiki-1.41.1/index.php?title=Particle_Accelerators&diff=3160&oldid=prev Lwcamp: /* Emittance */ 2025-04-09T00:36:28Z

<p><span dir="auto"><span class="autocomment">Emittance</span></span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:36, 8 April 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l424">Line 424:</td> <td colspan="2" class="diff-lineno">Line 424:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>It is intriguing to note that these beam cooling techniques can take <i>hours</i> to work!<ref name="RHIC emittance">[https://www.bnl.gov/isd/documents/86783.pdf M. Minty, R. Connolly, C. Liu,T. Summers, and S. Tepikian, "Absolute beam emittance measurements at RHIC using ionization profile monitors", Brookhaven National Laboratory formal report BNL-105970-2014-IR]</ref> However, without beam cooling methods like this, the emittance of ions or protons in storage rings can slowly increase on a time scale of hours.<ref>[https://arxiv.org/abs/1204.6022 Vladimir Shiltsev and Alvin Tollestrup. "Emittance growth mechanisms in the Tevatron beams", arXiv:1204.6022 [physics.acc-ph]]</ref></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>It is intriguing to note that these beam cooling techniques can take <i>hours</i> to work!<ref name="RHIC emittance">[https://www.bnl.gov/isd/documents/86783.pdf M. Minty, R. Connolly, C. Liu,T. Summers, and S. Tepikian, "Absolute beam emittance measurements at RHIC using ionization profile monitors", Brookhaven National Laboratory formal report BNL-105970-2014-IR]</ref> However, without beam cooling methods like this, the emittance of ions or protons in storage rings can slowly increase on a time scale of hours.<ref>[https://arxiv.org/abs/1204.6022 Vladimir Shiltsev and Alvin Tollestrup. "Emittance growth mechanisms in the Tevatron beams", arXiv:1204.6022 [physics.acc-ph]]</ref></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>But perhaps the simplest method to reduce emittance is just to speed the particles up. For particles moving at a fraction of light speed <del style="font-weight: bold; text-decoration: none;"><math>\beta</math> </del>and Lorentz factor <del style="font-weight: bold; text-decoration: none;"><math>\gamma</math></del>, the emittance of particles from the same source with the same temperature and same random motion will be proportional to <del style="font-weight: bold; text-decoration: none;"><math></del>1/(<del style="font-weight: bold; text-decoration: none;">\beta \, \gamma</del>)<del style="font-weight: bold; text-decoration: none;"></math></del>. To reflect this, it is convenient to define a normalized emittance <<del style="font-weight: bold; text-decoration: none;">math</del>><del style="font-weight: bold; text-decoration: none;">\varepsilon_n</del></<del style="font-weight: bold; text-decoration: none;">math</del>> such that for the usual geometric emittance <del style="font-weight: bold; text-decoration: none;"><math>\varepsilon</math> </del>that we have been talking about</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>But perhaps the simplest method to reduce emittance is just to speed the particles up. For particles moving at a fraction of light speed <ins style="font-weight: bold; text-decoration: none;">β </ins>and Lorentz factor <ins style="font-weight: bold; text-decoration: none;">γ</ins>, the emittance of particles from the same source with the same temperature and same random motion will be proportional to 1/(<ins style="font-weight: bold; text-decoration: none;">β γ</ins>). To reflect this, it is convenient to define a normalized emittance <ins style="font-weight: bold; text-decoration: none;">ε</ins><<ins style="font-weight: bold; text-decoration: none;">sub</ins>><ins style="font-weight: bold; text-decoration: none;">n</ins></<ins style="font-weight: bold; text-decoration: none;">sub</ins>> such that for the usual geometric emittance <ins style="font-weight: bold; text-decoration: none;">ε </ins>that we have been talking about</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><<del style="font-weight: bold; text-decoration: none;">math</del>></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><div class="center" style="width: auto; margin-left: auto; margin-right: auto;"></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">\varepsilon_n </del>= <del style="font-weight: bold; text-decoration: none;">\beta \, \gamma \, \varepsilon</del>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">ε</ins><<ins style="font-weight: bold; text-decoration: none;">sub>n</sub</ins>> = <ins style="font-weight: bold; text-decoration: none;">β γ ε</ins>.</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></math></del></div></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div></div></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Normalized emittance measures makes it convenient to get basic estimates of particle beam performance and compare emittance of different sources and beams. For example, various high-performing electron beams around the world have managed to push their normalized emittances down to about 2&times;10<sup>-6</sup> m&middot;rad to 3&times;10<sup>-6</sup> m&middot;rad. Facilities with proton or ion beams seem to have about an order of magnitude higher normalized emittance.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Normalized emittance measures makes it convenient to get basic estimates of particle beam performance and compare emittance of different sources and beams. For example, various high-performing electron beams around the world have managed to push their normalized emittances down to about 2&times;10<sup>-6</sup> m&middot;rad to 3&times;10<sup>-6</sup> m&middot;rad. Facilities with proton or ion beams seem to have about an order of magnitude higher normalized emittance.</div></td></tr> <tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l433">Line 433:</td> <td colspan="2" class="diff-lineno">Line 433:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tr><td width=200>Machine <td width=300>normalized emittance (m&middot;rad) <td width=100>particles<td>reference</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tr><td width=200>Machine <td width=300>normalized emittance (m&middot;rad) <td width=100>particles<td>reference</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tr><td>European XFEL <td>2.3&times;10<sup>-6</sup> <td> electrons <td><ref>[https://journals.aps.org/prab/pdf/10.1103/PhysRevAccelBeams.24.110702 T. Hara <i>et al.</i>, "Low-emittance beam injection for a synchrotron radiation source using an X-ray free-electron laser linear accelerator", Physical Review Accelerators and Beams <b>24</b>, 110702 (2021)]</ref>, using reported &epsilon; &thickapprox; 1.5&times; 10<sup>-10</sup> m&middot;rad and electrons at 8 GeV for &beta; &rarr; 1 and &gamma; = 15700.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tr><td>European XFEL <td>2.3&times;10<sup>-6</sup> <td> electrons <td><ref>[https://journals.aps.org/prab/pdf/10.1103/PhysRevAccelBeams.24.110702 T. Hara <i>et al.</i>, "Low-emittance beam injection for a synchrotron radiation source using an X-ray free-electron laser linear accelerator", Physical Review Accelerators and Beams <b>24</b>, 110702 (2021)]</ref>, using reported &epsilon; &thickapprox; 1.5&times; 10<sup>-10</sup> m&middot;rad and electrons at 8 GeV for &beta; &rarr; 1 and &gamma; = 15700.</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><tr><td>DESY PETRA III<td>0.93&times;10<sup>-6</sup> <td> electrons <td><ref>[https://www.desy.de/news/news_search/index_eng.html?openDirectAnchor=444&two_columns=0 New emittance world record at PETRA III]</ref>, using reported &epsilon; = 1.6&times;10<sup>-10</sup> m&middot;rad and electrons at 3 GeV for &beta; &rarr; 1<del style="font-weight: bold; text-decoration: none;"></math> </del>and &gamma; = 5870.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><tr><td>DESY PETRA III<td>0.93&times;10<sup>-6</sup> <td> electrons <td><ref>[https://www.desy.de/news/news_search/index_eng.html?openDirectAnchor=444&two_columns=0 New emittance world record at PETRA III]</ref>, using reported &epsilon; = 1.6&times;10<sup>-10</sup> m&middot;rad and electrons at 3 GeV for &beta; &rarr; 1 and &gamma; = 5870.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tr><td>SLAC <td>3&times;10<sup>-6</sup> <td> electrons <td><ref>[https://accelconf.web.cern.ch/pac97/papers/pdf/4W009.PDF M. Hernandez <i>et al.</i>, "Emittance Measurements for the SLAC Gun Test Facility", 1998 IEEE]</ref></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tr><td>SLAC <td>3&times;10<sup>-6</sup> <td> electrons <td><ref>[https://accelconf.web.cern.ch/pac97/papers/pdf/4W009.PDF M. Hernandez <i>et al.</i>, "Emittance Measurements for the SLAC Gun Test Facility", 1998 IEEE]</ref></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tr><td>LHC <td>3.75&times;10<sup>-5</sup> <td> protons <td><ref>[https://www.lhc-closer.es/taking_a_closer_look_at_lhc/0.beta___emittance Beta & Emittance; Taking a closer look at LHC]</ref></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tr><td>LHC <td>3.75&times;10<sup>-5</sup> <td> protons <td><ref>[https://www.lhc-closer.es/taking_a_closer_look_at_lhc/0.beta___emittance Beta & Emittance; Taking a closer look at LHC]</ref></div></td></tr> </table>

Lwcamp https://www.galacticlibrary.net/mediawiki-1.41.1/index.php?title=Particle_Accelerators&diff=3159&oldid=prev Lwcamp: /* Emittance */ 2025-04-09T00:33:42Z

<p><span dir="auto"><span class="autocomment">Emittance</span></span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:33, 8 April 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l392">Line 392:</td> <td colspan="2" class="diff-lineno">Line 392:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Or, at least we could if all the particles came from a point, or if they traveled on perfectly parallel lines. Unfortunately, they don't. When particles are made, they come from a source with a finite spot size and a finite temperature. The random motion of the particles moving around from their initial thermal motion means that they are never moving perfectly parallel to each other. And when you try to focus them, instead of all converging onto a single point they make a spot of finite size. And if your focusing and beam bending elements are not perfect, you add additional imperfections to the beam that further hinder your ability to focus it.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Or, at least we could if all the particles came from a point, or if they traveled on perfectly parallel lines. Unfortunately, they don't. When particles are made, they come from a source with a finite spot size and a finite temperature. The random motion of the particles moving around from their initial thermal motion means that they are never moving perfectly parallel to each other. And when you try to focus them, instead of all converging onto a single point they make a spot of finite size. And if your focusing and beam bending elements are not perfect, you add additional imperfections to the beam that further hinder your ability to focus it.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Your ability to focus can be quantified by a measure called <b>emittance</b>. Emittance is measured in units of a length scale times an angle - nominally meters-radians (m&middot;rad) but the technical literature more commonly gives units of millimeters-milliradians (or mm&middot;mrad) or even 2 &pi; mm&middot;mrad. The divergence angle of a beam that you try to make parallel will be the the emittance divided by the width of the beam. If you focus a beam down so it converges at an angle, the minimum spot size you can get at the focal point is the emittance divided by the beam angle. If you try to focus a beam with emittance <del style="font-weight: bold; text-decoration: none;"><math>\varepsilon</math> </del>on a spot a distance <del style="font-weight: bold; text-decoration: none;"><math></del>R<del style="font-weight: bold; text-decoration: none;"></math> </del>away with an initial beam diameter of <del style="font-weight: bold; text-decoration: none;"><math></del>D<del style="font-weight: bold; text-decoration: none;"></math></del>, the spot size <del style="font-weight: bold; text-decoration: none;"><math></del>S<del style="font-weight: bold; text-decoration: none;"></math> </del>on the target you can achieve is </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Your ability to focus can be quantified by a measure called <b>emittance</b>. Emittance is measured in units of a length scale times an angle - nominally meters-radians (m&middot;rad) but the technical literature more commonly gives units of millimeters-milliradians (or mm&middot;mrad) or even 2 &pi; mm&middot;mrad. The divergence angle of a beam that you try to make parallel will be the the emittance divided by the width of the beam. If you focus a beam down so it converges at an angle, the minimum spot size you can get at the focal point is the emittance divided by the beam angle. If you try to focus a beam with emittance <ins style="font-weight: bold; text-decoration: none;">ε </ins>on a spot a distance R away with an initial beam diameter of D, the spot size S on the target you can achieve is </div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><div class="center" style="width: auto; margin-left: auto; margin-right: auto;"<del style="font-weight: bold; text-decoration: none;">><math</del>></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><div class="center" style="width: auto; margin-left: auto; margin-right: auto;"></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>S = R <del style="font-weight: bold; text-decoration: none;">\frac{\varepsilon}{</del>D<del style="font-weight: bold; text-decoration: none;">}</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>S = R <ins style="font-weight: bold; text-decoration: none;">ε / </ins>D</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></math></del></div></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div></div></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>It is interesting to note that this is very nearly the same spot size you get from a laser due to [[Diffraction#Spot_Size|diffraction]] with a wavelength equal to the particle beam's emissivity. And, like lasers, the more you can expand the beam to large diameters at the focusing equipment, the tighter the spot you can achieve on the target.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>It is interesting to note that this is very nearly the same spot size you get from a laser due to [[Diffraction#Spot_Size|diffraction]] with a wavelength equal to the particle beam's emissivity. And, like lasers, the more you can expand the beam to large diameters at the focusing equipment, the tighter the spot you can achieve on the target.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l404">Line 404:</td> <td colspan="2" class="diff-lineno">Line 404:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>There are a lot of more complicated mathematics you can do to find how this works. But one result is that with perfect lenses and other beam transport equipment, the emittance is always conserved, so that the product of the uncertainty in transverse position and uncertainty in transverse speed (which is all that normalized emittance is, in a sense) remains the same.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>There are a lot of more complicated mathematics you can do to find how this works. But one result is that with perfect lenses and other beam transport equipment, the emittance is always conserved, so that the product of the uncertainty in transverse position and uncertainty in transverse speed (which is all that normalized emittance is, in a sense) remains the same.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tr><td></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><tr><td></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>We can blithely wave our hands and say that emittance is somehow related to the random motions of the beam particles and thus the beam temperature, and that cooling the beam will reduce the emittance, but what is the actual relation between beam temperature and emittance? Fortunately reference <ref>[http://uspas.fnal.gov/materials/08UCSC/Lecture%202%20Slides_Emission%20and%20cathode%20emittance.pdf D.H. Dowell, S. Lidia, J.F. Schmerge, "Lecture 2: Electron Emission and Cathode Emittance", High Brightness Electron Injectors for Light Sources - January 14-18 2007]</ref> can give us an answer. Interested readers can follow their derivation, we'll just give the result here. For a particle source of diameter <del style="font-weight: bold; text-decoration: none;"><math></del>d<del style="font-weight: bold; text-decoration: none;"></math> </del>and temperature <del style="font-weight: bold; text-decoration: none;"><math></del>T<del style="font-weight: bold; text-decoration: none;"></math></del>, emitting particles of mass <del style="font-weight: bold; text-decoration: none;"><math></del>m<del style="font-weight: bold; text-decoration: none;"></math></del>, the normalized emittance is </div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>We can blithely wave our hands and say that emittance is somehow related to the random motions of the beam particles and thus the beam temperature, and that cooling the beam will reduce the emittance, but what is the actual relation between beam temperature and emittance? Fortunately reference <ref>[http://uspas.fnal.gov/materials/08UCSC/Lecture%202%20Slides_Emission%20and%20cathode%20emittance.pdf D.H. Dowell, S. Lidia, J.F. Schmerge, "Lecture 2: Electron Emission and Cathode Emittance", High Brightness Electron Injectors for Light Sources - January 14-18 2007]</ref> can give us an answer. Interested readers can follow their derivation, we'll just give the result here. For a particle source of diameter d and temperature T, emitting particles of mass m, the normalized emittance is </div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><div class="center" style="width: auto; margin-left: auto; margin-right: auto;"><<del style="font-weight: bold; text-decoration: none;">math</del>></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><div class="center" style="width: auto; margin-left: auto; margin-right: auto;"></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;">\varepsilon_n </del>= <del style="font-weight: bold; text-decoration: none;">\frac{</del>d<del style="font-weight: bold; text-decoration: none;">}{</del>4<del style="font-weight: bold; text-decoration: none;">} \, \sqrt{ \frac{k_B \, </del>T<del style="font-weight: bold; text-decoration: none;">}{</del>m <del style="font-weight: bold; text-decoration: none;">\, </del>c<del style="font-weight: bold; text-decoration: none;">^2} }</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">ε</ins><<ins style="font-weight: bold; text-decoration: none;">sub>n</sub</ins>> = <ins style="font-weight: bold; text-decoration: none;">(</ins>d<ins style="font-weight: bold; text-decoration: none;">/</ins>4<ins style="font-weight: bold; text-decoration: none;">) &radic;[ k<sub>B</sub> </ins>T <ins style="font-weight: bold; text-decoration: none;">/ (</ins>m c<ins style="font-weight: bold; text-decoration: none;">&sup2;) ]</ins></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></math></del></div></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div></div></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>where <<del style="font-weight: bold; text-decoration: none;">math</del>><del style="font-weight: bold; text-decoration: none;">k_B </del>= 1.380649 <del style="font-weight: bold; text-decoration: none;">\</del>times 10<del style="font-weight: bold; text-decoration: none;">^{</del>-23<del style="font-weight: bold; text-decoration: none;">}</del></<del style="font-weight: bold; text-decoration: none;">math</del>> J/K is the Boltzmann constant and <del style="font-weight: bold; text-decoration: none;"><math></del>c = 299792458<del style="font-weight: bold; text-decoration: none;"></math> </del>m/s is the speed of light.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>where <ins style="font-weight: bold; text-decoration: none;">k</ins><<ins style="font-weight: bold; text-decoration: none;">sub>B</sub</ins>> = 1.380649 <ins style="font-weight: bold; text-decoration: none;">&</ins>times<ins style="font-weight: bold; text-decoration: none;">; </ins>10<ins style="font-weight: bold; text-decoration: none;"><sup></ins>-23</<ins style="font-weight: bold; text-decoration: none;">sup</ins>> J/K is the Boltzmann constant and c = 299792458 m/s is the speed of light.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Similarly, if you have a beam whose diameter is <del style="font-weight: bold; text-decoration: none;"><math></del>d<del style="font-weight: bold; text-decoration: none;"></math> </del>at some given point, and you know its normalized emittance, the closest approximation to temperature you can get at that point is</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Similarly, if you have a beam whose diameter is d at some given point, and you know its normalized emittance, the closest approximation to temperature you can get at that point is</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><div class="center" style="width: auto; margin-left: auto; margin-right: auto;"<del style="font-weight: bold; text-decoration: none;">><math</del>></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><div class="center" style="width: auto; margin-left: auto; margin-right: auto;"></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>T = <del style="font-weight: bold; text-decoration: none;">\frac{</del>m <del style="font-weight: bold; text-decoration: none;">\, </del>c<del style="font-weight: bold; text-decoration: none;">^2}{k_B} \left</del>[ <del style="font-weight: bold; text-decoration: none;">\frac{</del>4 <del style="font-weight: bold; text-decoration: none;">\, \varepsilon_n}{</del>d<del style="font-weight: bold; text-decoration: none;">} \right</del>]<del style="font-weight: bold; text-decoration: none;">^2</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>T = <ins style="font-weight: bold; text-decoration: none;">(</ins>m c<ins style="font-weight: bold; text-decoration: none;">&sup2; / k<sub>B</sub>) </ins>[ 4 <ins style="font-weight: bold; text-decoration: none;">ε<sub>n</sub> / </ins>d ]<ins style="font-weight: bold; text-decoration: none;">&sup2;</ins></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></math></del></div></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div></div></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></table></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></table></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table>

Lwcamp https://www.galacticlibrary.net/mediawiki-1.41.1/index.php?title=Particle_Accelerators&diff=3158&oldid=prev Lwcamp: /* Synchrotron radiation */ 2025-04-09T00:06:53Z

<p><span dir="auto"><span class="autocomment">Synchrotron radiation</span></span></p> <table style="background-color: #fff; color: #202122;" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:06, 8 April 2025</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l322">Line 322:</td> <td colspan="2" class="diff-lineno">Line 322:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>δE(MeV) = 8.85 &times; 10<sup>-2</sup> [E (GeV)<sup>4</sup> / (r (m))] &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; energy loss for a full revolution, electrons, β &rarr; 1.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>δE(MeV) = 8.85 &times; 10<sup>-2</sup> [E (GeV)<sup>4</sup> / (r (m))] &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; energy loss for a full revolution, electrons, β &rarr; 1.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></div></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></div></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>If your particle only gets deflected a bit and does not go a full revolution, but is only deflected by an angle <del style="font-weight: bold; text-decoration: none;"><math>\theta</math></del>, multiply the energy loss per revolution by <del style="font-weight: bold; text-decoration: none;"><math>\theta</del>/<del style="font-weight: bold; text-decoration: none;">360^{\circ}</math> </del>(degrees) or <del style="font-weight: bold; text-decoration: none;"><math>\theta</del>/(2 <del style="font-weight: bold; text-decoration: none;">\pi</del>)<del style="font-weight: bold; text-decoration: none;"></math> </del>(radians) to find the energy loss for being bent by that amount.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>If your particle only gets deflected a bit and does not go a full revolution, but is only deflected by an angle <ins style="font-weight: bold; text-decoration: none;">θ</ins>, multiply the energy loss per revolution by <ins style="font-weight: bold; text-decoration: none;">θ</ins>/<ins style="font-weight: bold; text-decoration: none;">360° </ins>(degrees) or <ins style="font-weight: bold; text-decoration: none;">θ</ins>/(2 <ins style="font-weight: bold; text-decoration: none;">π</ins>) (radians) to find the energy loss for being bent by that amount.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>If your don't know the radius of revolution <del style="font-weight: bold; text-decoration: none;"><math></del>r<del style="font-weight: bold; text-decoration: none;"></math></del>, but you do know the angle of deflection <del style="font-weight: bold; text-decoration: none;"><math>\theta</math> </del>and the distance over which the deflection happens <del style="font-weight: bold; text-decoration: none;"><math></del>d<del style="font-weight: bold; text-decoration: none;"></math></del>, then</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>If your don't know the radius of revolution r, but you do know the angle of deflection <ins style="font-weight: bold; text-decoration: none;">θ </ins>and the distance over which the deflection happens d, then</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><div class="center" style="width: auto; margin-left: auto; margin-right: auto;"<del style="font-weight: bold; text-decoration: none;">><math</del>></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><div class="center" style="width: auto; margin-left: auto; margin-right: auto;"></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>r = <del style="font-weight: bold; text-decoration: none;">\frac{</del>d<del style="font-weight: bold; text-decoration: none;">}{\</del>sin (<del style="font-weight: bold; text-decoration: none;">\theta</del>)<del style="font-weight: bold; text-decoration: none;">}</del>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>r = d <ins style="font-weight: bold; text-decoration: none;">/ </ins>sin(<ins style="font-weight: bold; text-decoration: none;">θ</ins>).</div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"></math></del></div></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div></div></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>For small <del style="font-weight: bold; text-decoration: none;"><math>\theta</math> </del>measured in radians, this simplifies to <del style="font-weight: bold; text-decoration: none;"><math></del>r = d/<del style="font-weight: bold; text-decoration: none;">\theta</math></del>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>For small <ins style="font-weight: bold; text-decoration: none;">θ </ins>measured in radians, this simplifies to r = d / <ins style="font-weight: bold; text-decoration: none;">θ</ins>.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><blockquote></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><blockquote></div></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><i>Example</i>: We have a 50 GeV electron beam coming out of a plasma accelerator with a beam radius of 0.1 mm. We want to expand the beam radius out to 10 cm so we can focus it better on a distant target. We will use one plasma mirror where the beam comes out to expand it, and a second plasma mirror 10 m away to finish the focusing. In that 10 m, the beam has to drift 5 cm in each direction before it hits the second lens, so that is an angle of approximately <del style="font-weight: bold; text-decoration: none;"><math></del>0.05/10 = 0.005<del style="font-weight: bold; text-decoration: none;"></math> </del>radians in the small angle limit. If the plasma lenses are 1 meter long each, then plugging in <del style="font-weight: bold; text-decoration: none;"><math></del>d = 1<del style="font-weight: bold; text-decoration: none;"></math> </del>m into the formula above, we get a radius of curvature of <del style="font-weight: bold; text-decoration: none;"><math></del>r = 200<del style="font-weight: bold; text-decoration: none;"></math> </del>m. Putting this in to our energy loss formula, we get an energy loss per revolution of approximately 2770 MeV. We only go through a fraction <del style="font-weight: bold; text-decoration: none;"><math></del>0.005/(2 <del style="font-weight: bold; text-decoration: none;">\pi</del>)<del style="font-weight: bold; text-decoration: none;"></math> </del>of a revolution, though, so our energy loss is 2.2 MeV. But we then lose another 2.2 MeV as our diverging beam is re-focused at the second plasma lens 10 m away. So this focusing procedure loses roughly 5 MeV out of 50 GeV, or 0.01% of the beam energy, which is deemed by your engineering supervisor to be an acceptable loss.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><i>Example</i>: We have a 50 GeV electron beam coming out of a plasma accelerator with a beam radius of 0.1 mm. We want to expand the beam radius out to 10 cm so we can focus it better on a distant target. We will use one plasma mirror where the beam comes out to expand it, and a second plasma mirror 10 m away to finish the focusing. In that 10 m, the beam has to drift 5 cm in each direction before it hits the second lens, so that is an angle of approximately 0.05/10 = 0.005 radians in the small angle limit. If the plasma lenses are 1 meter long each, then plugging in d = 1 m into the formula above, we get a radius of curvature of r = 200 m. Putting this in to our energy loss formula, we get an energy loss per revolution of approximately 2770 MeV. We only go through a fraction 0.005/(2 <ins style="font-weight: bold; text-decoration: none;">π</ins>) of a revolution, though, so our energy loss is 2.2 MeV. But we then lose another 2.2 MeV as our diverging beam is re-focused at the second plasma lens 10 m away. So this focusing procedure loses roughly 5 MeV out of 50 GeV, or 0.01% of the beam energy, which is deemed by your engineering supervisor to be an acceptable loss.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></blockquote></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></blockquote></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l337">Line 337:</td> <td colspan="2" class="diff-lineno">Line 337:</td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><i>Example</i>: You are part of the planetary defense force in low orbit around Earth. Your space warcraft are equipped with the electron beams in the above example. An invading alien force is attacking! You aim your electron beam at an alien spacecraft 100 km away and begin zapping it.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><i>Example</i>: You are part of the planetary defense force in low orbit around Earth. Your space warcraft are equipped with the electron beams in the above example. An invading alien force is attacking! You aim your electron beam at an alien spacecraft 100 km away and begin zapping it.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> <tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The magnetic field around Earth is approximately 0.0001 tesla. For the shot you need to take, your beam will be moving perpendicular to the magnetic field. At 50 GeV, your electrons have a <del style="font-weight: bold; text-decoration: none;"><math>\gamma</math> </del>of nearly 100,000. With <<del style="font-weight: bold; text-decoration: none;">math</del>><del style="font-weight: bold; text-decoration: none;">v_\</del>perp = 300 000<del style="font-weight: bold; text-decoration: none;"></math> </del>m/s (so close to the speed of light as to make no difference), an electron mass of <del style="font-weight: bold; text-decoration: none;"><math></del>9.11 <del style="font-weight: bold; text-decoration: none;">\</del>times 10<del style="font-weight: bold; text-decoration: none;">^{</del>-31<del style="font-weight: bold; text-decoration: none;">}</del></<del style="font-weight: bold; text-decoration: none;">math</del>> kg, and an electron charge of <del style="font-weight: bold; text-decoration: none;"><math></del>1.6 <del style="font-weight: bold; text-decoration: none;">\</del>times 10<del style="font-weight: bold; text-decoration: none;">^{</del>-19<del style="font-weight: bold; text-decoration: none;">}</del></<del style="font-weight: bold; text-decoration: none;">math</del>> C the gyroradius of your beam in the Earth's field is 1,700 km. The angle of deflection of your beam will be very close to <del style="font-weight: bold; text-decoration: none;"><math>\theta </del>= 100 <del style="font-weight: bold; text-decoration: none;">\mbox{</del>km<del style="font-weight: bold; text-decoration: none;">} </del>/ 1700 <del style="font-weight: bold; text-decoration: none;">\mbox{</del>km<del style="font-weight: bold; text-decoration: none;">} </del>= 0.059<del style="font-weight: bold; text-decoration: none;"></math> </del>radians, so the fraction of a full revolution your electrons will take en route to the target is 0.0094. The energy loss for a full revolution at 1700 km gyroradius and 50 GeV energy is 0.33 MeV; because your beam only goes through 0.0094 of a revolution your beam only loses 0.003 MeV on its way to deliver hot radioactive death to your enemy.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The magnetic field around Earth is approximately 0.0001 tesla. For the shot you need to take, your beam will be moving perpendicular to the magnetic field. At 50 GeV, your electrons have a <ins style="font-weight: bold; text-decoration: none;">γ </ins>of nearly 100,000. With <ins style="font-weight: bold; text-decoration: none;">v</ins><<ins style="font-weight: bold; text-decoration: none;">sub</ins>><ins style="font-weight: bold; text-decoration: none;">&</ins>perp<ins style="font-weight: bold; text-decoration: none;">;</sub> </ins>= 300 000 m/s (so close to the speed of light as to make no difference), an electron mass of 9.11 <ins style="font-weight: bold; text-decoration: none;">&</ins>times<ins style="font-weight: bold; text-decoration: none;">; </ins>10<ins style="font-weight: bold; text-decoration: none;"><sup></ins>-31</<ins style="font-weight: bold; text-decoration: none;">sup</ins>> kg, and an electron charge of 1.6 <ins style="font-weight: bold; text-decoration: none;">&</ins>times<ins style="font-weight: bold; text-decoration: none;">; </ins>10<ins style="font-weight: bold; text-decoration: none;"><sup></ins>-19</<ins style="font-weight: bold; text-decoration: none;">sup</ins>> C the gyroradius of your beam in the Earth's field is 1,700 km. The angle of deflection of your beam will be very close to <ins style="font-weight: bold; text-decoration: none;">θ </ins>= 100 km / 1700 km = 0.059 radians, so the fraction of a full revolution your electrons will take en route to the target is 0.0094. The energy loss for a full revolution at 1700 km gyroradius and 50 GeV energy is 0.33 MeV; because your beam only goes through 0.0094 of a revolution your beam only loses 0.003 MeV on its way to deliver hot radioactive death to your enemy.</div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></blockquote></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div></blockquote></div></td></tr> <tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr> </table>

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