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Intrabeam scattering
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<h2 id="bjorkenmtingwa-formulation">Bjorken–Mtingwa formulation</h2> <p>The betatron growth rates for intrabeam scattering are defined as, </p> 1 T p = d e f 1 σ p d σ p d t {\displaystyle {\frac {1}{T_{p}}}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {1}{\sigma _{p}}}{\frac {d\sigma _{p}}{dt}}} , 1 T h = d e f 1 ϵ h 1 / 2 d ϵ h 1 / 2 d t {\displaystyle {\frac {1}{T_{h}}}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {1}{\epsilon _{h}^{1/2}}}{\frac {d\epsilon _{h}^{1/2}}{dt}}} , 1 T v = d e f 1 ϵ v 1 / 2 d ϵ v 1 / 2 d t {\displaystyle {\frac {1}{T_{v}}}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {1}{\epsilon _{v}^{1/2}}}{\frac {d\epsilon _{v}^{1/2}}{dt}}} . <p>The following is general to all bunched beams, </p> 1 T i = 4 π A ( log ) ⟨ ∫ 0 ∞ d λ λ 1 / 2 [ det ( L + λ I ) ] 1 / 2 { Tr L i Tr ( 1 L + λ I ) − 3 Tr [ L i ( 1 L + λ I ) ] } ⟩ {\displaystyle {\frac {1}{T_{i}}}=4\pi A(\operatorname {log} )\left\langle \int _{0}^{\infty }\,d\lambda \ {\frac {\lambda ^{1/2}}{[\operatorname {det} (L+\lambda I)]^{1/2}}}\left\{\operatorname {Tr} L^{i}\operatorname {Tr} \left({\frac {1}{L+\lambda I}}\right)-3\operatorname {Tr} \left[L^{i}\left({\frac {1}{L+\lambda I}}\right)\right]\right\}\right\rangle } , <p>where T p {\displaystyle T_{p}} , T h {\displaystyle T_{h}} , and T v {\displaystyle T_{v}} are the momentum spread, horizontal, and vertical are the betatron growth times. The angle brackets <...> indicate that the integral is averaged around the ring. </p> ( log ) = ln b m i n b m a x = ln 2 θ m i n {\displaystyle (\operatorname {log} )=\ln {\frac {b_{min}}{b_{max}}}=\ln {\frac {2}{\theta _{min}}}} A = r 0 2 c N 64 π 2 β 3 γ 4 ϵ h ϵ v σ s σ p {\displaystyle A={\frac {r_{0}^{2}cN}{64\pi ^{2}\beta ^{3}\gamma ^{4}\epsilon _{h}\epsilon _{v}\sigma _{s}\sigma _{p}}}} L = L ( p ) + L ( h ) + L ( v ) {\displaystyle L=L^{(p)}+L^{(h)}+L^{(v)}\,} L ( p ) = γ 2 σ p 2 ( 0 0 0 0 1 0 0 0 0 ) {\displaystyle L^{(p)}={\frac {\gamma ^{2}}{\sigma _{p}^{2}}}{\begin{pmatrix}0&0&0\\0&1&0\\0&0&0\end{pmatrix}}} L ( h ) = β h ϵ h ( 1 − γ ϕ h 0 − γ ϕ h γ 2 H h β h 0 0 0 0 ) {\displaystyle L^{(h)}={\frac {\beta _{h}}{\epsilon _{h}}}{\begin{pmatrix}1&-\gamma \phi _{h}&0\\-\gamma \phi _{h}&{\frac {\gamma ^{2}{\mathcal {H}}_{h}}{\beta _{h}}}&0\\0&0&0\end{pmatrix}}} L ( v ) = β v ϵ v ( 0 0 0 0 γ 2 H v β v − γ ϕ v 0 − γ ϕ v 1 ) {\displaystyle L^{(v)}={\frac {\beta _{v}}{\epsilon _{v}}}{\begin{pmatrix}0&0&0\\0&{\frac {\gamma ^{2}{\mathcal {H}}_{v}}{\beta _{v}}}&-\gamma \phi _{v}\\0&-\gamma \phi _{v}&1\end{pmatrix}}} H h , v = [ η h , v 2 + ( β h , v η h , v ′ − 1 2 β h , v ′ η h ) 2 ] / β h , v {\displaystyle {\mathcal {H}}_{h,v}=[\eta _{h,v}^{2}+(\beta _{h,v}\eta '_{h,v}-{\frac {1}{2}}\beta '_{h,v}\eta _{h})^{2}]/\beta _{h,v}} ϕ h , v = η h , v ′ − 1 2 β h , v ′ η h , v / β h , v {\displaystyle \phi _{h,v}=\eta '_{h,v}-{\frac {1}{2}}\beta '_{h,v}\eta _{h,v}/\beta _{h,v}} <p>Definitions: </p> r 0 2 {\displaystyle r_{0}^{2}} is the classical radius of the particle c {\displaystyle c} is the speed of light N {\displaystyle N} is the number of particles per bunch β {\displaystyle \beta } is velocity divided by the speed of light γ {\displaystyle \gamma } is energy divided by mass β h , v {\displaystyle \beta _{h,v}} and β h , v ′ {\displaystyle \beta '_{h,v}} is the betatron function and its derivative, respectively η h , v {\displaystyle \eta _{h,v}} and η h , v ′ {\displaystyle \eta '_{h,v}} is the dispersion function and its derivative, respectively ϵ h , v {\displaystyle \epsilon _{h,v}} is the emittance σ s {\displaystyle \sigma _{s}} is the bunch length σ p {\displaystyle \sigma _{p}} is the momentum spread b m i n {\displaystyle b_{min}} and b m a x {\displaystyle b_{max}} are the minimum and maximum impact parameters. The minimum impact parameter is the closest distance of approach between two particles in a collision. The maximum impact parameter is the largest distance between two particles such that their trajectories are unaltered by the collision. The maximum impact parameter should be taken to be the minimum beam size. See <a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> for some analysis of the Coulomb log and support for this result. θ m i n {\displaystyle \theta _{min}} is the minimum scattering angle. <h2 id="equilibrium-and-growth-rate-sum-rule">Equilibrium and growth rate sum rule</h2> <p>IBS can be seen as a process in which the different "temperatures" try to equilibrate. The growth rates would be zero in the case that </p> <ul><li> σ δ γ = σ x ′ = σ y ′ {\displaystyle {\frac {\sigma _{\delta }}{\gamma }}=\sigma _{x'}=\sigma _{y'}} </li></ul> <p>which the factor of γ {\displaystyle \gamma } coming from the Lorentz transformation. From this equation, we see that due to the factor of γ {\displaystyle \gamma } , the longitudinal is typically much "colder" than the transverse. Thus, we typically get growth in the longitudinal, and shrinking in the transverse. </p><p>One may also the express conservation of energy in IBS in terms of the Piwinski invariant </p> <ul><li> ϵ x β x + ϵ y β y + η s ϵ z β z {\displaystyle {\frac {\epsilon _{x}}{\beta _{x}}}+{\frac {\epsilon _{y}}{\beta _{y}}}+\eta _{s}{\frac {\epsilon _{z}}{\beta _{z}}}} </li></ul> <p>where η s = 1 γ 2 − α c {\displaystyle \eta _{s}={\frac {1}{\gamma ^{2}}}-\alpha _{c}} . Above transition, with just IBS, this implies that there is no equilibrium. However, for the case of radiation damping and diffusion, there is certainly an equilibrium. The effect of IBS is to cause a change in the equilibrium values of the emittances. </p> <h2 id="inclusion-of-coupling">Inclusion of coupling</h2> <p>In the case of a coupled beam, one must consider the evolution of the coupled eigenemittances. The growth rates are generalized to 1 τ 1 , 2 , 3 = 1 ϵ 1 , 2 , 3 d ϵ 1 , 2 , 3 d t {\displaystyle {\frac {1}{\tau _{1,2,3}}}={\frac {1}{\epsilon _{1,2,3}}}{\frac {d\epsilon _{1,2,3}}{dt}}} </p> <h2 id="measurement-and-comparison-with-theory">Measurement and comparison with Theory</h2> <p>Intrabeam scattering is an important effect in the proposed "ultimate storage ring" light sources and lepton damping rings for International Linear Collider (ILC) and Compact Linear Collider (CLIC). Experimental studies aimed at understanding intrabeam scattering in beams similar to those used in these types of machines have been conducted at KEK,<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> CesrTA,<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> and elsewhere. </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>A. Piwinski, in Proceedings of the 9th International Conference on High Energy Accelerators, Stanford, CA, 1974 (SLAC, Stanford, 1974), p. 405 <a href="http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-r-839-c.pdf" target="_blank">http://www.slac.stanford.edu/cgi-wrap/getdoc/slac-r-839-c.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>J. Bjorken and S. Mtingwa, Part. Accel. 13, 115 (1983) https://s3.cern.ch/inspire-prod-files-a/a7d86ec1529ba6512d446523cd88c2d5 <a href="https://s3.cern.ch/inspire-prod-files-a/a7d86ec1529ba6512d446523cd88c2d5" target="_blank">https://s3.cern.ch/inspire-prod-files-a/a7d86ec1529ba6512d446523cd88c2d5</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>K. Kubo et al., Phys. Rev. ST Accel. Beams 8, 081001 (2005) https://journals.aps.org/prab/abstract/10.1103/PhysRevSTAB.8.081001 <a href="https://journals.aps.org/prab/abstract/10.1103/PhysRevSTAB.8.081001" target="_blank">https://journals.aps.org/prab/abstract/10.1103/PhysRevSTAB.8.081001</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>B. Nash et al., "A New analysis of intrabeam scattering", Conf.Proc. C030512 (2003) 126, http://inspirehep.net/record/623294 <a href="http://inspirehep.net/record/623294" target="_blank">http://inspirehep.net/record/623294</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>"SLAC-R-820 -- Analytical Approach to Eigen-Emittance Evolution in Storage Rings". Archived from the original on 3 April 2013. Retrieved 20 February 2013. <a href="https://archive.today/20130403115006/http://www.slac.stanford.edu/pubs/slacreports/slac-r-820.html" target="_blank">https://archive.today/20130403115006/http://www.slac.stanford.edu/pubs/slacreports/slac-r-820.html</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>K. L. F. Bane, H. Hayano, K. Kubo, T. Naito, T. Okugi, and J. Urakawa, Phys. Rev. ST Accel. Beams 5, 084403 (2002). http://prst-ab.aps.org/abstract/PRSTAB/v5/i8/e084403 Archived 20 May 2009 at the Wayback Machine <a href="http://prst-ab.aps.org/abstract/PRSTAB/v5/i8/e084403" target="_blank">http://prst-ab.aps.org/abstract/PRSTAB/v5/i8/e084403</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>M. P. Ehrlichman, et al., Phys. Rev. ST Accel. Beams 16, 104401 (2013). http://prst-ab.aps.org/abstract/PRSTAB/v16/i10/e104401 <a href="http://prst-ab.aps.org/abstract/PRSTAB/v16/i10/e104401" target="_blank">http://prst-ab.aps.org/abstract/PRSTAB/v16/i10/e104401</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> </ol>