Intrabeam scattering

The two principal methods for calculating the effects of intrabeam scattering were done by Anton Piwinski in 1974 and James Bjorken and Sekazi Mtingwa in 1983. The Bjorken-Mtingwa formulation is regarded as being the most general solution. Both of these methods are computationally intensive. Several approximations of these methods have been done that are easier to evaluate, but less general. These approximations are summarized in Intrabeam scattering formulas for high energy beams by K. Kubo et al.

Intrabeam scattering rates have a ${\displaystyle 1/{\gamma }^4}$ dependence. This means that its effects diminish with increasing beam energy. Other ways of mitigating IBS effects are the use of wigglers, and reducing beam intensity. Transverse intrabeam scattering rates are sensitive to dispersion.

Intrabeam scattering is closely related to the Touschek effect. The Touschek effect is a lifetime based on intrabeam collisions that result in both particles being ejected from the beam. Intrabeam scattering is a risetime based on intrabeam collisions that result in momentum coupling.

Bjorken–Mtingwa formulation

The betatron growth rates for intrabeam scattering are defined as,

${\displaystyle \frac{1}{T_p}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{1}{{\sigma }_p}\frac{d{\sigma }_p}{dt}}$,

${\displaystyle \frac{1}{T_h}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{1}{{ϵ}_h^{1/2}}\frac{d{ϵ}_h^{1/2}}{dt}}$,

${\displaystyle \frac{1}{T_v}\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{1}{{ϵ}_v^{1/2}}\frac{d{ϵ}_v^{1/2}}{dt}}$.

The following is general to all bunched beams,

${\displaystyle \frac{1}{T_i}=4\pi A(\log )⟨{\int }_0^{\mathrm{\infty }}d\lambda \frac{{\lambda }^{1/2}}{[\det (L+\lambda I){]}^{1/2}}\left\{\mathrm{Tr}{L}^i\mathrm{Tr}\left(\frac{1}{L+\lambda I}\right)-3\mathrm{Tr}\left[L^i\left(\frac{1}{L+\lambda I}\right)\right]\right\}⟩}$,

where ${\displaystyle T_p}$, ${\displaystyle T_h}$, and ${\displaystyle T_v}$ are the momentum spread, horizontal, and vertical are the betatron growth times. The angle brackets <...> indicate that the integral is averaged around the ring.

${\displaystyle (\log )=\ln \frac{b_{min}}{b_{max}}=\ln \frac{2}{{\theta }_{min}}}$

${\displaystyle A=\frac{r_0^{2}cN}{64{\pi }^2{\beta }^3{\gamma }^4{ϵ}_h{ϵ}_v{\sigma }_s{\sigma }_p}}$

${\displaystyle L=L^{(p)}+L^{(h)}+L^{(v)}}$

${\displaystyle L^{(p)}=\frac{{\gamma }^2}{{\sigma }_p^2}\left(\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right)}$

${\displaystyle L^{(h)}=\frac{{\beta }_h}{{ϵ}_h}\left(\begin{array}{ccc}1& -\gamma {\varphi }_h& 0\\ -\gamma {\varphi }_h& \frac{{\gamma }^2{\mathcal{H}}_h}{{\beta }_h}& 0\\ 0& 0& 0\end{array}\right)}$

${\displaystyle L^{(v)}=\frac{{\beta }_v}{{ϵ}_v}\left(\begin{array}{ccc}0& 0& 0\\ 0& \frac{{\gamma }^2{\mathcal{H}}_v}{{\beta }_v}& -\gamma {\varphi }_v\\ 0& -\gamma {\varphi }_v& 1\end{array}\right)}$

${\displaystyle {\mathcal{H}}_{h,v}=[{\eta }_{h,v}^2+({\beta }_{h,v}{\eta }_{h,v}^{\prime }-\frac{1}{2}{\beta }_{h,v}^{\prime }{\eta }_h{)}^2]/{\beta }_{h,v}}$

${\displaystyle {\varphi }_{h,v}={\eta }_{h,v}^{\prime }-\frac{1}{2}{\beta }_{h,v}^{\prime }{\eta }_{h,v}/{\beta }_{h,v}}$

Definitions:

${\displaystyle r_0^2}$ is the classical radius of the particle

${\displaystyle c}$ is the speed of light

${\displaystyle N}$ is the number of particles per bunch

${\displaystyle \beta }$ is velocity divided by the speed of light

${\displaystyle \gamma }$ is energy divided by mass

${\displaystyle {\beta }_{h,v}}$ and ${\displaystyle {\beta }_{h,v}^{\prime }}$ is the betatron function and its derivative, respectively

${\displaystyle {\eta }_{h,v}}$ and ${\displaystyle {\eta }_{h,v}^{\prime }}$ is the dispersion function and its derivative, respectively

${\displaystyle {ϵ}_{h,v}}$ is the emittance

${\displaystyle {\sigma }_s}$ is the bunch length

${\displaystyle {\sigma }_p}$ is the momentum spread

${\displaystyle b_{min}}$ and ${\displaystyle b_{max}}$ 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 for some analysis of the Coulomb log and support for this result.

${\displaystyle {\theta }_{min}}$ is the minimum scattering angle.