Intrabeam scattering (IBS) is an effect in accelerator physics where collisions between particles couple the beam emittance in all three dimensions. This generally causes the beam size to grow. In proton accelerators, intrabeam scattering causes the beam to grow slowly over a period of several hours. This limits the luminosity lifetime. In circular lepton accelerators, intrabeam scattering is counteracted by radiation damping, resulting in a new equilibrium beam emittance with a relaxation time on the order of milliseconds. Intrabeam scattering creates an inverse relationship between the smallness of the beam and the number of particles it contains, therefore limiting luminosity. 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 1 / γ 4 {\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,
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}}} . The following is general to all bunched beams,
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