A new algorithm for the kinetic analysis of intra-beam scattering in storage rings

For the kinetic analysis of multiple intra-beam scattering (IBS) in storage rings the solution of the Fokker–Planck equation (FPE) or equivalent kinetic equations is needed. The exact solution of the FPE for a beam presented by a set of the macro-particles can be found using the “binary collision” model (BCM). However this method requires a large number of macro-particles (>104). Here we suggest a simple approximate method based on the following assumptions: (1) the friction force has a linear dependence on momentum; (2) the components of the diffusion tensor are constant. The friction coefficients and components of the diffusion tensor are chosen in order to provide the correct growth rates of the invariants describing the motion in storage rings. The integration over the distribution function (which is assumed to be Gaussian) is performed for strong focusing lattice according to the Bjorken–Mtingwa (BM) scheme. Change of the particle momentum is calculated using Langevin equations with account of correlation between the horizontal and longitudinal momentum changes. This algorithm allows us to present IBS effects as a “collective map” in the momentum space acting on each macro-particle. A numerical code using this algorithm is validated for the ITEP ring by comparing the BM code with results of the code based on the BCM algorithm. The algorithm allows including the IBS process in “turn by turn” macro-particle methods, which usually operate with a small number of macro-particles (about 100–1000).