Last advances in analysis of intra-beam scattering in the hadron storage rings

Multiple intra-beam scattering (IBS) appears because of Coulomb scattering of charged particles on each other. In hadron storage rings, it results in re-distribution of the energy between different degrees of freedom, growth of six-dimensional emittance, development of non-Gaussian tails in distribution function and sometimes to the loss of particles. Traditional method of IBS analysis is based on “Gaussian model”, which is assumed that distribution functions are Gaussian ones for all degrees of freedom. The basic theorems derived from the Gaussian model are shortly discussed. More detailed IBS analysis can be done using the Fokker-Planck equation (FPE) or equivalent tools. The FPE is written in “coordinate-momentum” space and in “invariant space” with account of the multiple IBS. Here, we consider two numerical methods of three-dimensional FPE solution developed at the last time: (1) the use of the “Langevin equations map” (LEM) and (2) the simulation of three-dimensional IBS using “the binary collision map” (BCM). The basics of both methods and examples of their application to storage rings are given. Alternative approach to the IBS analysis is named “molecular dynamics” (MD) based on the straight-forward integration of the particle trajectories taking into account the external electromagnetic fields and the Coulomb forces. Two modifications of MD are described: (1) a two-dimensional “wire model” and (2) a three-dimensional model of “periodical cells”, which is fruitful in the case of small density beams, for example, the “crystalline beams”. The basic problems of the molecular dynamics and results of its application to IBS analysis are shortly discussed.