Application of the Fokker-Planck equation to particle-beam injection into e^- storage rings

Nonlinear forces in the longitudinal accelerating field—or in the transverse magnetic fields—lead to filamentation of the injected emittance and to the decoherence of the center-of-mass motion. The dynamics of the particle distribution function in the presence of synchrotron radiation is governed by the Fokker-Planck equation. We derive the time evolution of the distribution function after injection as an approximate solution to the Fokker-Planck equation. The approximation assumes the injected emittance to be considerably larger than the equilibrium emittance, which is fulfilled for a certain class of storage rings, e.g., damping rings. In the limit of no quantum excitation, this distribution function will then be an exact solution. Higher moments of the distribution can be expressed in combinations of elementary functions and agree very well with multiparticle simulations.