Fokker-Planck approach to the dynamics of mismatched charged-particle beams

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Abstract
A semianalytic formalism is constructed for investigating the transverse dynamics of intense, mismatched charged-particle beams which are centered on, and propagate through, focusing channels. It uses the Fokker-Planck equation to account for the rapid evolution of the coarse-grained distribution function in the phase space of a single beam particle. It also incorporates the space-charge potential, which is calculated from Poisson’s equation using the coarse-grained density. A simple phenomenological model of dynamical friction and diffusion represents the effects of turbulence triggered by charge redistribution. Sheet beams and fully two-dimensional beams are both considered in detail. In addition, closed-form solutions are presented for beams in which the space charge is negligible and noise arises from other stochastic processes.