Halo formation in mismatched, space-charge-dominated beams

A semianalytic formalism was recently developed for investigating the transverse dynamics of a mismatched, space-charge-dominated beam propagating through a focusing channel. 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. A simple model of dynamical friction and diffusion represents the effects of turbulence resulting from charge redistribution. The initial application was to sheet beams, In this paper, the formalism is generalized to fully two-dimensional beams