Electromagnetic (Darwin) model for three-dimensional perturbative particle simulation of high intensity beams

A nonlinear perturbative (δf) electromagnetic particle simulation scheme has been developed for studying the propagation of intense charged particle beams in high-intensity accelerators and transport systems. The scheme is based on the Darwin approximation of Ampere´s law, in which the transverse displacement current is neglected, resulting in the elimination of high-frequency transverse electromagnetic effects and, consequently, the associated numerical restrictions in the simulations. In this paper, a novel approach of replacing the mechanical momentum, p, by the canonical momentum, P = p + qA/c, in the equations of motion is used for the purpose of eliminating the numerically troublesome ∂A/∂t term. The present scheme, cast in the δf formalism, is ideal for studying two-stream and filamentation instabilities, which may cause deterioration of the beam quality in heavy ion fusion drivers and the fusion chamber