Near equilibrium distributions for beams with space charge in linear and nonlinear periodic focusing systems

Abstract

A procedure to obtain a near equilibrium phase space distribution function has been derived for beams with space charge effects in a generalized periodic focusing transport channel. The method utilizes the Lie transform perturbation theory to canonically transform to slowly oscillating phase space coordinates. The procedure results in transforming the periodic focusing system to a constant focusing one, where equilibrium distributions can be found. Transforming back to the original phase space coordinates yields an equilibrium distribution function corresponding to a constant focusing system along with perturbations resulting from the periodicity in the focusing. Examples used here include linear and nonlinear alternating gradient focusing systems. It is shown that the nonlinear focusing components can be chosen such that the system is close to integrability. The equilibrium distribution functions are numerically calculated, and their properties associated with the corresponding focusing system are discussed. V C 2015 AIP Publishing LLC.

Dynamics near equilibrium for intense beams in a nonlinear integrable focusing channel

“…The Vlasov equilibria can be determined, and these equilibria possess features not observed in the case of linear external focusing, including the appearance of local density minima and unusual density contours. (See [18] for additional examples.) At high intensity, the presence of space charge leads to a mixture of (bounded) regular and chaotic orbits within the equilibrium beam, and the -8 - strong nonlinearity of the external focusing fields drives rapid relaxation to Vlasov equilibrium.…”

Spectral Galerkin solver for intense beam Vlasov equilibria in nonlinear constant focusing channels

“…With the exception of singular KV-type equilibria [6,7], such studies have also assumed rotational symmetry about the direction of the beam centroid motion. A small number of authors have considered the case of nonlinear focusing forces with this symmetry [8,9]. In this paper, we describe a numerical method for producing families of beam equilibria in a general nonlinear focusing potential in two degrees of freedom, without symmetry restrictions.…”