An RMS particle core model for rings

Abstract

A self-consistent set of equations for the azimuthal variation of rms betatron oscillation amplitudes, including the effects of dispersion and space charge, is derived. These effective envelope equations can be integrated over the beam energy distribution to provide space charge forces in a particle core model for rings. The derivation of the envelope equations involves an accelerator ordering scheme for the beam dynamics and a statistical moments analysis of the canonical distribution function in the six-dimensional phase space of the beam Hamiltonian. The azimuthal variation of the second moments of the transverse canonical coordinates, xβ and zβ, integrated over the kinetic distribution function of the beam, provides the rms equations. These equations, at fixed beam energy, are integrated over the beam energy distribution to provide the overall space charge distribution and force. Because the envelope equations and dispersion function both depend upon and determine the space charge forces, the consistency of the particle core model requires either analytic closure assumptions or numerical iteration.