Guiding-center Vlasov-Maxwell description of intense beam propagation through a periodic focusing field

Abstract

This paper provides a systematic derivation of a guiding-center kinetic model that describes intense beam propagation through a periodic focusing lattice with axial periodicity length S, valid for sufficiently small phase advance (say, σ < 60°). The analysis assumes a thin (a, b ≪ S) axially continuous beam, or very long charge bunch, propagating in the z direction through a periodic focusing lattice with transverse focusing coefficients κX (s + S) = κX (s) and κy (s + S) = κy (s), where S = const is the lattice period. By averaging over the (fast) oscillations occurring on the length scale of a lattice period S, the analysis leads to smooth-focusing Vlasov-Maxwell equations that describe the slow evolution of the guiding-center distribution function f̄b,(x̄,ȳ,x̄′, ȳ′,s) and (normalized) self-field potential ψ̄ (x̄,ȳ,s) in the four-dimensional transverse phase space (x̄,ȳ,x̄′,ȳ′). In the resulting kinetic equation for f̄b (x̄,ȳ,x̄′,ȳ′, s), the average effects of the applied focusing field are incorporated in constant focusing coefficients κx sf > 0 and κ y Sf > 0, and the model is readily accessible to direct analytical investigation. Similar smooth-focusing Vlasov-Maxwell descriptions are widely used in the accelerator physics literature, often without a systematic justification, and the present analysis is intended to place these models on a rigorous, yet physically intuitive, foundation. © 2001 The American Physical Society.