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

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,bS) 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)0ex0ex=0ex0exκₓ(s) and κy(s+S)0ex0ex=0ex0exκy(s), where S0ex0ex=0ex0exconst 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 ψ(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 κxsf>0 and κysf>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.