Periodically-focused solutions to the nonlinear Vlasov-Maxwell equations for intense charged particle beams

摘要

An intense nonneutral ion beam propagates in the z-direction through a periodic focusing quadrupole field with transverse focusing force, F_(foc) = -#kappa#_q(s)(xe_x - ye_y), on the beam ions. Here, the oscillatory lattice coefficient satisfies #kappa#_q(s + S) = #kappa#_q(s), where S = const. is the axial periodicity length. The model employs the Vlasov-Maxwell equations to describe the nonlinear evolution of the distribution function f_b(x,y,x',y',s) and the normalized self-field potential #psi#(x,y,s) in the transverse laboratory-frame phase space (x,y,x',y'). Using a third-order Hamiltonian averaging technique, a canonical transformation is employed with an expanded generating function which transforms away the rapidly oscillating terms. and leads to a Hamiltonian in the 'slow' transformed variables (X-tilde,Y-tilde,X'-tilde,Y'-tilde), with constant focusing coefficient #kappa#_(fq) = const.