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

Abstract

An intense nonneutral ion beam propagates in the z-direction through a periodic focusing quadrupole field with transverse focusing force, F foc =−κ q(s)(x e ̂ x−y e ̂ y) , on the beam ions. Here, the oscillatory lattice coefficient satisfies κ q ( s+ S)= κ 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 ψ( 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 ̃ , Y ̃ , X ̃ ′, Y ̃ ′) , with constant focusing coefficient κ fq=const.