Kinetic description of intense nonneutral beam propagation through a periodic focusing field

This article makes use of the nonlinear Vlasov–Poisson equations to investigate intense nonneutral beam propagation through a periodic solenoidal focusing field Bsol(x)=Bz(s)êz−(12)Bz′(s)(xêx+yêy), where Bz(s+S)=Bz(s), and S the axial length periodicity is a constant. The analysis is carried out for a thin beam with characteristic beam radius rb⪡S, and the transverse momentum components and axial momentum spread of the beam particles are assumed to be small in comparison with the directed axial momentum γbmβbc. The general formalism is applied to a periodically focused Vlasov equilibrium with step-function radial density profile and average azimuthal motion of the beam corresponding to a rigid rotation (in the Larmor frame) about the axis of symmetry. A variety of beam equilibrium properties are presented, such as the average flow velocity, and the transverse temperature profile.