Self-consistent Vlasov-Maxwell description of the longitudinal dynamics of intense charged particle beams

Abstract

This paper describes a self-consistent kinetic model for the longitudinal dynamics of a long, coasting beam propagating in straight (linear) geometry in the z direction in the smooth-focusing approximation. Starting with the three-dimensional Vlasov-Maxwell equations, and integrating over the phase-space (x_{⊥},p_{⊥}) transverse to beam propagation, a closed system of equations is obtained for the nonlinear evolution of the longitudinal distribution function F_{b}(z,p_{z},t) and average axial electric field ⟨E_{z}^{s}⟩(z,t). The primary assumptions in the present analysis are that the dependence on axial momentum p_{z} of the distribution function f_{b}(x,p,t) is factorable, and that the transverse beam dynamics remains relatively quiescent (absence of transverse instability or beam mismatch). The analysis is carried out correct to order k_{z}^{2}r_{w}^{2} assuming slow axial spatial variations with k_{z}^{2}r_{w}^{2}≪1, where k_{z}∼∂/∂z is the inverse length scale of axial variation in the line density λ_{b}(z,t)=∫dp_{z}F_{b}(z,p_{z},t), and r_{w} is the radius of the conducting wall (assumed perfectly conducting). A closed expression for the average longitudinal electric field ⟨E_{z}^{s}⟩(z,t) in terms of geometric factors, the line density λ_{b}, and its derivatives ∂λ_{b}/∂z,… is obtained for the class of bell-shaped density profiles n_{b}(r,z,t)=(λ_{b}/πr_{b}^{2})f(r/r_{b}), where the shape function f(r/r_{b}) has the form specified by f(r/r_{b})=(n+1)(1-r^{2}/r_{b}^{2})^{n} for 0≤r<r_{b}, and f(r/r_{b})=0 for r_{b}<r≤r_{w}, where n=0,1,2,…. The general kinetic formalism developed here is valid for the entire range of beam intensities (proportional to λ_{b}) ranging from low-intensity, emittance-dominated beams, to very-high-intensity, low-emittance beams.