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

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 Fb(z,pz,t) and average axial electric field ⟨Ezˢ⟩(z,t). The primary assumptions in the present analysis are that the dependence on axial momentum pz of the distribution function fb(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 kz²rw² assuming slow axial spatial variations with kz²rw²1, where kz~∂/∂z is the inverse length scale of axial variation in the line density λb(z,t)=∫dpzFb(z,pz,t), and rw is the radius of the conducting wall (assumed perfectly conducting). A closed expression for the average longitudinal electric field ⟨Ezˢ⟩(z,t) in terms of geometric factors, the line density λb, and its derivatives ∂λb/∂z, is obtained for the class of bell-shaped density profiles nb(r,z,t)=(λb/πrb²)f(r/rb), where the shape function f(r/rb) has the form specified by f(r/rb)=(n+1)(1-r²/rb²)ⁿ for 0≤r<rb, and f(r/rb)=0 for rb<r≤rw, where n=0,1,2,. The general kinetic formalism developed here is valid for the entire range of beam intensities (proportional to λb) ranging from low-intensity, emittance-dominated beams, to very-high-intensity, low-emittance beams.