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 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_z²r_w² assuming slow axial spatial variations with k_z²r_w²≪1, where k_z∼∂/∂z is the inverse length scale of axial variation in the line density λ_b(z,t)=∫dp_zF_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_b²)f(r/r_b), where the shape function f(r/r_b) has the form specified by f(r/r_b)=(n+1)(1-r²/r_b²)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.