Chaotic particle trajectories in high-intensity finite-length charge bunches

A Vlasov-Maxwell equilibrium for a charged particle bunch is given in the beam frame by the distribution function that is a function of the single-particle Hamiltonian f=f(H), where in an axisymmetric cylinder H=p2/2m+κ⊥r2/2+κzz2/2+qφ(r,z), the kinetic energy is p2/2m, κ⊥ and κz are the external focusing coefficients in the transverse and longitudinal directions, and φ is the electrostatic potential determined self-consistently from Poisson's equation ∇2φ=- 4πq∫d3pf(H). The self-field potential φ introduces a coupling between the otherwise independent r and z motions. Under quite general conditions, this leads to chaotic particle motion. Poisson's equation is solved using a spectral method in z and a finite-difference method in r, and a Picard iteration method is used to determine φ self-consistently. For the thermal equilibrium distribution f=Aexp(-H/T), the single-particle trajectories display chaotic behavior. The properties of the chaotic trajectories are characterized.