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 = p 2 / 2 m + κ ⊥ r 2 / 2 + κ z z 2 / 2 + q φ ( r , z ) , the kinetic energy is p 2 / 2 m , κ ⊥ and κ z are the external focusing coefficients in the transverse and longitudinal directions, and φ is the electrostatic potential determined self-consistently from Poissonʹs equation ∇ 2 φ = - 4 π q ∫ d 3 pf ( 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. Poissonʹs 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 = A exp ( - H / T ) , the single-particle trajectories display chaotic behavior. The properties of the chaotic trajectories are characterized.