Head-tail instability and Landau damping in bunches with space charge

In this paper, a simplified theoretical model that allows prediction of the final stationary state attained by an initially mismatched beam is presented. The proposed stationary state has a core-halo distribution. Based on the incompressibility of the Vlasov phase-space dynamics, the core behaves as a completely degenerate Fermi gas, where the particles occupy the lowest possible energy states accessible to them. On the other hand, the halo is given by a tenuous uniform distribution that extends up to a maximum energy determined by the core-particle resonance. This leads to a self-consistent model in which the beam density and self-fields can be determined analytically. The theory allows one to estimate the emittance growth and the fraction of particles that evaporate to the halo in the relaxation process. Self-consistent N-particle simulation results are also presented and are used to verify the theory.