Three-dimensional thermal distributions for bunched beams

The Maxwell-Boltzmann (“thermal”) distribution of a bunched charged particle beam is the state toward which every other distribution will relax. For beams with lifetimes shorter than the time required for relaxation to equilibrium, it is the distribution at injection which minimizes the emittance growth due to relaxation toward equilibrium. Three-dimensional thermal distributions are found numerically for the case of linear external focusing forces acting on an axially symmetric bunched beam in a conducting pipe. Meta-equilibria are also found for beams with unequal transverse and longitudinal temperatures. Parameters such as perveance, emittance, space-charge tune depression, rms radius and rms bunch length are related to each other for each profile, and are given over a wide range of conditions from short to long bunches and from space-charge dominated to emittance dominated beams. Comparison is made with previous results of radial density profiles for unbunched beams and line-charge density profiles for bunched beams [1].