Thermal equilibrium of bunched charged particle beams

The Maxwell–Boltzmann distribution of a bunched charged particle beam is the state toward which every other distribution will relax. For beams with lifetimes much shorter than the time required for relaxation to equilibrium, it is the distribution at injection that 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. Equations are derived for current loss into the conducting channel due to particle thermal motion in the equilibrium distribution. Relations between parameters such as perveance, emittance, space‐charge tune depression, bunch radius, and bunch length 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 [Phys. Rev. Lett. 71, 2911 (1993)].