Thermal Equilibrium of Charged Particle Beams.

The thermal equilibrium distribution of a charged particle beam is the natural state toward which every other distribution will relax. While beam lifetimes in linear accelerators are normally much shorter than the time required for relaxation to equilibrium via Coulomb collisions (interparticle scattering), instabilities and coupling between the transverse and longitudinal space charge forces can drastically speed up the process of relaxation. Beams which are injected and maintained in a state which is as close as possible to equilibrium will minimize the emittance growth and reduction in beam quality which results from the process of relaxation. Measurements were made on the evolution of the transverse density profile of a continuous electron beam, showing that the beam maintains a profile which is near thermal equilibrium. Thermal equilibria are found in one dimension for the longitudinal profiles of bunched beams, in two dimensions for the radial profiles of unbunched beams, and in three dimensions for bunched beams with axial symmetry. Equilibrium profiles are used to predict the current loss rate in the transverse direction in linear accelerators, and the longitudinal current loss rate in the nonlinear focusing field of a radio-frequency linear accelerator. Measurements were made on the energy spread of a continuous electron beam in a two-lens solenoidal focusing channel; the results are consistent with a rapid increase in the longitudinal temperature. A simulation which includes thermal motion, lens nonlinearities and nonuniformities in the density profile is used to predict individual particle motion. Energy spread resulting from several sources is found and compared with the experimental results.