Space-charge calculation for bunched beams with 3-D ellipsoidal symmetry

A method for calculating 3-D space-charge forces are developed that is suitable for bunched beams of either ions or relativistic electrons. The method is based on the analytical relations between charge-density and electric fields for a distribution with 3-D ellipsoidal symmetry in real space. At each step a Fourier-series representation is used for the smooth particle-density function obtained from the distribution of the macroparticles being tracked through the elements of the system. The resulting smooth electric fields reduce the problem of noise from artificial collisions, associated with small numbers of interacting macroparticles. Example calculations are shown for comparison with other methods.<>