Analysis of Longitudinal Accelerator Instabilities

A sufficient stability condition for uniform coasting beams in cyclic particle accelerators is derived from Neil and Sessler´s dispersion equation. Specific conditions are worked out for various injection patterns. The instability threshold depends fairly strongly upon the particle density distribution across the tank, and upon whether operation is below or above transition energy. For the latter case, the optimum distribution (allowing the largest number of particles in the beam) is derived. The possibility is investigated, using Nyquist diagrams, of injecting more particles by operating in a superstable state - one which depends for its stability on the low resistance of the tank walls. It is shown that any practical improvement by this means is probably illusory.