Longitudinal Stability of a Coasting Beam in a Corrugated Vacuum Chamber

Widenings of a vacuum chamber may function as resonant cavities, which can promote longitudinal instabilities. We have reexamined the stability question in such a situation, following earlier work of Keil and Zotter. We solve the full linearized Vlasov-Maxwell equations for a model vacuum chamber, using an improved computational method which is efficient for virtually any choice of geometrical parameters, in particular for large ring radius R. Because of mode coupling induced by the corrugations, the dispersion relation that determines the growth time of an instability involves an infinite-dimensional impedance matrix, rather than the usual single impedance. We give a method to compute the growth time ¿, and find an explicit formula for ¿ as an oscillating function of the mean revolution frequency ¿, with sharp minima at points where the real part of a cavity resonance frequency is equal to an harmonic of ¿ (plus a small shift). An illustration for parameters of interest in ion beam fusion is included.