On the Instability of a Coasting Beam

Abstract

The time-dependent Boltzmann equation is used to study longitudinal motion of particles in a circulating beam of particles. The equation is linearized about the stationary solution corresponding to a coasting beam of almost monoenergetic particles to determine the effect of small density fluctuations. An integral equation is obtained and solved, leading to a dispersion relation quite similar to that well known in the theory of plasma oscillations. It is shown that if the accelerator is operating above the transition energy, then the motion can be unstable. Expressions are given for the rate of growth of longitudinal waves, and it is shown that if the unperturbed beam satisfies certain criteria, then the motion is either stable or it grows at such a slow rate as to be of no consequence. Numerical examples for MURA panticle accelerators show that essentially there is stability. (auth)

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