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The electromagnetic interaction of an intense relativistic coasting beam with itself, including the effect of a nonperfectly conducting vacuum tank, or a quiescent rf cavity, is investigated theoretically. It is shown that the resonances that may occur between harmonics of the particle circulation frequencies and the electromagnetic modes of the cavities can lead to a longitudinal instability of the beam. A criterion for stability of the beam against such longitudinal bunching is obtained as a restriction on the shunt impedance of the rf cavity, or the Q of the vacuum tank. This criterion contains the energy spread and intensity of the coasting beam, as well as the parameters of the accelerator. Numerical examples are given which indicate that, in general, the resonances with the vacuum tank will not cause instabilities, while those with an rf cavity can be prevented from causing instabilities by choosing the shunt impedance at a sufficiently low but still convenient value.
Laslett et al. (1961) studied this question.