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The effect of finite resistance in the vacuum-tank walls on the longitudinal stability of an intense beam of particles in an accelerator is investigated theoretically. We show that even if the particle frequency is an increasing function of particle energy, the wall resistance can render the beam unstable against longitudinal bunching. In the absence of frequency spread in the unperturbed beam, the instability occurs with a growth rate that is proportional to (N/σ)½, where N is the number of particles in the beam and σ is the conductivity of the surface material. By means of the Vlasov equation a criterion for beam stability is obtained. In the limit of highly conducting walls the criterion involves the frequency spread in the unperturbed beam, the number of particles N, the beam energy, geometrical properties of the accelerator, but not the conductivity σ. A numerical example presented indicates that certain observations of beam behavior in the MURA 40 MeV electron accelerator may be related to the phenomenon we investigated.
Neil et al. (1965) studied this question.