Longitudinal instability of space-charge dominated beams in transport channels with complex wall impedances

The longitudinal instability of space-charge dominated beams in transport channels with complex wall impedances is studied. The dispersion equation, derived from the one-dimensional Vlasov equation, characterizes the instability with all the relevant parameters including the space charge, the beam energy spread, the real and imaginary parts of the complex wall impedances. For a beam without energy spread, the growth rates of the instability in the different parameter ranges are discussed. The application of the theory to a resistive-wall channel and to a channel with induction gaps is presented. For a beam with energy spread, Lorentzian and Gaussian distributions are treated and the results presented illustrate the dependence of the stable regions on the beam perveance, the energy spread and the wall properties.