Effects of electron collisions on the resistive hose instability in intense charged particle beams propagating through background plasma

The dispersion relation for the resistive hose instability in a charged particle beam with a flattop density profile is derived from the linearized Vlasov-Maxwell equations. Stability properties of the resistive hose instability where the perturbations are initiated at the beam entrance are investigated. In particular, the complex eigenfrequency Ω in the dispersion relation is expressed as a function of the real oscillation frequency ω of the excitation at the beam entrance. As expected, the growth rate ImΩ=Ω_i decreases rapidly as the conducting wall approaches the beam (r_w/r_b→1). The growth rate also decreases substantially as the frequency ratio ω/ν_c increases, where ν_c is the electron collision frequency. Stability properties for perturbations propagating through the beam pulse from its head to tail are also investigated. In this case, the growth rate Imω is calculated in terms of the real oscillation frequency Ω of each beam segment. It is shown that the resonance frequency Ω=Ω_r corresponding to the infinite growth rate detunes considerably from the betatron frequency ω_β of the beam particles. It is also found that the bandwidth corresponding to instability is narrow when the plasma electron collision time (1/ν_c) is long compared with the magnetic decay time (τ_d).