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

Abstract

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 $\ensuremath{\Omega}$ in the dispersion relation is expressed as a function of the real oscillation frequency $\ensuremath{\omega}$ of the excitation at the beam entrance. As expected, the growth rate $\mathrm{Im}\ensuremath{\Omega}={\ensuremath{\Omega}}_{i}$ decreases rapidly as the conducting wall approaches the beam (${r}_{w}/{r}_{b}\ensuremath{\rightarrow}1$). The growth rate also decreases substantially as the frequency ratio $\ensuremath{\omega}/{\ensuremath{\nu}}_{c}$ increases, where ${\ensuremath{\nu}}_{c}$ 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 $\mathrm{Im}\ensuremath{\omega}$ is calculated in terms of the real oscillation frequency $\ensuremath{\Omega}$ of each beam segment. It is shown that the resonance frequency $\ensuremath{\Omega}={\ensuremath{\Omega}}_{r}$ corresponding to the infinite growth rate detunes considerably from the betatron frequency ${\ensuremath{\omega}}_{\ensuremath{\beta}}$ of the beam particles. It is also found that the bandwidth corresponding to instability is narrow when the plasma electron collision time ($1/{\ensuremath{\nu}}_{c}$) is long compared with the magnetic decay time (${\ensuremath{\tau}}_{d}$).