Slipping-instability of a relativistic electron beam

We investigate the slipping-instability of a relativistic electron beam taking the own azimuthal magnetic field into account. A uniformly dense beam partially compensated in charge by a uniform steady ion background is emitted from an equipotential cathode into a vacuum drift tube in an external longitudinal magnetic field. The problem is solved under the assumption of a strong external field and the beam current small compared to the Alfven current. We study the stability of the initial equilibrium with respect to small and, in a general case, helical perturbations solving the linearized equations for the electron fluid and wave equations for the electromagnetic potentials. The equations for the generalized potential can be reduced to Bessel´s equation in the vicinity of the cherenkov resonance of the helical wave. Natural boundary conditions for the potential give rise to the dispersion relation, which determines instability regions and growth rates. We obtained expressions for the maximum temporal and spatial growth rates and for the threshold current of the instability excitation.