Finite-Temperature Relativistic Electron Beam

Abstract

Moments of the relativistic Vlasov equation were taken over a Maxwellian distribution. The resulting fluid equations were coupled with Maxwell's equations and solved for an axially uniform cylindrically symmetric electron beam in the steady state. Magnetic neutralization and thermal conduction were neglected, and the beam was assumed to be charge neutralized in the observer's reference frame. The equilibrium was supported by radial gradients in particle pressure balancing against magnetic forces. With no externally applied magnetic field, or for a beam propagating parallel to a magnetic field with no rotation, the total current carried by the beam was found to be significantly less than the Alfv\'en critical current ${I}_{A}$, the ratio of beam current to ${I}_{A}$ increasing with temperature and decreasing with increasing $\ensuremath{\beta}$. For a diamagnetic beam in an external field, a rigid-rotator model was assumed. Solutions of the steady-state equations were obtained for various values of rotation frequency $\ensuremath{\omega}$ and beam temperature. The radial density profiles were peaked on axis for small $\ensuremath{\omega}$ and became hollow as $\ensuremath{\omega}$ increased, because of centrifugal forces. For all $\ensuremath{\omega}$, the beam density was observed to exhibit radial oscillations which grew in amplitude and decreased in wavelength with decreasing temperature. Equilibria with magnetic field reversal were possible for all values of $\ensuremath{\omega}$. Purely rotational self-consistent equilibria of the rigid-rotor type were not possible for a charge-neutralized beam of the type considered.