Equilibria and stability of an electron beam confined in a torus

Relativistic, cylindrical, rigid-rotor equilibria for electrons are developed in the beam frame, and the corresponding laboratory frame equilibria are obtained by Lorentz transformation. Limitations of the two-mass approximation are thus illuminated. A toroidal equilibrium is developed that is based on the two-mass approximation and expansion in terms of the ratio of minor to major radius. The toroidal equilibrium is shown to have no unstable kink-modes or no Kruskal-Shafranov current limit as in a Tokamak. Small variations in. the toroidal magnetic field around the torus result in trapped electrons if they are injected with sufficient transverse energy. The presence of both trapped and untrapped electrons leads to two-stream instabilities. Some comparisons are made with HIPAC experiments and the recent torus experiments at MLI.