Linear two-stream instability of warm relativistic electron beams

The linear dispersion relation for a warm relativistic electron beam propagating through a cold plasma is studied. The beam electrons have the same energy, but there is a spread in their directions of motion. Numerical solutions have been obtained for the fastest growth rate delta for wave vectors k at various angles to the direction of propagation of the beam. It is found that even for very strong beams ( alpha 1/3 gamma >>1; alpha =n b /n p ) the hydrodynamic phase, when delta varies as alpha 1/3 , during which waves parallel to the beam axis grow in the same fashion as if the beam were cold, plays an important role. The kinetic phase ( delta varies as alpha ) occurs only after the angular spread reaches large values.