Generalization of the Bennett equilibrium condition for a relativistic electron beam propagating in the Ohmic plasma channel and ion focusing regime along an external magnetic field

The problem of formulating the generalization of the Bennett equilibrium condition is considered for a relativistic electron beam propagating in the Ohmic plasma channel, as well as in the ion focusing regime in the presence of an external longitudinal uniform magnetic field. We assume that the electron component of the background plasma is not completely removed from the region occupied by the beam. This equilibrium condition is derived using the mass and momentum transport equations obtained for a paraxial monoenergetic beam from the Fokker–Planck kinetic equation.