Axially-Dependent Equilibrium of an Unneutralized Electron Beam in an Axial Magnetic Field,

Virtual cathode formation in vacuum-propagating beams may be more widespread than previously recognized. The spin death of parallel beam energy by E sub r x B sub z rotation in front of a density pileup, may assist or steepen the axial density gradient, since the rotation speed tends toward Brillouin flow, omega omega sub c2. It may be that the propagation limit on the beam density is not omega-squared sub p approx. equal omega-squared sub c in the applied or total magnetic field, but rather omega-squared sub p max in the presence of self-field pileup small enough that cE sub rB sub z V sub 0 V sub 0 beam electron speed, with E sub r given self-consistently from the charge distribution. In what follows, self-consistent BGK-like axially dependent equilibria will be described, in which the space charge effects determine the axial extent and profile of the beam. Generalization to axially nonuniform magnetic fields is relatively straightforward. Sections III and IV treat the problem in a purely one-dimensional way, keeping the electrostatic effects of E sub z electric fields but not the effects of beam rotation caused by E sub r. Section V generalizes these results to include E sub r.