Exact solutions for the electron distribution in plane magnetrons

The old problem of electron distribution in crossed electric and magnetic fields, such as exist in magnetrons, has in the past proved full of pitfalls, owing to the decisive influence which even very small initial electron velocities can have on the character of the solution. A complete analysis of the plane magnetron is presented, with a thermal emitter, i.e. with Maxwellian distribution of the initial velocities. Instead of looking for self-consistent solutions, which vary strongly with the space charge, the solution is given for three simple types of prescribed electric potentials, zero, linear and parabolic. The first two are mainly for orientation, the third is of practical interest as it is approximately self-consistent. For zero or weak electric fields the distribution is ‘triangular’, i.e. the function decreases monotonically as we move away from the cathode. For strong electric fields, the distribution has a peak away from the cathode and strongly resembles that obtained in the so called ‘double-stream’ flow. Finally, for a parabolic potential distribution (linear field variation) the space charge density exhibits a pronounced plateau which is highly reminiscent of the conditions in a Brillouin or ‘single-stream’ flow, although the electron motion is anything but rectilinear.