A re-examination of the Buneman–Hartree condition in a cylindrical smooth-bore relativistic magnetron

The Buneman–Hartree condition is re-examined in a cylindrical, smooth-bore, relativistic magnetron using both the conventional, single particle model, and the Brillouin flow model. These two models yield the same result for the Buneman–Hartree condition only in the limit of a planar magnetron. When b/a=1.3, where a is the cathode radius and b (>a) is the anode radius, the difference in the two models becomes significant. When b/a=4 the difference is acute, the Buneman–Hartree magnetic field at a given voltage in the Brillouin flow model exceeds four times that in the single particle model. Such a difference is always present, whether the voltage is relativistic or not. These results are quantified for b/a⪢1 using Davidson’s model, conveniently cast in terms of the normalized gap voltage and normalized magnetic flux imposed on the cylindrical magnetron. A comparison with the University of Michigan/L-3 relativistic magnetron experiment is given.