Buneman-Hartree condition re-visited

Summary form only given. The Buneman-Hartree (B-H) condition is re-examined in a cylindrical 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 >> using Davidson's model [1], conveniently cast in terms of the normalized gap voltage and normalized magnetic flux imposed on the cylindrical magnetron [2]. Comparison with experiments will be reported.