Buneman-Hartree Condition Revisted

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 B-H 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 B-H 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 [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.\\[4pt] [1] R. C. Davidson \textit{et al}., Proc. SPIE \textbf{1061}, 186 (1989). \\[0pt] [2] Y. Y. Lau \textit{et al}., Phys. Plasmas \textbf{17}, 033102 (2010).