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 , where a is the cathode radius and b is the anode radius, the difference in the two models becomes significant. When 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 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.
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