An unnoticed property of the cylindrical relativistic Brillouin flow

It is shown that the vector potential and the scalar potential satisfy a Buneman-Hartree like (BHL) condition and a Hull-cutoff like (HCL) condition everywhere within the Brillouin flow of a cylindrical, relativistic magnetron, when the phase velocity in the Buneman-Hartree condition is replaced by the laminar, local flow velocity of the Brillouin flow. The vector potential and the scalar potentials include the Brillouin flow’s self magnetic field and the self electric fields. The HCL condition reduces to the conventional Hull cutoff condition derived from single particle orbit theory when the Brillouin hub extends to the anode. However, the BHL condition reduces to the conventional Buneman-Hartree condition only in the planar magnetron limit but may be substantially different for a cylindrical magnetron, as demonstrated recently [Lau et al., Phys. Plasmas 17, 033102 (2010)]. These conclusions apply also to the inverted magnetron configuration. The effects of ions are discussed.