Some unusual properties of the cylindrical Brillouin flow

It was recently found that the Buneman-Hartree (B-H) condition in a cylindrical relativistic magnetron assumes a very different form depending on the single particle model or the Brillouin flow model [1]. Such a difference is always present, whether the voltage is relativistic or not. These two models yield the same result only in the limit of a planar magnetron. As the difference between the conventional and the inverted magnetron arises only in a cylindrical geometry, the cylindrical Brillouin flow is of renewed interest because the inverted magnetron exhibits the negative mass instability [2], which is utilized in our recent invention of the recirculating planar magnetron [3,4]. We find that there is yet another novel property of the cylindrical Brillouin flow. If we replace the gap voltage by the spatially varying scalar potential, and the magnetic field by the spatially varying vector potential, both the B-H condition and the Hull cutoff condition are satisfied at all radii within the cylindrical Brillouin flow if the phase velocity of the wave in the B-H condition is replaced by the local electron flow velocity. This property does not seem to have been noted previously, and its implications are being explored. The negative mass effects in the Brillouin flow in the inverted magnetron configuration, as well as the role played by the slow wave structure at the anode, will also be studied.