A thermal instability in the non-relativistic planar magnetron

It is well known that an anisotropic temperature distribution gives rise to thermal instability in many Systems. In their numerical simulations of a two-vane, non-relativistic planar magnetron, Thomas et al. reported a temperature anisotropy and a concomitant formation and growth of a swirling electron cloud bunch (convective cell). In this paper we present an analytic calculation of the shift in the eigenfrequencies by treating the thermal effects as a perturbation, while taking due account of the temperature anisotropy. We find the magnitude of the resulting rate to be of the order of that seen in the simulations.