Slow cyclotron instability due to surface modulation of an annular beam

The Cherenkov instability used in slow-wave devices has been well studied in the literature. However, in previous analyses, the beam motion is restricted to the longitudinal direction assuming an infinitely strong magnetic field. For the finite strength magnetic field, the transverse beam perturbation cannot be ignored and leads to the slow cyclotron instability. Recently, a new version of self-consistent, field theory considering three-dimensional perturbation has been developed based on a solid beam. in which the effect of the transverse perturbation appears as a surface charge at a fixed boundary. In the case of a thin annular beam. the boundary is modulated and is essentially different from the solid beam case. We propose a self-consistent field theory considering the moving modified boundary surface. The slow cyclotron instability due to the modulation of an infinitesimally thin annular electron beam is presented.