Instabilities in a rotating hollow beam

An analysis is made to determine the nature of the Weibel mode instability, if any, in the Radiatron. The eigenmodes are derived by small signal analysis for a thin, hollow beam rotating at the cyclotron frequency in a uniform dc axial magnetic field. All six rf field components are matched at the beam surface. It is found that the effect of the transverse magnetic fields of the TE wave, which leads to the "Weibel" instability in large cycloiding beams, is destroyed by the large axial electric fields related to space charge repulsion. Thus, the Radiatron based on the Weibel instability is not suitable for high power mm-wave applications. However, for larger beam densities, the transverse electric fields of the TM wave lead to the Kyhl and Webster instability, which may have mm-wave applications.