Effects of a finite axial magnetic field on the beam loading on a cavity [klystron cavity example]

The beam loading of a cavity is mostly computed under the assumption of an infinite axial magnetic field. In practice, a finite focusing magnetic field is almost always present. In this paper, we extend Branch´s classic paper (IEEE Trans. Electron Devices, vol.8, p.193, 1961) on ballistic bunching in that both the conductive and reactive components of the beam-loaded admittance are computed, and for general values of axial magnetic field. Also included is a comparison of the analytic formulation with a 2D particle-in-cell simulation. This work suggests that the finite axial magnetic field used in linear beam tubes (typically exceeding 1.5 times the Brillouin field) would modify the beam-loaded admittance by about 20%, or less, from that computed under the assumptions of an infinite axial magnetic field.