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.