Electron beams in axially-symmetric crossed fields

M-type traveling-wave tubes use electron beams that drift in crossed electric and magnetic fields. One such tube, the axiotron described by Warnecke and Doehler in 1950, used a hollow beam drifting parallel to the tube axis in a radial electric field crossed by an azimuthal magnetic field. The addition of an axial magnetic field to the azimuthal one adds another degree of complication and flexibility to the beam equations, yet maintains their symmetry about the tube axis. It gives, in effect, a helical magnetic field crossed by a radial electric field. This report examines the behavior of hollow electron beams drifting in laminar flow through fields of the latter configuration. It defines a stability index for electron paths, and four fairly general types of beam. It determines the stability index and the distribution of space charge obtainable in each type as functions of the amplitudes and directions of the fields and drift velocities. In general, the density tends to be greatest at the inner beam radius, but it is possible to approach uniform density in stable beams. This report does not consider beam launching nor the "gun" problem; nor does it consider over-all beam instabilities such as scalloping.