The effect of crossed fields on the velocity distribution of hollow electron beams

An analysis of the effect of radial electric and longitudinal magnetic fields on the axial velocity distribution of electrons in a hollow beam was carried out on a simplified model. Two results, both consistent with experimental observations, were obtained. First, the longitudinal velocity spread (or equivalent temperature) increases with distance from the axis. Second, the retarding potential at which all electrons are collected also increases with distance from the axis; this indicates that the energy of the slowest electrons at the beam edge is decreased as a result of the crossed fields. The radial electric field, which is greatest near the beam edge and zero on the axis, converts some of the longitudinal energy of the electrons into rotational energy. This kind of scattering is analyzed by a perturbation method whereby the transverse motion of the incident beam is considered harmonically bound by the magnetic field. The spectrum of the energy absorbed in the rotational mode is shown to be velocity dependent and thus to give rise to distortion in the Maxwellian distribution of the incident beam.