Wave types in vortical and in irrotational electron beams

The theory of wave propagation on electron beams focused by a finite magnetic field -though frequently treated in literature -is still unsatisfactory:1)On Brillouin beams waves with radial periodicity are thought to be physically nonexistent [l], [2] while experimental results show the existence of higher ranking waves of this type [3]. No attempt has been made to calculate the excitation of Brillouin beams by gridded gaps since this is impossible if waves with radial periodicity are excluded.2)The influence of the magnetic flux threading the cathode and thus the basic dynamical difference between vortical (cathode flux) and irrotational (no cathode flux) beams is not understood. It is unknown which modes carry the vortical part of the noise on an irrotational beam which is excited by initial thermal velocities at the cathode.3)Transverse waves propagating the center-of-gravity motion of the beam have not been shown to appear in the set of modes resulting from a general first order theory.