Breakup of hollow cylindrical electron beams

The experimental breakup of a hollow electron beam focused by a longitudinal magnetic field into a discrete set of vortex filaments is described. Agreement with the theory of J. R. Pierce is satisfactory. The theory is extended to include the cases of Harris flow and hollow beams focused by a negative center conductor and a magnetic field. It is predicted that growing waves are possible in Harris flow beams. General considerations of energy and momentum conservation support the detailed analysis.