Space Charge Waves in Inhomogeneous Electron Beams

Abstract

A magnetically confined sheet electron beam filling the space between two plane parallel electrodes is discussed. It is assumed that the average electron space charge is not neutralized by positive ions so that the potential at the center is depressed, and the beam is inhomogeneous in velocity and charge density. The objective of this theoretical treatment is to determine whether or not exponentially growing space charge waves can be supported by such a beam. For small inhomogeneity and continuous behavior of all quantities as functions of the inhomogeneity, it is shown that growing waves are not possible. This conclusion is supported by discussion of the analogous adjacent beam problem and the analogous velocity distributed filamentary beam. Although the adjacent beams can support growing waves, it is argued that the analogy is not valid because the adjacent beams cannot be obtained by a continuous perturbation of a single homogeneous beam. The analogy of the velocity distributed beam does not suffer from this deficiency. In this case it is shown that growing space charge waves cannot exist unless the distribution function has more than one relative maximum.