Diocotron Instability in a Cylindrical Geometry

TLDR

The diocotron instability of low‑density electron beams in crossed fields is examined in cylindrical geometry, showing that normal modes form a continuum plus two discrete eigenvalues, yet the continuous spectrum is largely irrelevant to stability and the problem is analogous to the stability of an ideal rotating fluid. The authors derive a stability condition based solely on the discrete modes, demonstrating that under suitable geometrical and electrical parameters these modes can be stable, and they note that axial perturbations influence the stability criteria, with implications for the electron‑beam problem briefly discussed.

Abstract

The diocotron (or slipping stream) instability of low density (ωp « ωc) electron beams in crossed fields is considered for a cylindrical geometry. For a simple density distribution, the normal modes of the electron beam correspond to a continuum of eigenvalues, plus two discrete eigenvalues. Work due to Case and Dikii appears to show that the continuous spectrum is not important in stability studies of this type. The condition for stability considering the discrete modes only is derived; under suitable geometrical and electrical conditions, it is shown that these modes can be stable. The analogy between the electromagnetic problem considered here and the problem of the stability of an ideal rotating fluid is discussed. It is shown that stability conditions derived for the latter problem depend on the possibility of axial perturbations; what this implies for the electron beam problem is briefly discussed.