The diocotron stability properties of a relativistic hollow electron beam are investigated within the framework of the macroscopic, cold fluid model, in which the electron motion is assumed to be laminar. The stability analysis is carried out for the case of a sharp boundary density profile, including the important influence of relativistic effects and fractional charge neutralization on the stability behavior. It is found that the growth rate of instability is proportional to 1/γb2, thereby being reduced considerably by increasing the electron energy γbmc2. Moreover, the fractional charge neutralization plays a significant role in the stability behavior, particularly for the relativistic electron beam.
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Uhm et al. (1979) studied this question.