A simplified model is discussed that captures the basic physics of the phenomenon of oscillatory virtual cathodes in electron beams. A monoenergetic non-relativistic one-dimensional electron beam is injected through a conducting grid into a semi-infinite drift space. Attraction from image charges (and, possibly, an adverse externally applied electric field) cause particle reflection and the formation of a caustic where the charge density has an integrable singularity. The steady-state solution of the Vlasov equation describing the flow is known from numerical simulation to be unstable, but analytical demonstration of this instability has proved intractable. Here we derive an integral-delay equation describing the time-dependent evolution of the electron beam under the assumption that the caustic accelerates much more slowly than the electrons in its neighbourhood and thus at most two streams are present at each point. Under this assumption we show that the charge singularity is ~|x-xc|-\xbd the presence of an external field, exactly what it would be for non-interacting particles, but in the absence of applied field it is weaker, ~|x-xc|-\u2153 Our methods can be used to estimate the charge singularity, and thus the collisionless shock conditions' for virtual cathodes in any geometry. The importance of delay effects for the onset of beam oscillations is demonstrated in an exactly solvable version of the model in which the interaction between the two streams is ignored. This solution, although unphysical, can provide a means for testing the performance of numerical schemes, which have difficulties in problems of this type due to the charge singularity.'
Cambridge University Press
Journal of Plasma Physics
Cambridge University Press http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=4747580&fulltextType=RA&fileId=S0022377800013349
Journal of Plasma Physics, 40(2): 369-384