Dynamic behavior of a long thin electron beam beyond critical perveance

A discrete mathematical model is derived for the case of a thin infinite planar electron beam injected with symmetry between two parallel semi-infinite plane conductors separated by a distance d and joined by a third equipotential plane through which the beam is injected. The computer model is first shown correctly to reproduce the calculable space-asymptotic beam velocity for subcritical perveance and is then employed to simulate the dynamic behavior accompanying operation beyond critical perveance. Rejection of a fraction of the injected charge is shown to be an essential feature of this mode of operation, and the oscillating potential minimum is seen to exist at a distance less than from the injection plane. A relatively simple modification of the model portrays the unstable behavior of a terminated beam only long. Longitudinal potential profiles for this terminated beam are plotted at selected points in time. Oscillation frequency at a given supercritical perveance is noted to be an inverse function of transit time defined with respect to any chosen stable perveance and beam length in units of .