The Effect of Velocity Distribution in a Modulated Electron Stream

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Abstract
A method of solving electron beam problems is described which takes into account the thermal velocity spread. The method is based on the Liouville theorem. Two such problems are solved by means of a power series. In the case of small-signal velocity modulation of a drifting stream, the velocity spread distorts the shape of the standing wave of current which is produced, in addition to lengthening the plasma wavelength and the electronic wavelength. For a drifting stream with full shot noise in each velocity class at the input, a standing wave of noise current is produced with deep minima. The size of the first minimum is found to increase as the square of frequency and with increasing velocity spread. That full uncorrelated shot noise can produce a standing wave of convection current is somewhat surprising. The present analysis demonstrates that space charge is the important factor in determining the behavior of noise currents in electron streams even at high frequencies, and that velocity spread plays a less important role.