Analysis of Multivelocity Electron Beams by the Density-Function Method

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Abstract
Some mathematical techniques are presented which assist in the analysis of one-dimensional multivelocity accelerated electron streams by the density-function method. These techniques include methods for removing mathematical singularities from the equations. The correspondence between the density-function method and the usual single-velocity analysis is discussed; and a new definition of the equivalent ac voltage in terms of the density function is given. This definition is meaningful even for large velocity spread. The density-function method and the single-velocity theory agree for small velocity spread while, for a strongly multivelocity beam, no simple transmission-line type of equations are to be found. Because of the latter fact, a common assumption about the noise conservation properties of the low-voltage region of an electron gun is not valid, and the beam's ``noisiness'' is not invariant with distance in this region. These conclusions are verified by detailed noise calculations reported in an accompanying paper.