The brightness of electron beams

Abstract

An expression is derived for the brightness or directional beam intensity of an electron beam, for a general velocity distribution at the emitting cathode. The result is then specialized to the case of the brightness on the axis in the direction of the axis for a cylindrically symmetric system. In this case the Langmuir law B/J=eV/πkT is shown to hold for emission in the Schottky region even though the velocity distribution is not Maxwellian; and an analogous expression B/J=eV/πd, where d is a function only of the applied field, is shown to hold for field-emitted beams. It is shown that detailed ray tracing is necessary to evaluate the brightness for points not on the axis and directions not in the axial direction.