Conservation principles in multivelocity electron flow

This paper considers the extension of Poynting\´s theorem to an electron gas with a continuous distribution of velocities. In particular, an extension of the concept of "kinetic potential" is attempted, since this concept has proved itself very useful in the investigation of single-velocity flow. It is found that in three dimensions the electrokinetic-flow vector cannot be expressed as the product of the convection-current density and a single scalar quantity of the dimension potential. In one-dimensional applications, however, this circumstance is immaterial. Another difficulty is encountered when a small perturbation component on a steady state is considered. The nonlinear Bolzmann transport equation gives a linear equation between the first-order perturbations, but the nonlinearity makes possible a conversion of part of the perturbation power flow to dc power flow. In other words, the power flow associated with a single-frequency perturbation is not necessarily conserved in multivelocity flow, even in the absence of an ac Poynting vector. In the case of space-charge-limited flow in an electron gun, the consequence is that the cooling of an electron stream by adiabatic expansion can proceed beyond the potential minimum and very low noise temperatures are attainable in principle if the accelerating field is maintained small over an appreciable length of the electron gun.