Velocity Distributions for Elastically Colliding Electrons

The form of the function giving the distribution in velocity of electrons in a gas is determined by a pair of equations which correspond to the detailed balancing of energy and momentum, and take into account the variation of collision cross section with velocity. The kinetic energy $\epsilon$ of the electrons is supposed to be larger than that of the gas atoms, yet small enough so that the majority of the energy lost is by elastic collisions with the atoms. The equations are solved in detail for two cases. One case is that of electrons in a uniform electric field, where the distribution is independent of position. The distribution function is found to be proportional to $\exp \left(\frac{{-\epsilon }^2}{a^2}\right)$, instead of to $\exp \left(\frac{-\epsilon }{a}\right)$, as is the case for the Maxwell distribution, when the electrons are in temperature equilibrium with the gas atoms. The average energy of the electrons, the drift current, etc., are computed as a function of the field strength. The other case considered is that of a homogeneous beam of electrons of energy ${\epsilon }_0$, shot into a field free space, where they lose energy to the gas atoms by collisions. The distribution function depends on $z$, the distance along the beam in mean free paths, and on $t$, the average number of collisions the electron has had before its energy decreases from ${\epsilon }_0$ to $\epsilon$. This distribution is also not Maxwellian, but depends on a solution of an equation in $z$ and $t$ having the form of the heat flow equation. The solution has been tested experimentally, and a quantitative check is obtained.