New Theory of Electron Drift Velocity in Gases

Abstract

The electron drift velocity is usually obtained on the assumptions that (i) the velocity variation due to the electric field is small compared to the thermal velocity, and (ii) the inelastic collision rate is negligible with respect to the elastic collision rate. By introducing the distribution function $f_0\left(c_0\right)$ of the velocities $c_0$ immediately after a collision ("initial" distribution function), we develop a new theory which obviates the above assumptions and is therefore particularly suitable in the case of high values of the ratio between the electric field and the gas pressure. The drift velocity is obtained by two successive steps. First we obtain a rigorous expression for the drift velocity $w\left(c_0\right)$ of electrons having initial velocity $c_0$. By expanding this expression to first order under assumption (i) only, the usual expression for $w\left(c_0\right)$ is found again, and its validity is therefore extended even to the case of inelastic collisions, since assumption (ii) has not been used. In order to obtain the drift velocity $W$, the expression for $w\left(c_0\right)$ must be averaged over the initial distribution function $f_0\left(c_0\right)$. The initial velocity distribution, not being affected by the electric field, is isotropic when the differential collision cross section is isotropic. Therefore, the resulting integral equation for $f_0\left(c_0\right)$, though rigorous, has the same simplicity as the usual first-order expansion of the Boltzmann equation.