Solution with Third-order Accuracy for the Electron Distribution Function in Weakly Ionized Gases (or in Intrinsic Semiconductors): Application to the Drift Velocity

Abstract

The first four components 10, I" 12 and 13 of the expansion in Legendre polynomials of the electron distribution function I are shown to be of order t:D, et, e2 and e3 respectively, with e = (m/M)'/2 where m and M are the masses of the electron and molecule respectively. This allows the solution of the so-called P3 approximation to the Boltzmann equation applied to a weakly ionized gas (or to an intrinsic semiconductor) in steady-state and uniform conditions and for dominant elastic collisions. However, nonphysical divergences appear in 10 and in the drift velocity W. This can be understood by the equivalence of the Boltzmann-Legendre formulation and the mean free path formulation in which a Taylor expansion is performed around the 'origin', i.e. for a -+ 0, where a = eE/m is the acceleration due to an external electric field E. Indeed, one sees that the expansion under the integral sign (integrals appear in the evaluation of transport quantities) leads to divergent integrals if the expansion is around a = O. Fortunately, it is easy to perform a Taylor expansion around a oft 0 in the mean free path formulation and then to find the corresponding expansion in Legendre polynomials outside the origin. In this way, explicit convergent expressions are found for 10, I" 12, 13 and W, with third-order accuracy in e = (m/M)'/2. This is better than the best preceding expression, that by Davydov-Chapman-Cowling, which has first-order accuracy only (it is the solution of the P, approximation to the Boltzmann equation).