Electron energy distribution functions and transport coefficients for the rare gases

Abstract

The previous method of finding the distribution function of electrons in weakly ionized gases (Lewis 1958) in which elastic loss was neglected is now extended so that all collision processes may be taken into account, including those due to elastic, rotational and vibrational collisions. The Smit (1936) derivation, which is related to the more usual treatment based on the Boltzmann transport equation, is used as the starting-point. Distribution functions are calculated for helium, neon and argon over ranges of E/p from 5 to 250 V cm$^{-1}$ mm Hg$^{-1}$ and agree well with the few isolated cases reported in the literature. At E/p = 5, these bear some resemblance to the standard Maxwellian, Druyvessteyn and Townsend-Ramsauer forms but show significant differences in the range of excitation and ionization energies. The way in which the inclusion of the elastic collision loss affects the distribution function at low E/p and depends on the atomic weight of the gas is shown clearly. From the distribution functions, the diffusion coefficients, electron mobilities, mean energies and Townsend `$\alpha$' coefficients are determined. While no experimental values of the diffusion coefficients are available for comparison, the derived electron mobilities agree well with other theoretical values and with those experimental ones which have been obtained from the use of a time of flight measurement. Comparison of the calculated mean electron energies with the experimental Townsend values which imply a Maxwellian distribution, shows that great care has to be exercised in the use of the Einstein relation for those gases in which the distributions are far from Maxwellian. The calculated Townsend $\alpha$ values agree with the experimental ones to the degree of accuracy that can be expected from the measured cross-sections and this is a sensitive test of the soundness of the theory. It is concluded for the rare gases in which no major energy loss occurs below the onset of optical excitation that the energy distributions are far from Maxwellian. Further calculations for molecular gases show that rotational and vibrational collisions tend to produce electron energy distributions of Maxwellian form.