Comparative calculations of electron-swarm properties inN2at moderateENvalues

Abstract

The recently developed density gradient and multiterm spherical harmonic expansion technique for the numerical solution of the electron Boltzmann equation is evaluated by comparison of results with those obtained using the conventional two-term spherical harmonic technique and using the Monte Carlo technique. Comparisons are made of electron energy distributions, transport coefficients, and excitation coefficients for electrons in ${\mathrm{N}}_{2}$ at moderate electric-field to gas-density ratios $\frac{E}{N}$ where the large cross section for vibrational excitation leads to significant errors when conventional solutions of the Boltzmann equation are used. The $\frac{E}{N}$ values were varied from (1 - 200)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}21}$ V ${\mathrm{m}}^{2}$, corresponding to mean electron energies from 0.3 to 5 eV. The first two terms of the density-gradient expansion are used. As the number of terms $n$ in the spherical harmonic expansion is increased from the conventional two terms to $n\ensuremath{\ge}4$, the spherically symmetric component of the electron energy distribution and the transport and excitation coefficients become independent of $n$ and close to results obtained from the Monte Carlo calculation. The errors resulting from the use of two spherical harmonics at $\frac{E}{N}=7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}20}$ V ${\mathrm{m}}^{2}$, for example, are approximately 1, 5, and 30% for the drift velocity, the transverse diffusion coefficient, and the electronic excitation coefficients, respectively. For the lower $\frac{E}{N}$ values the errors in the transport coefficients are approximately proportional to an energy-loss-per-collision parameter. The variation of the coefficients of the lower-degree terms in the spherical-harmonic expansion with $n$ is examined through a comparison with an analytical solution of the Boltzmann equation for a model atom valid in the case of low $\frac{E}{N}$ and high electron energies. Monte Carlo techniques are used to show that the effects of electrodes are negligible for the conditions of recent measurements of electron excitation coefficients in ${\mathrm{N}}_{2}$.