Energy-diffusion calculation of the electron-swarm distribution function.

Abstract

An approximate treatment of the Boltzmann collision integral for electrons in a gas, valid for small fractional average energy loss, is studied. It is essentially a Fokker-Planck expansion in energy space, including mean energy loss (dynamical friction) and energy straggling (coefficient of diffusion). When applied to electron swarms in weakly ionized gases, treating angle variables in the two-term Legendre series, there results a useful, physically meaningful, differential equation for the time evolution of the energy spectrum in a time-dependent electric field. Elastic scattering, and inelastic and superelastic energy-transfer collisions are included. The time-independent solution in a constant field is a simple approximate expression for the steady-state energy spectrum of swarm electrons. The physical meaning of its functional form is made clear by showing its relation to ordinary diffusion-convection theory. Previous spectra by Pidduck [Proc. R. Soc. London Ser. A 88, 296 (1913); Proc. London Math. Soc. 15, 89 (1916); Q. J. Math. 7, 199 (1936)]; Druyvesteyn [Physica 10, 61 (1930)]; Davydov [Phys. Z. Sowjetunion 8, 59 (1935)]; Morse, Allis, and Lamar [Phys. Rev. 48, 412 (1935)]; Chapman and Cowling [The Mathematical Theory of Non-Uniform Gases, 2nd ed. (Cambridge University Press, Cambridge, 1952), p. 350]; and Wannier [Am. J. Phys. 39, 281 (1971)] are special cases.The reasons for the inadequacy of the continuous-slowing-down approximation (CSDA) become apparent. The new spectrum is exact in the limit of small quantum transition energies. It is further shown that the CSDA violates detailed balance. Consequences of detailed balance on the loss function and related functions are investigated, and the Boltzmann H theorem is studied. During non-steady-state behavior, collisions may increase or decrease the swarm entropy, but the effect of the electric field is always to increase entropy. The spectrum is used with experimental cross sections to compute transport coefficients in ${\mathrm{O}}_{2}$ and ${\mathrm{N}}_{2}$, in both of which fractional average energy loss is acceptably small over most energy ranges. Agreement with compiled swarm data is excellent over more than four orders of magnitude in E/N for most coefficients, except at certain energies in ${\mathrm{N}}_{2}$ that strain the approximation's validity. In the absence of an electric field, the inclusion of energy straggling provides a treatment of spectral relaxation valid for arbitrary energies that is an improvement over common mean-stopping-power formulas.