PHYSICAI. REVIEW A VOLUME 32, NUMBER 6 DECEMBER 1985 Electric field dependence of transient electron transport properties in rare-gas moderators B. Shizgal and D. R. A. McMahon* Department of Chemistry, University of British Columbia, Vancouver, British Columbia, Canada V6T1F6 (Received 29 May 1985) A discrete-ordinate method of solution of the time-dependent Boltzmann-Fokker-Planck equation for electron swarms in rare-gas moderators is employed in the study of the time dependence of the average electron energy, mobility, and transverse diffusion coefficient versus the strength of an externally applied electric field. The solution of the Fokker-Planck equation is based on the expan- sion of the solution in the eigenfunctions of the Lorentz-Fokker-Planck operator. With the transfor- mation to an equivalent Schrodinger eigenvalue problem, the eigenvalue spectrum is shown to be en- tirely discrete, thereby validating the eigenfunction-expansion approach. The effects studied include the effect of an electric field on the thermalization times, a comparison of the effects of moderators with and without Ramsauer minima in the momentum-transfer cross sections, and the effect of an external electric field on the transient negative-mobility phenomena predicted in an earlier paper. A comparison with experimental results for Xe shows good agreement with the calculations. I. INTRODUCTION The study of the transient behavior of a nonequilibrium ensemble of electrons in different moderators has impor- tant applications in many different fields, and the theoret- ical description of the approach to equilibrium is an im- portant endeavor. Examples of important applications in- clude the interpretation of electron-swarm experi- ments, ' delayed luminescence in gases, radiation chem- istry and biology, laser physics, ' discharge devices, and many other applications. The present paper is a continuation of the authors' re- cent works ' on the thermalization of low-energy elec- trons in rare-gas moderators. The earlier work was con- cerned with the transient behavior of the transport proper- ties of the electron population in the limit of zero external electric field. The present paper extends the earlier work to include a study of the effect of finite external electric field on the transient behavior of the distribution func- tion, the average electron energy, the mobility, the trans- verse diffusion coefficient, and the corresponding thermalization times. The determination of the electron- distribution function and the transient behavior involves the solution of the appropriate Boltzmann or Fokker- Planck (FP) equation. This electron-therm alization problem has been con- sidered by several authors with different methods of analysis. Olaussen and Hemmer" have carried out an analytical study of the asymptotic short-time transient mobility of a hard-sphere cross section. Mozumder' and Tembe and Mozumder' have assumed that the electron- distribution function is a pseudo-Maxwellian character- ized by a time-dependent temperature. A discussion and critique of this approach has been presented in the earlier papers. ' Knierem et a/. ' have employed traditional moment methods of solution of the FP equation. This ap- proach is based on the expansion of the distribution func- tion about a Maxwellian characterized with the time- dependent electron temperature. %'ith this expansion, dif- ferential equations for the lower-order moments are de- rived from the FP equation. Although the FP equation is linear, the resulting moment equations are characterized by 'nonconstant coefficients, due to the collision operator being parameterized with a time-dependent temperature. Consequently, a numerical integration of the moment equations is required. Monte Carlo simulations of the thermalization of electrons have also been carried out by Koura. ' Pitchford and Green' have studied the zero- field thermalization and the effect of an electric field for model systems characterized by constant cross sections and constant collision frequency. The present paper employs the discrete-ordinate (DO) method introduced in earlier papers by Shizgal, ' and Shizgal and Blackmore, ' and employed in the earlier study of electron therrnalization. ' The solution of the FP equation appropriate to the present problem is solved with a standard eigenfunction expansion of the distribu- tion function. This is a useful approach, since the re- ciprocals of the eigenvalues are the characteristic relaxa- tion times of the system. The DO method provides an ex- tremely efficient numerical procedure for the evaluation of the eigenvalues and the corresponding eigenfunctions, as well as the transport coefficients. The transport coeffi- cients are given as integrals over the distribution function. Their evaluation is facilitated due to the fact that the dis- tribution function is determined at the set of quadrature points for which rapid convergence of the integrations in- volved is obtained. The connection is made between the eigenvalue problem associated with the FP equation that occurs for the electron-thermalization problem, and an equivalent eigen- value problem based on a Schrodinger equation as described by Garrett, ' Braglia et al. , and in a recent paper on FP equations for bistable systems by Blackmore and Shizgal. ' The equivalence with a Schrodinger prob- lem is important, as it provides useful information with regard to the nature of the eigenvalue spectrum and, in particular, whether it has a continuum portion. We 32 3669 1985 The American Physical Society