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 expansion of the solution in the eigenfunctions of the Lorentz-Fokker-Planck operator. With the transformation to an equivalent Schrodinger eigenvalue problem, the eigenvalue spectrum is shown to be entirely 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.

An efficient discrete-ordinate method of solution of the time-dependent Boltzmann equation is employed in the calculation of the zero-field electron mobility and diffusion coefficients for hotelectron thermalization in rare-gas moderators. The discrete-ordinate method is modified to permit a rescaling of the quadrature points. This procedure is somewhat analogous to the twotemperature-moment methods employed in the theoretical analysis of electron swarms. The timedependent transport coefficients are given as a sum of exponential decay terms characterized by the discrete eigenvalues of the Lorentz-Fokker-Planck operator for elastic electron-atom collisions. For argon, krypton, and xenon, the time dependence is strongly influenced by the Ramsauer-Townsend minimum and leads to maxima in the transient mobility and diffusion coefficient. Helium and neon with hard-sphere-like cross sections exhibit transient mobilities which initially are below the thermal zero-field mobility and then increase to the thermal mobilities as the electron distribution approaches equilibrium.The transient mobility for cross sections with Ramsauer minima are sufficiently sensitive to the details of the cross sections such that it may be feasible to distinguish between different cross sections experimentally.The calculations also indicate that the transient mobility is insensitive to the initial distribution function. A nonequilibrium phenomenon not previously recognized is the possibility of a negative transient mobility which occurs provided that the momentumtransfer cross section increases sufficiently rapidly with energy.

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In the Franck–Hertz experiment one observes the effect of inelastic collisions in which fixed quanta of energy are exchanged between electrons and atoms. It is shown here that one can also readily demonstrate with a Franck–Hertz apparatus energy-dependent features of the elastic collision cross section. For mercury vapor of sufficiently high pressure, elastic electron–atom collisions between the grid and the anode are able to energy analyze the electrons so that the characteristic peaks and troughs in the anode current are still observed without the traditional retarding field to separate off the lowest-energy electrons. This is because in mercury vapor the most energetic electrons have the longest mean free path, are more penetrating through the gas, and are the most likely electrons to reach the anode. The electron transport theory for this effect is developed and applied to a crude determination of the electron energy distribution. Not surprisingly, the electron energy distribution in this experiment consists of two electron groups separated in energy by 4–5 eV consistent with the known 61S-63P energy-level separation in mercury of 4.9 eV.