Diffusion theory of electrons in a uniform electric field: Time-of-flight analysis

A different approach to time-of-flight (TOF) analysis is described in this paper. The view is adopted that average energy must obey a conventional continuity equation like that obeyed by electron density, leading to a pair of coupled equations consisting of an isotropic diffusion equation with two nonuniform transport coefficients and an isotropic-heat-flow equation with three nonuniform transport coefficients. By a finite-difference technique, these equations are solved simultaneously for density and average energy, both of which vary in time and space. This approach is based on energy moments of the scalar equation for the isotropic part of the velocity distribution function ${\mathit{f}}_0$ that results from the well-known two-term Legendre expansion. The nonuniform transport coefficients are expressed as functionals of the average energy of the electrons. These functionals are obtained by evaluating certain integrals involving ${\mathit{f}}_0$, which must be assumed, and the energy-dependent momentum transfer collision frequency. The method is applicable primarily to those gases for which the collision frequency can be expressed as a power of electron energy. In the special case of constant cross section, the predicted transit time of a TOF pulse agrees with the accepted value obtained by the density-gradient-expansion technique of solving the Boltzmann equation, while the predicted pulse width is 11% larger than the accepted value obtained by the same technique. Possible reasons for this difference are discussed.