Relaxation constants in electron thermalization: Comparison of WKB and SWKB eigenvalues with exact results

The relaxation to equilibrium of a nonequilibrium distribution of electrons in gases is governed by a linear–Planck equation. The decay in time of the electron average energy as well as other transport properties can be expressed as a sum of exponential terms with each term characterized by an eigenvalue of the Fokker–Planck operator. This eigenvalue problem can be transformed into a Schrödinger equation with a potential function for which the Hamiltonian factors and belongs to the class of potentials encountered in supersymmetric quantum mechanics. The eigenvalues are calculated with the standard Wentzel–Kramers–Brillouin (WKB) semiclassical quantization condition as well as with a modified semiclassical quantization condition based on supersymmetric quantum mechanics (SWKB). The eigenvalues calculated in these ways are compared with the exact values obtained by the diagonalization of the operator in a large basis set. The applications considered are for the relaxation of electrons in the inert gases for which the electron–atom momentum transfer cross sections are available. The SWKB quantization condition gives results in much better agreement with the exact result (often within much less than 1%) than the WKB approximation.