Improved age-diffusion model for low-energy electron transport in solids. I. Theory

We have developed in this paper a semianalytical electron transport model designed for parametric studies of secondary-electron emission induced by low-energy electrons (keV range) and by fast light ions (100 keV range). The primary-particle transport is assumed to be known and to give rise to an internal electron source. The importance of the nearly isotropic elastic scattering in the secondary-electron energy range (50 eV) and the slowing-down process strongly reduce the influence of the anisotropy of the internal electron source, and the internal electron flux is nearly isotropic as is evidenced by the experimental results. The differential energy behavior of the inelastic scattering kernel is very complicated and the real kernel is replaced by a synthetic scattering kernel of which parameters are obtained by energy and angle moments conservation. Through a $P_1$ approximation and the use of the synthetic scattering kernel, the Boltzmann equation is approximated by a diffusion–slowing-down equation for the isotropic part of the internal electron flux. The energy-dependent partial reflection boundary condition reduces to a Neumann-Dirichlet boundary condition. An analytical expression for the Green’s function of the diffusion–slowing-down equation with the surface boundary condition is obtained by means of approximations close to the age-diffusion theory and the model allows for transient conditions. Independently from the ‘‘improved age-diffusion’’ model, a correction formula is developed in order to take into account the backscattering of primary electrons for an incident-electron problem.