Inelastic scattering of electrons in a crystalline environment may be
represented by a complex non-Hermitian potential. Complete generalized
expressions for this inelastic-electron-scattering-potential matrix,
including virtual-inelastic scattering, are derived for outer-shell el
ectron and plasmon excitations. The relationship between these express
ions and the general anisotropic dielectric-response matrix of the sol
id is discussed. These generalized expressions necessarily include the
off-diagonal terms representing effects due to departure from transla
tional invariance in the interaction. Results are presented for the di
agonal band-structure-dependent inelastic and virtual-inelastic scatte
ring potentials for Si, from a calculation of the inverse dielectric m
atrix in the random-phase approximation. Good agreement is found with
experiment as a function of incident energies from 10 eV to 100 keV. A
nisotropy effects, and, hence, the interaction delocalization represen
ted by the off-diagonal scattering-potential terms are found to be sig
nificant below 1 keV.