Theory of electron scattering from solids

We present a theory which describes effects of the losses which determine the electron's mean free path, and which is valid not only for elastic, but also for inelastic, scattering. The results are given in terms of one-electron expressions involving damped one-electron functions and optical potentials. The operators in these expressions are expectation values over correlated many-electron states for the solid. The development follows similar lines as in recent work on photoemission and extended x-ray-absorption fine-structure spectroscopy by Bardyszewski and Hedin. Some of the present results were obtained by Fujikawa and Hedin in a study of scattering theory based on the Low formulation and a quasiboson representation. The theory involves an expansion in the diagonal (or coherent) operators introduced by Van Hove, and uses Feshbach projection-operator techniques. The theory is developed in three steps. First, we regard the scattering electron as an external particle, and we can then work with a product space of the scattering electron and the electrons in the solid. Second, we account for bare exchange between the scattering and target electrons, but keep different sets of one-electron states for the two types of electrons, and do not allow any coupling terms that annihilate the scattering electron. Third, we sketch a theory allowing intermediate, virtual states, where the scattering electron has been absorbed by the solid into a resonant state. We argue in detail that excited-state and ground-state optical potentials should be simply related in the case of extended excitations. All our results are quite general, i.e., they are valid not only for solids, but also for atoms and molecules. The central objects in our theory, the optical potentials, are discussed in some detail. In particular, they are related to the Van Hove, the Francis-Watson, and the Bell-Squires potentials