The spherical-solid model: An application to x-ray edges in Li, Na, and Al

In order to treat highly localized excitations in a solid a simple spherical model has been developed. The Coulomb potential from the nucleus at the site of the excitation is treated exactly, while the potentials from the surrounding sites are turned into pseudoion pontentials. Only the spherical average of these potentials is retained and separate self-consistent calculations for the ground state and for the excited state are made. These states are mistreated by the model but the difference accurately describes the large effects of polarization involved. The positions of the x-ray edges are obtained with an accuracy of 0.5 eV for singly ionized levels and 1 eV for doubly ionized levels. The phase shifts relevant to the Nozières-De Dominicis theory of x-ray absorption and emission edges and x-ray photoemission line shapes are also calculated. We obtain a noticeable many-body enhancement of the $L_{2,3}$ edge in Na while the many-body effect in Li and Al is negligible. Our exponents agree with those extracted from recent x-ray photoemission experiments, but they do not agree with those extracted from x-ray emission or absorption spectra. We argue that the exponents taken from the photoemission experiment are more reliable but no definite conclusion concerning the validity of the present many-body theory should be drawn until the experimental situation has become more clear.