Atoms in jellium

Abstract

We report one-electron calculations of the soft-x-ray $L_{2,3}$ absorption and emission of Na, Mg, and Al and the soft-x-ray $K$ absorption and emission of Li. The absorbing ion is placed at the center of a metal consisting of a valence band of electrons and of a uniform positive background which has a spherical hole removed from around the ion. Self-consistent solutions for states with and without a core hole are found using the density - functional approach. However, we eliminate self-interaction effects from the bound-state wave equation in order to get more realistic wave functions. This was originally proposed by Cowan but ignored subsequently. We think it is essential for getting good wave functions. In our solution both the core electrons bound to the central ion and the valence electrons are allowed to relax into the self-consistent solution. We find self-consistent potentials, charge densities, and threshold energies for soft-x-ray transitions. Scattering states in the relaxed potentials are used in spectrum calculations. We report cross sections found for Li, Na, Mg, and Al and compare them with experimental results. Moreover, we calculate the exponents $\alpha$, ${\alpha }_0$, and ${\alpha }_1$ essential to the threshold theory of Mahan and Nozières and de Dominicis (MND). For Li, Mg, and Al we get agreement with experimental results for $\alpha$ and ${\alpha }_0$. We get no agreement for Na but we attribute this not to a failure in MND theory but to an inadequacy in our model when used to find low-energy zero-angular-momentum valence-electron wave functions.