Scattered impurity states in transition metals. II. Approximate solutions using phase shifts.

Abstract

To solve the problem of a transition metal with an impurity, the one-electron linear-combination-of-atomic-orbitals wave functions are constructed in terms of spherical harmonics multiplied by spherical Bessel or spherical Neumann functions. These solutions are approximated afterwards by the standing-wave-like projected-coefficient (SWLPC) functions, calculated in the preceding paper, and their modifications. At large distances from the impurity area the impurity wave function has the form of a scattered wave. Any solution for an imperfect crystal differs from that obtained for a perfect crystal by the presence of a phase shift. In each case the electron states can be quantized according to the requirement that their wave functions vanish at the crystal boundary, which is assumed to be spherical. This makes the analysis of the impurity problem in a crystal very similar to that done by Friedel in the free-electron case. However, any present phase shift depends on the symmetry index of the SWLPC function, the branch of the solution of the secular problem, and the wave vector belonging to the irreducible part of the Brillouin zone. The phase shifts, which can be expressed in terms of the perturbation matrix, are proportional to corresponding components of the density of states of the unperturbed metal. An expression for the change in the number of metal electrons due to scattering is obtained. In its derivation the phase shifts found in the present paper enter into a modified Friedel's formula given originally for the free-electron case.