Phase-shift analysis of the impurity problem in cubic metals. I.

Abstract

The original phase-shift analysis of the impurity problem in metals, done first by Friedel, was confined mainly to crystals having spherical energy band. The present paper extends the phase-shift approach to any band of a cubic metal. At the first step the band of states $s$ is considered but then the method is generalized to states $p$ and $d$. The electron wave functions are the non-Bloch linear combinations of atomic orbitals extended throughout the crystal volume which is a large but finite sphere. It is shown that the electron density of a crystal perturbed by an isolated impurity can be represented by the wave functions of the perfect crystal whose coefficient functions are changed in their argument by a phase shift. The approach done in terms of phase shifts is shown to be equivalent to the Green's-function approach. For an impurity potential which is weak and confined to one lattice site the phase shift of any wave function is equal to the density of states contributed by this function at the position of the impurity times the potential strength. Therefore phase shifts depend on the position of the impurity. An analog of Friedel's sum rule can be derived also in the present theory. The total number of charge displaced in the crystal is proportional to the total density of states at the Fermi level. The scattering cross section due to an impurity can be expressed with the aid of phase shifts and in terms of parameters which define the unperturbed coefficient functions.