Theory of Positron Annihilation in Real Metals

A theory of positron annihilation applicable to real metals is developed. It is based on a band model for a solid and the ladder approximation for the electron-positron Green's function. The Block character of the positron as well as the conduction-band single-particle wave functions are included. The core electrons are treated within the tight-binding approximation. Throughout, the effect of the direct Coulomb coupling between the annihilating pair is stressed. The general formulas are then used to derive a theory of core annihilation. In the course of the derivation it was necessary to make a number of simplifying assumptions which restrict the theory to simple metals. We find that the contribution to the partial annihilation rate $R\left[\mathbf{p}\right]$ coming from a core electron is proportional to the square of the sum of two terms: the usual pth Fourier component of the product of the positron and core function plus another term accounting for the polarization of the ion core by the positron. After studying core annihilation in detail using an idealized model for the unoccupied electron Bloch states in sodium, we conclude that the second term is just as important as the first, although it has never been included in past treatments of this problem. Unfortunately our model is too crude to give quantitative results. An accurate calculation of core annihilation is very much more difficult than the corresponding computation of annihilation in a conduction-electron gas.