Exact solution of the Kahana equation for a positron in an electron gas

Abstract

It is pointed out that the spherical approximation used by Kahana when solving the Bethe-Goldstone equation for a positron in an electron gas is not satisfactory. In this paper the Kahana equation is treated as a two-dimensional integral equation and solved by expressing the electron-positron wave function as a series of Legendre polynomials. From the solution obtained in this way, the new values of the enhancement factors and total annihilation rates are computed. It is shown that the shape of the enhancement curve for large momenta differs from that obtained by Kahana. Moreover, the total screening charge and its distribution are found. These are analyzed and compared with equivalent distributions for $r_s=2,3, \mathrm{and} 4$ obtained by Sjölander and Stott and also Arponen and Pajanne. The essential contribution of higher partial waves to the screening cloud neglected by previous authors is emphasized.