Polarization of the Electron Gas in Metals by Substitutional Impurities

The ground-state energy of the system of impurities and conduction electrons in a metal has been obtained in the high electron density limit. The procedure used is an extension of the Wentzel method applied to a reduced Hamiltonian which includes an electron-impurity interaction. It is reduced in the sense that the Coulomb interaction between electrons and the electron-impurity interaction are only effective in raising an electron in a state below the Fermi level to one above and vice versa. The ground-state energy is then obtained by a canonical transformation. The shift in energy of the ground state of the electron gas, due to the introduction of the impurities, is quadratic in the electron-impurity matrix element. Higher order processes in this matrix element do not contribute since they are represented by unlinked diagrams. Considering this shift in the ground-state energy expandable in powers of $r_s$, a measure of the average inter-electronic distance, we show the leading or lowest order term in this parameter to go as $\left(\frac{n}{n_e}\right){r_s}^{-\frac{1}{2}}$, where $n$ is the impurity density and $n_e$ the electron density. Impurity locations are assumed random. All processes omitted in the reduction of the Hamiltonian are shown to contribute to higher powers in $r_s$. The role of exchange is indicated.