Correlation of Electron Amplitudes in Impure Metals

The amplitude correlation function in an impure metal is the average over all possible arrangements of impurities of the correlation $\Sigma n^{}{{\psi }_n}^{*}\left({\mathrm{r}}^{\prime }\right){\psi }_n\left(\mathrm{r}\right)\delta (E-E_n)$, where the ${\psi }_n$ are eigenfunctions of a one-electron Hamiltonian including impurity potentials. The $\delta$ function restricts the sum to states $n$ on the energy shell $E$. This quantity, whose usefulness in studies of impure metals is pointed out, is evaluated from first principles by means of multiple scattering techniques. If the impurity density is not too high, the amplitude correlation is $\rho \mathrm{}\left(E\right)\mathrm{}sin\left[{\mathrm{}\left(2mE\right)\mathrm{}}^{\frac{1}{2}}R\right]\exp (-\frac{R}{2\lambda }){\left[{\mathrm{}\left(2mE\right)\mathrm{}}^{\frac{1}{2}}R\right]}^{-1\mathrm{}}$, where $\rho \mathrm{}\left(E\right)\mathrm{}$ is the density of states and $R=|\mathrm{r}-{\mathrm{r}}^{\prime }|$.