Positron annihilation in metals

We derive a nonlinear integral equation, which has three-body correlations built in it, for the Fourier transform of the polarization charge around a positron in an electron liquid. The calculated annihilation rate of the positron in the entire metallic density range, besides being in very good agreement with experiment, merges smoothly into the mean positronium lifetime for larger density. To illustrate the large nonlinear effects, the polarization charge around a fixed proton for a density ${r}_{s}=2$ has also been calculated.