Ladder-Graph Approximation in Positron Annihilation

The annihilation rate of the positron in a pure Fermi gas has been calculated (a) by using a ladder-graph approximation with an exponentially screened Coulomb interaction (which is nearly the static limit of the random-phase-approximation interaction), (b) by taking into account certain graphs up to third order in the same interaction, and (c) by using a better approximation for the interaction in a ladder-graph series. The contributions from graphs other than ladders was found to be 2-3% up to second order and 4-5% to third order. The results appear to establish that a ladder sum with a quite simple interaction approximates well the two-body propagator. The modification of the interaction has only a slight effect on the rate. At low electron densities, the ladder sum diverges because of the existence of bound states, which are proved to exist at any density as poles in the two-body propagator.