Effect of a finite energy spread of the positron beam on the threshold behaviour of the positron annihilation cross section

In a recent paper [Phys. Rev. Lett. 88, 163202 (2002)] we established the threshold behavior of the cross section of positron-atom annihilation into two gamma quanta near the positronium (Ps)-formation threshold. Here, the near-threshold behavior of the positron 3 gamma annihilation cross section and its relation to the ortho-Ps-formation cross section are determined. We also analyze the feasibility of observing these effects by examining the effect of the ?nite-energy resolution of a positron beam on the threshold behavior.