Electron Interaction and Positron Annihilation in Electron Gases

Abstract

An intuitive many-body treatment is presented of the distribution in the total momentum of two gamma rays emitted in annihilation of positrons in a gas of interacting electrons. This treatment results in a momentum distribution of photon pairs which is very close to that of the ideal Fermi distribution of quasi-electrons in the electron gas. This seems to be supported by Stewart's experiment for sodium metal. 1 ) After a careful analysis he has concluded that the momentum distribution of r-ray pairs is very similar to that of the ideal Fermi distribution of conduction electrons in sodium metal. Stewart's result can easily be understood if one adopts the Sommerfeld model discarding both correlation effects and lattice effects completely. The same model, however, fails to account for the observed variation in total annihilation rates of positrons in metals with the conduction electron densities as well as their absolute values. To remove this difficulty some account should inevitably be taken of the correlation effect between the positron and an annihilating electron, as was shown by several authors. 2 ) The most elaborate of these calculations will be due to Kahana. His calcu­ lation, however, resulted in a non-negligible deviation of the above momentum distribution from the ideal Fermi shape, as illustrated in Fig. 1. This curve was derived by the numerical integration of a Bethe-Goldstone equation for an annihilating electron-positron pair within the sea of electrons. In his approxi­ mation the two-body correlation between the positron and an annihilating electron is fully taken into account, but no higher-order correlation is considered except those which are responsible for the screening of the attractive electron-positron force. At the limit of high electron densities his approximation will be adequate, but in electron densities of real metals it cannot be warranted. On the contrary, if the correlation among electrons are fully taken into account, but instead the electron-positron correlation is completely disregarded, then the resulting momentum distribution of r-ray pairs would be just that of