Annihilation of the electron-positron pairs in the positronium ion Ps${}^{-}$ and bi-positronium Ps${}_2$

Abstract

Rates of the two-, three-, four- and five-photon annihilations of the electron-positron pairs are determined numerically for the three-body positronium ion Ps${}^{-}$ ($e^{-}{e}^{+}{e}^{-}$) and four-body bi-positronium `molecule' Ps${}_2$ ($e^{-}{e}^{+}{e}^{-}{e}^{+}$). The values obtained in our computations are ${\Gamma }_{2\gamma }($Ps${}^{-})\approx$ 2.08048530525$\cdot 10^9$ $se{c}^{-1}$, ${\Gamma }_{3\gamma }($Ps${}^{-})\approx$ 5.6364151550$\cdot 10^6$ $se{c}^{-1}$, ${\Gamma }_{4\gamma }($Ps${}^{-})\approx$ 3.075$\cdot 10^3$ $se{c}^{-1}$, ${\Gamma }_{5\gamma }($Ps${}^{-})\approx 5.383$ $se{c}^{-1}$ and ${\Gamma }_{2\gamma }($Ps${}_2)\approx$ 4.4385952$\cdot 10^9$ $se{c}^{-1}$, ${\Gamma }_{3\gamma }($Ps${}_2)\approx$ 1.202497$\cdot 10^7$ $se{c}^{-1}$, ${\Gamma }_{4\gamma }($Ps${}_2)\approx$ 6.562$\cdot 10^3$ $se{c}^{-1}$, ${\Gamma }_{5\gamma }($Ps${}_2)\approx$ 11.484 $se{c}^{-1}$. The four- and five-photon annihilation rates are significantly smaller than the corresponding two- and three-photon annihilation rates known for these systems. We also determine the rates of one- and zero-photon annihilation for the Ps${}^{-}$ ion and Ps${}_2$ system. The corresponding numerical values are ${\Gamma }_{1\gamma }($Ps${}^{-})\approx$ 3.82491$\cdot 10^{-2}$ $se{c}^{-1}$, ${\Gamma }_{1\gamma }($Ps${}_2)\approx$ 1.94188$\cdot 10^{-1}$ $se{c}^{-1}$ and ${\Gamma }_{0\gamma }($Ps${}_2)\approx$ 2.32197$\cdot 10^{-9}$ $se{c}^{-1}$. Pusblished almost `as is' in Phys. Rev. A {\bf 80}, 014502 (2009).

10 pages

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