Positronium negative ion

The positronium negative ion ( Ps − {\displaystyle {\text{Ps}}^{-}} ) is an exotic atom composed of three elementary particles: two electrons ( e − {\displaystyle e^{-}} ) and one positron ( e + {\displaystyle e^{+}} ). It is an unstable system due to electron-positron annihilation, with a lifetime of 479 ps. Observed for the first time in 1981, it is the simplest three-body system containing both matter and antimatter and composed exclusively of leptons, serving as a reference system for studying the quantum mechanical three-body problem. Additionally, the production of this ion is the initial step toward generating energy-tunable positronium beams for fundamental physics research.

History The existence of a bound state between two electrons and a positron was first theoretically predicted by John Archibald Wheeler in 1946. In his work Polyelectrons, he theorized the existence of exotic bound systems, including the di-positronium, observed for the first time in 2007, and the positronium positive ion. Since then, numerous theoretical works on Ps − {\displaystyle {\text{Ps}}^{-}} have been published, concerning its binding energy, lifetime and other characteristics. Nevertheless, given the low production efficiency and its short lifetime, the first experimental observation was achieved only in 1981 by Allen P. Mills. In 2007 a new experimental technique achieved the first production of this ion with increased conversion efficiencies (up to 1.5%). This approach to generate Ps − {\displaystyle {\text{Ps}}^{-}} provided a positron-to- Ps − {\displaystyle {\text{Ps}}^{-}} conversion efficiency significantly higher compared to the method used by Mills (less than 0.1%).

Energy levels and annihilation

The internal structure of Ps − {\displaystyle {\text{Ps}}^{-}} presents characteristics similar to an electron weakly bound to a positronium (Ps) atom. The expectation values for the positron-electron and electron-electron distances have been computed, obtaining 2.90 Å ( e + {\displaystyle e^{+}} - e − {\displaystyle e^{-}} distance) and 4.52 Å ( e − {\displaystyle e^{-}} - e − {\displaystyle e^{-}} distance). As a three-body quantum system, the Schrödinger equation for Ps − {\displaystyle {\text{Ps}}^{-}} cannot be solved analytically, and some approximation methods must be applied. Furthermore, the Born–Oppenheimer approximation cannot be applied, since the three bodies possess exactly the same mass. Therefore, theoretical determinations of the ground state energy have been performed through precision numerical computation, yielding a value of E g = − 7.13 eV {\displaystyle E_{\mathrm {g} }=-7.13\,{\text{eV}}} . The negative sign indicates that Ps − {\displaystyle {\text{Ps}}^{-}} is stable against dissociation into its constituent particles. Nevertheless, it is an unstable system against annihilation. This occurs, from a quantum mechanical point of view, when the wave functions of an electron and of the positron partially overlap in space. The positron can possess up or down spin orientation, without restrictions. Nevertheless, the two electrons must be in a singlet state, due to the Pauli exclusion principle. As a consequence, the annihilation process may occur either with an electron that possesses the same spin orientation of the positron or with an electron with opposite spin. These two cases present different properties due to the distinct selection rules for the processes. In the first case, an odd number of gamma rays is generated (predominantly three), as it happens for ortho-positronium. In the second case, an even number of gamma rays is generated (most probably two), as in the case of para-positronium. Taking the spin-average between the theoretical decay rates of ortho-Ps and para-Ps results in:

Γ = 3 4 ⋅ Γ ortho-Ps + 1 4 ⋅ Γ para-Ps ≈ 2.0871 ns − 1 {\displaystyle \Gamma ={\frac {3}{4}}\cdot \Gamma _{\text{ortho-Ps}}+{\frac {1}{4}}\cdot \Gamma _{\text{para-Ps}}\approx 2.0871\,{\text{ns}}^{-1}}

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