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Physical Review A

Internal structure of the positronium ion Ps−

Dario Bressanini

Published Aug 23, 2021• 6 citations
Publisher

Abstract

The positronium ion ${\mathrm{Ps}}^{--}$ is often qualitative described as a positronium Ps with a loosely bound second electron, since the value of the annihilation rate for ${\mathrm{Ps}}^{--}$ is nearly equal to that of the spin-averaged neutral positronium. This similarity however has never been fully explained. To study its internal structure, we decompose the electron-positron distribution function of ${\mathrm{Ps}}^{--}$ into the sum of two terms: one for the electron closest to the positron and the other for the farthest. We show that the inner electron behaves almost as if the outer electron were not present, forming a Ps substructure inside ${\mathrm{Ps}}^{--}$ and that the effect of electron correlation is such that the outer electron tries to stay as close as possible to the positron but without interfering too much with the distribution of the inner electron. We show that this phenomenon is closely related to the problem of the critical stability of the system $({Z}^{+},\phantom{\rule{0.28em}{0ex}}{e}^{--},\phantom{\rule{0.28em}{0ex}}{e}^{--})$ with varying $Z$ charge.

DOI
10.1103/physreva.104.022819

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Positronium scattering from closed-shell atoms and ions

Jim Mitroy, Igor A. Ivanov

The scattering of orthopositronium from He, Ne, Ar, ${\mathrm{Li}}^{+},$ ${\mathrm{Na}}^{+},$ and ${\mathrm{K}}^{+}$ is investigated with the fixed-core stochastic variational method. The scattering length for $\mathrm{Ps}\ensuremath{-}\mathrm{He}$ scattering was ${1.57a}_{0},$ a value consistent with the threshold cross sections derived from three positron lifetime experiments. The scattering lengths for the $\mathrm{P}\mathrm{s}\ensuremath{-}\mathrm{N}\mathrm{e}$ and $\mathrm{P}\mathrm{s}\ensuremath{-}\mathrm{A}\mathrm{r}$ systems were ${1.55a}_{0}$ and ${1.79a}_{0},$ respectively. That there was only a 15% variation among the scattering lengths is compatible with the experiment of Coleman et al. [J. Phys. B 27, 981 (1994)], who found the low-energy cross sections for He, Ne, and Ar to be about the same size. The scattering lengths for Ps scattering from ${\mathrm{Li}}^{+},$ ${\mathrm{Na}}^{+},$ and ${\mathrm{K}}^{+}$ were ${12.9a}_{0},$ ${28.5a}_{0},$ and $\ensuremath{-}{1.9a}_{0}.$ The relatively small magnitude of the ${\mathrm{P}\mathrm{s}\ensuremath{-}\mathrm{K}}^{+}$ scattering length strongly supports previous suggestions that the ${\mathrm{P}\mathrm{s}\ensuremath{-}\mathrm{K}}^{+}$ system does not support a bound state. The annihilation parameter ${}^{1}{Z}_{\mathrm{eff}}$ has also been computed as part of the analysis. The present values of ${}^{1}{Z}_{\mathrm{eff}}$ are about 2.5--3 times smaller than the accepted experimental values since short-range electron-positron correlations were not taken into consideration when the annihilation matrix element was evaluated.

Physical Review A•Dec 14, 2001•44 citations
The positronium ion Ps−−{\mathrm{Ps}}^{--}Ps−− is often qualitative described as a positronium Ps with a loosely bound second electron, since the value of the annihilation rate for Ps−−{\mathrm{Ps}}^{--}Ps−− is nearly equal to that of the spin-averaged neutral positronium. This similarity however has never been fully explained. To study its internal structure, we decompose the electron-positron distribution function of Ps−−{\mathrm{Ps}}^{--}Ps−− into the sum of two terms: one for the electron closest to the positron and the other for the farthest. We show that the inner electron behaves almost as if the outer electron were not present, forming a Ps substructure inside Ps−−{\mathrm{Ps}}^{--}Ps−− and that the effect of electron correlation is such that the outer electron tries to stay as close as possible to the positron but without interfering too much with the distribution of the inner electron. We show that this phenomenon is closely related to the problem of the critical stability of the system (Z+,e−−,e−−)({Z}^{+},\phantom{\rule{0.28em}{0ex}}{e}^{--},\phantom{\rule{0.28em}{0ex}}{e}^{--})(Z+,e−−,e−−) with varying ZZZ charge.
The positronium ion (Ps−\mathrm{Ps}^-Ps−), a coulombic three-body bound state of two electrons and a positron, predominantly decays via electron-positron annihilation into electromagnetic final states. While its radiative decay channels have been extensively studied, much less attention has been given to weak processes in this system. In this work, we investigate the rare decay Ps−→e−νμνˉμ\mathrm{Ps}^- \to e^- ν_μ\barν_μPs−→e−νμ​νˉμ​, obtained by replacing the photon in Ps−→e−γ\mathrm{Ps}^- \to e^- γPs−→e−γ with a virtual ZZZ boson. Treating the three-body process as an effective two-body transition, Ps−→e−Z∗(→νμνˉμ)\mathrm{Ps}^- \to e^- Z^*\left(\to ν_μ\barν_μ\right)Ps−→e−Z∗(→νμ​νˉμ​), we compute the decay rate by explicitly evaluating all spin configurations of the initial bound state and final particles. The result agrees with that obtained using the standard spin-summation formalism of quantum field theory. We find that the branching ratio is comparable to that of the weak decay of ortho-positronium, o-Ps→γννˉ\mathrm{o\text{-}Ps} \to γν\barνo-Ps→γννˉ.
Various geometrical and energetical (bound-state) properties of the positronium negative ion $({\mathrm{Ps}}^{\ensuremath{-}}$ or ${e}^{\ensuremath{-}}{e}^{+}{e}^{\ensuremath{-}})$ are determined by using highly accurate variational wave functions. These wave functions have been constructed by applying the advanced two-stage strategy proposed by Frolov [Phys. Rev. A 57, 2436 (1998)]. The determined total energy of the ground state $E=\ensuremath{-}0.2620050702329757\mathrm{a}.\mathrm{u}.$ is the lowest and most accurate value obtained for this system to date (the corresponding binding energy equals $\ensuremath{-}0.326674721317821\mathrm{eV}).$ The computation of the second-order relativistic corrections $(\ensuremath{\simeq}{\ensuremath{\alpha}}^{2})$ to the total energy is discussed also. The general form of the Breit-Pauli Hamiltonian in the relative coordinates for the ${\mathrm{Ps}}^{\ensuremath{-}}$ ion is presented.
Electron-positron annihilation rates calculated directly from the electron and positron densities are known to underestimate the true annihilation rate. A correction factor, known as the enhancement factor, allows for the local increase of the electron density around the positron caused by the attractive electron-positron interaction. Enhancement factors are given for positrons annihilating with the 1s1s1s electron in H, He+{\mathrm{He}}^{+}He+, He, Li2+{\mathrm{Li}}^{2+}Li2+, and Li+{\mathrm{Li}}^{+}Li+. The enhancement factor for a free positron annihilating with He+{\mathrm{He}}^{+}He+ and He is found to be close to that of ortho-positronium (i.e., Ps in its triplet state) annihilating with these atoms. The enhancement factor for Ps-He scattering is used in conjunction with the known annihilation rate for pickoff annihilation to derive a scattering length of 1.47a01.47{a}_{0}1.47a0​ for Ps-He scattering. Further, enhancement factors for e+{e}^{+}e+-Ne and e+{e}^{+}e+-Ar annihilation are used in conjunction with the pickoff annihilation rate to estimate scattering lengths of 1.46a01.46{a}_{0}1.46a0​ for Ps-Ne scattering and 1.75a01.75{a}_{0}1.75a0​ for Ps-Ar scattering.
Although it can probably only exist within a confining potential, a positronium (Ps) Bose-Einstein condensate (BEC) would be interesting because it could be viewed as the ultimate member of Wheeler’s polyelectron series, e+, Ps, Ps−, Ps+, Ps2, …, Ps-BEC [1], not counting high density neutral e+-e− plasmas. A Ps-BEC could be a source of very cold Ps atoms for precision spectroscopy, for gravitation experiments, and as the gain medium for an annihilation gamma ray laser [2]. It would also be interesting to look for other effects that are non-linear in the positron density, such as the production of Ps+ ions and highly polarized positrons at a metal surface. Specifically, a process in which one thermalized bulk positron passes into the vacuum accompanied by an electron and a surface positron (surface pseudo-Ps) to form a Ps+ ion in vacuum, e+solid + Pssurf → Ps+vac + □E, is energetically allowed for a Ni(100) remoderator surface, for which + □E = 1.00 eV. The positive ions could be used in an experiment in which one positron undergoes a process in a material and the other positron reports on the process via its subsequent Ps emission, etc. Since the two positrons involved in the formation of Ps+ and Ps2 must have opposite spins, at sufficiently high implanted positron densities the minority positron spins will be significantly depleted by these processes, resulting in the enhancement of the spin polarization of the reemitted positrons which might be useful for spin polarized angle-resolved positronium emission spectroscopy of the electronic structure of solids and surfaces.
When orthopositronium collides with a closed-shell atom or ion the ``pick-off'' process is the dominant process for positron annihilation. In pick-off annihilation the positron annihilates with electrons in the target but not with the electron that is part of the positronium atom. The rate for pick-off annihilation, expressed in terms of the dimensionless parameter 1Zeff,{}^{1}{Z}_{\mathrm{eff}},1Zeff​, for orthopositronium scattering from Li+,Na+,{\mathrm{Li}}^{+},{\mathrm{Na}}^{+},Li+,Na+, and K+{\mathrm{K}}^{+}K+ was computed with an L2{L}^{2}L2 variant of the fixed-core stochastic variational method. The threshold 1Zeff{}^{1}{Z}_{\mathrm{eff}}1Zeff​ for Li+{\mathrm{Li}}^{+}Li+ and Na+,{\mathrm{Na}}^{+},Na+, which can form a stable bound state with Ps were found to be 0.067 and 0.73, respectively, while that for K+,{\mathrm{K}}^{+},K+, which does not bind Ps, was 0.064.
The scattering of orthopositronium from He, Ne, Ar, Li+,{\mathrm{Li}}^{+},Li+, Na+,{\mathrm{Na}}^{+},Na+, and K+{\mathrm{K}}^{+}K+ is investigated with the fixed-core stochastic variational method. The scattering length for $\mathrm{Ps}\ensuremath{-}\mathrm{He}$ scattering was 1.57a0,{1.57a}_{0},1.57a0​, a value consistent with the threshold cross sections derived from three positron lifetime experiments. The scattering lengths for the $\mathrm{P}\mathrm{s}\ensuremath{-}\mathrm{N}\mathrm{e}$ and $\mathrm{P}\mathrm{s}\ensuremath{-}\mathrm{A}\mathrm{r}$ systems were 1.55a0{1.55a}_{0}1.55a0​ and 1.79a0,{1.79a}_{0},1.79a0​, respectively. That there was only a 15% variation among the scattering lengths is compatible with the experiment of Coleman et al. [J. Phys. B 27, 981 (1994)], who found the low-energy cross sections for He, Ne, and Ar to be about the same size. The scattering lengths for Ps scattering from Li+,{\mathrm{Li}}^{+},Li+, Na+,{\mathrm{Na}}^{+},Na+, and K+{\mathrm{K}}^{+}K+ were 12.9a0,{12.9a}_{0},12.9a0​, 28.5a0,{28.5a}_{0},28.5a0​, and $\ensuremath{-}{1.9a}_{0}.$ The relatively small magnitude of the ${\mathrm{P}\mathrm{s}\ensuremath{-}\mathrm{K}}^{+}$ scattering length strongly supports previous suggestions that the ${\mathrm{P}\mathrm{s}\ensuremath{-}\mathrm{K}}^{+}$ system does not support a bound state. The annihilation parameter 1Zeff{}^{1}{Z}_{\mathrm{eff}}1Zeff​ has also been computed as part of the analysis. The present values of 1Zeff{}^{1}{Z}_{\mathrm{eff}}1Zeff​ are about 2.5--3 times smaller than the accepted experimental values since short-range electron-positron correlations were not taken into consideration when the annihilation matrix element was evaluated.