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Physical Review A

Determination and applications of enhancement factors for positron and ortho-positronium annihilations

Jim Mitroy

Published Dec 16, 2005• 12 citations
Publisher

Abstract

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 $1s$ electron in H, ${\mathrm{He}}^{+}$, He, ${\mathrm{Li}}^{2+}$, and ${\mathrm{Li}}^{+}$. The enhancement factor for a free positron annihilating with ${\mathrm{He}}^{+}$ 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 $1.47{a}_{0}$ for Ps-He scattering. Further, enhancement factors for ${e}^{+}$-Ne and ${e}^{+}$-Ar annihilation are used in conjunction with the pickoff annihilation rate to estimate scattering lengths of $1.46{a}_{0}$ for Ps-Ne scattering and $1.75{a}_{0}$ for Ps-Ar scattering.

DOI
10.1103/physreva.72.062707

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The scattering of orthopositronium from Kr and Xe is investigated with the fixed core stochastic variational method. The scattering lengths for Ps-Kr and Ps-Xe scattering in the fixed core approximation are ${3.18a}_{0}$ and ${3.82a}_{0},$ respectively. Inclusion of core-polarization potentials leads to a significant reduction in the scattering lengths. The Ps-Kr and Ps-Xe scattering lengths were ${1.98a}_{0}$ and ${2.29a}_{0}$ for polarization potentials that were tuned to an average of the model electron-atom and positron-atom scattering lengths for the respective atoms. Other choices of the polarization potential resulted in scattering lengths that could be larger by ${0.3a}_{0}$ or smaller by ${0.8a}_{0}.$ The pick-off 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 3 to 10 times smaller than the accepted experimental values since short-range electron-positron correlations were ignored when the annihilation matrix element was evaluated.

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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.
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.
A two-parameter semiempirical theory of positron scattering and annihilation is developed and used to investigate the behavior of positrons interacting with the rare gases and metal vapors. The two-parameter theory is able to do a reasonable job of reproducing existing cross section and annihilation data for the rare gases. A model-potential calculation that correctly predicts the behavior of the phase shifts will also predict the energy dependence of Zeff(k){Z}_{\mathrm{eff}}(k)Zeff​(k) even if the magnitude is incorrect. Analysis of the Zeff{Z}_{\mathrm{eff}}Zeff​ versus temperature data of Kurz et al. [Phys. Rev. Lett. 77, 2929 (1996)] suggests scattering lengths of $\ensuremath{-}5.6\ifmmode\pm\else\textpm\fi{}{1.0a}_{0},$ $\ensuremath{-}10.3\ifmmode\pm\else\textpm\fi{}{2.0a}_{0},$ and $\ensuremath{-}56\ifmmode\pm\else\textpm\fi{}{15a}_{0}$ for Ar, Kr, and Xe, respectively. Existing bound-state calculations can be used to fix the values of the semi-empirical parameters for a number of metal vapors, resulting in predicted Zeff{Z}_{\mathrm{eff}}Zeff​ of 119, 36, and 94 for Be, Mg, and Cu at threshold. In addition to the calculations, expressions relating the threshold form of Zeff(k){Z}_{\mathrm{eff}}(k)Zeff​(k) to the complex scattering length are presented.
When positrons are injected into a gas, 75% of the positronium (Ps) is likely to be formed as long-lived ortho-Ps. The main decay mechanisms for the ortho-Ps have been assumed to be natural decay of ortho-Ps and pickoff annihilation of the positron during Ps-atom collisions. A third possibility for annihilation is $\mathrm{o}\mathrm{r}\mathrm{t}\mathrm{h}\mathrm{o}\mathrm{\text{\ensuremath{-}}}\mathrm{P}\mathrm{s}\ensuremath{\rightarrow}\mathrm{p}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{\text{\ensuremath{-}}}\mathrm{P}\mathrm{s}$ conversion due to the spin-orbit interaction between the atom and colliding Ps. This extra quenching mechanism may explain a number of phenomena observed in the annihilation spectrum of Kr and Xe, including the very small Ps fraction of 3% seen for Xe.
The scattering of orthopositronium from Kr and Xe is investigated with the fixed core stochastic variational method. The scattering lengths for Ps-Kr and Ps-Xe scattering in the fixed core approximation are 3.18a0{3.18a}_{0}3.18a0​ and 3.82a0,{3.82a}_{0},3.82a0​, respectively. Inclusion of core-polarization potentials leads to a significant reduction in the scattering lengths. The Ps-Kr and Ps-Xe scattering lengths were 1.98a0{1.98a}_{0}1.98a0​ and 2.29a0{2.29a}_{0}2.29a0​ for polarization potentials that were tuned to an average of the model electron-atom and positron-atom scattering lengths for the respective atoms. Other choices of the polarization potential resulted in scattering lengths that could be larger by 0.3a0{0.3a}_{0}0.3a0​ or smaller by 0.8a0.{0.8a}_{0}.0.8a0​. The pick-off annihilation parameter 1Zeff{}_{1}{Z}_{\mathrm{eff}}1​Zeff​ has also been computed as part of the analysis. The present values of 1Zeff{}_{1}{Z}_{\mathrm{eff}}1​Zeff​ are about 3 to 10 times smaller than the accepted experimental values since short-range electron-positron correlations were ignored when the annihilation matrix element was evaluated.