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Enhancement Factors for Positron Annihilation on Valence and Core Orbitals of Noble-Gas Atoms

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Concepts, Methods and Applications of Quantum Systems in Chemistry and Physics

Part of the book series: Progress in Theoretical Chemistry and Physics ((PTCP,volume 31))

Abstract

Annihilation momentum densities and vertex enhancement factors for positron annihilation on valence and core electrons of noble-gas atoms are calculated using many-body theory for s, p and d-wave positrons of momenta up to the positronium-formation threshold. The enhancement factors parametrize the effects of short-range electron-positron correlations which increase the annihilation probability beyond the independent-particle approximation. For all positron partial waves and electron subshells, the enhancement factors are found to be relatively insensitive to the positron momentum. The enhancement factors for the core electron orbitals are also almost independent of the positron angular momentum. The largest enhancement factor (\({\sim }10\)) is found for the 5p orbital in Xe, while the values for the core orbitals are typically \({\sim }1.5\).

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Notes

  1. 1.

    Positron annihilation with core electrons is also affected by exchange-assisted tunnelling [26, 27]. This is a manifestation of electron exchange, which increases the wavefunctions of inner electrons in the range of distances of the valence electrons. For this effect to be properly included in a calculation, one needs to use true nonlocal exchange potentials, e.g., at the Hartree-Fock level, as is the case in the present calculations.

  2. 2.

    Alternatively to the Doppler-shift spectrum, experiments measure the one-dimensional angular correlation of annihilation radiation (1D-ACAR), i.e., the small angle \(\varTheta \) between the direction of one photon and the plane containing the other. The corresponding distribution can be obtained from \(w(\varepsilon )\) using \(\varTheta =2\epsilon /mc^2\). Not also that if the positron wavefunction is constant, then the annihilation momentum density is proportional to the electron momentum density, and the \(\gamma \) spectrum becomes similar to the Compton profile [22, 23, 41].

  3. 3.

    In this and subsequent sections we make wide use of atomic units (a.u.).

  4. 4.

    It is also possible to develop a diagrammatic expansion for \(Z_\mathrm{eff}\) [19, 20, 44, 45, 48] that enables one to calculate the annihilation rate directly, rather than from Eq. (4).

  5. 5.

    The term in braces can also be compared with the expression for the natural geminal corresponding to the positron state \(\varepsilon \) and electron orbital n, \(\alpha _{\varepsilon n}(\mathbf{r},\mathbf{r})=\sqrt{\gamma _{\varepsilon n}(\mathbf{r})}\psi _{\varepsilon }(\mathbf{r})\varphi _n(\mathbf{r})\) (cf. Eq. (9) in Ref. [34]), which can be used to determine the position dependent EF \(\gamma _{\varepsilon n}(\mathbf{r})\), see Sect. 4.

  6. 6.

    For HF positron wavefunctions the values of the parameters are \(A=1.54~\text {a.u.}=42.0~\text {eV}\), \(B=0.92~\text {a.u.}=24.9~\text {eV}\), and \(\beta =2.54\). For Dyson positron wavefunctions the values are \(A=1.31~\text {a.u.}=35.7~\text {eV}\), \(B=0.83~\text {a.u.}=22.7~\text {eV}\), and \(\beta =2.15\) [24].

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Acknowledgements

DGG is supported by a United Kingdom Engineering and Physical Sciences Research Council Fellowship, grant number EP/N007948/1.

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Green, D.G., Gribakin, G.F. (2018). Enhancement Factors for Positron Annihilation on Valence and Core Orbitals of Noble-Gas Atoms. In: Wang, Y., Thachuk, M., Krems, R., Maruani, J. (eds) Concepts, Methods and Applications of Quantum Systems in Chemistry and Physics. Progress in Theoretical Chemistry and Physics, vol 31. Springer, Cham. https://doi.org/10.1007/978-3-319-74582-4_14

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