Positron annihilation with core and valence electrons

Abstract

γ-ray spectra for positron annihilation with the core and valence electrons of the noble gas atoms Ar, Kr and Xe is calculated within the framework of diagrammatic many-body theory. The effect of positron-atom and short-range positron-electron correlations on the annihilation process is examined in detail. Short-range correlations, which are described through non-local corrections to the vertex of the annihilation amplitude, are found to significantly enhance the spectra for annihilation on the core orbitals. For Ar, Kr and Xe, the core contributions to the annihilation rate are found to be 0.55%, 1.5% and 2.2% respectively, their small values reflecting the difficulty for the positron to probe distances close to the nucleus. Importantly however, the core subshells have a broad momentum distribution and markedly contribute to the annihilation spectra at Doppler energy shifts > ∼ 3 keV, and even dominate the spectra of Kr and Xe at shifts > ∼ 5 keV. Their inclusion brings the theoretical spectra into excellent agreement with the experimental γ-spectra across the full range of Doppler energy shifts. Additionally, the theory enables the calculation of the 'exact' vertex enhancement factors γn for individual core and valence subshells n . They are found to follow a simple and physically motivated scaling with the subshell ionization energy I n : γn = 1 + A/I n + (B/I n ) β , where A, B and β are positive constants. It is demonstrated that these factors can be incorporated in simple independent-particle-model calculations to successfully reconstruct the true many-body annihilation γ-spectra. This formula can be used to determine the enhancement factors for positron annihilation with core electrons of atoms across the periodic table, and with the localized atomic-like core electrons of condensed matter systems.

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

“…We conclude this section by noting that calculations of the corresponding γ spectra were reported in [24,43]. They showed excellent agreement with the measured spectra for Ar, Kr and Xe [3], and firmly established the fraction of core annihilation for these atoms.…”

“…The AMD are broader for the core orbitals for which the typical electron momenta are greater, leading to greater Doppler shifts. The core AMD (and the core annihilation probabilities [24,43]) also have noticeably smaller magnitudes compared with those of the valence electrons. For all electron orbitals whose radial wavefunctions have nodes (e.g., 2s in Ne, 2s, 3s and 3p in Ar, etc.)…”

“…1, where V is the electron-positron Coulomb interaction and χ is the propagator of the intermediate electron-positron state. Discretizing the electron and positron continua by confining the system in a spherical cavity reduces this to a linear matrix equation, which is easily solved numerically (see [19,21,41,43] for further details).…”

“…Refs. [42,43] present the partial-wave analysis and corresponding working analytic expressions for the matrix elements involving the vertex corrections.…”

“…We use an expotential knot sequence for the B-splines, which provides for an efficient spanning of the electron and positron continua in the sums over intermediate states [41]. The maximum angular momentum of the intermediate states is l max =15, and we extrapolate to l max → ∞ as in [19,21,41,43].…”