Positron annihilation with core and valence electrons
$\gamma$-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 $\gtrsim3$\,keV, and even dominate the spectra of Kr and Xe at shifts $\gtrsim5$\,keV. Their inclusion brings the theoretical spectra into excellent agreement with the experimental $\gamma$-spectra across the full range of Doppler energy shifts. Additionally, the theory enables the calculation of the `exact' vertex enhancement factors $\bar{\gamma}_{n\ell}$ for individual core and valence subshells $n\ell$. They are found to follow a simple and physically motivated scaling with the subshell ionization energy $I_{n\ell}$: $\bar{\gamma}_{n\ell}=1+\sqrt{A/I_{n\ell}}+(B/I_{n\ell})^{\beta}$, where $A$, $B$ and $\beta$ 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 $\gamma$-spectra. This formula can be used to determine the enhancement factors for positron annihilation with core electrons of atoms across the periodic table and condensed matter systems.