Auger widths of core levels in metallic sodium and lithium

The Auger decay rates of the $1s$ level in lithium and $2p$ level in sodium are calculated using a diagrammatic many-body technique. The lowest-order nonvanishing contribution to the self-energy of these core states gives Auger widths of the order of ${10}^{-3\mathrm{}}$ eV for Li and ${10}^{-4\mathrm{}}$ eV for Na. However, higher-order terms are found to diverge showing that a renormalized theory must be used for such calculations. We find that for systems with low-lying excitations, there is a many-body mechanism which enhances the Auger width. Using a self-consistent renormalized theory, we find widths of the order of 20 meV for both Li and Na. The sodium result is consistent with predictions based on observed soft-x-ray spectra, but for lithium it is still one order of magnitude too small to be able to attribute the observed threshold behavior as purely due to an Auger core-level width.