Inner-shell Photodetachment of Na$^{- }$

Calculations of the photodetachment of a 2p core electron in the Na$^{-}$ ion over the photon energy range 30-41 eV have been performed using R-matrix theory with a perturbative method in the asymptotic region. Our results show a very strong Feshbach resonance in the Na 1s$^{2}$2s$^{2}$2p$^{5}$2s$^{2 }(^{2}$P) channel at about 34 eV, just below the 1s$^{2}$2s$^{2}$2p$^{5}$3s3p$^{ }(^{2}$D) threshold, the 1s$^{2}$2s$^{2}$2p$^{5}$3s3p$^{2 }(^{1}$P) resonance. Since 3s and 3p orbitals are about the same ``size'', they have a significant attractive exchange interaction; this attraction pulls the resonance below the 1s$^{2}$2s$^{2}$2p$^{5}$3s3p$^{ }(^{2}$D) threshold, making it a Feshbach resonance. The Auger decay 1s$^{2}$2s$^{2}$2p$^{5}$3s$^{2 }\to $1s$^{2}$2s$^{2}$2p$^{6 }$ (Na$^{+ })+e$ leads to the production of Na$^{+}$. Therefore, we expect experiment to find this resonance around 34 eV. But recent experiment explored in this region and found nothing [1]. We are puzzled by this discrepancy. Another resonance in our calculation is located at 36.318 eV, just below the 1s$^{2}$2s$^{2}$2p$^{5}$3s4s ($^{ 2}$P$^{o})$ threshold. This resonance is confirmed by experiment [1] which is found at 36.213 eV and assigned as a 1s$^{2}$2s$^{2}$2p$^{5}$ 3s 4s \textit{nl} resonance. The situation remains under theoretical scrutiny Work was supported by DOE, NASA and NSF. [1] A. M. Covinton \textit{et al}., J. Phys. B \textbf{34}, L735 (2001) and D. J. Pegg, private communication (2005).