Effect of the atomic core on the fine-structure splitting for excitednpstates of the alkali-metal atoms

Abstract

We have calculated the second-order interaction between the spin-orbit coupling perturbation $H_{\mathrm{so}}\propto r^{-1\mathrm{}}\left(\frac{dV}{\mathrm{dr}}\right)\overrightarrow{1}·\overrightarrow{\mathrm{s}}$ acting on the core electrons and the exchange Coulomb interaction between the valence and core electrons. This calculation was previously carried out for two alkali $\mathrm{nd}$ states ($\mathrm{Na} 3d$ and $\mathrm{Rb} 4d$), and was shown to result in the observed inversion of the fine-structure interval $\Delta \nu$ for these $d$ states. This calculation is extended to excited $\mathrm{np}$ states, and it is shown that the radial integrals have opposite sign to those for $\mathrm{nd}$, resulting in a small increase of the calculated $\Delta \nu$ (by ∼10-20% in all cases). Specific results have been obtained for several excited $\mathrm{np}$ states of Na, K, Rb, and Cs, and give agreement to within 20-30% with the experimental fine-structure splittings $\Delta {\nu }_{\mathrm{expt}}$.