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

We have calculated the second-order interaction between the spin-orbit coupling perturbation $H_{\mathrm{SO}}\propto r^{-1\mathrm{}}\left(\frac{\mathrm{dV}}{\mathrm{dr}}\right)\overrightarrow{\mathrm{l}}·\overrightarrow{\mathrm{s}}$ 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 (Na $3d$ and Rb $4d$) and was shown to result in the observed inversion of the fine-structure interval $\Delta \nu$ for these $d$ states. In the present paper we have evaluated the second-order correction $\Delta {\nu }_2$ for the $\mathrm{nf}$ states of Na, and we have found that $\frac{\Delta {\nu }_2}{\Delta {\nu }_1\simeq -0.07\mathrm{}}$ for Na $11f$ and Na $14f$, in good agreement with the experiment of Gallagher et al., who have observed that the fine-structure interval of Na $11f$, $13f$, and $14f$ is normal, i.e., noninverted, and slightly smaller than the hydrogenic value $\Delta {\nu }_1$ (by ∼ 5%). Calculations have also been carried out for Na $3d$, $4d$, $5d$, $11d$, and $14d$, and the results for $\Delta {\nu }_1+\Delta {\nu }_2$ are negative, in good agreement with the observed inverted fine structure for these states. Additional calculations have been carried out for Cs $4f$, $5f$, and $6f$.