Effective Spin–Orbit Hamiltonian

Deviations from the Landé interval rule governing separations between the $J$ multiplets of atomic terms are known to stem from term mixing by the spin–orbit interaction. A phenomenological interaction of the form ${\mathbf{H}}_E\mathrm{\hspace{0.17em}=\hspace{0.17em}}{{\Sigma }}_n{\lambda }_n(\mathbf{L}{·}\mathbf{S}{)}^n$ with $\text{1\hspace{0.17em}≤\hspace{0.17em}n\hspace{0.17em}≤\hspace{0.17em}p}$ can account for any such separations if $\text{p\hspace{0.17em}=\hspace{0.17em}2S}$ , one less than the multiplicity of the term. The general theory of effective operators is applied to the spin–orbit interaction for electron configurations of the form $l^N$ , and it is shown that, through pth order in the perturbation expansion of the spin–orbit Hamiltonian $\mathbf{h}\mathrm{\hspace{0.17em}=\hspace{0.17em}\eta }{{\Sigma }}_i{{1}}_i·{\mathbf{s}}_i$ , the above form for ${\mathbf{H}}_E$ obtains where ${\lambda }_n\mathrm{\hspace{0.17em}=\hspace{0.17em}\eta }{{\Sigma }}_m{\mathrm{(\eta \hspace{0.17em}/\hspace{0.17em}F}}_2{)}^{\text{m−1}}{\lambda }_{\mathrm{nm}}\mathrm{,\; n\hspace{0.17em}\le \hspace{0.17em}m\hspace{0.17em}\le \hspace{0.17em}p}$ . Assuming hydrogenlike wavefunctions for $f$ electrons, the ${\lambda }_{\mathrm{nm}}$ are shown to be constants depending only on the term being considered and the number of equivalent electrons $N$ . Using $\eta$ and $F_2$ values given in the literature, the ${\lambda }_n$ are calculated for the ground terms of the triply ionized rare earths. Values for the ${\lambda }_n$ have been reported for Nd³⁺, Tb³⁺, Ho³⁺, and Er³⁺ in CaWO₄ where the Stark‐split energy levels of the ground terms were analyzed by treating the ${\lambda }_n$ and the crystal‐field parameters in ${\mathbf{H}}_X\mathrm{\hspace{0.17em}=\hspace{0.17em}}{{\Sigma }}_{\mathrm{km}}{{B}}_{\mathrm{km}}†{\mathbf{C}}_{\mathrm{km}}$ as adjustable. The theoretical and experimental values for ${\lambda }_1{\mathrm{,\; \lambda }}_2$ and ${\lambda }_3$ agree to within 2.6%, 7.0%, and 23%, respectively. The calculation is then inverted and $\eta$ and $F_2$ values for the four ions are determined by requiring the theoretical ${\lambda }_n$ parameters to equal those obtained empirically. The $\eta$ and $F_2$ values obtained in this manner differ from those reported by 2.3% and 5.1%, respectively. It is therefore feasible, through the use of the effective spin–orbit Hamiltonian ${\mathbf{H}}_E$ , to determine $\eta$ and $F_2$ values for the ground configuration of an ion by considering the ground term alone.