Multiplet Separations for Equivalent Electrons and the Roentgen Doublet Law

Expressions are derived for the separation of multiplets arising from configurations of equivalent electrons. It is explained why the Roentgen doublets obey the Sommerfeld formula, although this formula was derived formerly for a single electron only. The separations of more complicated multiplets are proved to be connected also with the Sommerfeld formula. According to Landé the multiplet separations can be characterized by factors, designated by $\Gamma$, which give the displacement of each level from the centre of gravity of the whole multiple level. The properties of these $\Gamma$ factors are studied in this paper. They are quite analogous to the well-known properties of the magnetic separation factors, $g$, introduced by Landé for the explanation of the Zeeman effect. It is pointed out that there is a $\Gamma$ sum rule, corresponding to the well-known $g$ sum rule, stating that, for given quantum numbers of the electrons, the sum of the $\Gamma$ values belonging to a certain total moment $j$ is a constant, independent of the type of coupling of the electrons. This rule is important especially because it makes it possible to obtain expressions for the multiplet separations in general cases.