Multiplet Splitting and Intensities of Intercombination Lines Part I

Abstract

Kramers' symbolic representation method for the treatment of those properties of free atoms which can be derived on the basis of irreducible representations of rotations in $R_3$ is outlined. Wave functions and operators are represented by functions of spinor quantities ($\xi , \eta$) whose properties are known from the theory of invariants. Integrations yielding matrix elements are only symbolic and the results, being determined by the rotation invariance properties, contain therefore undetermined constant factors of the nature of radial integrals. The method is particularly adapted to the problem of multiplets (interaction between two or more "vectors"—e.g., orbital angular momenta, spins—treated as a perturbation). The method is applied to configurations involving two valence electrons of which one is in an $s$ state. Such a configuration gives rise to a singlet and a triplet with the $l$ value of the second electron. The deviations from the normal $l$ to $l+1\mathrm{}$ interval ratio in the triplet are due to a repulsion between the singlet level and the center triplet level, their $j$ values being the same. The mixing up of the wave functions of these two levels gives rise to singlet-triplet intercombinations. The constants which represent the interaction energies (1) interchange, (2) spin-orbit, (3) spin of one electron—orbit of the other (assuming that other types of interaction may be neglected) are found in terms of the three intervals of the multiplet. Corrections to the Kronig-Hönl intensity formulas are found in terms of the intervals of the two multiplets between which the transitions occur. These formulas then give the relative intensities of combination and intercombination lines. The sum rules for intensity hold for the complete multiplet and not for the singlet and triplet separately. Part II will contain application of the theory to observed spectra.