X-ray-absorption sum rules injj-coupled operators and ground-state moments of actinide ions

Sum rules for magnetic x-ray dichroism, relating the signals of the spin-orbit split core level absorption edges to the ground-state spin and orbital operators, are expressed in jj-coupled operators. These sum rules can be used in the region of intermediate coupling by taking into account the cross term between the j=l±1/2 ground-state levels and are, therefore, particularly useful in the study of actinides. The calculated expectation values for the ground-state moments of the actinide ions in intermediate coupling show that the spin-orbit-induced operators, such as the magnetic-dipole term, differ strongly from their Hund's rule ground-state values. We also prove the general rule that, when there is a perturbing interaction which is weak compared to the spin-orbit interaction, the ratio of operators with the same total moment remains constant. This condition is usually fulfilled for the crystal-field interaction in the lanthanides and actinides. The values of the ground-state moments as a function of f count give rise to an interesting trend in the dichroism of the spin-orbit split-core edges. The branching ratio of the $3d$ and $4d$ circular dichroism signal gradually increases from nearly zero for 5f¹ to {~}0.4 for 5f⁵ and is close to unity for a more than half-filled shell. The unusual behavior of the branching ratio can be related to the higher (lower) value of the magnetic dipole term, Tz, for a less (more) than half-filled shell of the actinides in the presence of spin-orbit interaction. Uranium compounds will have a much larger value of Tz than the corresponding $4f$ compounds. Its precise value can be used as a measure for the f count.