Capability of the Free-Ion Eigenstates for Crystal-Field Splitting
Any electronic eigenstate of the paramagnetic ion open-shell is characterized by the three independent multipole asphericities for and 6 related to the second moments of the relevant crystal-field splittings by , where . The Ak as the reduced matrix elements can serve as a reliable measure of the state capability for the splitting produced by the k-rank component of the crystal-field Hamiltonian. These multipolar characteristics allow one to verify any fitted crystal-field parameter set by comparing the calculated second moments and the experimental ones of the relevant crystal-field splittings. We present the multipole characteristics Ak for the extensive set of eigenstates from the lower parts of energy spectra of the tripositive 4 f N ions applying in the calculations the improved eigenfunctions of the free lanthanide ions obtained based on the M. Reid f-shell programs. Such amended asphericities are compared with those achieved for the simplified Russell-Saunders states. Next, they are classified with respect to the absolute or relative weight of Ak in the multipole structure of the considered states. For the majority of the analyzed states (about 80%) the Ak variation is of order of only a few percent. Some essential changes are found primarily for several states of Tm3+, Er3+, Nd3+, and Pr3+ ions. The detailed mechanisms of such Ak changes are unveiled. Particularly, certain noteworthy cancelations as well as enhancements of their magnitudes are explained.