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 byThe A k 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 A k 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 A k in the multipole structure of the considered states. For the majority of the analyzed states (about 80%) the A k variation is of order of only a few percent. Some essential changes are found primarily for several states of Tm 3+ , Er 3+ , Nd 3+ , and Pr 3+ ions. The detailed mechanisms of such A k changes are unveiled. Particularly, certain noteworthy cancelations as well as enhancements of their magnitudes are explained.

A new measure of the crystal-field strength, complementary to the conventional one, is defined. It is based on the rotational invariants |B k0 | av or | k B k0 | av , k = 2, 4, 6, of the crystal-(ligand)-field (CF) Hamiltonian H CF parametrizations, i.e. on the axial CF parameters modules averaged over all reference frame orientations. They turn out to be equal to H (k) CF av and |H CF | av , respectively. While the traditional measure is established on the parametrization modules or on the second moment of the CF energy levels, the introduced scale employs rather the first moment of the energy modules and has better resolving power. The new scale is able to differentiate the strength of various iso-modular parametrizations according to the classes of rotationally equivalent parametrizations. Using both the compatible CF strength measures one may draw more accurate conclusions about the Stark levels arrays and particularly their total splitting magnitudes.PACS: 71.70.Ch

The second moment of the sublevels within the initial state | \alpha SLJ > which constitutes a natural and adequate measure of the crystal-field (CF) effect can be redefined as sigma^{2}=1/(2J+1)\sum_{k} S_{k}^{2} A_{k}^{2}, where S_{k}=[1/(2k+1)\sum_{q}|B_{kq}|^2]^{1/2} is the so-called 2^{k}-pole CF strength, whereas A_{k}= < \alpha SLJ||C^{(k)}||\alpha SLJ > the reduced matrix element of the k-rank spherical tensor operator. Therefore, the CF effect depends on the sum of products of the two factors representing the identical multipole components of two different charge distributions. The term A_{k} expresses the asphericity of the central ion open-shell, whereas the term S_{k} the asphericity of its surroundings. When these two distributions do not fit each other the observed CF splitting can be unexpectedly weak even for considerable values of the total S=(\sum_{k}S_{k}^{2})^{1/2} and A=(\sum_{k}A_{k}^{2})^{1/2}. The tabulated quantities of the A_{k}(|\alpha SLJ >), as the 2^{k}-pole type asphericities, are the intrinsic characteristics of the electron states revealing their multipolar structure and hence their potential susceptibility to CF splitting separately for each effective multipole.Comment: LaTex2e, 13 pages (2 tables) paper submitted to physica status solidi (b