A fundamental requirement for crystal‐field parametrization

Abstract

The physically correct parametrization of the energy levels of transition ions in crystals in terms of crystal-field (CF) Hamiltoniansq has to be based on the CF parameters B kq that lead to the correct CF splitting second moments, both the global one s and the partial ones s k . Only such parametrizations correspond to the appropriate multipolar structure of the surrounding CF. Each parametrization being characterized by its own multipolar CF strengths S k ¼ ½1=ð2k þ 1Þ P q jB kq j 2 1=2 , for k ¼ 2, 4, and 6, yields a definite second moment s, which can be derived from the additivity relationshipjjJiÞ 2 , and the known asphericities hCjjC ðkÞ jjCi of the central-ion eigenfunctions C. The condition s ¼ s exp must be satisfied to ensure the parametrization's correctness. However, our survey of literature indicates that there exists many other well-fitted CF parameter sets that do not obey this condition. Therefore, such sets are erratic and nonphysical, and should be re-examined or rejected. Having s for several (!3) eigenstates jCi along with the relevant hCjjC ðkÞ jjCi asphericities, one can estimate s k and S k , which are well founded experimentally. The above findings set up the parametrization process properly. Lack of consistency between the second moments representing various parametrizations and the pertinent second moments observed in experiments is presumably the main reason for deluge of formally accurate but accidental and inequivalent parametrizations.

Capability of the Free-Ion Eigenstates for Crystal-Field Splitting

“…For the corrected Tm 3+ free-ion eigenstates we have found the proper solution of (Equation (5) The role of the capabilities k A in the approach is readable. It is proper to add that there is no solution of Equation (5) [20]. One can therefore suspect a remarkable part of published CF H parametrizations to suffer from this type of physical shortcoming.…”

Three Localized f Electrons in UPd2Al3and in UGe2Intermetallics

“…Such good agreement, according to Refs. [26,27], is strong argument for the localization of 5f electrons.…”

“…With known from the atomic physics the ion aspherities of the rank k of the multiplet 4 I 9/2 , given in Refs. [26,27] Fig. 1.…”

Terminological confusions and problems at the interface between the crystal field Hamiltonians and the zero-field splitting Hamiltonians—Survey of the CF=ZFS confusion in recent literature

“…Moreover, the symbols [63] B k q are confusingly akin to the CFPs in the ESO notation B q k . As an aside, we note that concerning the physical validity of a particular CF parameterization, Mulak and Mulak have proposed the method based on the second moments [64] and the criterion whether CF parameterization reproduces or not the actual multipolar structure of the surrounding CF [65]. It is worthwhile to note that the orthorhombic standardization may also be employed for the electric quadrupole 'interaction' Hamiltonians of the type H Q ¼ I Á Q Á I for systems with the nuclear spin quantum number IZ 1.…”