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) Hamiltonians ${\cal H}_{\rm CF}=\sum_{k}\sum_{q}B_{kq}C_{q}^{(k)}$ has to be based on the CF parameters $B_{kq}$ that lead to the correct CF splitting second moments, both the global one $\sigma$ and the partial ones $\sigma_{k}$. Only such parametrizations correspond to the appropriate multipolar structure of the surrounding CF. Each parametrization being characterized by its own multipolar crystal-field strengths $S_{k}=(\frac{1}{2k+1}\sum\limits_{q}|B_{kq}|^{2})^{1/2}$, for $k=2,4$ and 6, yields a definite second moment $\sigma$, which can be derived from the additivity relationship $\sigma^{2}=\sum_{k}\sigma_{k}^{2}$ and the known asphericities $< \Psi||C^{(k)}||\Psi >$ of the central-ion eigenfunctions $\Psi$. The condition $\sigma=\sigma_{\rm exp}$ must be satisfied to ensure the parametrization's correctness. However, our survey of literature indicates that there exists many other well-fitted crystal-field parameter sets that do not obey this condition. Therefore, such sets are erratic and non-physical, and should be re-examined or rejected. Having $\sigma$ for several $(\geq3)$ eigenstates $|\Psi>$ along with the relevant $< \Psi||C^{(k)}||\Psi >$ asphericities, one can estimate $\sigma_{k}$ 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.