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Verification of the fitted crystal-field Hamiltonian parametrization by the crystal-field splitting second moment



A physically valid parametrization of the crystal-field (CF) Hamiltonian HCF has to reproduce not only the energy of Stark sublevels but also the multipolar structure of the central ion surroundings. The square of the second moment equation image of equation image electronic state CF splitting must be a superposition of squares of the second moments of the partial splittings produced separately by the individual multipoles: equation image, where equation image represents the modulus of a k-rank multipole of the central ion electronic eigenstate equation image, whereas equation image is the k-rank CF strength. The relationship between equation image known from experiment and equation image obtained from fitting procedure allows us to verify whether the parametrization is physically valid. It is demonstrated by several examples of CF splittings of electronic states of trivalent lanthanide ions in different crystal matrices. The approach is confined only to the states with good quantum numbers J. Nevertheless, by finding such states in any analyzed spectrum, the method can provide the actual multipole characteristics of HCF, and plays the role of a preparametrization. In general, the stepwise fitting procedure can never lead to parametrizations with a correct multipolar structure due to the limitation of the initial eigenfunctions. In consequence, the pertinent crystal-field parameters (CFPs) are burdened with a methodical inherent fault. The loss of physical meaning of fitted CFPs, its source and consequences, constitute the main theme of the paper.