Trends in the crystal (ligand) field parameters and the associated conserved quantities for trivalent rare-earth ions at S4 symmetry sites in LiYF4

Abstract

Our recent application of the Noether’s theorem for the crystal (ligand) field (CF/LF) Hamiltonians invariant under continuous rotational symmetry has revealed existence of several conserved quantities. In this paper we utilize the conserved quantities stipulated by the Noether’s theorem and other associated quantities to analyse the CF parameter (CFP) datasets for the trivalent rare-earth (RE3+) ions at tetragonal type II S4 symmetry sites in LiYF4. This host is selected since sufficient amount of data is available to warrant a comparative CFP analysis across the whole 4fN series. Other ion/host systems with S4 symmetry are also briefly dealt with. Large number of CFP datasets for RE3+:LiYF4 together with the associated quantities calculated by us are tabularised and their variations are presented in graphs. This systematics facilitates consideration of trends among the CFPs and the derived quantities across the 4fN series. Knowledge of such trends would enable identification of unreliable CFP datasets and CFP predictions for ions for which no or little data are available. The major observations for RE3+:LiYF4 are: (i) the spread of the original CFPs is very large for most RE3+ ions, which indicates several inconsistent CFP datasets; and (ii) the values averaged over the available RE3+ data points show no well-defined trends across the 4fN series for the CFPs with q = 0, whereas weak diminishing trends with increasing N for the conserved quantities |v44| and |v64|. Various aspects bearing on the erratic CFP behaviour are discussed. Our considerations show the advantages of the Noether’s theorem stipulated conserved quantities over other derived quantities used in the literature and have implications for interpretation of the published CFP datasets and their reliability.