Optical Absorption Intensities of Rare-Earth Ions

Electric dipole transitions within the 4f shell of a rare-earth ion are permitted if the surroundings of the ion are such that its nucleus is not situated at a center of inversion. An expression is found for the oscillator strength of a transition between two states of the ground configuration 4f^N, on the assumption that the levels of each excited configuration of the type 4f^Nn^′d or 4f^Nn^′g extend over an energy range small as compared to the energy of the configuration above the ground configuration. On summing over all transitions between the components of the ground level ψ_J and those of an excited level ψ^′_J^′, both of 4f^N, the oscillator strength P corresponding to the transition ψ_J→ψ^′_J^′ of frequency ν is found to be given by P=ΣT_λν(ψ_J∥U^(λ)∥ψ^′_J^′)², where U^(λ) is a tensor operator of rank λ, and the sum runs over the three values 2, 4, and 6 of λ. Transitions that also involve changes in the vibrational modes of the complex comprising a rare-earth ion and its surroundings, provide a contribution to P of precisely similar form. It is shown that sets of parameters T_λ can be chosen to give a good fit with the experimental data on aqueous solutions of NdCl₃ and ErCl₃. A calculation on the basis of a model, in which the first hydration layer of the rare-earth ion does not possess a center of symmetry, leads to parameters T_λ that are smaller than those observed for Nd³⁺ and Er³⁺ by factors of 2 and 8, respectively. Reasons for the discrepancies are discussed.