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 $4f^N{n}^{\prime }d$ or $4f^N{n}^{\prime }g$ 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 ${\psi }_J$ and those of an excited level ${{\psi }^{\prime }}_{J^{\prime }}$, both of $4f^N$, the oscillator strength $P$ corresponding to the transition ${\psi }_J\to {{\psi }^{\prime }}_{J^{\prime }}$ of frequency $\nu$ is found to be given by $P=\Sigma T_{\lambda }\nu {\left({\psi }_J\parallel U^{\left(\lambda \right)}\parallel {{\psi }^{\prime }}_{J^{\prime }}\right)}^2,$ where ${\mathrm{U}}^{\left(\lambda \right)}$ is a tensor operator of rank $\lambda$, and the sum runs over the three values 2, 4, and 6 of $\lambda$. 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 $T_{\lambda }$ can be chosen to give a good fit with the experimental data on aqueous solutions of Nd${\mathrm{Cl}}_3$ and Er${\mathrm{Cl}}_3$. 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 $T_{\lambda }$ that are smaller than those observed for ${\mathrm{Nd}}^{3+}$ and ${\mathrm{Er}}^{3+}$ by factors of 2 and 8, respectively. Reasons for the discrepancies are discussed.