Oscillator strength

Abstract: 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 $4{f}^{N}$, on the assumption that the levels of each excited configuration of the type $4{f}^{N}{n}^{\ensuremath{'}}d$ or $4{f}^{N}{n}^{\ensuremath{'}}g$ 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 ${\ensuremath{\psi}}_{J}$ and those of an excited level ${{\ensuremath{\psi}}^{\ensuremath{'}}}_{{J}^{\ensuremath{'}}}$, both of $4{f}^{N}$, the oscillator strength $P$ corresponding to the transition ${\ensuremath{\psi}}_{J}\ensuremath{\rightarrow}{{\ensuremath{\psi}}^{\ensuremath{'}}}_{{J}^{\ensuremath{'}}}$ of frequency $\ensuremath{ u}$ is found to be given by $P=\ensuremath{\Sigma}{T}_{\ensuremath{\lambda}}\ensuremath{ u}{({\ensuremath{\psi}}_{J}\ensuremath{\parallel}{U}^{(\ensuremath{\lambda})}\ensuremath{\parallel}{{\ensuremath{\psi}}^{\ensuremath{'}}}_{{J}^{\ensuremath{'}}})}^{2},$ where ${\mathrm{U}}^{(\ensuremath{\lambda})}$ is a tensor operator of rank $\ensuremath{\lambda}$, and the sum runs over the three values 2, 4, and 6 of $\ensuremath{\lambda}$. 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 ${T}_{\ensuremath{\lambda}}$ 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 ${T}_{\ensuremath{\lambda}}$ 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.