Anisotropic exchange effects in optical spectra:Er3+in ferromagnetic Tb(OH)3

The theory of spontaneous splittings of impurity levels in magnetic insulators below their ordering temperatures has been examined, with particular consideration of the anisotropies of the interactions and possible collective excitation effects. Under appropriate realistic conditions these splittings are accurately described by a first-order analysis, with the result that the contribution of the exchange interaction may be isolated and its orbital anisotropy determined empirically. As an illustration of the effectiveness of this method for studying the nature of the exchange interaction, the splittings of isolated ${\mathrm{Er}}^{3+}$ impurities have been investigated in ferromagnetic Tb${\mathrm{}\left(\mathrm{OH}\right)\mathrm{}}_3$. This is a system for which a number of the simplifying conditions can be shown to apply quite accurately, so that the exchange contribution can be isolated unambiguously from the competing effects of the crystal field, electric multipole, virtual phonon exchange, and magnetic dipole-dipole interactions. An effective exchange operator for the ${\mathrm{Er}}^{3+}$ ion has been derived in terms of single-electron spherical tensor operators which are applicable to all levels of the ground $4f^{11}$ configuration. A least-squares fit of 11 observed splittings using eight parameters has provided a good description of both the signs and magnitudes of the splittings. The results show that the contributions of the anisotropic terms are generally one order of magnitude larger than the contribution of the isotropic term, indicating that any realistic analysis of ion-ion interaction effects involving ions with large orbital admixtures must consider the effects of anisotropy in the exchange interaction as a major factor.