Dispersion of the Magnon and Optical-Exciton States of GdCl3and Gd(OH)3and Their Effect upon the Single-Ion-Induced Absorption Line Shapes

The high-resolution optical-absorption spectra of the magnetic insulators Gd${\mathrm{Cl}}_3$ and Gd${\mathrm{}\left(\mathrm{OH}\right)\mathrm{}}_3$ in a 35-kG magnetic field are used to demonstrate, for the first time, the influence of the magnon dispersion on the line shapes of single-ion-induced optical transitions. The absorption line shapes for transitions from the first thermally populated spin state to several single-ion states of the ${}_{}{}^6{}_{}{}^{}P_{\frac{7}{2}}^{}{}_{}{}^{}$ and ${}_{}{}^6{}_{}{}^{}P_{\frac{5}{2}}^{}{}_{}{}^{}$ manifolds of the ${\mathrm{Gd}}^{+3\mathrm{}}$ ion are calculated. The initial and final states are properly treated as a magnon and exciton, respectively, including the dispersion of both states. The single-ion transition-moment operator is transformed to a basis in the crystal eigenstates and the resulting absorption line shape is shown to depend on a $\overrightarrow{\mathrm{k}}$-dependent transition-moment operator weighted by a combined exciton-magnon density of states and the magnon occupation number. The magnon dispersion is calculated from the known ground-state exchange interactions while the exciton dispersion is varied to give a best fit to the observed line shapes. The resulting excited-state exchange parameters are shown to be reasonable. The exciton dispersion is found to be quite significant (up to 2 ${\mathrm{cm}}^{-1\mathrm{}}$) for one of the excited states. These two materials thus represent the first examples for which the exciton dispersion of the electronically excited states of a rare-earth salt have been directly observed.