Fundamental two-electron tensor-operator description of the ${\mathrm{Gd}}^{3+}$ $4f^7$ configuration exchange splittings in ${\mathrm{GdCl}}_3$

The applicability and usefulness of a fundamental two-electron description of the exchange interaction in magnetic rare-earth insulators, has been tested by applying it in spherical tensor-operator form to the analysis of the observed exchange splittings of ${\mathrm{GdCl}}_3$. The experimental splittings were obtained by extrapolating the optical transition energies from the ground state to five excited levels as a function of large magnetic fields back to zero field, thereby determining the splittings in a nearly completely ordered ${\mathrm{Gd}}^{3+}$ environment. An analysis of the mechanisms which can give rise to the splittings shows that only the exchange and magnetic dipole-dipole interactions can contribute. The contributions from the latter and from the dynamic exchange, which we show must be included in the analysis of the $\overrightarrow{\mathrm{k}}=0\mathrm{}$ exciton exchange splittings in concentrated materials, are isolated, allowing the determination of the static exchange contribution. An analysis of the selection rules on the exchange Hamiltonian in this host for the states of the ${}_{}{}^6{}_{}{}^{}P$ manifold allows us to eliminate all but two parameters in the two-electron exchange operator which is applied to the observed static exchange contributions. The two parameters were determined from data on five exciton states and the resulting isotropic parameter is shown to be identical to that determined from magnetic studies of the ground state. Further studies indicate that the fitted two-electron exchange parameters are applicable not only to other levels of ${\mathrm{Gd}}^{3+}$ in ${\mathrm{GdCl}}_3$, but also to those of other rare-earth ions.