Calorimetric Determination of Energy Levels in Rare-Earth and Yttrium-Iron Garnets

It is shown that low-temperature calorimetry can be a sensitive method for determining the lowest excited energy levels in the rare-earth iron garnets. From Pauthenet's magnetization data one expects the lowest excited levels of several rare-earth ions to become populated at temperatures well below 20°K and to contribute a large specific heat. This property offers the possibility of testing the validity of the Weiss molecular field and spin-wave approximations for this isomorphic series of oxides. After a discussion of the specific heat in terms of the Weiss molecular-field approximation, a spin-wave treatment for the garnets is then presented and the dispersion equation for the acoustical branch is derived. It is shown by a perturbation calculation that in garnets with magnetic rare-earth ions, there are twelve low-lying optical modes that will contribute to the specific heat below 20°K. Heat-capacity measurements between 1.3 and 20.6°K on the iron garnets of Y, Sm, Gd, Tb, Dy, Ho, Er, Yb, and Lu are presented and interpreted in terms of the two theoretical models. The energy levels so obtained are compared to those measured by optical absorption and deduced from magnetic data. For YIG and LuIG, where only the acoustical mode contributes to the magnetic specific heat, the result is compared to other heat-capacity and magnetic measurements. With the exception of TbIG and SmIG the magnetic specific heat of the garnets can be satisfactorily interpreted. Reasonable agreement is obtained in particular between the energy levels as deduced from specific heat data and those observed directly by optical absorption on YbIG and ErIG. In general it is found that for temperatures lower than$\sim \frac{E_1}{2k_B}$, where $E_1$ is the energy of the lowest excited level and $k_B$ the Boltzmann constant, the spin-wave approximation can be used to interpret the results, while for temperatures larger than about $\frac{E_1}{6k_B}$ the Weiss molecular-field treatment is valid. In the overlapping temperature range, both approximations are equally good. The nuclear magnetic specific heat for TbIG and HoIG is observed and is found to be consistent with the predictions from resonance measurements in other rare-earth compounds.