Thermal conductivity of disordered garnets from infrared spectroscopy

The thermal conductivity (k) of garnets is calculated from a formula originating with Debye, adapted to treat the vibrations in a solid as a collection of damped harmonic oscillators. Our model utilizes phonon lifetimes obtained from Kramers-Kronig analyses of existing IR reflectivity spectra, as well as new data presented here, and calculates k at ambient conditions within a nominal uncertainty of 6% of the experimental values for eight natural samples with well-constrained chemical compositions. Agreement is good for the remaining garnets with uncertain compositions. Two series of natural mixed crystals were studied: (1) $X_3{\mathrm{Al}}_2{\mathrm{Si}}_3{\mathrm{O}}_{12},$ where the X site has varying amounts of Mg, ${\mathrm{Fe}}^{2+},$ and Ca and (2) ${\mathrm{Ca}}_3{Y}_2{\mathrm{Si}}_3{\mathrm{O}}_{12},$ where the Y site has Al and/or ${\mathrm{Fe}}^{3+}.$ The model predicts k at ambient conditions for the end members. The occurrence of a minimum in k near the midpoint of each compositional series correlates with the maximum widths of the IR peaks. Thus, disorder on crystallographic sites largely controls the thermal conductivity of mixed crystals. We also tested the model using available data on synthetic yttrium garnets. Agreement is good for YAG. Observation of $\sim 30\%$ discrepancy between calculated and experimental k of YGG and YIG suggests that these unconfirmed measurements should be repeated.