Evaluation of Debye-Waller factors in rare-earth monosulfides: Evidence for softening of optic phonons in mixed valentSm0.7Y0.3S

The temperature-dependent Debye-Waller factors for SmS, GdS, ${\mathrm{Eu}}_{0.8}$ ${\mathrm{Y}}_{0.2}$S, and ${\mathrm{Sm}}_{0.7}$ ${\mathrm{Y}}_{0.3}$S have been measured by single-crystal x-ray diffraction technique. From these data, we have determined the mean-square displacements $〈u_i^2\mathrm{}\left(T\right)\mathrm{}〉$ for the sulfur and rare-earth atoms. $〈u_{\mathrm{S}}^2〉$ in ${\mathrm{Sm}}_{0.7}$ ${\mathrm{Y}}_{0.3}$S shows a strong increase at lot $T$ near the continuous gold-black phase transition. Yet $〈u_{\mathrm{Sm}}^2〉$ decreases normally with $T$, as do all $〈u^2〉$ values of the other materials. The anomaly in $〈u_{\mathrm{S}}^2\mathrm{}\left(T\right)\mathrm{}〉$ strongly suggests a softening of optic phonons. We believe that this is caused by the fluctuating valence of the Sm ion, and propose a model for the coupling of the Sm ion to nearest-neighbor S ions, very similar to the breathing-shell model of Schröder. Our model leads to an understanding of both the softening of optic phonons and the decrease of the bulk modulus, observed recently by Melcher et al. We present $〈u_i^2\mathrm{}\left(T\right)\mathrm{}〉$ calculations for ${\mathrm{Sm}}_{0.7}$ ${\mathrm{Y}}_{0.3}$S and for the other materials, and give an estimate of static disorder effects on $〈u^2〉$.