Critical Scattering of Sound in Rare-Earth Metals

We present ultrasonic attenuation measurements in the vicinity of the Néel temperature for terbium, dysprosium, and holmium. Longitudinal sound waves along the $c$ axis exhibit a large critical attenuation, whereas shear waves show only a small but measurable effect in dysprosium. The spin-phonon coupling responsible for this critical attenuation is predominantly of volume-magnetostrictive character for longitudinal waves and of linear-magnetostrictive (single-ion-type) character for shear waves. Longitudinal wave attenuation in the paramagnetic region gives the following critical exponent $\eta \left[\alpha \propto {\omega }^2{(T-T_N)}^{-\eta }\right]:\mathrm{terbium} 1.24\pm 0.1, \mathrm{dysprosium} 1.37\pm 0.1, \mathrm{holmium} 1.0\pm 0.1$. Shear waves in dysprosium give $\eta =0.8\mathrm{}\pm 0.15$. These exponents, with the exception of the shear-wave case, can be fairly well accounted for by recent theories of ultrasonic attenation at magnetic phase transitions. The critical exponent $\eta$ for shear waves in Dy, together with the magnitude of the effect, can tentatively be explained with present theories by considering higher-order terms in the spin-phonon coupling. For temperatures very close to $T_N$ the attenuation remains finite. We show this to be an impurity effect.