Critical Scattering of Sound in Rare-Earth Metals

We present ultrasonic attenuation measurements in the vicinity of the N\'eel 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 ${η}[{α}{∝}{{ω}}²{(T{-}{T}N)}^{{-}{η}}]:terbium 1.24±{}0.1, dysprosium 1.37±{}0.1, holmium 1.0±{}0.1$. Shear waves in dysprosium give ${η}=0.8±{}0.15$. 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 ${η}$ 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 TN the attenuation remains finite. We show this to be an impurity effect.