Ultrasonic dispersion due to the rattling motion in the filled skutterudite La(Os$_{0.5}$Ru$_{0.5})_{4}$Sb$_{12}$

The filled skutterudites$ R$M$_{4}$Sb$_{12}$ ($R$: rare-earth, M: transition metal) show a considerable reduction of thermal conductivity, which is caused by the rattling motion of the weakly bonded rare-earth ion in an oversized icosahedral cage of antimony. Ultrasound measurements are useful for observing the rattling motion of the rare-earth ion. Our recent ultrasonic study* demonstrated the ultrasonic dispersion in the elastic constant ($C_{11}-C_{12})$/2 due to the rattling motion of Pr in PrOs$_{4}$Sb$_{12}$, which is the first Pr-based heavy fermion superconductor with $T_{C }$= 1.85 K. In order to examine the rattling motion in a system without 4f-electrons, we have performed ultrasonic measurements on La(Os$_{0.5}$Ru$_{0.5})_{4} $Sb$_{12}$. The longitudinal $C_{11}$ mode shows Debye-type dispersion and, at around 50 K, the ultrasonic echo shows the highest attenuation. The rattling motion of La has thermally activated behavior according to the relaxation data. In contrast, the transverse $C_{44 }$mode shows only a monotonic increase. These results suggest that, like PrOs$_{4}$Sb$_{12}$, the rattling motion in La(Os$_{0.5}$Ru$_{0.5})_{4}$Sb$_{12 }$has $\Gamma _{3}$ symmetry. T. Yanagisawa is supported by a Research Fellowship of the JSPS Young Scientists. The research at UCSD was supported by U.S. DOE., [*Phys. Rev. \textbf{B 69}, 180511 (R) 2004]