Ferroquadrupole ordering and ${\Gamma }_5$ rattling motion in the clathrate compound ${\mathrm{Ce}}_3{\mathrm{Pd}}_{20}{\mathrm{Ge}}_6$

Lattice effects in a cerium based clathrate compound ${\mathrm{Ce}}_3{\mathrm{Pd}}_{20}{\mathrm{Ge}}_6$ with a cubic ${\mathrm{Cr}}_{23}{\mathrm{C}}_6$-type structure have been investigated by ultrasonic and thermal expansion measurements. Elastic softenings of ${(C}_{11}-C_{12})/2\mathrm{}$ and $C_{44}$ proportional to the reciprocal temperature $1/T$ above $T_{\mathrm{Q}1\mathrm{}}=1.25\mathrm{}$ K are well described in terms of the quadrupole susceptibility for the ground state ${\Gamma }_8$ quartet. A huge softening of 50% in ${(C}_{11}-C_{12})/2\mathrm{}$ and a spontaneous expansion $\Delta L/L=1.9\mathrm{}\times {10}^{-4}$ along the [001] direction in particular indicate the ferroquadrupole ordering of $O_2^0$ below $T_{\mathrm{Q}1\mathrm{}}.$ The elastic anomalies associated with the antiferromagnetic ordering at $T_{\mathrm{N}2\mathrm{}}=0.75\mathrm{}$ K and the incommensurate antiferromagnetic ordering are also found. Notable frequency dependence of $C_{44}$ around 10 K is accounted for by the Debye-type dispersion indicating a ${\Gamma }_5$ rattling motion of an off-center Ce ion along the [111] direction with eight fractionally occupied positions around the 4a site in a cage. The thermally activated ${\Gamma }_5$ rattling motion obeying a relaxation time $\tau ={\tau }_0\exp {(E/k}_{B}T)$ with an attempt time ${\tau }_0=3.1\mathrm{}\times {10}^{-11}$ sec and an activation energy $E=70\mathrm{}$ K dies out with decreasing temperature, and then the off-center tunneling state of Ce ion in the 4a-site cage will appear at low temperatures.