Ferroquadrupole ordering and Gamma_5 rattling motion in clathrate compound Ce_3Pd_20Ge_6

Lattice effects in a cerium based clathrate compound Ce_3Pd_20Ge_6 with a cubic Cr_23C_6-type structure have been investigated by ultrasonic and thermal expansion measurements. Elastic softenings of (C_11-C_12)/2 and C_44 proportional to the reciprocal temperature 1/T above T_Q1 = 1.25 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 and a spontaneous expansion DL/L = 1.9x10^-4 along the [001] direction in particular indicate the ferroquadrupole ordering of O_2^0 below T_Q1. The elastic anomalies associated with the antiferromagnetic ordering at T_N2 = 0.75 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 t = t_0exp(E/k_BT) with an attempt time t_0 = 3.1x10^-11 sec and an activation energy E = 70 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.