  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.
