Quadrupole Ordering and Rattling Motion of Clathrate Compound Pr3Pd20Ge6

We have investigated the quadrupole ordering of 4f-electrons and the rattling motion of guest rare-earth ions in the clathrate compound Pr 3 Pd 20 Ge 6 using ultrasonic measurements. The elastic constants $(C_{11} - C_{12})/2$ and $C_{44}$ exhibit considerable softening at low temperatures. A minimum of $(C_{11} - C_{12})/2$ at $T_{\text{Q1}} = 0.25$ K indicates antiferroquadrupole ordering associated with a $\Gamma_{3}$ doublet of a crystal-field ground state at the 8c site, while a minimum of $C_{44}$ at $T_{\text{Q2}} = 60$ mK indicates the ferroquadrupole ordering of a $\Gamma_{5}$ triplet of a crystal-field ground state at the 4a site. The antiferroquadrupole phase depending on magnetic field direction shows a highly anisotropic behavior. The field-induced ordered phase due to the level crossing of the CEF ground state at the 8c site in magnetic fields along the [001] direction has been found. Furthermore, for the elastic constant $C_{44}$ and the attenuation coefficient $\alpha_{44}$ we found a frequency dependence at two different temperatures of 15 and 25 K, which we describe in terms of the Debye-type dispersion with an Arrhenius-type relaxation time $\tau = \tau_{0}\exp(E/k_{\text{B}} T)$ for two different activation energies of $E_{\text{L}} = 102$ K and $E_{\text{H}} = 198$ K. These ultrasonic dispersions are caused by the thermally activated rattling motions of guest rare-earth Pr 3+ ions at the 4a site of Pr 3 Pd 20 Ge 6 .