Quadrupolar effect and rattling motion in the heavy-fermion superconductor ${\mathrm{PrOs}}_4{\mathrm{Sb}}_{12}$

The elastic properties of a filled skutterudite ${\mathrm{PrOs}}_4{\mathrm{Sb}}_{12}$ with a heavy fermion superconductivity at $T_{\mathrm{C}}=1.85\mathrm{}\mathrm{K}$ have been investigated. The elastic softening of ${(C}_{11}-C_{12})/2\mathrm{}$ and $C_{44}$ with lowering temperature down to $T_{\mathrm{C}}$ indicates that the quadrupolar fluctuation due to the CEF state plays a role for the Cooper paring in superconducting phase of ${\mathrm{PrOs}}_4{\mathrm{Sb}}_{12}.$ A Debye-type dispersion in the elastic constants around 30 K revealed a thermally activated ${\Gamma }_{23}$ rattling due to the off-center Pr-atom motion obeying $\tau ={\tau }_0\exp {(E/k}_{\mathrm{B}}T)$ with an attempt time ${\tau }_0=8.8\mathrm{}\times {10}^{-11}\mathrm{s}$ and an activation energy $E=168\mathrm{}\mathrm{K}.$ It is remarkable that the charge fluctuation of the off-center motion with ${\Gamma }_{23}$ symmetry may mix with the quadrupolar fluctuation and enhance the elastic softening of ${(C}_{11}-C_{12})/2\mathrm{}$ just above $T_{\mathrm{C}}.$