Quadrupolar effect and rattling motion in the heavy-fermion superconductorPrOs4Sb12

The elastic properties of a filled skutterudite ${\mathrm{PrOs}}_{4}{\mathrm{Sb}}_{12}$ with a heavy fermion superconductivity at ${T}_{\mathrm{C}}=1.85\mathrm{K}$ have been investigated. The elastic softening of ${(C}_{11}\ensuremath{-}{C}_{12})/2$ 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 ${\ensuremath{\Gamma}}_{23}$ rattling due to the off-center Pr-atom motion obeying $\ensuremath{\tau}={\ensuremath{\tau}}_{0}\mathrm{exp}{(E/k}_{\mathrm{B}}T)$ with an attempt time ${\ensuremath{\tau}}_{0}=8.8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}11}\mathrm{s}$ and an activation energy $E=168\mathrm{K}.$ It is remarkable that the charge fluctuation of the off-center motion with ${\ensuremath{\Gamma}}_{23}$ symmetry may mix with the quadrupolar fluctuation and enhance the elastic softening of ${(C}_{11}\ensuremath{-}{C}_{12})/2$ just above ${T}_{\mathrm{C}}.$