The elastic properties of a filled skutterudite PrOs 4 Sb 12 with a heavy fermion superconductivity at T C ϭ1.85 K have been investigated. The elastic softening of (C 11 ϪC 12 )/2 and C 44 with lowering temperature down to T C indicates that the quadrupolar fluctuation due to the CEF state plays a role for the Cooper paring in superconducting phase of PrOs 4 Sb 12 . A Debye-type dispersion in the elastic constants around 30 K revealed a thermally activated ⌫ 23 rattling due to the off-center Pr-atom motion obeying ϭ 0 exp(E/k B T) with an attempt time 0 ϭ8.8ϫ10 Ϫ11 s and an activation energy Eϭ168 K. It is remarkable that the charge fluctuation of the off-center motion with ⌫ 23 symmetry may mix with the quadrupolar fluctuation and enhance the elastic softening of (C 11 ϪC 12 )/2 just above T C .The rare-earth cubic compounds based on Pr 3ϩ ions have received much attention because various unusual properties are expected at low temperatures. The system with a non-ϪJ y 2 favors a quadrupole ordering. We refer the ⌫ 3 ground state systems as a metallic compound PrPb 3 showing the antiferro-quadrupole ordering at T Q ϭ0.4 K 1 and a semiconductor PrPtBi undergoing ferroquadrupole ordering at T Q ϭ1.2 K. 2 PrSb is known as a singlet ground state system. 3 The elastic constant (C 11 ϪC 12 )/2 is responsible for the quadrupolar susceptibility of O 2 0 and O 2 2 with ⌫ 3 symmetry, while C 44 is for the susceptibility of O yz ϭJ y J z ϩJ z J y , O zx ϭJ z J x ϩJ x J z , O xy ϭJ x J y ϩJ y J x with ⌫ 5 symmetry. The softening of (C 11 ϪC 12 )/2 and C 44 is useful proof to clarify the quadrupolar effects of Pr-based compounds.Recently, Bauer et al. have found a new-type of the heavy-fermion superconductor in a filled skutterudite PrOs 4 Sb 12 with space group T h 5 (Im3). 4 The heavy fermion state with a large specific heat coefficient ␥ϭ750 mJ/mol K 2 of PrOs 4 Sb 12 exhibits the superconducting transition at T C ϭ1.85 K associated with a large jump ⌬C/T C ϳ500 mJ/mol K 2 . A sign of the double transition in the specific heat has been found. 5 The thermal transport measurement in fields suggests the two distinct superconducting phases in PrOs 4 Sb 12 . 6 The nuclear spin relaxation rate 1/T 1 of Sb indicates unconventional superconductivity possessing neither a coherence peak nor a T 3 -power law. 7 The muon spin relaxation in PrOs 4 Sb 12 yields a penetration depth indicating a new type of energy gap. 8 The odd-parity Cooper pairing mediated by the quadrupole fluctuation is argued as unconventional heavy-fermion superconductivity in PrOs 4 Sb 12 . 9 Because the magnetic susceptibility is rather silent to distinguish non-magnetic ⌫ 23 doublet from ⌫ 1 singlet, it has not been settled whether the CEF ground state of PrOs 4 Sb 12 is ⌫ 23 doublet or ⌫ 1 singlet. 4,10 The measurement of (C 11 ϪC 12 )/2 and C 44 responsible for the quadrupolar sus-ceptibility in PrOs 4 Sb 12 is a central issue to clarify the CEF state and the interplay of the quadrupolar fluctuation to the superconductivity in PrOs 4 Sb 12 .

Elastic constant measurements on iron pnictide Ba(Fe 0:9 Co 0:1 ) 2 As 2 with an optimal superconducting transition temperature of T SC ¼ 23 K have been performed by the ultrasonic pulse echo method. The shear elastic constant of C 66 associated with elastic strain " xy reveals considerable softening of 21% below 300 K down to T SC and becomes increasing in the superconducting phase below T SC , while other shear elastic constants of ðC 11 À C 12 Þ=2 and C 44 show no sign of softening below 80 K down to 4.2 K. The electric quadrupole O v 0 existing in the degenerate d y 0 z and d zx 0 bands participates in the superconductivity of the present iron pnictide system. KEYWORDS: layered iron pnictide, superconductor, Ba(Fe 0:9 Co 0:1 ) 2 As 2 , elastic softeningThe discovery of the superconductor LaFeAs(O 1Àx F x ) with a high transition temperature of T SC ¼ 26 K by Hosono's group triggered thousands of research works pursuing FeAs-based superconductors with various chemical composites and different structures.1) Five crystal structures with a common iron-based layer structure are now known.2)The high transition temperatures of up to 55 K and high critical magnetic fields beyond 50 T have already been realized.3,4) Among these FeAs-based crystals, the tetragonal compound of Ba(Fe 1Àx Co x ) 2 As 2 , in particular, has received much attention, because large single crystals are available for the investigation of physical properties depending on crystal anisotropy. 5)The end-material BaFe 2 As 2 exhibits a simultaneous transition of structural transition from a tetragonal TrCr 2 Si 2 -type structure of space group D orbitals by Co 2þ with 3d 7 orbitals in Ba(Fe 1Àx Co x ) 2 As 2 reduces both the structural transition temperature of T s and the antiferromagnetic one of T N .8) The superconductivity emerges in the systems above x ¼ 0:03 at the temperatures of T SC below the structural point T s and the antiferromagnetic one T N . The quantum critical point, x QCP , where both structural and antiferromagnetic phases die out, exists in the proximity of x ¼ 0:07. The optimum superconducting temperatures of T SC ¼ 23 K are achieved with x ¼ 0:10, which is a slightly excess of the quantum critical concentration x QCP . The superconductivity vanishes in the highly doped region of x ¼ 0:17.The Fermi surfaces of iron pnictides arise from Fe-3d orbitals and have three hole bands at a À-point and two electron bands at an X-point. These bands consist primarily of d y 0 z and d zx 0 with À 5 under tetragonal symmetry and d x 02 Ày 02 with À 3 .9) The increasing of the Fermi level upon electron doping due to the substitution of Fe with Co reduces the three sheets of hole bands around the À point in the endmaterial BaFe 2 As 2 to two sheets of hole bands in the present Ba(Fe 0:9 Co 0:1 ) 2 As 2 . We adopt the electric quadrupole associated with the degenerate d y 0 z and d zx 0 orbital states to describe elastic properties of the compound Ba(Fe 1Àx Co x ) 2 -As 2 . It is noted that the coordinates x 0 and y 0 of the quadrupole are ori...

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Lattice effects in a cerium based clathrate compound Ce 3 Pd 20 Ge 6 with a cubic Cr 23 C 6 -type structure have been investigated by ultrasonic and thermal expansion measurements. Elastic softenings of (C 11 ϪC 12 )/2 and C 44 proportional to the reciprocal temperature 1/T above T Q1 ϭ1.25 K are well described in terms of the quadrupole susceptibility for the ground state ⌫ 8 quartet. A huge softening of 50% in (C 11 ϪC 12 )/2 and a spontaneous expansion ⌬L/Lϭ1.9ϫ10 Ϫ4 along the ͓001͔ 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 ϭ0.75 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 ⌫ 5 rattling motion of an off-center Ce ion along the ͓111͔ direction with eight fractionally occupied positions around the 4a site in a cage. The thermally activated ⌫ 5 rattling motion obeying a relaxation time ϭ 0 exp (E/k B T) with an attempt time 0 ϭ3.1ϫ10 Ϫ11 sec and an activation energy Eϭ70 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.

We present tunneling and rattling motions of an off-center guest atom in cage referring to a clathrate crystal La_3Pd_{20}Ge_6. The elastic constant C_{44} of La_3Pd_{20}Ge_6 shows a Debye-type dispersion around 20 K obeying a relaxation time tau = tau_0exp(E/k_{B}T) with an attempt time tau_0 = 2.0*10^{-12} sec and an activation energy E = 197 K. At low temperatures below 3 K down to 20 mK, the C_{44} shows a softening of C_{44} = C_{44}^0(T-T_C^0)/(T-Theta) with T_C^0 = -337.970 mK and Theta = -338.044 mK. These facts are attributed to two different types of the off-center motions with Gamma_5 symmetry in 4a-site cage of La_3Pd_{20}Ge_6, a thermally activated rattling motion over the potential hill and a tunneling motion through the potential hill at low temperatures.Comment: 5 pages, 4 figures, to be published Phys. Rev.

In challenging a direct observation of the vacancy in crystalline silicon, we have carried out lowtemperature ultrasonic measurements down to 20 mK. The longitudinal elastic constants of non-doped and B-doped crystalline silicons, which were grown by a floating zone (FZ) method in commercial base, reveal the elastic softening proportional to the reciprocal temperature below 20 K. The applied magnetic fields turn the elastic softening of the B-doped FZ silicon to a temperature-independent behavior, while the fields up to 16 T do not affect the elastic softening of the non-doped FZ silicon. We present a plausible scenario for this result. Namely the vacancy with the non-magnetic charge state V 0 in the nondoped silicon and the magnetic V þ in the B-doped silicon is responsible for the low-temperature softening of the shear elastic constants ðC 11 À C 12 Þ=2 and C 44 , which can be described in terms of the quadrupole susceptibility due to the Jahn-Teller effect.

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