蒋敏强----中国科学院力学研究所.pdf
蒋敏强----中国科学院力学研究所.pdf
PHYSICAL REVIEW B 76, 054204 共2007兲 Intrinsic correlation between fragility and bulk modulus in metallic glasses Minqiang Jiang1 and Lanhong Dai1,2,* 1State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing 100080, People’s Republic of China State Key Laboratory of Explosion Science and Technology, Beijing 100081, People’s Republic of China 共Received 23 January 2007; revised manuscript received 10 April 2007; published 14 August 2007兲 2 A systematic study on the available data of 26 metallic glasses shows that there is an intrinsic correlation between fragility of a liquid and bulk modulus of its glass. The underlying physics can be rationalized within the formalism of potential energy landscape thermodynamics. It is surprising to find that the linear correlation between the fragility and the bulk-shear modulus ratio exists strictly at either absolute zero temperature or very high frequency. Further analyses indicate that a real flow event in bulk metallic glasses is shear dominant, and fragility is in inverse proportion to shear-induced bulk dilatation. Finally, extension of these findings to nonmetallic glasses is discussed. DOI: 10.1103/PhysRevB.76.054204 PACS number共s兲: 64.70.Pf, 62.20.Dc, 61.43.Dq I. INTRODUCTION Understanding of the material-dependent slowing down dynamics is one of the central subjects in glass physics. In terms of the concept of fragility introduced by Angell,1 the glass-forming systems can be classified into “strong” or “fragile” pattern. The kinetic fragility m is defined as m= 冏 冏 log共/⬁兲 , 共Tg/T兲 T=Tg 共1兲 where ⬁ is the limiting high-temperature shear viscosity and Tg is the glass transition temperature. Here, the dimensionless fragility index, a measure of deviation from the Arrhenius behavior in the temperature-dependent viscosity, opens a new window into the understanding of glass transition and its slowing down dynamics. The kinetic fragility was found to be correlated with other properties characterizing the liquid side of glass transition2 and low-temperature properties of glasses such as vibration.3,4 An unexpected linear correlation between bulk-shear modulus ratio 共K⬁ / G⬁兲 and fragility in nonmetallic glass was reported by Novikov and Sokolov.5 Due to the technological and the scientific significance of this surprising finding, it has attracted a lot of discussions.6–10 However, the question whether such a linear correlation universally exists still remains open up to now. As for metallic glasses, a number of important correlations among their properties have been revealed.11–16 Recently, Novikov and Sokolov17 discovered that there is also a linear correlation between the bulk-shear modulus ratio and fragility in metallic glasses, and the deviation from the linear correlation is due to the material-specific free electron gas. In this paper, a systematic study on the correlation between fragility and elastic modulus in metallic glasses is done, and further, its underlying physics is discussed. II. EXPERIMENTAL OBSERVATIONS Table I lists the measured physical properties of 26 metallic glasses,18 such as density , molar mass M, bulk modulus K, shear modulus G, glass transition temperature Tg, and fragility m. It is well known that values of m can be directly 1098-0121/2007/76共5兲/054204共7兲 taken by using the Vogel-Fulcher-Tamman 共VFT兲 equation fits to viscosity or relaxation time data.1 This is the kinetic method. On the other hand, the liquid fragility can be determined from a purely thermodynamic way, i.e., the heating rate dependent Tg.15,20 Let us note that the fragility values of metallic glasses numbered as 1–8, 14, 15, 17, and 18 in Table I are given by the thermodynamic method, whereas the others are obtained from viscosity or relaxation data. Besides, we also note that the fragility values are mostly derived from the latest literature such as Refs. 9, 15, and 17. Previous studies have demonstrated that the heating rate dependent Tg describes the fragility equally well as complementary viscosity or relaxation time measurements.29–31 Thus, the following results according to these values in Table I are believable, and the unavoidable slight error due to different measurements cannot significantly change the intrinsic relationship among the parameters listed in Table I. As shown in Fig. 1, there seems no clear linear relation between fragility and K 关or G, K / G, and the Poisson ratio, = 1 / 2 − 3 / 共6K / G + 2兲兴 just as Battrzzati7 and Johari9 observed. However, the scrupulous examination shows that a parameter combination, mRTg / 共M / 兲, exhibits a striking linear correlation with K 共see Fig. 2兲 regardless of chemical components, structural details, etc., and the best fit of data is given by mRTg/共M/兲 − 0.729 = 0.198K, 共2兲 where M / is the molar volume Vm. That is, the correlation between m and K does not follow a simple linear relation, and the influence of material parameters such as Tg, M, and should be involved. III. THEORY Potential energy landscape 共PEL兲 is an ideal method to describe the behavior of disordered systems,32,33 which can be used to reveal the underlying physics of the linear correlation in Eq. 共2兲. For simplicity, let us assume that the potential energy of a system with N pointlike constituent particles only depends on the spatial location ri for each particle in a 共3N + 1兲-dimensional hypersurface. Intuitively, the state of 054204-1 ©2007 The American Physical Society PHYSICAL REVIEW B 76, 054204 共2007兲 MINQIANG JIANG AND LANHONG DAI TABLE I. Physical properties of 26 metallic glasses, such as density , molar mass M, bulk modulus K, shear modulus G, glass transition temperature Tg, and fragility m. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 Metallic glasses 共g / cm3兲 M 共g/mol兲 K 共GPa兲 G 共GPa兲 Tg 共K兲 m Refs. Ce70Al10Ni10Cu10 Cu46Zr46Al8 Cu46Zr46Al7Gd1 Fe70P10C5B5Si3Al5Ga2 Ho39Al25Co20Y16 La55Al25Cu10Ni5Co5 Nd60Fe20Co10Al10 Pd39.1Ni10.1Cu29.9P20.9 Pd39Ni10Cu30P21 Pd40Ni10Cu30P20 Pd40Ni40P20 Pd48Ni32P20 Pd64Ni16P20 Pd77.5Cu6Si16.5 Pr60Al10Ni10Cu20 Pt60Ni15P25 Mg65Cu25Y10 Mg65Cu25Gd10 Mg65Cu25Tb10 Ni60Nb35Sn5 Ni64Pd16P20 Zr41Ti14Cu12.5Ni10Be22.5 Zr41.2Ti13.8Cu12.5Ni10Be22.5 Zr46.75Ti8.25Cu7.5Ni10Be27.5 Zr55Al19C019Cu7 Zr65Al10Ni10Cu15 6.67 7.23a 7.40 6.24b 6.50b 6.00 7.00 9.2 9.152 9.3 9.405 9.83 10.1 10.4 6.90 15.7 3.978 4.04 3.98 8.64 8.75 6.12 5.9 6.01 6.2 6.75f 113 73.3 74.6 46.9 97.1 95.4 106 73.0 72.9 73.7 72.2 76 83.7 90.9 106 133 40.6 47.4 47.4 74.3 61.3 59.9 60.0 59.5 101.7 77.4 27 116.4 123.7 113.4 63.60 44.2 46.54 158.7 159.1 146 185 176.7 166 167 45.2 202 44.71d 46.3 44.7 267 169.8 114.7 114.1 113.38 114.9 106.65 11.5 34.3 32.9 58.5 26.2 15.6 19.44 35.2 35.1 33 38.6 36.2 32.7 31.5 13.6 33.8 19.6d 18.6 19.6 66.32 37.9 37.4 34.1 35.2 37.6 30.27 359 701 720 736c 630 430 493 576 586 560 602 590 582 635 409 500 402 417 414 885 587 627 618 590 733 653 21 43 29 34 49 28 33 52 55 52 54 48 51 52 31 86 41 41 47 70 50 50 50 44 72e 30 7, 12, and 13 15 15 15 15 13, 15, and 19 15, 17, and 19 15 15 and 17 13, 20, and 21 15, 19, and 22 15 and 17 13, 21, and 23 7, 23, and 24 20 and 25 13 and 26 7, 21, and 23 13 and 15 15 and 17 15 and 17 15 and 17 7, 17, and 21 13 and 27 15, 19, and 27 15 and 17 15, 19, and 28 a d Value for Cu46Zr42Al7Y5. bCalculated by atom mole ratio. Values for Mg65Cu25Tb10. eValue for Zr Al 55 22.5Co22.5. f Value for Zr65Cu17.5Al7.5Ni10. c Value for Fe80P13C7. such a system can be represented by a point s on or above the hypersurface. By analogy to Earth’s topographic maps, Stillinger and Weber provided a formally exact portioning of the configurational space as a sum of distinct basins, associating with each local minimum of the potential energy surface, namely, an inherent structure 共IS兲. The purpose is to assign any configuration of atoms uniquely to one local minimum by the steepest descent path; if it is not at a minimum, the displacement exhibited by the system is simply regarded as a “vibrational,” possibly anharmonic in character, displacement. Thus, packing and vibration effects can be cleanly separated. The system free energy in an IS can be expressed as a sum of a configurational contribution, accounting for the number of the available basins, and a vibrational one, expressing the free energy of the system when constrained in one of the basins.34,35 Here, the Helmholtz free energy F共T , V兲 of a metallic glass 共unit mass兲 at the temperature below Tg can also be written as F共T,V兲 = Fconf 共T,V兲 + Fvib共T,V兲, 共3兲 where the first term on the right hand side is the configurational portion denoted as Fconf = U共V兲 − TSconf , with U共V兲 the average specific potential energy of internal energy and −TSconf the configurational contribution to Fconf that switches off due to T ⬍ Tg. The second term, Fvib = Uvib − TSvib, indicates the vibrational part, where Svib is the vibrational entropy due to departure from the local minimum and the kinetic part of internal energy Uvib can be negligible. Since a metallic glass is an isotropic body, its potential energy can be assumed as36 U共V兲 = − A/V␣/3 + B/V/3 共␣ ⬎ 兲, 共4兲 where A and B are constants. Here, the first term represents the energy of the attractive interaction and the second term is the energy of repulsion. The values of ␣ and  are determined by material composition and structure. A glass at tem- 054204-2 PHYSICAL REVIEW B 76, 054204 共2007兲 INTRINSIC CORRELATION BETWEEN FRAGILITY AND… FIG. 1. Fragility versus 共a兲 bulk modulus K, 共b兲 shear modulus G, 共c兲 bulk-shear modulus ratio, K / G, and 共d兲 Poisson ratio v. perature below Tg is at or near a potential energy minimum or an IS. Its potential energy is U0 = U共V0兲 due to 共U / V兲V0 = 0. According to the thermodynamics,37 the isothermal bulk modulus at T ⬍ Tg can be obtained by K = − ␣U0/9 − 共T/兲共2Svib /V2兲T,V0 . Note that Eqs. 共3兲 and 共4兲 are used in deriving this equation. Many studies38 have demonstrated that the viscositytemperature relation of supercooled liquids for metallic glass formers, especially approaching Tg, can be described well by the VFT equation.39 In terms of U0, the VFT equation can be expressed as40,41 = ⬁ exp兵− lMU0/关R共T − T0兲兴其, FIG. 2. Correlation between mRTg / 共M / 兲 and bulk modulus K in the glassy state, where the straight line is the best fitting results by Eq. 共2兲. 共5兲 共6兲 where l is a numerical factor, R is a gas constant, and T0 is the Vogel temperature. Compared to the usual expression of the VFT equation, = ⬁ exp关B / 共T − T0兲兴, where B is the activation energy of viscous flow, we find B = −lMU0 / R. So, Eq. 共6兲 implies that the activation energy of viscous flow when T ⬎ T0 is proportional to the IS potential energy U0. Based on this assumption, Gemant41 established a relation between bulk modulus and viscosity of plastics. Applying the obtained relation to glass materials exhibits a good correlation between room-temperature elastic modulus and viscosity at high temperature.41 Also, the assumption is consistent with the basic tenet of PEL, namely, “the existence of potential energy barriers large compared to thermal energy are intrinsic to the occurrence of the glassy state, and dominate flow, at least at low temperature.32 Inserting Eq. 共6兲 into Eq. 054204-3 PHYSICAL REVIEW B 76, 054204 共2007兲 MINQIANG JIANG AND LANHONG DAI m = a共K/G兲 + b共T;m,G兲, FIG. 3. Correlation between RTg / 共M / 兲 and shear modulus G in the glassy state, where the straight line is the best fitting results by Eq. 共8兲. 共1兲 yields U0 = −mR共Tg − T0兲2 ln 10/ 共lMTg兲. Further, according to Eq. 共5兲 and Tg = T0关1 + B / 共17T0 ln 10兲兴,1 we have the following correlation between fragility and bulk modulus: mRTg/共M/兲 + L共T兲 = CK, 共7兲 where L共T兲 = C关−共T / 兲共2Svib兲 / V2兴T,V0 and C = 9l关1 + 17共T0 / B兲ln 10兴 / 共␣ ln 10兲. Surprisingly, Eq. 共7兲 is totally identical to the empirical fitting, i.e., Eq. 共2兲, so long as L共T兲 = −0.729 GPa and C = 0.198. It is important to point out that Eq. 共7兲 is valid if temperature is below Tg, and Eq. 共6兲 is only valid when T ⬎ T0. The term L共T兲 reflects the temperature softening effect on bulk modulus, and C is weakly material dependent for metallic glasses according to the fit to experimental data. IV. DISCUSSION A. Fragility and bulk-shear modulus ratio To shed an insight into whether fragility m is correlated linearly with K / G, we examined the data in Table I. A striking linear correlation between RTg / 共M / 兲 and shear modulus G is observed, as shown in Fig. 3. This linear correlation can be best fitted by RTg/共M/兲 + H共T兲 = DG, 共8兲 where H共T兲 = – 0.063 GPa, which is temperature dependent like the term L共T兲 in Eq. 共7兲, and D = 0.014. It is worth noting that this intrinsic relation for metallic glasses is consistent with those obtained by Johnson and Somwer13 and Yang et al.16 Now, we realize that although the correlation between fragility m and bulk modulus K is characterized explicitly by Eq. 共7兲, the correlation between fragility m and shear modulus G is characterized implicitly by this equation through the parameter RTg / 共M / 兲 that is scaled linearly with G, as shown in Fig. 3. Combining Eq. 共7兲 with Eq. 共8兲, we have the following correlation: 共9兲 where a = C / D is a material-independent constant and b共T ; m , G兲 = 关mH共T兲 − L共T兲兴 / 共DG兲 indicates the effect of temperature on modulus. It is easy to see that the linear correlation between m and K / G in Eq. 共9兲 exists only when b共T ; m , G兲 is a material-independent constant. However, this condition is too harsh given that both m and G are highly material dependent. Obviously, m must be strictly proportional to K / G for solid glasses at absolute zero temperature or very high frequency. Each of these two conditions leads to L共T兲 = 0 and H共T兲 = 0. In fact, the former condition, i.e., the observed correlation at absolute zero temperature, is consistent with the finding of Scopigno et al.3 共vide post兲, while the latter implies that fragility is linearly linked to the instantaneous bulk-shear modulus ratio.5 Both the conditions require that measurements of the elastic constants of glasses be made either at a temperature low enough that molecular motion in structure is kinetically frozen or at a frequency used for ultrasonic measurements high enough that the temperaturedependent fast  relaxational contributions 共not only the structure relaxation5兲 are negligible. However, it is almost impossible that the above conditions are strictly satisfied, because, in a real experiment, sound velocity measurements on modulus in the frequency range of 20– 50 Hz may not yield elastic properties attributable to vibrations alone, while measurements on solid glasses are usually performed at room temperature. This is why the experimental data do not obey the linear correlation.7,9 Our finding that fragility is linearly related to the zero temperature or much higher frequency bulk-shear modulus ratio implies that liquid viscosity is determined by its zero temperature or short-time-scale elastic properties. This seems to be at odds with the prevailing paradigm of glass science,10 which is embodied in the Adam-Gibbs model.42 Nevertheless, according to elastic models of glass forming, the barrier transition for a “flow event” 共an atomic rearrangement兲 does take place on a very short time scale.43 Furthermore, both Egami’s atomistic theory8 and the microstructural model of Ichitsubo et al.6 can perfectly explain the Novikov-Sokolov observations. Actually, in their models, the temperature-dependent modulus was neglected, which corresponds to b共T ; m , G兲 ⬅ 0 in our case. B. Shear-induced dilatation Next, let us discuss the physical pictures behind Eqs. 共7兲 and 共8兲. In soil mechanics, it has been known for years that shear of randomly close-packed grains causes dilatation. The same phenomenon can be applied at the atomic scale. Crystals can deform at constant volume because the periodicity along slip planes provides identical atomic positions for sheared materials. A sheared portion of an amorphous material, on the other hand, does not find such a perfect fit and thus will leave some holes.44 It has been recognized that macroscopic flow of an amorphous material occurs as a result of a number of individual atomic jumps or flow events. Thus, a real flow event in bulk metallic glasses must result in shear deformation and an accompanying bulk deformation 054204-4 PHYSICAL REVIEW B 76, 054204 共2007兲 INTRINSIC CORRELATION BETWEEN FRAGILITY AND… 共or dilatation兲. Actually, according to Dyre’s shoving model, spherical symmetry in real flow events is probably violated, leading to some compression of surroundings and a contribution to activation energy that is proportional to bulk modulus.45 The equivalent PEL of the shoving model also confirmed this point; thereby, activation energy must involve both shear and bulk moduli.46 However, which of them controls the activation energy? The elastic models such as rate theory, shoving model, etc., give a theoretical answer and clarify that activation energy is proportional to shear modulus that is a controlling parameter in a real flow event.43 It is of interest to note that our results are consistent with the elastic models from the viewpoint of experimental data for bulk metallic glasses 共BMGs兲. Equation 共8兲 characterizes shear deformation in real flow events, while Eq. 共7兲 determines shear-induced bulk deformation. Based on the physical analogy between the shear transformation zone deformation and glass transition, Yang et al. found that the shear yield strength of BMGs at ambient temperature T0 can be predicted well by a unified parameter, R共Tg − T0兲 / Vm.16 Compared to Eq. 共8兲, we find that a compound parameter, RTg / Vm, can be regarded as a scale on shear yield strength at zero temperature, although H共T兲 is weakly dependent on T0. Data fitting shows that D is a material-independent constant with magnitude of 10−2, which is consistent with the average shear yield strain ␥C observed by Johnson and Samwer.13 Thus, D can be considered as an apparent shear yield strain ␥ˆ C. The linear correlation between RTg / Vm and G implies that G is a controlling quantity in real flow events. In parallel with this consideration, it is natural to investigate the relationship between RTg / Vm and K. Although mRTg / Vm is proportional to K, RTg / Vm does not scale with K, i.e., 共RTg / Vm兲 / K ⬃ ˆ c ⫽ const, where ˆ C = C / m is shear-induced dilatation 共with the order of magnitude of 10−3兲 while shear yielding occurs. It is noted that ˆ C is associated with m, and thus highly material specific. Since C is a materialindependent constant for metallic glasses, it is easily seen that fragility m is in inverse proportional to ˆ C. This means that, subjected to same hydrostatic compression, strong glasses are of higher bulk dilatation, while fragile ones have smaller bulk deformation.47 The explicit correlation between bulk 共not shear兲 modulus and fragility may result from this highly material-dependent bulk deformation in BMGs.17 Using a model developed by Knuyt et al. based on a Gaussian distribution for the nearest-neighbor distance in an ideal unicomponent metallic glass,48 we can approximately calculate the ratio of bulk to shear elastic energy in a flow event with the following equation: Eelastic,bulk 共1/2兲ˆ 2K . = Eelastic,shear 共3/2兲␥ˆ 2G 共10兲 Note that this ratio increases with decreasing m and increasing K / G. However, there is a tendency for fragility m to increase with increasing K / G, and vice versa. For BMGs 共see Table I兲, m mostly ranges from 30 to 80, and K / G is in the range 2–5; thus, the range of this ratio of bulk to shear elastic energy in a flow event is usually less than about 10%. Our result is consistent with the result of 8% calculated by Dyre and Olsen based on PEL46 or the 10% calculated recently by Dyre on a standard dipole expansion.49 It indicates that the bulk modulus’ contribution to the activation energy is much less than that due to shear deformation in a real flow event. Consequently, shear 共not bulk兲 modulus controls the activation energy 共that is shear dominant兲 in a real flow event. C. Comparison with related works At this point, we briefly discuss related works. Based on local topological instability analysis and the assumption that bulk modulus is almost independent of temperature, Egami14 predicts that there is a correlation between Tg and bulk modulus K for metallic glasses, Tg = 6.14⫻ 10−3K具⍀典 / kB. Actually, this assumption is equivalent to the case that the temperature-dependent term L共T兲 → 0 in Eq. 共7兲. Thus, Eq. 共7兲 can be converted into Tg = 共C/m兲共K具⍀典/kB兲, 共11兲 where 具⍀典 = M / 共N0兲 is the average atomic volume of a metallic glass with N0 the Avogadro constant. It is interesting to find that if the fragility value in Eq. 共11兲 is equal to 32.25 共the fragility value of metallic glasses ranges from 21 to 86 in this study兲, Egami’s prediction is recovered. Also, Eq. 共7兲 leads us directly to an explanation of the striking finding of Scopigno et al.3 They found that fragility is proportional to a dimensionless quantity ␣, i.e., the temperature steepness of the nonergodicity factor at T → 0.12 The low-temperature ␣ only depends on the interaction potential and disordered structure. In the harmonic approximation for vibrational dynamics, ␣ can be expressed as3 ␣ 2 = 共MN兺 p−2 p 兲 / 共kBTgQ 兲, where M is the molecular mass, kB is the Boltzmann constant, N is the number of particles, Q is the wave vector, and p is summed over 3N normal modes. In a system consisting of N0 共the Avogadro constant兲, ␣ can be converted into ␣ = 共M / 兲兵 / 关Q2共1 / N0兲兺 p−2 p 兴其 / 共RTg兲, where −2 −2 2 Q 共1 / N0兲兺 p p ⬀ vl with vl the longitudinal sound velocity.50 Because K ⬀ v2l , where vl is proportional to the transverse sound velocity vt,3 ␣ reduces to ␣ ⬀ 兩共M/兲K/共RTg兲兩T→0 . 共12兲 At T → 0, Eq. 共8兲 becomes RTg / 共M / 兲 ⬀ G. Substituting it into Eq. 共12兲 gives ␣ ⬀ 共K / G兲T=0. Due to m ⬀ 共K / G兲T=0, as discussed above, the finding of Scopigno et al., i.e., m ⬀ ␣, is also recovered. It is easy to see that the low-temperature ␣ can also be determined by elastic modulus at zero temperature. However, modulus measurements on metallic glasses are usually performed at room temperature. So, it is impossible to calculate the low-temperature ␣ by using the data listed in Table I, whereas Scopigno et al.3 cleverly derived the parameter ␣ from the IXS data available for glasses at low temperature. Finally, to survey the universality of two correlations in Eqs. 共7兲 and 共8兲, let us consider 11 nonmetallic glasses, including BeF2, B2O3, SiO2, GeO2, AS2S3, CKN, glycerol, salol, m-toluidine, OTP, and m-TCP.3,5 As shown in Fig. 4, 054204-5 PHYSICAL REVIEW B 76, 054204 共2007兲 MINQIANG JIANG AND LANHONG DAI Thus, further work is needed to check the universality of these intrinsic correlations. V. CONCLUSIONS FIG. 4. mRTg / 共M / 兲 versus K and RTg / 共M / 兲 versus G for metallic and nonmetallic glasses. Black and open stars: metallic glasses; black and open circles: nonmetallic glasses. the linear relationship indicates that the observed intrinsic correlations might be more general although the fitting parameters are not constants. Unfortunately, there are not enough glass formers whose relevant data are all known. ACKNOWLEDGMENTS L.H. is indebted to F. Spaepen for valuable discussions. This work was supported by the National Science Foundation of China 共Nos. 10472119 and 10232040兲, the key project of the Chinese Academy of Sciences 共Nos. KJCXSW-L08 and KJCX2-YW-M04兲, and the K. C. Wong Education Foundation, Hong Kong. 13 *Corresponding author; lhdai@lnm.imech.ac.cn 73, 1 共1985兲; 131-133, 13 共1991兲; R. Böhmer, K. L. Ngai, C. A. Angell, and D. J. Plazek, J. Chem. Phys. 99, 4201 共1993兲; C. A. Angell, Science 267, 1924 共1995兲. 2 A. P. Sokolov, E. Rössler, A. Kisliuk, and D. Quitmann, Phys. Rev. Lett. 71, 2062 共1993兲; K. Ngai and O. Yamamuro, J. Chem. Phys. 23, 10403 共1999兲; L.-M. Martinez and C. A. Angell, Nature 共London兲 410, 663 共2001兲; S. Venkataraman, K. Biswas, B. C. Wei, D. J. Sordelet, and J. Eckert, J. Phys. D 39, 2600 共2006兲. 3 T. Scopigno, G. Ruocco, F. Sette, and G. Monaco, Science 302, 849 共2003兲; U. Buchenau and A. Wischnewski, Phys. Rev. B 70, 092201 共2004兲. 4 Yong Li, H. Y. Bai, W. H. Wang, and K. Samwer, Phys. Rev. B 74, 052201 共2006兲. 5 V. N. 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