Role of low-frequency vibrations on sound propagation in glasses at intermediate temperature

Role of low-frequency vibrations on sound propagation in glasses at intermediate temperature A. Criado, M. Jime ´nez-Ruiz, C. Cabrillo, and F. J. Bermejo Consejo Superior de Investigaciones Cientı ´ficas, Serrano 123, E-28006 Madrid, Spain and Departamento de Fı ´sica de la Materia Condensada, Universidad de Sevilla, P.O. Box 1065, E-41080 Seville, Spain M. Grimsditch Argonne National Laboratory, Argonne, Illinois 60439 H. E. Fischer Institut Laue Langevin, Boı ˆte Postale 156x, F-38042 Grenoble Cedex 9, France S. M. Bennington and R. S. Eccleston Rutherford Appleton Laboratory, Chilton Didcot, Oxon OX11 0QX, Great Britain 共Received 19 April 1999; revised manuscript received 3 January 2000兲 We report measurements of the temperature dependence of the sound attenuation and the fractional change in sound velocity for the glass 共G兲and orientational-glass 共OG兲phases of polymorphic ethanol. Strikingly similar behaviors are found for both phases despite the OG’s underlying crystal 共bcc兲lattice. Such similarity, which is also revealed in dielectric spectroscopy and inelastic neutron scattering measurements, suggests whole molecule small-angle librations as a common microscopic origin for a wide variety of ‘‘glassy’’ phenomena. I. INTRODUCTION The thermal,1elastic,2and dielectric3behaviors of glasses largely differ from those of bulk crystalline solids at low and moderately high 共tens of kelvins兲temperature. In particular, some elastic properties such as sound velocity and attenuation in vitreous materials are known to exhibit remarkable anomalies if compared with those exhibited by the parent crystals. If expressed in terms of dimensionless quantities such as the fractional change in sound velocity with temperature ␦ v/vand the internal friction Q⫺1, the former is found to increase with Tbelow 1 K and strongly decrease at higher T,4whereas Q⫺1shows an initial increase up to a few hundreds of millikelvins, then reaches a plateau of roughly constant Q0 ⫺1up to a few kelvins 共a temperature referred usually as T*), followed by an increase until a temperature dubbed Tmax and finally, in some cases, a strong drop5at even higher Thas been reported. In addition, the measurements of sound attenuation in glasses are found to depend quite strongly on the measurement frequency, which is not the usual behavior for well-ordered crystals. The above phenomena are explained on a phenomenological basis using concepts such as two-level systems6 共TLS兲, thermally activated relaxation,7or some interpolation between the two such as the ‘‘soft-potential’’ model.8The picture that emerges from most of these approaches portrays the dynamics of glasses below5Kasdominated by coherent motion of TLS’s weakly damped by relaxational and resonant interactions with elastic waves. The phase coherence of such motion is lost for T⬎T*mainly because of the onset of thermal motion. Finally, above T*thermal occupation of excited levels of the interaction potential leads to a breakdown of the two-level approximation, and the dynamics becomes governed by thermally activated processes, which usually show an Arrhenius dependence of the relaxation rate.5 It has recently become clear that a more microscopic basis is required for a deeper understanding of the dynamics of glassy matter. This results from puzzling observations made on polycrystalline metallic solids,9films or even bulk, deformed metallic solids10 that show that these materials behave as glasses and, even more, a comparison of such quantities carried out in metallic polycrystals in their normal and superconducting states9suggests that the TLS’s, if present in such materials, do not interact with the conduction electrons. In addition, other classes of partially ordered systems such as rotator-phase 共or ‘‘plastic’’兲crystals,11 mixed crystals,12 or even quasicrystals13 have shown dynamics that resemble that of amorphous matter. The ideal benchmark to clarify such apparently contradictory observations would then be constituted by a material that could be studied under controlled conditions of disorder. Such an endeavor has, however, not been carried out mainly because of the extreme difficulty in preparing glass samples having the same compositions as those employed for studies on polycrystals or rotator phases. Here we consider a material that can easily be prepared in thermodynamically well-characterized phases such as the fully ordered, orientationally disordered, and amorphous phases of solid ethanol. A structural glass 共G兲is formed upon quenching the supercooled liquid below Tg⬃97 K. The amorphous state can also be achieved using easily achievable cooling rates 共⬇6 K/min兲. Heating the glass above Tgyields a supercooled liquid that can be annealed to yield a rotator phase 共RP, also referred in the literature as plastic or glassy crystal兲phase. Once this is formed, cooling below Tg OG ⬃97 K leads to a sluggish freezing of molecular rotations and an orientational glass 共OG, or orientationally disordered crystal兲. The rotational freezing preserves the same crystal symmetry 共an Im3 ¯ mbcc lattice兲of the RP, and from the structural point of view only a jump in the volume expansivity coefficient that amounts ⬇4⫻10⫺4K⫺1accompanies PHYSICAL REVIEW B 1 APRIL 2000-IVOLUME 61, NUMBER 13 PRB 61 0163-1829/2000/61共13兲/8778共6兲/$15.00 8778 ©2000 The American Physical Society the transition.15 The latter is understood as a genuine glasstransition phenomenon of a purely dynamical origin.14 Previous studies have compared the structure15 and dynamics16,17 of the RP/OG and G phases, all showing very similar glassy behavior in their specific heats, vibrational frequency spectra, and dielectric spectroscopy to the stable fully ordered 共monoclinic兲crystal phase. Our purpose here is thus to carry a comparative study on the dynamics of a material in its amorphous and orientationally disordered forms at meso- and microscopic scales. Since both the glass and OG solids have the same chemical composition as well as rather close densities, the differences in dynamic behavior, if any, will be entirely attributable to the presence in the OG of long-range periodicity. The present study will thus explore time and length scales in addition to those within the macroscopic realm as are those already examined using dielectric relaxation17 that have witnessed the dynamic proximity of the ␣ and ␤ relaxations of the glass 共or supercooled liquid兲and OG 共or RP兲phases. II. LIGHT SCATTERING We have measured the frequency and attenuation of hypersonic waves by Brillouin light scattering.18 The technique probes phonons with wavelengths comparable to that of light, the measured frequency shift ␻ Bis proportional to the sound velocity, and the width ⌫of the Brillouin peak is related, through a convolution with the instrument function, to the attenuation. The sample was contained in a sealed copper cuvettete attached to the cold finger and the temperature was controlled using a continuous flow cryostat. The explored temperature interval was 5 K–120 K, which covers the range of both glassy phases 共G and OG兲as well as that of the RP crystal. As in previous experiments, all the phases were prepared in situ and carefully monitored. Below 5 K the low scattering intensity precluded the measurement. The temperature range investigated also covers the lowtemperature anomaly 关i.e., the excess in Cp(T)/T3specific heat with respect to the ordered crystal兴and the region where the heat conductivity shows its ‘‘plateau.’’16 The measured spectra show Brillouin peaks arising from longitudinal elastic waves. Transverse sound modes are too weak to be observed. In addition, data already reported for fully hydrogenated glass and supercooled liquid samples at temperatures near the structural glass transition19 are included for comparison purposes. Within the TLS framework, ␻ Band ⌫are related to the dispersive and dissipative parts of the susceptibility ␹ ( ␻ ) by5 ␦ ␻ B ␻ B ⫽ ␦ v v ⫽⫺ A 2 ␹ ⬘共 ␻ 兲,Q⫺1⫽⌫ ␻ B ⫽A ␹ ⬙共 ␻ 兲, A⫽g2 ␳ v2, where Aincludes the ‘‘deformation potential’’ g共i.e., a measure of the coupling strength of TLS with elastic waves兲, mass density ␳ , and sound velocity v. Our data for both glass and OG phases are shown in Fig. 1. Both graphs reveal the presence of two rather different regimes for both solids. Below ⬇30 K, the variation with Tof ␦ v/vas well as Q⫺1is rather mild, whereas a stronger Tdependence is observed up to the glass transition 共⬃100 K兲, and finally even more abrupt changes are seen in the supercooled-liquid or RP temperature range. Our data for Q⫺1at T⬇5 K should be taken as an upper bound since they may be resolution limited. Taken at its face value the ⬇5 K data are 2.7⫻10⫺3共G兲and 3.0⫻10⫺3共OG兲. If we take these as representative of the ‘‘plateau’’ value of Q0 ⫺1, then they are above the average of those compiled by White and Pohl13 but compare with data for some metal films.10 If these values are used to make an estimate of the dimensionless constant5 C⫽2Q0 ⫺1 ␲ ⫽P ¯ g2 ␳ v2,共1兲 then one gets C⬇3.6⫻10⫺3and 5⫻10⫺3. These are certainly larger than those reported for oxide glasses. However, our values come closer to those measured for electrolyte glasses as studied by Reichert et al.21 Those reported in Table 1 of Ref. 21 reach values up to (9–9.6)⫻10⫺4de- pending upon the salt content. In particular, the dependence of the constant Cupon the content of salt 共LiCl and ZnCl2) given in percentage units follows a quasilinear law with a zero-salt limit of 1.34⫻10⫺3. This is the value for Cexpected for an amorphous material chemically and structurally FIG. 1. 共a兲Fractional change of sound velocity with temperature for the glass 共open symbols兲, OG/RP phases 共filled symbols兲, and the data from Ref. 19 共vertical bars兲. The inset shows data covering the whole glass-transition region. The solid line shows an estimation using parameters given in the text. 共b兲Internal friction as a function of temperature 共same symbols as above兲. The straight line depicts a prediction based on a thermal-activation model 共Ref. 7兲. PRB 61 8779ROLE OF LOW-FREQUENCY VIBRATIONS ON SOUND . . . not very dissimilar to ours such as amorphous water. In other words, our estimate for Ccomes close to others derived previously. On the other hand, our values can also be understood on the basis of the linear correlations found by Hunklinger20 and Reichert et al.21 between the quantities entering Eq. 共1兲and the glass transition temperature. In this respect, it is worth mentioning that both glassy ethanol and water show the highest densities of TLS states while the oxide glasses are found in the opposite extreme. From ⬇30 K up to Tg,Q⫺1for both solids is well approximated by a linear temperature dependence with coefficients 1.68⫻10⫺4K⫺1共G兲and 1.53⫻10⫺4K⫺1共OG兲. The implication is that data within this range pertain to a regime of thermally activated relaxation, which is governed by an Arrhenius relaxation rate. If such values are taken together with the previous estimate for Cto make a guess for the zero point energy of the tunneling particle E0from Ref. 7Q⫺1⫽ ␲ CkBT/E0, one gets E0⫽18 共19.9兲KforG共OG兲, which is amazingly close to the 15 K derived for oxide glasses.5 Our data for Q⫺1do not show maxima at Tmax such as that found for oxide glasses by acoustic5or Brillouin7spec- troscopies, probably because the appearance of both glass transitions at temperatures lower than that expected for Tmax. From the inset of Fig. 1共a兲one can, however, set the bounds 120 K⭐Tmax⭐140 K. III. RELATIONSHIP WITH MEASUREMENTS CARRIED OUT AT MACROSCOPIC SCALES The ␤ relaxation in both G and OG phases above ⬇40 K has been recently studied by dielectric spectroscopy.17 The temperature dependence was describable in terms of an activation energy V0, a width of the energy-barrier distribution ␴ 0, and the inverse of the attempt frequency ␶ 0, having values of about 1000 K, 600 K, and 0.5⫻10⫺13 s, respectively. The parameters for both phases are the same within statistical precision. From data for ␶ 0, as well as from the estimate of Tmax given above, one obtains an upper bound for Vmax , which is the upper limit of the distribution function of the barrier heights for states that fulfill the condition ␻ ␶ ⫽1, as7 kBTmax⫽⫺Vmax /ln( ␻ ␶ 0) with ␻ being the observation frequency. The result yields Vmax⫽720 K for a temperature of 130 K, in the midpoint of the interval given above. If we now consider how ␦ v/vis expected to behave in a thermally activated regime fulfilling ␻ ␶ 0Ⰶ1, namely, ␦ v/v ⫽CkBTln( ␻ ␶ 0)/E0, we find that, using the parameters referred to above, the description becomes valid up to T⬇60 K as Fig. 1 shows. The stronger decrease in ␦ v/vabove this temperature is probably a signature of the increasing importance of thermal expansion effects not accounted for by the model. Our previous study17 enabled the identification of thermally activated low-angle reorientations 共librations兲, which are strongly coupled to the collective 共phonon兲modes, as the main microscopic entity giving rise to the dielectric signal of both G and OG.17 Moreover, such motions show a peaked density of states that accounts for the excess in the C(T)/T3 specific heat plots at about 5–8 K as well as to the ‘‘plateau region’’ in thermal conductivity.16 From the closeness of the thermal activation parameters at both macro- 共dielectric兲and mesoscopic 共Brillouin兲scales, we infer that mechanisms similar to those causing dielectric relaxation should be responsible for the observed behavior of ␦ v/vand Q⫺1. Such motions should then be considered akin to those ‘‘additional harmonic excitations’’ comprising correlated motions of a few structural units in the well studied case of vitreous SiO2. IV. DYNAMICS AT MICROSCOPIC SCALES To explore the nature of the motions referred to above that have frequencies commensurate with maxima in C(T)/T3, that is, 2 meV⭐ ␻ ⭐4 meV, a number of measurements by neutron spectroscopy were carried out using the MARI spectrometer at the ISIS pulsed neutron source 共Rutherford Appleton Laboratory, U.K.兲. The sample was fully deuterated ethanol, which allowed us to monitor continuously its state by inspection of the diffraction patterns, and was contained in a standard aluminum sample can. The measurements were carried using incident energies energy of Ei⫽100 meV and Ei⫽15 meV. This was done in order to cover a large kinematic region so that the dependence of the dynamic structure factor S(Q, ␻ ) with wave vector can be safely explored and also to detect the possible existence of excitations occurring at finite frequency and constant wave vector. In view of the strong anharmonic effects displayed in Fig. 1, the measurements were carried at relatively low temperatures 共about 5 K兲that are close to those where C(T)/T3also show their maxima.16 As an added bonus, carrying out the measurement at such temperatures diminishes the importance of multiexcitation contributions. All three solids were prepared in situ after an initial quench of the hightemperature liquid into the deep glass phase. The spectra corresponding to the amorphous solids were the ones first measured. The orientationally disordered crystals were prepared by raising the temperature 10 K above Tgand a subsequent annealing under such conditions. Formation of the RP crystal was well monitored by the appearance of a strong Bragg peak in the diffraction pattern measured using the elastic I(Q, ␻ ⫽0) channels 共integrated over resolution width兲. Finally, the stable monoclinic phase was prepared by annealing at temperatures close to its melting Tm⫽159 K for a few hours. The diffraction patterns of the three solids are shown as insets of Fig. 3. Data reduction and analysis followed standard routes. Because of the high transmission of our sample multiple-scattering contributions to the measured spectral patterns were estimated to be rather small. Finally, the multiphonon contributions were evaluated following the procedure described in Ref. 22 and subtracted from the bare intensity. Figure 2 displays a representative set of spectra measured with Ei⫽15 meV, which enables one to compare directly the intensity patterns of the three solids. This allows us to see that there is no measurable difference in the spectra of the two glassy solids, while a substantially different behavior is followed by the ordered 共monoclinic兲crystal. The largest difference in intensities concern a range of frequencies that stretches up to frequencies commensurable with those where the monoclinic crystal shows its first intense phonon band (⬇6 meV兲. It concerns an easily visible ‘‘excess’’ of inelas- 8780 PRB 61A. CRIADO et al. tic intensity of both glass and OG with respect to the ordered crystal, which reaches a maximum at frequencies within the range 2–3 meV. Such ‘‘excess modes’’ once averaged over momentum transfers will give rise to vibrational frequency distributions such as those reported in some of our previous studies.23 With respect to the wave-vector dependence of the inelastic intensities shown in Fig. 2 the following qualifications should be taken into account. Below Q⬇0.5 Å⫺1the inelastic intensities of the glassy solids show a rather structureless shape. That of the ordered crystal grows steeply from the tail of the elastic line and crosses those of the glassy solids at about 5 meV. A small maximum starts to develop at about 2.5 meV in the spectra of both glassy solids for Q ⫽0.7 Å⫺1. From there to the largest explored wave vectors such a maximum appears as a dispersionless feature and can be identified with the ‘‘boson peak’’ found in studies employing incoherent-scattering samples.16,11 What seems worth exploring is the observation that while the center of such a peak does not show any clear dependence with wave vector, the amount of excess intensity shown by the two glasses with respect to the crystalline ground state shows some modulation with Q. For wave vectors corresponding to the first Brillouin zone of the ordered crystal 共e.g., below 0.85 Å⫺1) the excess intensities are maximal. Such a difference gets almost suppressed when approaching the Brillouin zone boundary and reappears as one proceeds further. Having measured S(Q, ␻ ) for relatively large Q’s enables us to consider in some detail the quantity S(Q, ␻ ⫽const), that is, the wave-vector dependence of the inelastic intensity for the range of frequencies where the excess modes commented on above is maximal. Such quantities are, in fact, inelastic structure factors, giving information about the vector displacements of the atoms taking place in such motions. A model-free assessment of the nature of motions being sampled is provided by comparison of the phases of oscillations in S(Q, ␻ ⫽const) with those shown by the static S(Q) or elastic S(Q, ␻ ⫽0) structure factors. Purely translational motions 共i.e., long-wavelength acoustic phonons兲leave all distances unaltered and therefore S(Q, ␻ ⫽const) should show oscillations in phase with those of S(Q)orS(Q, ␻ ⫽0). Cuts of S(Q, ␻ ⫽const) for 2.5 meV⭐ ␻ ⭐3.5 meV, an interval free from any substantial contamination by the elastic line, are shown in Fig. 3. Both glassy solids 共G and OG兲 show a strikingly similar intensity pattern, with a shoulder at about 3 Å⫺1, a broad maximum at 7 Å⫺1, and a minimum at about 11.2 Å⫺1. Only a small peak is seen for G and OG at Qvalues corresponding to the main peak in the diffraction pattern Qp⬇1.7 Å⫺1. In contrast, sharp peaks at 1.7 Å⫺1 and 2.5 Å⫺1identify the presence of propagating excitations in the spectrum for the monoclinic crystal, a feature also reproduced from a crystal lattice dynamics calculation. This shows that even at these relatively low frequencies, most of the intensity in spectra of the glassy solids arises from motions not involving in-phase displacements of the molecular centers of mass 共COM兲. In other words, most of the intensity at frequencies comparable to that of the ‘‘boson peak’’ seem to arise from motions that should have a molecular reorientational component as dominant, as explained. Now we take advantage of the presence within the OG phase of a crystal bcc cell as well as the availability of the absolute frequency scale provided by the monoclinic crystal. This enables the separation of purely translational and rotational motions since the molecular centers of mass sit at the nodes of the bcc lattice. We have carried out a number of computer simulations for such a model of the OG crystal. Our point of departure was to assume that relative molecular orientations all have the same statistical weight, a fact grounded on previous results.15 A random sampling strategy FIG. 2. Constant-Qspectra as measured using an incident energy Ei⫽15 meV for values of the momentum transfer given in the insets. The solid line depicts the spectra of the monoclinic crystal, the open symbols show the spectra for the glass, and the filled symbols the spectra of the orientationally disordered crystal. PRB 61 8781ROLE OF LOW-FREQUENCY VIBRATIONS ON SOUND . . . was devised to generate the initial molecular orientations as well as the small librational displacements. Molecular COM motions were simulated by displacing the molecules along the Cartesian displacement coordinates of the bcc lattice. A further average was then taken over the different displacement modes and the final atomic displacement vectors ui were calculated. The calculated inelastic structure factor I(Q, ␻ ⫽const) was finally evaluated from the one-phonon partial structure factors Fij(Q), I共Q, ␻ ⫽const兲⬀兺 i兺 j⫽iFij共Q兲G共 ␻ 兲,共2兲 Fij共Q兲/Q2⫽1 冑 MiMj 冋 1 3共ui•uj兲j0共Qdij兲 ⫹ 冉 1 3共ui•uj兲⫺1 dij 2共dij•uj兲 冊 j2共Qdij兲 册 ,共3兲 where Miare atomic masses, uare vector displacements, dij are vectors joining two atoms, G( ␻ ) are spectral functions 共combination of Bose factors and ␦ peaks兲, and jx( ) are spherical Bessel functions. The calculated curve is compared with experiment after addition of a self-scattering component24 and is shown in Fig. 3. The result shows that purely translational motions contribute to the observed intensity at such frequencies with only a small feature at Q⫽Qp. In contrast, reorientational excursions with an average angle of about 0.3° account for the shape of I(Q, ␻ ⫽const). Because of the very close similarity of the experimental inelastic structure factor for the OG and the fully disordered solid, one expects that the results for the former, here considered as a reference ‘‘glass,’’ should also be of relevance for the latter. In other words, they serve to quantify the contribution to I(Q, ␻ ), on Å and picosecond scales, coming from purely acoustic motions certainly involving the displacements of molecular COM’s, versus those arising from additional 共librational兲excitations. From the very small weight of the peak at Q⫽Qpwe infer that a high-frequency cutoff for the existence of well-defined soundlike excitations should not be far beyond ␻ co⭐3 meV, a frequency that becomes comparable to that of the ‘‘boson peak.’’ V. CONCLUSIONS The findings reported here are of relevance to our understanding of glassy dynamics since they enable a coherent description of the mechanisms driving the dynamics of disordered matter at widely different spatiotemporal scales as explored by dielectric 共macro-兲, Brillouin 共meso-兲, and neutron 共microscopic兲spectroscopies. A number of characteristic phenomena can be understood as resulting from the interaction of long-wavelength acoustic waves with smallamplitude molecular librations. The latter are known to give rise to the peak in C(T)/T3, constitute an efficient mechanism for sound attenuation via relaxational and resonant scattering mechanisms, and explain the concomitant reduction in sound velocity with temperature. The present results also suggests a set of weakly damped 共TLS兲librators resonantly interacting with sound waves as the most feasible entities giving rise to the linear term in the specific heat below 2 K found for the bcc crystal and glass.16 ACKNOWLEDGMENTS We acknowledge support from DGICYT 共Spain兲Grant No. PB95-0075-C03-01 and U. S. Department of Energy, Basic Energy Sciences–Materials Sciences under Contract No. W-31-109-ENG-38. FIG. 3. Neutron spectroscopy data at constant frequency I(Q,2.5 meV⬍ ␻ ⬍3.5 meV兲for the glass, OG, and monoclinic crystal phases at T⫽5K关frames 共a兲–共c兲, respectively兴. The solid line shown in frame 共b兲corresponds to the calculation made for the bcc crystal on the basis of Eq. 共2兲. The dashed line shown in frame 共c兲depicts the result from a harmonic lattice dynamics calculation for the monoclinic powder 共see Ref. 16兲. Insets display the elastic I(Q, ␻ ⫽0) data. 8782 PRB 61 A. CRIADO et al. 1R. C. Zeller and R. O. Pohl, Phys. Rev. B 4, 2029 共1971兲. 2K. A. 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B 57, R13 977 共1998兲; M. Jime ´nez-Ruiz, M. A. Gonza ´lez, F. J. Bermejo, M. A. Miller, N. O. Birge, I. Lendoya, and A. Alegria, ibid. 59, 9155 共1999兲; M. A. Gonza ´lez et al., Phys. Rev. E 共to be published兲. 18J. Sandercock, in Light Scattering in Solids III, Topics in Applied Physics Vol. 51 共Springer, New York, 1979兲. 19A. Srinivasan, F. J. Bermejo, and A. Bernabe, Mol. Phys. 87, 1439 共1996兲. 20S. Hunklinger, in Heidelberg Colloquium on Glassy Dynamics, edited by J. L. van Hemmen et al. 共Springer, Berlin, 1987兲,p. 110. Such correlations would yield about 0.05 eV for gand ⬇1034 erg⫺1cm⫺3for P ¯ , which together with v⫽2.6 ⫻105cm s⫺1and ␳ ⫽0.98 g cm⫺3leads to C⫽9.7⫻10⫺4. 21U. Reichert, M. Schmidt, and S. Hunklinger, Solid State Commun. 57, 315 共1986兲. The article includes a data point for amorphous water in Fig. 3. An alternative view on the meaning of linear correlations between the TLS parameters and the glasstransition temperature is given by J. F. Berret and M. Meissner, Z. Phys. 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