PHYSICAL REVIEW B VOLUME 57, NUMBER 5 1 FEBRUARY 1998-I Sound-wave scattering in silica A. Wischnewski and U. Buchenau Institut fu¨r Festko¨rperforschung, Forschungszentrum Ju¨lich, D­52425 Ju¨lich, Federal Republic of Germany A. J. Dianoux Institut Laue-Langevin, Boi te Postale 156, F-38042 Grenoble Cedex 9, France W. A. Kamitakahara National Institute of Standards and Technology, Reactor Radiation Division, Gaithersburg, Maryland 20899 J. L. Zarestky Ames Laboratory, Iowa State University, Ames, Iowa 50011 Received 4 April 1997 The combination of x-ray Brillouin scattering data with coherent inelastic neutron scattering results in vitreous silica allows us to identify the scattering mechanisms for high-frequency sound waves in the glass. Below one THz, one has resonant scattering from low-frequency vibrations, above one THz Rayleigh scatter- ing from the atomic disorder. S0163-1829 98 01705-6 One of the most controversial issues in the field of disor- lower temperature,15 the neutron data give convincing evi- dered solids is the scattering of sound waves at higher fre- dence for non-sound-wave modes at the boson peak. The quencies, above 100 GHz. The existence of a very strong number of additional modes can be determined to give a scattering mechanism at these frequencies is evidenced by solid basis for an estimate of the scattering of the sound the universal plateau in the thermal conductivity of glasses waves by the local modes. Together with the evidence from around 5 K, where the thermal conductivity of the glass is the x-ray Brillouin technique,13 that estimate leads to a clear often many orders of magnitude lower than that of the cor- and simple picture for the phonon scattering in glasses. The neutron scattering data for silica at higher tempera- responding crystal.1­3 The many different theoretical expla- tures were taken at two different spectrometers. The first was nations for that plateau can be grouped into two fractions. the time-of-flight spectrometer IN6 at the cold source of the The first assumes a strong scattering of the sound waves by high-flux reactor of the ILL at Grenoble. On the IN6 with 4.1 some form of disorder frozen-in free volume,4 clusters,3,5,6 Å neutron wavelength and measuring in energy gain of the fractals,7 and disorder in the force constants,8 to name only a neutron, one has excellent resolution and intensity, but only a few . The second group of explanations attributes the scat- limited range in elastic momentum transfer Q up to about 2.6 tering to a resonant interaction of the sound waves with soft Å 1. Therefore, the measurements were complemented by a local vibrational modes, similar in eigenvector to the tunnel- second experiment on a triple axis spectrometer with thermal ing states and to low-barrier classical relaxations of the neutrons of 2.3 Å wavelength at the Oak Ridge High Flux glass.9­11 Reactor.16 This measurement was taken with a constant final In both cases, the explanations tend to link to the boson wave vector kf in order to more easily relate the elastic to the peak, another controversial feature of glasses, an excess of inelastic intensities. On the IN6, data were taken up to 1673 low-frequency modes over the Debye model expectation in K, 200 K above the glass transition temperature. The triple the same frequency region. For those who believe in the axis measurements extended up to 1273 K. Here we concen- scattering of the sound waves from some kind of disorder, it trate on the measurements near the temperature of 1050 K of is a consequence of the strong localization of the sound the x-ray Brillouin measurement.13 For the IN6, we took a waves. For the soft mode proposers, it demonstrates the ex- measurement at 1104 K. On the Oak Ridge triple axis spec- istence of additional modes coexisting with the high- trometer, we summed two measurements at 841 and 1273 K frequency sound waves. to obtain data at the average temperature of 1057 K. The recent development of a high-resolution x-ray Bril- Figure 1 shows a comparison of these two sets of data. louin technique12 for the measurement of sound waves in the Figure 1 a compares the elastic intensity in the intensity THz region has further stimulated the interest in the disper- scale of the triple axis spectrometer. Figure 1 b compares sion and damping of high-frequency sound waves in glasses. the inelastic intensities at an energy transfer corresponding to In the particular case of silica, there are two opposing inter- 1.3 THz, the position of the boson peak at that temperature. pretations for the x-ray Brillouin data, one of them in terms Both inelastic data sets have frequency windows of about of damped sound waves13 and the other one in terms of a 300 GHz width. The same scaling factor was used for both crossover from sound waves to boson peak modes.14 cases, showing a very satisfactory agreement between the The present paper combines the x-ray Brillouin data13 two measurements. The data look like a mirror image of with inelastic neutron scattering measurements of vitreous earlier low-temperature data,15 but have a higher counting silica at higher temperatures. As earlier measurements at rate and better statistics. 0163-1829/98/57 5 /2663 4 /$15.00 57 2663 © 1998 The American Physical Society 2664 BRIEF REPORTS 57 FIG. 2. Density of states of vitreous silica for four temperatures in the representation g( )/ 2 vs frequency. The solid lines repre- sent the Debye density of states for the lowest upper line and highest lower line temperature. clusters of about 20 to 30 Å diameter, i.e., effective wave- lengths of the same order. Assuming plane waves with the corresponding average wave vector of 0.3 Å 1 and adapting the number of modes to give the best agreement, one finds the dashed curve in Fig. 1 b , which still gives a very bad fit. The weakness of the first peak rather requires a mode eigenvector where second-nearest oxygen neighbors move out of phase. Earlier results at lower temperatures15 show a FIG. 1. Q dependence of the neutron scattering from vitreous reasonable description of the data by the model of coupled silica at elevated temperature a elastic the line is a guide to the SiO4-tetrahedra libration shown in the inset of Fig. 1 b . eye b at 1.3 THz between 1.15 and 1.45 THz , together with Since there is perfect agreement between these high- calculated curves showing the failure of the sound wave picture. temperature data and the low-temperature measurement, one The inset of b shows the motional model proposed in Ref. 15. can take over this explanation, which has been also recently In order to assess the meaning of the results in Fig. 1, corroborated in numerical work on silica.20 Using such a imagine that one had only long wavelength sound waves in motional model, one can calculate the vibrational density of silica at 1.3 THz. The wavelength of the longitudinal modes states from the neutron data on an absolute scale. The earlier sound velocity 6370 m/s at 1.3 THz is 49 Å, that of the work15 determined a vibrational density of states from 51-K transverse modes with 3950 m/s is 30 Å, corresponding to data that agreed within experimental error with specific heat phonon wave vectors q of 0.13 and 0.21 Å 1, respectively. data around 10 K. A similar evaluation was done for the These q values are still small compared to the momentum high-temperature data reported here, with only a slight transfers at the first sharp diffraction peak at 1.6 Å 1, so the change. That change was to first subtract the calculated De- atomic neighbors giving rise to that peak still essentially bye signal calculated on the basis of the sound move in phase. If they did that for all modes in the frequency velocities18,19 from the data, in order to obtain the density of window, one should find the very marked first sharp diffrac- states of the additional modes. The same kind of evaluation tion peak of the elastic scattering reproduced in the inelastic was repeated for the older low-temperature data. Figure 2 scattering. Since there is only a weak trace of that peak in the shows the resulting total vibrational density of states. Note inelastic data at the boson peak frequency, one is forced to the unusual temperature dependence of the boson peak in conclude that one does not deal with pure sound waves. silica, a shift to higher frequency with increasing tempera- The argument can be put on a quantitative basis using the ture. formulas of Carpenter and Pelizzari17 for the scattering from With that density of states of additional modes, one then sound waves in glasses in the one-phonon approximation. In turns to the second possible explanation of the plateau in the the long-wavelength limit, these formulas yield an inelastic thermal conductivity at low temperatures, assuming resonant dynamic structure factor proportional to Q2S(Q,0), where scattering of the sound waves from these additional modes. S(Q,0) denotes the elastic intensity. For somewhat shorter The coupling parameter describing the bilinear coupling be- wavelengths, one gets a broadening and a shift of the peaks tween the sound wave strain and the displacement of an ad- of S(Q,0). The continuous line in Fig. 1 b shows the expec- ditional mode can be taken from soft potential fits21 of the tation using the Debye density of states on the basis of the low-temperature glass anomalies. One then gets11 light Brillouin scattering sound velocities.18,19 As expected, the signal is much too small to explain the data. g W 2 1 2 add But even if we assume that sound waves of higher wave lres,vib 6 2 0 , h 0 2 , 1 Mv2 vector q get down to the boson peak frequency, say, by some vibrational localization mechanism as envisaged by the first where denotes the wavelength of the sound wave, v the group of theoretical models for the boson peak,3,5­8 one can- sound velocity, and the bilinear coupling coefficient be- not explain the data. These models suggest vibrations of tween the strain of the sound wave and the displacement of 57 BRIEF REPORTS 2665 data.13 The right scale of the data shows the ratio / of the damping for small damping the full width at half- maximum of a Lorentzian to the frequency . One has 2 l 1 2 the factor 2 in this equation was erroneously omitted in Ref. 16 .The low-temperature data at 51 K compare favorably with estimates from the thermal conductivity3 and with phonon spectroscopy data at about 1 K.23 That good agreement has been pointed out before21 and serves to validate the whole procedure. The 155-K data extrapolate to light scattering Brillouin values,24,25 and agree within a factor of 2 with those of a picosecond optical technique POT .26 The 1104-K data also extrapolate to the three light Brillouin val- ues available18,19,13 at the low-frequency end. The comparison to the x-ray Brillouin data shows that the resonant scattering from low-frequency vibrations fails to de- scribe the strong scattering above 2 THz. In view of this discrepancy, we return to old ideas on the scattering of sound waves from the static atomic disorder in the glass.1,2,27 Zait- lin and Anderson2 were the first to give a quantitative esti- mate for Zeller and Pohls1 simple idea that each atom in a glass, not sitting at an ordered place, gives rise to Rayleigh scattering of the sound waves. They derived a mean free path l 1 static B 4 3 with a parameter B of the order of 2.6 10 3Å 1THz 4. If one adds that inverse mean free path to the one calculated from the resonant scattering via the neutron measurement, FIG. 3. Inverse mean free path times wavelength vs frequency at one finds the continuous line in Fig. 3 c , in reasonable three different temperatures in silica. Comparison of values calcu- agreement with the high-frequency x-ray Brillouin results.13 lated from neutron data at a 51 K to fits of the thermal conduc- That interpretation is further supported by a recent analy- tivity (T) Ref. 3 and to phonon spectroscopy data at 1 K Ref. sis of low-frequency modes in a simple model glass,28 which 23 ; b 155 K to light scattering Brillouin data at 155 K Refs. showed that about half of the damping of the sound waves at 24,25 , and to data obtained by POT between 100 and 300 K Ref. the boson peak comes from the interaction with the quasilo- 26 ; c 1104 K to light scattering Brillouin data at 1050 K Refs. calized modes, while the other half had to be attributed to 13,18,19 and to the x-ray Brillouin results at 1050 K Ref. 13 . scattering from the atomic disorder. the additional mode these three quantities have to be taken We conclude that the combination of the x-ray Brillouin separately for longitudinal and transverse sound waves; how- results with the interpretation of neutron data in terms of ever, the coefficient 2/v2 is very nearly equal for the two local vibrational modes reveals two scattering mechanisms kinds of modes . M is the average atomic mass of the glass, for sound waves in silica. The first at low frequencies below the frequency, g 1 THz is the scattering from local modes coexisting with the add( ) the density of additional modes, and W the crossover energy between tunneling states and reso- sound waves. Earlier papers11,22 have shown that this mecha- nant vibrations of the soft potential model. nism is not only able to explain the sound wave scattering at For silica, one finds from specific-heat and ultrasonic at- the plateau in the thermal conductivity, but also at lower tenuation data21,22 W 0.34 meV and 0.65 eV for the frequencies, assuming tunneling states and classical relax- longitudinal modes, yielding ation over low barriers related to the additional vibrations. 0 1.09 THz. Figure 3 shows a comparison of the mean free path deter- The second mechanism above 1 THz is the static Rayleigh mined in that way from neutron data to other results in this scattering from the atomic disorder as proposed by Zeller heavily studied glass. Since the mean free path at lower fre- and Pohl1 and evaluated in the classical papers of Zaitlin and quencies is temperature dependent, the comparison is carried Anderson2 and Ja¨ckle.27 out at three different temperatures, 51 K for the low- Stimulating and controversial discussions with A. P. temperature regime, 155 K for intermediate temperatures, Sokolov are gratefully acknowledged. We thank R. Vacher and 1104 K to compare with the 1050 K x-ray Brillouin for pointing out the initial error in Eq. 2 to us. 2666 BRIEF REPORTS 57 1 R. C. Zeller and R. O. Pohl, Phys. Rev. B 4, 2029 1971 . and W. A. Phillips, Phys. Rev. B 34, 5665 1986 . 2 M. P. Zaitlin and A. C. Anderson, Phys. Rev. B 12, 4475 1975 . 16 A. Wischnewski, U. Buchenau, A. 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