VOLUME 77, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 1 JULY 1996 Transition from Normal to Fast Sound in Liquid Water F. Sette,1 G. Ruocco,2 M. Krisch,1 C. Masciovecchio,1 R. Verbeni,1 and U. Bergmann1 1European Synchrotron Radiation Facility, B.P. 220, F-38043 Grenoble, Cedex France 2Universitá di L'Aquila and Istituto Nazionale di Fisica della Materia, I-67100, L'Aquila, Italy (Received 1 April 1996) Inelastic x-ray scattering data from water at 5 ±C show a variation of the velocity of sound from 2000 to 3200 m s in the momentum transfer range 1 4 nm21. The transition occurs when, at 4 meV, the energy of the sound excitations equals that of a second weakly dispersing mode. This mode is reminiscent of a phonon branch in ice Ih crystals, which is shown here to be of optical transverse character. The present work accounts for most of the highly debated difference between hydrodynamic 1500 m s and high-frequency 3200 m s velocities of sound in water. [S0031-9007(96)00500-5] PACS numbers: 61.10.Eq, 63.50.+x, 78.70.Ck Water, in its liquid and solid forms, continues to of this transverse dynamics in ice, although its observation fascinate for its complicated and quite unique properties. in the present experiment implies that it does not have a An interesting and still highly debated issue is the pure transverse symmetry as in the crystal. Consequently, phenomenon of fast sound in the liquid. The velocity we propose that it is the coupling between the acousticlike of acousticlike excitations, which is 1500 m s in the and opticlike dynamics that gives rise to at least most of hydrodynamic limit up to momentum transfers Q 2 3 the change from normal to fast sound. 1022 nm21, reaches the value of 3200 m s (fast sound) The experiment was carried out at the very high en- at Q above 4 nm21. This large velocity increase was ergy resolution inelastic x-ray scattering beam line (BL21- first predicted in a molecular dynamics (MD) calculation ID16) at the European Synchrotron Radiation Facility. [1], which generated a great number of MD simulations The undulator x-ray source was monochromatized us- aiming to understand the microscopic origin of this ing a Si(111) double crystal monochromator and a high process [2,3]. The fast sound was first shown to exist energy resolution backscattering monochromator [7], op- using coherent inelastic neutron scattering (INS) [4]. erating either at the Si(999) or at the 11 11 11 Bragg Recently, it was confirmed in a wider Q region using reflections. The scattered photons were collected by a inelastic x-ray scattering (IXS) [5], and, still using IXS, grooved spherical silicon crystal analyzer operating also it was found that the velocity of sound is equivalent in at the same Bragg backreflections, and in Rowland geom- liquid and solid (ice Ih) water in the whole investigated etry [8]. The total energy resolution function was mea- 4 14 nm21 Q region [6]. These results confirm the sured from the elastic scattering of a plastic sample at its existence of fast sound, show that at high frequencies the maximum in the static structure factor. The energy reso- dynamical response of the liquid becomes solidlike, and lutions were 3.2 and 1.4 meV full width at half maximum imply that in the liquid the processes responsible for the (FWHM) at the Si(999) and 11 11 11 reflections, respec- large variation in the speed of sound must take place in the tively. The scattering angle for momentum transfers be- mesoscopic (Q-v) region with Q in the 0.05 4 nm21 tween 2 and 21 nm21 was selected by rotating the 7 m range. This region, too high for light scattering, is also analyzer arm in the horizontal scattering plane. The Q experimentally difficult to reach for both INS and IXS resolution was set by an aperture in front of the analyzer. techniques. Energy scans were done by varying the relative temper- In this Letter we report IXS data on water and ice Ih. ature between the monochromator and analyzer crystals. The data on water were taken down to a Q transfer of Further experimental details are given elsewhere [5,7,8]. 1 nm21 using 21.75 keV x rays with 1.4 meV energy The water sample was high purity H2O kept at 5 ±C in resolution. They show that the velocity of sound of the the liquid measurements and at 210 ±C in the solid. The longitudinal acousticlike dynamics goes from 2000 m s solid samples were in the Ih crystal structure. The disper- at Q 1 nm21 to the fast sound value of 3200 m s sion was measured in a polycrystal, and in a single crystal at Q $ 4 nm21. This transition is observed when the along the (10-11) crystallographic direction. The ice sam- longitudinal acoustic dispersion crosses a second branch, ples were grown in situ following established procedures observed at 4 meV at Q larger than 4 nm21, and [9]. The samples' thicknesses were 18 mm in the di- no longer visible at Q smaller than 2 nm21. We also rection of the incident beam. report IXS data on ice Ih crystals where we identify Inelastic x-ray scattering spectra from liquid water at at 5 7 meV a transverse optical phonon branch. We 5 ±C, taken with 1.4 meV energy resolution, are shown suggest that the mode at 4 meV in water is reminiscent in Fig. 1 at selected Q together with the results of the 0031-9007 96 77(1) 83(4)$10.00 © 1996 The American Physical Society 83 VOLUME 77, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 1 JULY 1996 1.0 6 0.5 meV FWHM, it is possible to observe inelastic scattered intensity at energies changing with Q. The data have been fitted with a model function consisting of a Lorentzian for the central line and a damped harmonic oscillator (DHO) [10] for the two side bands. The detailed balance has been imposed. The DHO model was used to derive the spectroscopic parameters V Q and G Q , as defined in [5], independently of specific theories, and to compare them with previous studies also using the same model [4­6]. The fitting function is the convolution of the model with the experimentally determined resolution function. The fit was based on x2 minimization, and the results were always within its standard deviation. Similarly, in Fig. 1(b), we show examples of data taken at higher Q values. Beside the inelastic scattered intensity dispersing with Q and already observed in [4,5], thanks to the increased energy resolution it is now possible to observe a new weakly dispersing feature with 4­6 meV energy transfer. This is emphasized in the insets of Fig. 1(b). This feature is observed only in the spectra with Q larger than 4 nm21, and is no longer detected in the spectra at small Q. This observation is consistent with previous neutron data taken at Q values between 6 and 25 nm21, where the same excitation was found at 4 6 meV [11,12]. Similar to the data in Fig. 1(a), the data at Q $ 4 nm21 were fitted by a Lorentzian for the central peak and a DHO model for each of the two (dispersing and weakly dispersing) features. The energies of the excitations, determined from the fit and corresponding to the DHO fitting parameter V Q , are reported in Fig. 2. The data taken in the present experiment are shown together with those determined by IXS with 3.2 meV [6] and 5 meV [5] energy resolutions. The different measurements of the dispersing excitations are consistent among each other in the common Q range, and correspond to a sound velocity of y 3200 m s (dotted line in Fig. 2). This branch is due to the acousticlike modes propagating at these Q values with a sound velocity equivalent to that of the solid [6]. FIG. 1. The IXS spectra of water at T 5 ±C ± shown The values of the widths are also consistent among each together with the total fits and the individual components, as other, and are reported elsewhere [13]. At Q smaller than explained in the text, at the indicated Q values. The data 4 nm21, as emphasized in the inset of Fig. 2, V Q no are normalized to their maximum intensity corresponding to longer follows the dispersion with slope y , and shows (a) 1.4, 1.2, 1.1, 1.0 counts s and total counts of 450, 360, a bend down towards the hydrodynamic dispersion law 360, 350 at Q 1.0, 1.5, 2.0, 2.5 nm21, respectively; (b) 2.8 and 2.4 counts s (total counts 1100 and 500) at Q 4 and with slope y0 1500 m s. In Fig. 2 are also reported 10 nm21, respectively. The insets of (b) emphasize the weakly the peak positions of the weakly dispersing excitations. It dispersing features at 4 5 meV together with the total fit is important to notice that the bend down of the acoustic and the individual components. The Q resolution was 0.2 and branch takes place when the energy of the acoustic 0.4 nm21 for the data in (a) and (b), respectively. mode equals the one of the weakly dispersing feature. This observation, together with the disappearance of the excitation at 4 meV at Q smaller than 2 nm21, strongly supports the idea that the transition from the fast sound fits discussed in the following. In Fig. 1(a) we report the toward the normal sound is triggered by the degeneracy data taken at the lowest investigated Q values. The data between these two modes, observed at Q 4 nm21. are normalized to the maximum of the central peak. At The Q dependence of the sound velocity, y Q the sides of the quasielastic central line, with a width of V Q Q, derived from the data of Fig. 2, is reported in 84 VOLUME 77, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 1 JULY 1996 [14]. These authors derived the mode dispersing with fast sound, as well as a second one with very small dispersion at 4 6 meV. This last mode was associated to the O- O-O bending. In view of the strong similarities in the longitudinal dynamics of liquid and solid water [6], a study of the ice crystal can shed further information on the nature of the weakly dispersing feature, and on its role in the transition from normal to fast sound. In Fig. 4 we report IXS data of the S Q, v for an ice polycrystal at selected Q values and the corresponding fits. Data taken along the (10-11) direction in an ice single crystal are almost identical to those of Fig. 4. The measurements were performed at 210 ±C, with 3.2 meV energy resolution. At low Q the spectra are dominated FIG. 2. Excitation energies, V Q from the DHO model, by a feature dispersing with Q, which is identified with for the new high resolution data ( , de 1.4 meV) and the longitudinal acoustic phonon branch, LA [6,15]. The from previous IXS experiments ( , de 3.2 meV [6]; ±, de 5.0 meV, [5]). The open symbols refer to the dispersing excitation and the full diamond to the weakly dispersing ones. The dotted line, with a slope of 3200 m s results from a fit for Q $ 4 nm21. The inset shows an enlargement of the low Q region, where the transition from fast toward normal sound takes place, as emphasized by the two lines corresponding to the fast and normal sound branches. Fig. 3 together with INS data [4] and MD simulations data [2,3]. The speed of sound decreases from y to 2000 m s at Q 1 nm21. This reduction is well reproduced by the MD calculation of Sciortino and Sastry [3], although at Q larger than 6 nm21 the agreement is much less satisfactory. In this region the potential model used by Balucani et al. [2] shows a better consistency. A possible relation between the dispersion of the longitudinal dynamics and the feature at 4 5 meV was first pointed out by Sastry, Sciortino, and Stanley in a MD calculation performed at Q values larger than 3.3 nm21 FIG. 4. The IXS spectra of polycrystalline H2O ice Ih at 210 ±C, taken at the indicated Q values with 3.2 meV and 0.4 nm21 energy and momentum resolutions, are reported together with their fits. The longitudinal acoustic and the transverse optical phonons are labeled with LA and TO, respectively. The experimental data are normalized to the maximum of the LA Stokes peak. On this point, the count rates were 1 count s at all the investigated Q values. The integration time for each data point was 180 s at Q 2 and 6 nm21 and 90 s at Q 10 and 14 nm21. The data ± , shown with the error bars, are superimposed to the fit (solid line). The fit was made by the convolution of the experimental resolution function with two pairs of Lorentzians representing the LA and TO Stokes and anti-Stokes peaks. A fifth Lorentzian was FIG. 3. Values of the sound velocity y Q V Q Q deter- used to account for the small elastic intensity probably due to mined from the present ¶ , previous IXS ( [6], ± [5]), previ- residual disorder in the polycrystal. In the inset we report the ous INS data (full ), and MD simulations (1 [2], 3 [3]). The dispersion relation for the LA and TO branches. The speed of full and dashed line represent the infinite- and zero-frequency sound of the LA branch is 3200 m s. Note that the TO peak limit as derived in Ref. [2]. can be observed only at Q larger than 7 nm21. 85 VOLUME 77, NUMBER 1 P H Y S I C A L R E V I E W L E T T E R S 1 JULY 1996 dispersion of the LA phonons is shown in the inset of difference between the lowest value measured here of Fig. 4. The peak at 5 7 meV, labeled TO, appears only 2000 m s and y0 can be due either to the present limit at Q larger than 7 nm21, sharply increases in intensity in reaching lower Q values or to the presence of other with increasing Q, and dominates the spectrum at Q larger relaxation processes as suggested in a recent Brillouin than 10 nm21. The Q value of 7.5 nm21 corresponds light scattering experiment [17]. Further investigations to the smallest size of the reduced Brillouin zone in ice. on the possible temperature dependence of the Q value Therefore, the intensity behavior of the TO mode indicates characterizing the transition may resolve this issue. that this branch has a very strong transverse symmetry We acknowledge B. Gorges, for his technical assis- through the whole Brillouin zone [16]. Moreover, the tance, and U. Balucani and M. Sampoli for useful dis- weak dispersion of the TO peaks, as seen in the inset of cussion. Fig. 4, allows us to identify them as the lowest transverse optic phonon branch. This assignment is consistent with lattice dynamics calculations [15], but disagrees with the proposed interpretation of neutron data taken in the second [1] A. Rahman and F. H. Stllinger, Phys. Rev. A 10, 368 and higher Brillouin zones [12]. These neutron data (1974). extend in energy only up to the TO peaks and are fully [2] U. Balucani, G. Ruocco, A. Torcini, and R. Vallauri, Phys. consistent with the present IXS data. The TO branch, Rev. E 47, 1677 (1993). however, was interpreted as the ice longitudinal acoustic [3] F. Sciortino and S. Sastry, J. Chem. Phys. 100, 3881 branch [12]. This confusion, which also contributed to the (1994). erroneous assignment of the mode observed at a similar [4] J. Teixeira, M. C. Bellissant-Funel, S. H. Chen, and energy in water to the high Q limit of the normal sound B. Dorner, Phys. Rev. Lett. 54, 2681 (1985). excitations in the liquid [11,12], is now removed by the [5] F. Sette, G. Ruocco, M. Krisch, U. Bergmann, strong polarization dependence of the mode reported here. C. Masciovecchio, V. Mazzacurati, G. Signorelli, and R. Verbeni, Phys. Rev. Lett. 75, 850 (1995). The transverse optical character of the branch observed [6] G. Ruocco, F. Sette, U. Bergmann, M. Krisch, in ice at 5 7 meV suggests that the feature observed in C. Masciovecchio, V. Mazzacurati, G. Signorelli, and water at approximately the same energy has the same ori- R. Verbeni, Nature (London) 379, 521 (1996). gin, i.e., is reminiscent of a transverse optical dynamics. [7] R. Verbeni, F. Sette, M. Krisch, U. Bergmann, B. Gorges, The absence of translational invariance in the liquid, how- C. Halcoussis, K. Martel, C. Masciovecchio, J. F. Ribois, ever, does not allow modes with pure symmetry. There- G. Ruocco, and H. Sinn, J. Synch. Rad. 3, 62 (1996). fore, differently from ice where the two modes cannot [8] C. Masciovecchio, U. Bergmann, M. Krisch, G. Ruocco, mix, in liquid water they can interact as soon as their en- F. Sette, and R. Verbeni, Nucl. Inst. Methods B111, 181 ergies become comparable. This interaction is likely to (1996). be responsible for the determination of two regions with [9] P. V. Hobbs, Ice Physics (Clarendon Press, Oxford, 1974). distinctively different longitudinal dynamics: (i) a region [10] B. Fak and B. Dorner, Institute Laue Langevin, Grenoble, France, Report No. 92FA008S, 1992 (unpublished). including the hydrodynamic regime with a longitudinal dy- [11] P. Bosi, F. Dupre', F. Menzinger, F. Sacchetti, and M. C. namics characterized by the normal sound and (ii) a high Spinelli, Nuovo Cimento Lett. 21, 436 (1978). Q region with a longitudinal-like and transverselike dy- [12] F. J. Bermejo, M. Alvarez, S. M. Bennington, and namics very similar to those of the solid. The transition R. Vallauri, Phys. Rev. E 51, 2250 (1995). between the two regions takes place when the excitation [13] F. Sette, G. Ruocco, M. Krisch, U. Bergmann, energies of the longitudinal-like propagating dynamics be- C. Masciovecchio, V. Mazzacurati, G. Signorelli, and comes comparable to that of the transverselike dynamics R. Verbeni, Phys. Rev. Lett. 76, 3657 (1996). with its much more localized nature. [14] S. Sastry, F. Sciortino, and H. E. Stanley, J. Chem. Phys. In conclusion, thanks to the extension of the accessible 95, 7775 (1991). Q region down to 1 nm21, it has been possible to [15] P. Bosi, R. Tubino, and G. Zerbi, J. Chem. Phys. 59, 4578 demonstrate experimentally that the fast sound branch (1973). [16] This is due to the selection rules in the one-phonon ap- arises from a bend-up of the normal sound, and that this proximation, which allows one to detect in first Brillouin process takes place at the crossing of the acoustic branch zone only longitudinal modes, while transverse modes can with an opticlike dynamics. These findings account for be detected in higher zones. almost the whole difference between y and y0. The [17] A. Cunsolo and M. Nardone (to be published). 86