The vibrational modes of glasses

PERGAMON

Solid State Communications 117 (2001) 187±200

www.elsevier.com/locate/ssc

The vibrational modes of glasses E. Courtens*, M. Foret, B. Hehlen, R. Vacher Laboratoire des Verres, UMR 5587 CNRS, Universite Montpellier 2, F-34095 Montpellier, France

Abstract Our recent experimental observations on the harmonic vibrational modes of glasses are presented. Emphasis is placed on normal and densi®ed silica. These results are discussed within the broader current knowledge, including thermal properties and other spectroscopic data that are critically assessed. We ®nd that propagating acoustic modes enter a regime of strong scattering as their wavelength is reduced, and that this leads to an Ioffe±Regel crossover at frequencies of the order of the terahertz, corresponding to wavelengths of several nanometers. At similar frequencies, an excess in the density of states of optical modes, generally called the Boson peak, is observed. Hyper-Raman spectroscopy on these modes clearly shows that in silica they are due to the rocking of small groups of tetrahedra. These ®ndings provide unique and unexpected information on the structure of glasses at the extended length scale, about which so little is known otherwise. The strong elastic inhomogeneity found at this scale might be decisive in determining glass properties, and even stability, and this will justify further studies. q 2001 Elsevier Science Ltd. All rights reserved. Keywords: A. Disordered systems; D. Acoustic properties; D Phonons; E. Inelastic light scattering PACS: 63.50. 1 x; 78.30.Ly; 78.40.Pg

1. Introduction The problem of glasses has been somewhat neglected in solid state physics over the last century, as compared to the amount and quality of work performed on crystals. The early frontiers of the solid state have been in the understanding of periodic materials, for which progressively more re®ned experiments could be performed, which were explained by elegant theories. As a result, the structural, mechanical, thermal, electronic, optical, and magnetic properties of perfect crystals became known in considerable details. Speci®cally, their vibrational motions are now to a large extent well described and understood, in particular owing to some seminal contributions such as those of Huang [1,2]. It also became possible for these systems to calculate many macroscopic properties either on the basis of spectroscopic data, or from the mere knowledge of inter-atomic potentials, or even ab initio from the electronic wave functions. Perfect crystals are but the exception. Hence, the frontiers moved to the study of the many kinds of defects. These often effectively control the macroscopic and mesoscopic properties of solids, and this led to well known major applications. * Corresponding author. E-mail address: [email protected] (E. Courtens).

To the extent that defects could be treated as perturbations of otherwise periodic structures, experiments and related theories met with considerable success. This also applies to the vibrational properties of defective and mixed crystals, worked out by Lifshitz and Maradudin, among others (for an extensive review, see Ref. [3]). This happy situation changes radically in glasses, owing to the lack of the helpful periodicity. All at once one encounters enormous dif®culties, in experiments, in analytical theories (for an early review of analytical approaches and their dif®culties, see Ref. [4]), and in simulations. This might explain why the old ®eld of structural glasses remained relatively dormant for so many decades. However, this situation is changing very rapidly. Indeed, there now exist tools to address microscopic and mesoscopic scales in random systems. On the one hand, new experimental methods have been invented, and on the other hand the size and speed of numerical computers have reached the level where meaningful simulations can be envisaged. There is also a matching demand from industry, as glasses are important materials for advanced applications such as the production of highly transparent ®bers for communications. From the point of view of theory, glasses present the great challenges of complexity. In a way, they are the solid state paradigm of complex systems.

0038-1098/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved. PII: S 0038-109 8(00)00434-8

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The central issue in glasses remains that of their structure (for a brief review, see Ref. [5]). While they are isotropic at long length scales, X-ray diffraction reveals quasi-crystalline arrangements on the atomic and molecular scales. Clearly there must exist a scale, which we shall call the extended length scale, where the quasi-crystalline structure seen in the intermediate range randomizes. How does this happen, and at which scale, is still not understood in any signi®cant detail. Owing to the absence of Bragg periodicity, direct structural information are restricted to sizes below ,1 nm. However, it progressively emerges that it is at the extended length scale of maybe 5±10 nm that the structure of glasses in¯uences in a decisive way their properties and their stability. These questions will clearly be a frontier for the coming years. The absence of Brillouin zones also complicates the description of the vibrational excitations. However, some of the features found in periodic solids do remain. To the extent that glasses can be approximated by a continuum, one can speak of plane-wave acoustic phonons. Up to which wavevector value q such a description is meaningful is a central question, easily overlooked, but directly related to the extended structure. On the other hand, the structural units can collectively execute their own vibrations, which in the periodic crystals produces near q ˆ 0 the onset of the optic branches. In glasses, something quite analogous to optic modes should be expected at suf®ciently small q. Finally, owing to the structural disorder, one can anticipate the existence of non-harmonic motions, usually described by two-level states (TLS) and tunneling [6±10]. The discussion in this article will be restricted to harmonic or quasi-harmonic modes, as opposed to TLS and relaxations. The early evidence for anomalies on these modes, in particular the strong departure from the Debye predictions [11] observed in studies of thermodynamic properties [8], are brie¯y discussed in Section 2. These anomalies are largest for the so-called ªstrong glassesº [12,13], like SiO2, GeO2, B2O3, and their modi®cations. These pure materials are also known as ªglass formersº. The examples shown in this article mostly refer to SiO2. The harmonic modes have been extensively studied for a long time by light and neutron spectroscopies. However, this subject is becoming frontier as the vibrations do give information about the extended length scale. In the case of acoustic modes, one can now perform coherent scattering (i.e. Brillouin scattering conserving energy and momentum) from excitations in that frequency region [14], using either inelastic X-ray scattering (IXS) [15±17] or inelastic neutron scattering (INS) [18,19]. Such measurements can in principle give information about the end of acoustic branches. As explained in Section 3, our results show dramatic changes in properties at mode frequencies V=2p , 1 THz [20±25]. For typical sound velocities v , 5000 m=s this corresponds to a wavelength l ˆ 2pv=V , 5 nm; which indeed falls within the extended scale. This issue is currently very much debated.

The optic modes of glasses, discussed in Section 4, can be observed with light using infrared, Raman, or hyper-Raman [26] spectroscopies. One ®nds TO and LO modes that super®cially look quite similar to those observed in the corresponding crystals [27]. However, in these experiments incoherent scattering from vibrating regions much smaller than the probing wavelength can be seen. One also discovered in the same experiments a large density of unusually low frequency excitations which are harmonic in nature, i.e. whose spectrum simply obeys Bose±Einstein statistics, and which for this reason were named the ªBoson peakº [28±30]. Another approach to observe optic modes is to use INS [31], in which case one obtains at large scattering vectors Q a one-phonon contribution that re¯ects the vibrational density of states (DOS) [14,32], Z(v ), where v is the running angular frequency. 1 The Boson peak excitations are also seen with INS [33±35], and they indeed show up as a broad peak in the normalized DOS, Z…v†=v 2 : Typically, one ®nds the peak maximum at V BP =2p , 1 THz: This glass speci®c feature clearly must relate to disorder. Its exact nature, and its possible connection with the end of acoustic branches in the same v -region have been subjects of much discussion, and recent progress is presented in Section 5. In particular, we show in the case of vitreous silica that the excess modes correspond to collective rotations (or rocking) of the SiO4 tetrahedra at scales small compared to the optical wavelength [36]. In parallel with these experimental efforts, one witnesses a rapid development of simulations. For such a complex situation, these are particularly helpful to hopefully gain a correct mental picture. It is technically easier to simulate strong perturbations on a lattice rather than topologically disordered systems. It turns out that the former already provides considerable information on the vibrations. On the other hand, the latter also became possible, albeit at higher costs and with smaller maximum sample sizes. The work on topologically disordered systems principally explored three kinds of situations: (1) frozen assemblies of spheres interacting through a Lennard-Jones potential, like in the case of Ar; (2) random networks, like for amorphous Si; (3) or the prototypical oxide glass, SiO2. This does not exhaust the long list of simulated models, and we cannot review all that work, as this alone could easily take the space of a book. Simulations of topological disorder can address the issues of the structure [5]. This has been done with hand-made methods, with Monte Carlo perturbations, with molecular dynamics (MD) using ®xed potentials, or also with full ab initio MD. The criterion for a ªgoodº structure is generally 1 The reader will have noticed the systematic distinction which is being made between the scattering vector Q in a scattering experiment and the wavevector q of plane-wave excitations. The latter only equals the former for plane-wave eigenmodes. Similarly, the running frequency v will be distinguished from the mode frequencies V .

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these distinguish in an essential way the elasticity of glasses from that of the crystals [42]. The structures and the potentials directly control harmonic vibrations. It appears that harmonic vibrations do have some universal features in random systems that strongly differ from those of crystals. This universality allows to compare simulations on model systems to experiments on real glasses. In the course of this article references to simulations will be given when needed to clarify debated fundamental issues related to the extended length scale.

2. Thermodynamic anomalies Fig. 1. The reduced speci®c heat, Cp/T 3, of v-SiO2 compared to that of crystal quartz (after Ref. [43]). The dashed lines show the corresponding Debye values, CD/T 3.

to compare it to experimental measurements, the main information being that on pair-distribution functions obtained by X-ray and neutron diffraction. As pointed out by several authors, e.g. in Ref. [37], the information content of a pair distribution function g(r) is not a stringent test to validate a structure. In particular it hardly says anything at the extended length scale. The validity of the simulation results at that scale depends on the quality of the potentials, on the size of the models, and on the time allowed for quenching and equilibration. These are delicate issues, especially that one will often want to extract from simulations predictions at extended scales that are not really accessible to direct experimentation. One of the most interesting features that are coming out is the existence of strongly ¯uctuating internal pressures [38,39]. This property was postulated long ago [40]. Alexander has shown the importance of stress-induced contributions to the elastic energy of amorphous systems [41], and he re-emphasized recently that

Fig. 2. The viscosity of three glass-forming liquids, illustrating from top to bottom the strong, intermediate, and fragile behaviors (after Refs. [12,13]).

The early evidences for vibrational anomalies in glasses were derived from measurements of the low temperature (T ) speci®c heat, Cp or Cv, and thermal conductivity, k [43,44]. These depart from the Debye predictions for insulators [45] in manners that appear rather universal. Although these facts have been known for a long time, and have been thoroughly reviewed [8], it is useful to explain them brie¯y here. They are indeed crucial to questions relating to the nature of the Boson peak, as well as to the changes in the acoustic modes occurring in the region of V BP. Some of these aspects are far from being well understood. The Cp anomaly of interest is illustrated in Fig. 1 for vitreous silica, v-SiO2, and compared to the Cp of crystal quartz [43]. In v-SiO2, Cp/T 3 is very much above the dashed horizontal line calculated from the known acoustic velocities and which corresponds to the Debye value C D =T 3 ˆ constant [45]. There are in fact two types of disagreements. At very low T (below 1 K), one ®nds Cp / T; not shown in Fig. 1, whereas around 10 K there is a broad hump, with an excess, …Cp =CD 2 1†; as large as 4 near its maximum. Both features indicate extra modes. The ones seen below 1 K are the TLS [8], which are not discussed here. Those around 10 K correspond to the Boson peak. In quartz there is also a hump, but it is much weaker, it is located at higher T, and it is fully explained by the end of the low lying phonon branches whose ¯atness produces a high Z(v ). In that case the Debye value perfectly applies at suf®ciently low T, as seen in Fig. 1. To understand to what extent this excess in Cp is universal in glasses, it is useful to introduce a classi®cation due to Angell, and which is empirically based on the T-dependence of the shear viscosity, h (T ) [12,13]. For glass formers, like v-SiO2, one observes in the melt that h…T† / exp…A=T†: The temperature where h reaches 10 13 P is conventionally de®ned as the glass transition temperature, Tg. The exact value of Tg increases with the quenching rate, as the structural arrest is more effective when relaxations do not have time to occur. We do not need to enter into these details here. The above h (T ) gives a straight line when plotted as log10 h vs. Tg =T; illustrated in Fig. 2 for v-SiO2 …Tg , 1450 K†: In the same ®gure, values are shown for glycerol …Tg ˆ 186 K†; and for CKN which is the mixed nitrate

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Fig. 3. The excess speci®c heat, Cp/CD 2 1, vs. the reduced temperature T/TD, where TD is the Debye value for v-SiO2 (triangles), glycerol (line), and CKN (squares). The inset shows the maximum excess in function of the fragility (after Ref. [47]).

Ca2K3(NO3)7 …Tg ˆ 333 K†: Glasses behaving like v-SiO2 are called strong, while those behaving like CKN are called fragile, and glycerol is said to be intermediate. The strong glasses are characterized by covalent bonding, the intermediate ones by weaker bonds, like hydrogen bonds for glycerol, and the fragile ones are often dominated by van der Waals forces, as in the many organic glasses such as orthoterphenyl. One can characterize the degree of fragility F by the slope at Tg in the Angell plot (Fig. 2), or F ˆ R‰2ln h=2…Tg =T†ŠTˆTg ; where R is the gas constant [46]. Thus F is the effective activation energy at Tg divided by Tg.

Fig. 4. The thermal conductivity of v-SiO2. The line illustrates the T 2-law at low T [49].

The point here is that the Cp anomaly changes dramatically in magnitude on going from strong to fragile glasses [47]. This is illustrated in Fig. 3 for the three glasses in Fig. 2. The inset shows the maximum excess, as de®ned above, in function of the fragility, and this general behavior is rather universal [47]. For fragile glasses, considering the contributions of intermolecular modes, one might wonder whether one should speak of an excess at all [48]. For this reason the discussion below will be restricted to strong glasses, taking as an example v-SiO2. The other thermal anomaly is in the thermal conductivity, k (T ) [43,44]. The k of v-SiO2 is illustrated in Fig. 4 [49]. It departs from the usual k of insulating crystals in two ways. First, at low-T it grows like ,T 2, while the normal case was T 3. At low-T, one usually writes k ˆ P 1 Cv`; where the sum is over the heat conducting 3 modes [45]. C is their speci®c heat, which at low-T is the CD of the acoustic phonons of velocity v, since the TLSs do not conduct. Therefore, the phonon mean free path ` must be anomalous at low T. For very low T this is explained by the interactions of acoustic waves with TLS [8]. The second anomaly is the plateau that occurs around 10 K. It implies that at high V the mean free path ` must tend to 0. In fact, performing detailed estimates, one ®nds that in order to produce a plateau either ` must decrease fast with the increase of V , e.g. like / V 24 [49], or else there must be a frequency where a crossover occurs beyond which the excitations cease completely to propagate, i.e. beyond which ` ˆ 0 [50]. This is a very important issue. There is no doubt that there are k plateaus in glasses [43,44], but a currently proposed interpretation of acoustic mode spectra, discussed in the next section, is inconsistent with this universal observation.

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Fig. 5. Brillouin X-ray scattering signals obtained on densi®ed silica glass, d-SiO2 (from Ref. [25]). The elastic peak was subtracted. The solid lines show ®ts with the EMA model in frame (a) and (b), to be compared to the DHO model in frames (c) and (d).

3. Acoustic modes In the continuum limit, i.e. at large sound wavelengths, l q a; where a is a typical size of the constituents, the concept of acoustic waves in glasses is unchallenged. There is a large body of literature concerning ultrasonic [51] and hypersonic 2 [22] measurements. At these frequencies, quasi-plane waves, slightly damped in space or time, do propagate. The propagation up to V=2p , 300 GHz was actually demonstrated for v-SiO2 by pulse-echo experiments [52]. These waves have a dispersion characterized by the couple (q,V ) and in view of the small attenuation exhibit narrow peaks at Q ù q and v ù V in the inelastic structure factor S(Q,v ). The latter corresponds to the spectrum measured in scattering experiments. The situation becomes much more complex at suf®ciently high q-values. Basically, the questions are: 1. Up to which (q,V ) can plane waves exist, or in other words is there a crossover at some frequency V co and wavevector qco beyond which q is no more de®ned? 2. What happens then to acoustic-like excitations at V . V co ? 3. Is there a situation comparable to strong localization [53±56], where the eigenmodes decay exponentially in space? 2

These refer to Brillouin scattering experiments performed with visible light for which the sound frequencies are typically in the 1±100 GHz range.

4. Is there a relation of the acoustic crossover with the Boson peak? So far, experiments are not able to de®nitely settle any of these issues. However, simulations performed on suf®ciently large samples, whether random lattices [57,58] or topological glasses [37,59], do show indeed quite generally that propagating plane waves are not valid approximations of eigenmodes beyond a certain crossover, generally called the Ioffe±Regel crossover, V co [54±56,60]. For v-SiO2, the value V co =2p , 1 THz has been found in simulations [59], in agreement with the value that can be inferred from thermal conductivity [49], as well as with that obtained from combined INS and IXS experiments [20±22]. Fig. 5 illustrates some experimental results obtained in IXS. The data points shown in frames (a) and (b) are the inelastic parts of scattered spectra obtained from permanently densi®ed silica glass, d-SiO2, at the indicated Q-values [25]. This material was selected preferentially to v-SiO2 for technical reasons related to the capabilities of the spectrometer, as brie¯y explained at the end of this section and in full detail elsewhere [25]. One immediately notices the change in peak position, and the large change in width, upon increase of Q from 1 to 4 nm 21. In fact, the spectrum in frame (a) is not resolved. It corresponds to the spectral function of the instrument. Hence, the real change in width is much stronger than what ®rst appears from Fig. 5. The relation between the scattered spectra, S(Q,v ), and the eigenmodes of the glass is far from trivial at large Q, and the different lines in frames (b) and (d) correspond to two

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different models. We return to these towards the end of this section. First one must discuss what are reasonable expectations. The literature became so confusing in the use of words and concepts that the authors of Ref. [61] felt it necessary to propose a new vocabulary. Any eigenmode of frequency V is called a vibron rather than a phonon, as the latter word might carry from crystal physics the idea of plane waves and propagation. The extended vibrons are called extendons. Each one of these is distributed rather evenly over the entire sample of number size N. In other words, the participation ratio [62] of the jth mode, !2 , N N X X j4 pj ˆ uuji u2 uui u ; …1† =N iˆ1

uji

iˆ1

where is the eigenvector at site i for mode j, is a sizeable fraction of 1 for extendons. The extendons can be either propagative …V , V co † or diffusive …V . V co †: The former are called propagons and the latter diffusons. As V is raised further, diffusons reach a mobility edge beyond which the modes become truly localized [56], with their participation ratios dropping nearly to zero [37]. These modes are then called locons. The eigenvectors of a given locon decay exponentially away from its center, like in Anderson localization [53±55]. In simulations, these locons are always at very high frequencies, and they have a rather small contribution to the total density of vibrational states [37]. The propagons are approximately the plane-wave modes of large l discussed above. Indeed, one should note that strictly speaking plane waves are not eigenmodes. In the absence of anharmonicity, the ªnormal scatteringº of plane waves by static ¯uctuations produces a linewidth G / q d11 [56,63,64]. Here d is the space dimension, which enters this problem via the density of states that the plane waves can scatter into. Although this Rayleigh law has so far only been observed in few 3d-simulations, e.g. in Refs. [58,65], presumably owing to dif®culties with ®nite sizes, there is no microscopic reason for which the scattering of a wave by static defects should produce G / q 2 ; as recently claimed on the basis of phenomenology [66]. The Rayleigh law applies up to the limit wavevector qco, corresponding to excitations at V co. Beyond qco, the modes become diffusons and q loses precise meaning. The diffusive nature of the excitations has been established by several authors [37,67]. Interestingly, the largest majority of ªacousticº modes in amorphous Si for example [37] are diffusons. For these, there is no velocity v and no mean free path `. The modes however contribute to the thermal conductivity in another manner, which produces the rise of k for T above the plateau [37], as observed for example in v-SiO2 in Fig. 4. The depletion of the propagons near V co in v-SiO2 is so severe that a dip sometimes appears in k towards the end of the plateau, and this has been seen in v-SiO2. Non-propagating modes contribute to S(Q,v ) in a manner which is quite different from propagons. At a given V these

modes have Fourier components at many Q-values, and the sum of these modes gives at this V a broad line in a presentation of S(Q,v ) at constant v ˆ V [24]. This is observed in simulations up to very high Q-values (e.g. in Ref. [28, Fig. 10]) and it has been reported for experiments as well [68,69], although interpreted differently. On checking these references, the reader will notice that the simulated and measured pro®les are remarkably similar, so that in no case the measurement of such pro®les can be used to claim that one has observed propagons rather than diffusons, as done, e.g. in Refs. [68,69]. We now consider more speci®cally the linewidths. For long wave propagons …V p V co † the attenuation found experimentally is always strongly T-dependent [22] and it has a dynamical origin not included in the above description. This is due to viscous effects that originate from relaxation or anharmonic processes, for example through the Akieser mechanism [63,70,71]. This generally produces in the spectrum a half-width at half-height G / q 2 in the v and T region of interest here, i.e. outside that dominated by TLS [51]. It can be described phenomenologically by the effect of a dissipation function in the Langevin equation of motion. This leads typically to damped harmonic oscillator (DHO) spectra for S(Q,v ). In particular, for optical Brillouin spectra, the peaks of S(Q,v ) at constant Q closely correspond to the frequency of the eigenmodes at q ˆ Q; and this can be plotted as a meaningful dispersion relation V (q). As the frequency of the propagons is increased, the Rayleigh law, which for d ˆ 3 is G / q 4 ; should eventually become dominant. This produces a crossover in the planewave damping from a dynamical regime at low V to a static scattering regime at high V . A dynamic to static crossover, already discussed in Refs. [20±22] for v-SiO2, was recently rediscovered for glycerol [72]. Beyond that crossover, the damping of each plane wave becomes dominated by static local ¯uctuations in the glass. In other words, plane waves are then mostly inhomogeneously broadened since different v -components in the spectrum corresponds to different, identi®able, V -eigenmodes. We already said that the latter situation cannot be described by a viscous-like damping, as proposed in Ref. [66]. We just note additionally that two different laws G / q 2 cannot even crossover. In view of the inhomogeneous broadening, S(Q,v ) strictly cannot be derived from standard linear response theory which implies translational invariance. It can also not be described by a DHO. As already explained, S(Q,v ) does not even give access to single eigenmodes but only to some projection of a family of eigenmodes. In the limit where the homogeneous part of the broadening is negligible, spectra in function of Q and at constant v give the sum of the spatial Fourier transform of all the modes at V ˆ v: Spectra at constant Q are even more complicated as they correspond to a cut of all the above sums at that particular Q. This has already been explained in considerable detail elsewhere [24]. Since the growth G / q 4 is very fast, the propagons

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Fig. 6. A comparison of the homogeneously broadened DHO spectral function in (a), with the inhomogeneous EMA spectral function in (b). All frequencies have been reduced by vQ. Small G values in (a) correspond to high V co in (b).

rapidly reach the limit G , V; which marks the Ioffe±Regel crossover. 3 From this point one enters the diffuson regime [20,21,37,59], which is fully dominated by static local ¯uctuations. At these higher Q-values it becomes technically easier to obtain the spectra both in experiments and in simulations. Broad peaks are observed, as shown in Fig. 5(b), for which one can measure the position of the maximum, v max, as a function of Q. Of course v max(Q) will approximately extrapolate the V (q) of the low V propagons [74,75]. Although the shape of the v max(Q) curve might be reminiscent of the phonon dispersion of crystals, in the diffuson 3 More precisely, one ®nds both in experiments [25,73] and in simulations [58,59] that the crossover appears at G ˆ GV; where G is a number between , 1 and , 0:3: This constant seems larger for systems that are structurally more random [58,73].

regime it cannot anymore be interpreted in any reasonable manner as a dispersion curve since there is not even any wave velocity. While the latter has already been stated very clearly, also by others (see, e.g. Ref. [76]), the interpretation in terms of dispersion must remain appealing to many workers as it is still found repeatedly in the literature. To analyze the experimental spectra, it is very convenient to have at one's disposal some analytical form for S(Q,v ). Fortunately there exist approximations, such as the ªcoherent potentialº one (CPA) [4] or the ªeffective mediumº one (EMA) [77,78]. They generally amount to reintroducing translational invariance by way of a mean over a large number of excitations. In the EMA this leads to effective v -dependent complex force constants. The analytic form of S(Q,v ) essentially remains that obtained from a Green-function analysis, but with a v -dependent linewidth G (v ) and velocity v(v ). One particular example

Fig. 7. Brillouin X-ray scattering signals at constant v -values from v-SiO2 (after Ref. [69]). An elastic contribution was subtracted. The solid lines show the EMA described in the text, where only the amplitude was adjusted. One should note that the v -values are here too high (above 1 in the reduced units of Fig. 6) to provide the sensitive comparison of EMA and DHO claimed in Ref. [69].

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of such a spectral function [78] was successful at ®tting highly anomalous Brillouin scattering spectra obtained across the Ioffe±Regel crossover on very porous aerogels [79]. In these experiments the length scale near crossover can be of the order of optical wavelengths, and the spectra were obtained with excellent statistics, resolution, and contrast. The results could be interpreted in full agreement with all other scaling and crossover properties of these fractal aerogels, as reviewed in detail elsewhere [80]. As this particular EMA spectral function is not speci®c to fractals, but rather represents the crossover from normal scattering to strong scattering, and in view of its success, we used a similar expression to adjust IXS and INS spectra obtained on glasses in the crossover region [20,21,23,25]. To illustrate the difference between this EMA and the DHO, typical shapes of these spectra are shown in reduced units in Fig. 6. Frequencies and dampings are normalized to Qv, which equals the mode V in the case of the DHO in Fig. 6a. Spectra at small G values in Fig. 6a are to be compared to spectra at V co . Qv in Fig. 6b. In that limit, the spectral shapes in Fig. 6a and b are quite similar. The main difference arises when G is large in the DHO, corresponding to V co , Qv in the EMA, so that the frequency Qv is in the diffuson regime. Then the DHO, which is a homogeneous spectrum, shows a high intensity for v tending to 0, whereas the EMA, which is intrinsically inhomogeneous, shows an intensity going to zero together with v . Indeed, in the latter model all eigenmodes are in®nitely narrow, d…v 2 V†; and at small V s there exist only propagons of quite speci®c q which have no spectral weight for Q well above this q. An experimental example of constant v spectra of v-SiO2 at 1200 K, taken from Ref. [69], is shown in Fig. 7. The lines in Fig. 7 represent the S(Q,v ) that we calculate from the EMA assuming V co =2p , 1 THz; in agreement with our previous determinations at 300 K [20,21], but with G , 0:5 instead of 1, which seems to account well for the homogeneizing effect of the elevated T. 3 In Ref. [69] the same experimental points were not adjusted very well to a DHO, while one sees from Fig. 7 that the EMA is remarkably successful. The disagreement between the experimental points and the solid line observed at low Q-values in Fig. 7 is also found in the DHO ®ts [69], and we presume that it might result from dif®culties in exactly subtracting the elastic line in that region. Although this subtraction is a technical problem, it seriously affects the possibility of a de®nite interpretation, as there is always an elastic peak S(Q,0) that is rather strong and whose wings, in the current IXS instrument, tend to mask the details of the inelastic spectrum, especially at small v . As explained in Fig. 6, it is precisely near the center of the spectrum, that the strongest difference between the two interpretations is expected. Incidentally, the reader should note that constant v -scans at high energy, such as those illustrated in Fig. 7, are outside the interesting range of small v -values where the two models could be distinguished as explained in Fig. 6. It is to achieve such an experimental distinction that we

investigated d-SiO2, which is a material where V co is located higher in v than in v-SiO2, and where the elastic peak S(Q,0) is weaker. The inelastic contributions shown in Fig. 5 are obtained after ®ts of the total spectrum to the sum of both elastic and inelastic line shapes, using either model, and subsequent subtraction of the elastic component. The EMA was used for frames (a) and (b), and the DHO for (c) and (d). As the adjustments allow for independent elastic strengths in either case, the remaining inelastic contribution which is the part shown in Fig. 5 is not exactly the same in (a) and (c), or in (b) and (d). For very small Q, no real difference can be seen between DHO and EMA, as seen in comparing frames (a) and (c). However, at high Q it becomes clear that the DHO in (d) gives a poor representation while the EMA in (b) is perfect. From the EMA ®ts one ®nds a higher V co =2p , 2 THz [25], which very well agrees with the higher position in T of the plateau observed in k (T ) [81]. It is unfortunate that the DHO is still used so extensively for the adjustment of experimental inelastic X-ray scattering data in glasses. In doing so, the authors systematically reach to the conclusion that acoustic modes propagate up to very high q values. They use as de®nition of propagation the existence of a pseudo-dispersion curve v max(Q) that approximately follows the slope given by the low frequency V (q). As explained above, this amounts to postulate that plane waves are approximate eigenmodes up to very high q ˆ Q; which disagrees with simulation results, with thermal conductivity measurements, and also with some inelastic neutron scattering results, as recently reviewed by Dorner [82]. Carefully interpreted experiments agree with simulations in giving good support to a transition into a regime where there is strong scattering of acoustic waves. In normal silica glass this occurs for V co =2p , 1 THz; or at a crossover wavevector qco ˆ vV co , 1 nm21 : This is an important conclusion for what concerns the structure. How can one, in a material as homogeneous as a glass, obtain strong scattering at an extended length ,2p/qco , 6 nm? We return to this profound question in the ®nal discussion of Section 6.

4. The optic modes The systematic observation of optical spectra in glasses started in the 1950s using both Raman scattering (RS) and infrared (IR) spectroscopies. An early review is given in Ref. [83]. Hyper-Raman scattering (HRS), in which two incident photons generate a scattered photon, was experimentally demonstrated by Terhune et al. a few years later [84]. That seminal paper already reported a clear observation of two optical bands in v-SiO2, one of them de®nitely distinct from IR absorption. However, for technical reasons, it took about a decade until HRS also became a practical spectroscopy, as reviewed in Ref. [26]. A ®rst inelastic

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Fig. 8. A comparison of spectroscopic results on the optical modes of v-SiO2: (a) the transverse dielectric constant divided by v , from IR re¯ectivity measurements in Refs. [86,87] (line), together with the low-v values calculated from IR absorption data in Ref. [89] (dots); (b) the longitudinal dielectric constant calculated from the IR re¯ectivity data in Refs. [86,87]; (c) T-reduced polarized Raman scattering data where the low-v part is our own measurement, while the high-v region is taken from Ref. [88]; (d) T-reduced hyper-Raman scattering data (V 1 H) obtained in 908 scattering, where the low-v region is from Ref. [36] and the high-v one is from Ref. [26]; (e) the one-phonon vibrational DOS from Ref. [88] (line), together with a curve divided by v 2 that shows the low-v Boson peak (dots). The vertical bars in (a), (b) and (d) are the TO and LO assignments of Ref. [90], while for (c) the main Raman peak R and the defect lines D1 and D2 are also indicated. The scales in (a,b) are in cm, as the frequency is expressed in cm 21 for the division by v , while (c±e) are in arbitrary units.

neutron scattering (INS) study of v-SiO2 was reported by Leadbetter who observed, besides the DOS of acoustic-like modes, contributions in the region of optic modes [31]. A simulation of the DOS of silica was then performed by Bell and Dean [85]. That pioneering work reproduced some major features of the DOS, and the many subsequent papers by the same authors started to clarify the origin of the modes and their localization. Unfortunately, these calculations neglected the long-range Coulomb forces. The importance of electric forces was ®rst recognized by Galeener and Lucovsky, who did report TO±LO splittings in silica and germania, v-GeO2, and thereby contributed considerably to the clari®cation of the optical spectra [27]. Researchers interested in the structure of glasses soon realized that ªstudies of the vibrational spectra have considerable potential for increasing our understanding of the

structureº, but also that the IR and RS spectra are ªdistressingly similar to a smeared-out version of the corresponding crystalº [86,87]. This is even more so for the density of states, as demonstrated in INS measurements of crystalline and glassy SiO2, GeO2, and BeF2 [88]. Comparing to the parent crystal, quite generally one anticipates in the optical spectra of glasses three types of contributions: 1. There must be bands directly related to these of the crystals (e.g. to these of quartz in the case of silica) which are active in the corresponding glass. These will be blurred by disorder. 2. Forbidden bands of the crystals might become active in the glass, as the selection rules are likely to be relaxed by disorder. 3. Finally, defects that are absent from the parent crystals

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might occur in glasses, e.g., small rings of repeated ±Si± O± units in the case of silica. If such defects are optically active, and suf®ciently numerous, they might contribute to the optical spectrum. It is clear that there is indeed a lot of information about the structure of glasses in the details of their optic mode spectra, but it is also a formidable task to fully disentangle it. As mentioned below, this can now be undertaken with modern high quality simulations. For illustration, Fig. 8 compares some experimental results obtained with various spectroscopies on v-SiO2. The top two frames show IR results, the next two frames are scattering spectra (RS and HRS), and the last frame presents the INS DOS. It is obvious that each spectroscopy provides different information. What is shown in (a) is the imaginary part of the transverse dielectric constant divided by v , e 00' : With this normalization, the transverse IR spectrum can be directly compared to temperature corrected scattering intensities, I…v†=v…n 1 1†; as shown for example in frames (c) and (d). Here, I(v ) is the scattered intensity, and n 1 1 is the Bose±Einstein factor on the Stokes side. The division by v (n 1 1) removes the temperature effects from I(v ) without loss of spectral information at low v values. A semi-log scale is used in frame (a) to emphasize regions of low e 00 ', in particular the one at low v . In the latter region, the spectrum is sensitive to polar impurities like ±OH groups. That shown in (a) is intrinsic [89], i.e. it is free from water contributions. Frame (b) is the imaginary part of the longitudinal dielectric constant divided by v , e 00k =v: It is calculated from the data plotted in Refs. [86,87]. Therefore, it is not as accurate as frame (a), and for this reason a linear scale has been used. That quantity is de®ned by e 00uu ˆ 2e21 Im…1=e†; where e 1 is the high frequency limit of e [91]. Hence, (b) corresponds to the LO spectrum. While the modes active in (a) and (b) produce ¯uctuations in the polarization, those in (c) are associated with ¯uctuations of the polarizability. For suf®ciently high symmetry one expects a mutual exclusion between IR and RS spectra. It is clear from the comparison of (c) and (a) that the strict exclusion is relaxed by structural disorder. One should note, however, that the very large contribution noted R in (c) is not to be confused with the peaks marked either TO4 or LO4 in (a, b). The near coincidence is speci®c to SiO2, and it does not occur in GeO2 or BeF2 as discussed in Ref. [88]. In HRS (d), one expects all IR-active modes to be seen, while the RS-active modes should be silent [26]. This is indeed the case: all TO and LO lines of (a) and (b) can be identi®ed in (d), while the strong R-line of (c) seems essentially absent. Before indexing the modes in Fig. 8 one should ®rst remark that peaks in the DOS (e) cannot generally be associated with speci®c features in the long wavelength spectra (a±d). This has been shown in detail in Ref. [92], where the transverse and longitudinal components of the DOS simulated ab initio are found to be identical in shape. In

particular, the splitting seen at the highest frequencies in Fig. 8(e) is not due to a TO±LO effect. The DOS of v-SiO2 has often been simulated using model potentials, as recently summarized in Ref. [93]. As shown, the shape of the DOS provides a rather stringent test for the quality of the potentials. In fact the available model potentials are so far not able to reproduce the peak which is seen in Fig. 8(e) near 350 cm 21, as discussed in Ref. [93], while direct ab initio calculations give very satisfactory results in that region [92]. To reveal the origin of the peaks, one can project the eigenmodes either on various typical vibrations of structural units, or on typical motions of certain atoms, for example rocking, bending, and stretching of the oxygens in the ±Si±O±Si± bonds. These motions were de®ned in Ref. [85]. One should recall that ±Si±O±Si± forms an angle of about 1408. Rocking is the displacement of O perpendicular to the ±Si±O±Si± plane, bending is the in-plane motion along the ±Si±O±Si± bisector, and stretching is in the third orthogonal direction. For example, one ®nds at low frequencies (up to the gap around 900 cm 21) that rocking and bending co-exist, but that rocking dominates the DOS up to ,400 cm 21, beyond which bending clearly takes over [76]. Projecting on the SiO4 vibrations, one ®nds that the higher frequency bending motions of the oxygens are really stretching motions of the SiO4 units [93]. In Refs. [76,92], the peaks near 800 and 1080 cm 21 are found to be asymmetric stretch, while the one near 1200 cm 21 corresponds to fully symmetric stretch. Turning to the modes of the optical spectra, the observations (a,b,d) agree with the proposal of Kirk [90] who identi®ed four LO±TO pairs. Their positions have been drawn in Fig. 8, and are labeled as in Ref. [90]. The higher frequency modes, LO1 and LO2, are coupled in the model of Kirk, and this produces the unusual frequency inversion in the TO2 ±LO2 pair. Whether two distinct modes are really involved, or rather a continuum, is not settled. Note that LO1 falls in the high frequency wing of the DOS, and this could actually be remarkably reproduced in the ab initio simulation [92]. Modes 1±3 are due to stretching, while mode 4 corresponds to a polar bending motion. The defect lines D1 and D2 are only seen in the RS spectrum (c). These are produced by small (SiO)n rings, with n ˆ 4 or 3, respectively. The breathing motion of oxygen in these rings practically decouples from the rest of the structure as shown in Ref. [94], and this produces narrow lines. Although these rings are rare, the Raman coupling is unusually strong, leading to the observed intensity [95]. Therefore, a similar motion for the higher order rings (n . 4) produces a strong Raman band at lower frequencies. As suggested long ago by Galeener [96], and as discussed in Ref. [97], this is essentially the origin of the strongest R line which is also unseen in the other spectroscopies. The only spectral region that has not yet been discussed is the one below 200 cm 21. It is the subject of the next section. Several authors have considered the possible localization of these modes. The local nature of the modes is the basis for

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homogeneous broadening or about the lifetimes of these excitations. This type of investigation became recently possible using pump±probe experiments [100]. It is likely that this will become another frontier in the study of glasses in the years to come. 5. The Boson peak

Fig. 9. The Boson peak region of v-SiO2, showing the comparison of the DOS to the T-reduced RS and HRS spectra. The ordinate scale corresponds to S…Q; v†="v…n 1 1† [36], which is proportional to Z(v )/v 2 (open dots). The HRS data, IHRS …v†=v…n 1 1†; has been scaled to the DOS by a multiplicative constant (full dots). The HV depolarized RS spectral shape, IRS …v†=v…n 1 1†; is shown by the solid line.

a model [98] in which the scattered intensity I(v ) is proportional to the corresponding density of states, Z(v ), I…v† ˆ C…v†…n 1 1†Z…v†=v:

…2†

The coef®cient C(v ) accounts for the coupling of the vibrations to the light. This formula, originally derived for RS, should also apply to HRS [26,36]. However, based on the fact that distinct LO and TO modes, with appropriate polarization selection rules, are observed in HRS, Denisov et al. [99] concluded that the optical modes must be ªdelocalized and have the same collective nature as in a crystalº. It should be remarked that owing to the internal ®eld associated with these vibrations, the direction of q might be suf®ciently well de®ned although its length could be meaningless. This would perfectly preserve the polarization selection rules observed in Ref. [99], and thus these facts do not prove the extended character of the modes. On the other hand, the size of simulation boxes that could be used so far does not really allow to settle this issue, except for the high frequency stretching modes which clearly have a small Ê linear size [93]. participation ratio in boxes of only ,20 A Another argument presented for ªdelocalizationº is the observation of polaritons [99]. It is well known that in HRS polaritons are active in centrosymmetric materials, and therefore also in liquids and glasses. In that case, it is a mixed electromagnetic±mechanical excitation that is observed at very long wavelengths, and one probes essentially e (v ) averaged over that wavelength. The coherence of the modes is produced by the electromagnetic ®eld, and this does not imply a delocalized nature of the individual mechanical vibrations. It is in fact reasonable to expect that the bands observed in the optical spectra of glasses have an inhomogeneous linewidth. However, very little is known so far about the

As shown in Section 2, there exists in some glasses a strong excess of low frequency modes that manifest themselves by a sizeable anomaly in the speci®c heat, i.e. by a hump at low T in Cp/T 3. As already mentioned in connection with Fig. 3, this feature is not universal. It is most prominent in strong glasses, and particularly in v-SiO2 [47]. Historically, an unexplained low frequency band was ®rst observed in the Raman spectra of many glasses, both inorganic and organic [28±30,101]. This ªBoson peakº (BP), generally more pronounced in the strong glass formers, and is then also located at higher frequencies than in fragile glasses [101]. This Raman signal shows no appreciable dependence on the scattering angle, and thus on Q. This was recently veri®ed in detail for v-SiO2 [102]. In this case one expects that Eq. (2) applies, the scattering modes being like ªmolecularº ones. This means that their size is so much smaller than the light wavelength that they scatter incoherently. These low frequency excess modes must of course be seen in the DOS measured in INS. This is illustrated in Fig. 8(e), where a low frequency peak in Z(v )/v 2 is shown for v-SiO2. According to the Debye model, Z(v )/ v 2 should be a constant whose value is determined by the sound velocities [45]. The observed peak in v-SiO2 is ,7 times stronger than the calculated Debye value [33,34]. This shows, for this particular case, that a density of acoustic modes cannot be at the origin of the BP, contrary to recent assertions [103,104], including our own earlier speculations [20,21]. Since Z(v ) and I(v ) are measured quantities, it is natural to try to extract C(v ) using Eq. (2). The expectation is that C…v† / v2 for acoustic modes [30,105], and that C…v† / v0 for ªsimpleº optical bands [98]. In many cases, neither of this is observed. For example for v-SiO2 C(v ) is approximately linear in v from ,10 to ,100 cm 21 [104,106]. Considering these observations more carefully, one should ask whether the I(v ) that is measured in RS really corresponds to the same modes on which the Z(v ) is measured in INS. There are two delicate questions: 1. Is the BP measured in INS produced only by the ªexcess modesº? 2. Are these ªexcess modesº really active in RS, so that I(v ) is a fair measurement of their density? The ®rst problem occurs in the case of B2O3. The structure is composed of ¯at BO3 units [107]. The lowest frequency

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modes, as always, are presumably the rocking of these units. This rocking modulates the polarizability and it is thus active in RS. However, the excess is not so large, and the INS measurement contains two contributions [108]. One is related to Umklapp scattering of acoustic modes via the ®rst sharp diffraction peak [14,32], which has been called an ªinphaseº component in Ref. [108]. The other has a purely random phase, leading to incoherent-like neutron scattering proportional to Q 2. If C(v ) is calculated using the full Z(v ), a strange v -dependence results [108]. However, if one remarks that the in-phase component cannot produce RS, only the incoherent part of Z should be used in Eq. (2). Then, a value C(v ) ˆ constant results [108], which nicely agrees with reasonable expectations [98]. The second problem is found in the case of v-SiO2. There the excess of modes is so large that it is dif®cult to extract an in-phase component in INS [33,34]. The INS signal is mostly due to the random-phase part. However, the rocking of regular SiO4 tetrahedra does not modulate the polarizability, and therefore it is not really RS active. Only the static distortion of the tetrahedra could produce by rocking a RS signal at BP frequencies. It is easy to imagine how this leakage of forbidden modes can introduce the observed increase of C(v ) with frequency. On the contrary, the rocking of regular tetrahedra is active in HRS [36]. It is thus the HRS signal that should be compared to the INS Z(v ). This is shown in Fig. 9. One sees that indeed a constant C(v ) gives excellent agreement, as indicated by the superposition of HV Z…v†=v2 to IHRS …v†=v…n 1 1†: The value of IRS =v…n 1 1† is HV also shown, where IRS corresponds to the depolarized RS signal. One immediately recognizes that the RS signal is very different from the INS one. It should be noted that the polarization and Q-selection rules of the BP observed in HRS are ªmolecular-likeº as opposed to average glass ones [36], which con®rms the local nature of the BP modes. Local modes involving the rocking motion of tetrahedra are thus at the origin of the BP signal in v-SiO2. This agrees with an early model used to describe the INS spectra [33,34]. This view is also con®rmed by more recent simulations [93,109]. In silicas, v-SiO2 and d-SiO2, the frequencies of the BP maximum, V BP [36], are nearly coincident with the observed Ioffe±Regel crossovers of acoustic modes, V co [20,21,25,59], which were discussed in Section 3. In boron oxide, B2O3, V co is also located in the frequency region of V BP [110]. For many other glasses, V co has not yet been measured, but an estimate can be made within the frame of the soft-potential model [9,10]. These calculated values are also found to be close to the corresponding V BP [111]. Finally, in calculations of model systems, it has been found that a distribution of coupled, nearly unstable harmonic oscillators produces a low frequency peak in Z(v )/ v 2 and simultaneously a Ioffe±Regel crossover for the acoustic modes [58]. It is thus probable that a distribution of low frequency local vibrations can couple rather strongly to ªacoustic-likeº modes at the scale where the translational

invariance is strongly perturbed by disorder. In this picture, the BP and the Ioffe±Regel crossover of strong glasses are obviously related. For fragile glasses, the role of the BP in the scattering of acoustic modes can be taken over by the low frequency collective molecular excitations [48].

6. Conclusions and outlook One has observed in recent years a substantial progress in the understanding of the structure and of the vibrations of glasses. A signi®cant breakthrough that concerns the vibrations does point to the importance of extended length scales, of maybe ,5 nm in silica. It seems that it is at that scale that the disorder actually produces a well-known universal macroscopic effect, namely the plateau observed in the thermal conductivity. It should be associated with the onset of strong scattering of acoustic-like excitations. In view of the close relation that appears to exist between the latter and the Boson peak, one might anticipate that the scale of one to several nm also should play a role in producing the low frequency excess modes. One needs an explanation for the occurrence of strong acoustic scattering at such large scales. This can only be produced by very strong ¯uctuations of either the density or the elastic constants, or both. The former is unlikely in dense glasses, as no corresponding signal is detected in small angle scattering. So one can only invoke large excursions of the local elasticity. Alexander has pointed out that in glasses, as opposed to periodic structures, there can exist large internal pressures (compressions and tensions) at individual sites [42]. This has recently been con®rmed by simulations, in particular for a Lennard-Jones glass [38,39]. It is found that regions of high coordinations (near 12) have a slightly smaller density, but an enormous local tension, close to the one that produces rupture. On the other hand, regions of low coordination have a slightly higher density, but are very much compressed. One effect of these internal pressures is that the local elastic constants ¯uctuate by as much as an order of magnitude between the different regions [38,39]. An indication that something like this might actually occur in real glasses was found quite long ago in macroscopic measurements: large changes in the macroscopic elasticity (,10%) can occur with small changes in density (,1%) that result from different quenching rates [112]. The observation of the strongest Boson peak signals for materials that are known as good ªglass formersº points to the possible relevance of that feature, and of the related disorder at extended length scales, to the actual stability of glasses. The study of this question presents real challenges for experiments, theory, and simulations alike. In experiments, much can be learned using coherent spectroscopy where simultaneous information is obtained on excitation energies and sizes. Unfortunately, there exists currently a wide gap in scattering vectors, between , 6 £ 10 22 and ,1 nm 21, which is technically inaccessible. This

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is precisely the region that seems to be of crucial interest for this problem. Other spectroscopies can also be used, in particular to learn, e.g. about selection rules or relaxation times. Finally, suitable microscopy could be employed to investigate directly the inhomogeneities. For theoretical developments, it is obvious that great care should be exerted in using effective medium approximations to describe such highly inhomogeneous situations. In simulations, it was very interesting to have results on large samples. For these to be signi®cant at the extended length scales, both potentials of high quality and a satisfactory approximative modeling of the thermal history will presumably be required. It is clear that considerable challenges do lie ahead.

Acknowledgements Much of the experimental work on which this article is based was performed at large facilities, in particular at the Institut Laue-Langevin and at the European Synchrotron Radiation Facility, both in Grenoble, France. The authors thank their colleagues H. Casalta, R. Currat, B. Dorner, C. Masciovecchio, and J.-B. Suck for stimulating discussions and/or actual help during experiments. Thanks are also addressed to K. Inoue and A. Yamanaka in Sapporo, Japan, for a fruitful collaboration using their hyperRaman-scattering instrument.

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