Raman scattering and Boson peaks in glasses: temperature and pressure effects

Journal of Non-Crystalline Solids 349 (2004) 88–97 www.elsevier.com/locate/jnoncrysol

Raman scattering and Boson peaks in glasses: temperature and pressure effects John Schroeder a,*, Weimin Wu a,1, Jacob L. Apkarian a,e, Mierie Lee b, Luu-Gen Hwa c, Cornelius T. Moynihan d a

Department of Physics, Applied Physics and Astronomy, Rensselaer Polytechnic Institute, 110 Eighth Street, Troy, NY 12180-3590, USA b Yonsei University, Seoul, South Korea c Department of Physics, Fu-Jen University, Taipei 24205, Taiwan, ROC d Materials Science and Engineering, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA e Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA Available online 28 October 2004

Abstract Inelastic light scattering as a function of temperature and pressure was carried out on various oxide and halide glasses. For the low frequency Raman spectra, all of our glasses show a broad feature in the frequency range from 20 to 110 cm
1 (0.60–3.30 THz); the
Boson peak. This Boson peak is associated with the existence of intermediate range order (IRO) in glass. The Boson peak is due to an increase in the vibrational density of states, over the Debye value, caused by localized excitations (phonon localizations). The low frequency Raman scattering containing this dominant spectral line (Boson peak) is interpreted in terms of its relationship to the amplitude and extent of the density fluctuations in glasses and is, thereby, considered a measure of the intermediate range order in these glasses. Phonon localization (the Ioffe–Regel criterion) was used to calculate correlation lengths for the measured glass samples. Characteristic correlation lengths for the glasses in this study were in the range of about 2–5 nanometer in size. We found that the Boson peak energies are highly pressure dependent but show very little change with temperature. The pressure effects in the Raman spectra will be discussed in terms of existing theories. The concept of intermediate range order (IRO) in glass and its applications will also be discussed. Ó 2004 Elsevier B.V. All rights reserved. PACS: 78.30.
j; 61.43.Fs; 63.50.+x; 78.55.Qr

1. Introduction The determination of structure in amorphous or glassy materials has always been a challenge of modern solid state physics. Thermodynamic considerations have played a large role in attempts to give a physical explanation of the liquid-to-glass transition. The dynamics of *

Corresponding author. Tel.: +1 518 276 8408; fax: +1 518 276 6680. E-mail address: [email protected] (J. Schroeder). 1 Present address: Electrical Engineering Department, Pennsylvania State University, University Park, PA 16802, USA. 0022-3093/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2004.08.265

various relaxation processes and their relation to the nature of the glass transition have been studied extensively in recent years, both from theoretical and experimental standpoints [1–7]. Many different relaxation techniques reveal the presence of two general features common to all super-cooled liquids [8]: the non-exponential behavior toward (meta-stable) equilibrium, and rapidly increasing relaxation time as the temperature is lowered. Despite the observation of these two signature features in numerous materials and much theoretical work, there is still no accepted fundamental theory of the glass transition. This behavior has usually been interpreted in two fundamentally different ways [8].

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First, one can imagine that a heterogeneous set of environments exists in a super-cooled liquid; relaxation in a given environment is nearly exponential but the relaxation time varies significantly among environments (a spatially heterogeneous distribution of correlation times: thus parallel processes). Alternatively, one can imagine that super-cooled liquids are homogeneous and that each molecule relaxes nearly identically in an intrinsically non-exponential manner (an intrinsically non-exponential loss of correlation in a homogeneous system: therefore serial processes). Of course, these two points are extreme positions and it is possible that elements of both pictures are applicable [3]. In the first case, the material a few degrees above Tg is thought to consist of clusters of molecules with different rates of motion [9] which remain virtually unchanged for a time sfl exceeding the longest correlation time. Such behavior is sometimes termed non-ergodic on a time scale less than sfl, because within this time a given unit does not probe all the regions in configuration space accessible to other units in the ensemble. The second concept is often related to the Kohlrausch–Williams–Watt (KWW) function [10], a stretched exponential exp[
(t/s0)b], 0 < b 6 1, which yields good fits to most relaxation data. It has been suggested that a natural non-exponential dependence upon time may result from the cooperative nature of the glass transition process [11]. Several theoretical models assume that some sort of heterogeneity is an essential feature of super-cooled liquids [2,9,12–18], however, it is not clear at present how much of the non-exponentiality of the relaxation functions of super-cooled liquids should be attributed to spatially heterogeneous dynamics. Several experiments [8,19–22] have been interpreted as indicating that super-cooled liquids are heterogeneous in some manner but other workers [23] have concluded that spatial heterogeneity is relatively unimportant. It remains to be proven what features of the structure (e.g. density fluctuations, configurational entropy) are responsible for non-linearity, but that must follow directly from any correct model. Experimental techniques, which measure properties of an entire ensemble of molecules, cannot directly distinguish whether non-exponential relaxation is a result of homogeneous or heterogeneous dynamics [3]. The concept of propagating collective modes implies the existence of a simple relation between the energy and the momentum of specific excitations [24]. This occurs as a consequence of translational invariance, and is typical in crystals and free particles. In disordered systems, like glasses, this concept may lose its physical significance when local interparticle interactions start to dominate and the topological disorder does not allow one to derive some kind of quasiparticle and its excitation spectrum [25]. In opposition to this picture, however, one may argue that the translational invariance becomes an increasingly less important aspect at large

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energies and momentum transfers, because one samples particle–particle correlations on a very short time and space scale. Therefore, if local properties become dominant over long range correlations among particles, the differences in the dynamics of liquid and glass, and crystalline states may disappear [24]. At low momentum transfer values, the existence of propagating collective excitations in glasses demonstrated by the sharp Brillouin lines (typically below 1 cm
1), are observable in light scattering experiments. This is straightforward from an intuitive point of view since at low momentum transfer values one samples particle–particle correlations on a long time and large space scales with respect to interatomic motions and distances [26]. In the mesoscopic time-space domain the situation is more complicated, and existence of collective dynamics is doubted. The evidence of modes in glasses in the mesoscopic region comes from incoherent neutron scattering and light scattering studies [25], where a broad band is found around 20–100 cm
1 (0.6–3 THz or 2.5– 12 meV), and with an almost universal shape. This band has been named the Boson peak because its intensity scales with temperature according to the Bose–Einstein distribution function. The Boson peak has been connected with collective motions of atoms (acoustic phonons) within the intermediate range order of glasses [27–29]. The intermediate range order implies that the arrangements of structural units in a glass are not completely random but have some correlations on a nanometer scale varying slightly between different glasses. The correlation length is considerably larger than the characteristic interatomic length [30]. A lively debate exists on the excitations giving rise to the Boson peak and, so far, conclusions on their vibrational (localized or propagating) or relaxational nature have not yet been reached [25].

2. Theoretical background Raman scattering in disordered systems differs from the scattering in crystals in that it is related to spectrum of the vibrations in the material. Experimentally Shuker and Gammon [31–33] showed that the entire Raman spectra of glass consisted of broadened intensity lines (not the sharp spectral lines, as seen in crystals). The crystal state follows precise selection rules due to k-conservation resulting in relatively narrow Raman lines. In the glass counterpart the selection rules are broken because the disorder in a glass allows momentum conservation to be of no consequence and the Raman lines are extremely broad. This is also valid for the low frequency
Boson peak – but remember its crystal counterpart does not exist. According to Shuker and Gammon [31,32], the intensity as a function of frequency for Raman scattering is given by Eq. (2.1). The intensity

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of the Raman spectra in disordered solids is proportional to the density of vibrational states g(x) multiplied by light vibration coupling coefficient (Pockels coefficient) C(x): IðxÞ ¼ gðxÞCðxÞ½1 þ nðxÞ=x;

ð2:1Þ

here [1 + n(x)] is the Bose–Einstein occupation number for the Stokes component. So it is possible to find g(x) from Raman spectra, if one knows the frequency dependence of C(x). In the extreme case of complete randomness of the atomic couplings or the atomic amplitudes in the vibrational modes C(x) would be independent of frequency [34]. Comparison of the Raman spectra with inelastic neutron scattering data (which yield in the incoherent approximation directly g(x)) have shown [35,36] that in the Boson peak region, g(x) exhibits a broad maximum in a number of glasses, while C(x) has no maximum and varies nearly linearly with frequency. As discussed by numerous other authors [37–40] the Boson peak is due to an increase of the vibrational density (phonon density) of states over the Debye value (g(x) = 3x2/xD3, xD = Debye cutoff frequency), and this origin is now generally accepted. The Boson peak is caused from vibrational excitations made up of acoustic phonons, most likely shear or transverse phonons, which scatter strongly from elastic inhomogeneities in the structure due to the topological disorder in glasses. The scattering leads to a drastic decrease of the mean free path of vibrations L(x) and can increase the vibrational density of states in a certain frequency range. These excess phonon states may now be considered to be localized by the strong scattering. Hence, phonon localization may be viewed in terms of the Ioffe–Regel rule [41,42]. The Ioffe–Regel criteria is given as kðxIR ÞLðxIR Þ  1;

ð2:2Þ

with k(xIR) – the magnitude of wave vector, L(xIR) – the mean free phonon path at the Ioffe–Regel frequency xIR. (Note: the phonon localization here is analogous to the Anderson localization of electrons.) The Ioffe–Regel rule allows us to calculate a correlation length in terms of xBP, the Boson peak frequency, and the sound velocity, vs. Expressing this rule thusly, Lc  vs =xBP :

ð2:3Þ

The excess density of phonon states can be attributed to the consequence of strong scattering [43]. Now, the position of the Boson peak maximum leads to correlation length that is an indication of the intermediate range order (IRO) in glasses. These correlation lengths or IROs will be on the order of several nanometers in size. Intermediate range order has also been interpreted as a nano-scale inhomogeneity with respect to the anomalous Rayleigh scattering (hysteresis effect in the Landau– Placzek ratio) [19] and here we have postulated that the

region are homogeneous (domains) up to nanometer size extent. A clear difference is shown from the accepted short range order concept, which has governed the behavior of amorphous solids for so long. The observations reported in this work that a typical correlation length or IRO for a glass extends over several nanometers in length gives new insight to understand the basic properties of glasses.

3. Experimental aspects The Raman light scattering studies were done with an argon-ion laser (at 488 nm) and a 0.85 m double grating monochromator (SPEX 1403) with photon counting electronics (resolution 2 cm
1) were employed to collect the low frequency Raman spectra at a 90° scattering angle. For measurements above the ambient temperature, the samples were placed on a glass plate and positioned in a temperature-controlled molybdenum furnace with a temperature stability of ±1 °C. The furnace had three optical windows, which made it possible to collect 90° scattering light. The temperature inside the furnace was monitored with a chromel–alumel thermocouple positioned just beneath the sample holder. The samples and their compositions used for the temperature study are given in Table 1. The pressure dependence of the Boson peaks was obtained from the low frequency Raman intensities employing a diamond-anvil pressure cell (DAPC). In order to improve the signal-to-noise ratio for the weak Raman spectra of the glass samples in the DAPC we used a 45° incident light micro-Raman system instead of the normal 180° backscattering geometry used in high pressure Raman scattering. In the case of 45° lumination, the laser light is reflected by a surface mirror and then focused by an objective lens onto a thin glass sample at 45°. The resulting Raman scattering with a 135° scattering angle is collected by a microscope objective at right angles to the sample surface, and then enters a 0.85 m SPEX double monochromator without passing through a semi-transparent beam splitter which is required for the 180° micro-Raman scattering system. Compared to the back scattering system, the present system has higher collection efficiency and higher spatial resolution as well as a higher signal-to-noise ratio, which has very important advantages for Raman scattering experiments in a DAPC. In order to further improve the signal-to-noise ratio, we chose a low luminescence diamond pair for the DAPC. Within the entire frequency region for Raman scattering the background due to the diamonds was found to be smaller than 50 counts per second at about 100 mW of laser power. The glass samples were polished plates (37 lm thick) and around 200 lm diameter, which were loaded into a gasketed diamond anvil cell along with a 4:1

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Table 1 The composition of the sample glasses used in this work Glass sample

Composition (mass%)

NBS-710 (Tg = 557 °C)

SiO2

K2O

Na2 O

CaO

SbO2

70.5

7.7

8.7

11.6

1.1

Composition (mol%) La2O3(x)

GeO2

Ga2O3(y)

LGG lanthanum-gallo-germanate

33 33 33 33

(100–xLa–yGa)

20 25 33 50

SiO2

CaO

Al2O3

CAS calcium-alumino-silicate

0 0.25 5 10 12.5 17.5 20

methanol–ethanol mixture used as the pressure medium. The pressure was determined by the standard ruby fluorescence technique. Up to 90 kbar the relative shift between R1 and R2 lines is no larger than one wave number, indicating that true hydrostatic pressure was well maintained in the diamond anvil cell. The laser power focused on a typical sample at a 45° angle was about 100 mW for the Raman light scattering experiments in the pressure cell. The Boson peak consists of a Raman line that is located from 20 to 110 cm
1 (0.6–3 THz) from the exciting line thus a double monochromator is required to give acceptable resolution and sufficient contrast to reject the background noise due to parasitic scattering (i.e. Rayleigh scattering, photoluminescence background, etc). To obtain the polarized spectral line (I(x)VV) and the depolarized spectral line (I(x)VH) two Glan-laser prisms (Karl Lambrecht, Inc.) were employed as polarizer and analyzer to give high quality VV and VH polarized spectral lines for the Boson peak measurements. The resolution accuracy achieved in our measurements was about 1–2 cm
1 for the Boson peak position. All samples were polished to optical quality surface finish and in the scattering cross-section selection for every sample (the sample volume that is illuminated by the strongly focused laser light, usually the shape of a small cylinder) all bubbles, nucleation seeds or other imperfections were avoided, assuring that the measured spectra contained the absolute minimum in parasitic scattering.

4. Results Low frequency Raman scattering was employed to look at the existence of intermediate range order

CaO/Al2O3 = 1.5

(IRO) on the ubiquitous
Boson peak. Fig. 1(a) and (b) show the VV and VH Raman spectra for the NBS710 oxide glass. Three dominant peaks are immediately evident: the one closest to the exciting frequency is the Boson peak, for NBS-710 centered around 70 cm
1 with a slight temperature effect visible in the figures. The other peaks at 550 cm
1 and 1090 cm
1 are highly polarized and not involved with this Boson peak study. Consequently we shall only deal with the low frequency Raman line. This low frequency (70 cm
1) vibrational line representing the Boson peak was analyzed from the measured IVV(x) and IVH(x) values. The calciumalumino-silicate glass (CAS) and lanthanum-gallo-germanate glass (LGG) also exhibited strong Boson peaks at frequency of (80 cm
1) and (110 cm
1), respectively. The typical spectrum is not given for the CAS and LGG samples in Figs. 2 and 3. Fig. 2 shows the Raman spectra for the LGG-glass as a function of temperature. The Raman line of interest that represents the Boson peak is at a frequency around 110 cm
1. This Raman line does show a slight temperature effect in that the Boson peak maximum shifts to lower frequency values with increasing temperatures. Fig. 3 gives the Raman intensity as a function of frequency and the amount of silica added to the CAS (calcium-alumino-silicate) glass. The Boson peak has a maximum at 64 cm
1 rises to a maximum of 75 cm
1 and then decreases to 68 cm
1 as the amount of silica starts at 0% and goes to a maximum of 20%. In Fig. 4 the Boson peak maximum intensity frequency is plotted as a function of temperature for three different specified values of silica in the CAS glass. All three compositions exhibit a decrease in the frequency until the glass transition temperature (Tg) is reached and then each composition shows a characteristic local peak. Some pressure data for the Boson peak for pure SiO2 and GeO2 glass

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Fig. 1. (a) Temperature dependence of polarized (VV) Raman scattering from NBS-710 glass during a heating run. The doted line shows the changes in the frequency of the Boson peak at maximum intensity with temperature. (b) Temperature dependence of depolarized (VH) Raman scattering of NBS-710 glass during a heating run. The dotted line passing through the Boson peaks shows the changes in the frequency of these Boson peaks with temperature.

Fig. 2. Raman spectra for lanthanum-gallo-germanate glass with emphasis on the Boson peak and temperature. The vertical line show the Boson peak intensity frequency maximum at 20 °C.

Fig. 3. Raman spectra for the calcium-alumino-silicate glass system as a function of mol% SiO2 in the glass. The numbers above the Boson peak are the individual frequency values as a function of the amount of SiO2 in the glass.

is given in Fig. 5. We compare our data with Hemley et al. [44] for SiO2 and Ishihara et al. [45] for GeO2. The Hemley results are done by in situ measurements, whereas the Ishihara results are for quenched and densified GeO2 glasses. Our results were done in a diamondanvil pressure cell with optical access as an in situ study. We agree with the Hemley data and differ at the higher pressure with Ishihara. The composition dependence for CAS and LGG glasses of the Boson peak frequency is portrayed in Table 2. In the CAS glass the mol%

of SiO2 is varied. We calculate the correlation length (Ltc ) from the Boson peak frequency and the high frequency transverse sound velocity. The LGG glass is shown in Table 2 with the Boson peak frequency versus the amount of Ga2O3 in the glass. The corresponding correlation length (Ltc ) is given in units of nanometer. Tables 3 and 4 show the Boson peak frequencies for various glasses used in this study other than CAS or LGG. Calculated values of the correlation lengths for

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93

Fig. 4. The maximum intensity Boson peak frequency for CaO Æ Al2O3 Æ ySiO2 glasses as a function of temperature and the amount of SiO2 in the glass.

Fig. 5. Boson peak frequency data as a function of pressure for SiO2 and GeO2 glasses. The closed points represent our data, while the open squares and circles are from Hemley et al. [44] for SiO2 and Ishihara et al. [45] for GeO2, respectively.

these glasses are also given. In Table 4 we compare some of our correlation lengths obtained from Boson peak

data with V AG , the cube root of a correlation volume obtained from Rayleigh–Brillouin measurements [19].

1=3

Table 2 Measured Boson peak frequencies, transverse sound velocities and calculated correlation Lengths for the CAS and LGG glass systems Sample

SiO2 (mol%)

xBP (cm
1)

V ts (m/s)

Ltc  V ts =xBP (nm)

CAS calcium-alumino-silicate

0 5 10 15 20

64 71 75 70 68

3861 3834 3810 3789 3743

2.0 1.8 1.7 1.80 1.83

(y)Ga2O3 20 25 33 50

xBP (cm
1) 68.6 67.8 67.9 69.0

V ts (m/s) 2668 2664 2705 2276

Ltc (nm) 1.28 1.30 1.31 1.32

LGG lanthanum-gallo-germanate h33La2O3-GeO2-yGa2O3

Table 3 Calculation of correlation lengths for various glasses from Boson peak data
1 xmax BP (cm ) (Boson peak maximum)

vts (m/s) (Transverse sound velocity)

vls (m/s) (Longitudinal sound velocity)

ltc  vts =xBP (nm) (Correlation length)

SiO2: 1 bar 16.8 kbar

49 69

3748 (a) 3519 (b)

5944 (a) 5259 (b)

2.55 1.77

GeO2: 6 kbar 41.1 kbar

50 79

2400 (c) 2180 (c)

3400 (c) 3600 (c)

1.60 0.92

B2O3: 1 bar 41.1 kbar

30 74

1803 (a) 3017 (c)

3376 (a) 5650 (c)

2.00 1.36

NBS-710a ZBLAa ZBLAN20a HBLAN20a

70 48 50 45

3303 2333 2384 2530

5760 3968 4270 4035

1.56 1.62 1.59 1.74

Sample

(d) (e) (e) (e)

The data of (a), (b), (c), (d), and (e) are taken from Ref. [52–56], respectively. a NBS-710 and the three halide glasses were measured at atmospheric pressure.

(d) (e) (e) (e)

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Table 4 Comparison of calculated correlation lengths from Boson peak data and thermodynamic approaches with Rayleigh–Brillouin data [19,50] at 1 atmosphere pressurea 1=3

Sample

ltc (nm)

V AG (nm)

Glass type

SiO2 GeO2a(6 kbar) B2O3 NBS-710 ZBLA ZBLAN20 HBLAN20

2.55 1.60 2.00 1.56 1.62 1.59 1.74

– – 2.30 2.53 3.89 3.68 –

Strong glass former Strong glass former Strong glass former Strong glass former Fragile glass former Less fragile glass former Less fragile glass former

a

Note: the GeO2 values are for pressures of 6 kbars, only.

5. Discussion The main part of vibrational excitations in amorphous solids has no well-defined wave vector because of the structural disorder. As a result, a conservation of momentum between the excitation and the phonon is no longer a restrictive selection rule for the process of Raman scattering and all vibrational modes contribute to the Raman spectra. Malinovsky and Sokolov [27] discovered the universal shape of the Boson peak (lowfrequency Raman scattering peak) in quite a broad range of amorphous materials and the weakness of the phonon model is supported by their results; they showed that amorphous germanium could not be fitted by the accepted phonon model. A number of attempts have been made to interpret the Boson peak [35,27,46–48]. Duval et al. [47] have suggested a model for the Boson peak in which they assume a random non-continuous structure of glasses. According to their assumption, the change in the density of vibrational states (DVS), which causes Raman scattering in the vicinity of the Boson peak, originates from the vibration of what Duval et al. call
blobs or we may call these domains with ill-defined boundaries. They have interpreted inelastic neutron scattering and Raman scattering Boson peak results for inorganic glasses and polymers as indicating discontinuities in the glass structure corresponding to regions (or blobs) of 2–5 nm in size [47,48]. In this sense the
blobs are somewhat analogous to the micro-crystallites except that the
blobs (domains) are amorphous. In typical glasses periodicity is violated after 10–20 molecular diameters comprising the coordination sphere and this gives us the concept of intermediate range order. The Boson peak is attributed to the cooperative vibrations of these micro-domains of nanometer extent. Malinovsky and Sokolov [27] and Sokolov et al. [49] have estimated for them from measurements to be in the range of about 1.5–3.0 nm for structural density fluctuations responsible for the low frequency Boson peak in the Raman spectra of polymers and various oxide glasses, including B2O3. The detection of a low frequency Boson peak in the Raman scattering in glasses may correlate with the anomalous behavior found in

the Rayleigh and Brillouin light scattering measurements of certain glasses in the glass transition region [19,50]. The Shuker–Gammon theory demands that the only way Boson peaks are possible is if phonon localization exists. Using the cluster model and the subsequent Ioffe–Regel criteria for phonon localization gives direct evidence from the Boson peak measurements that any glass has intermediate range order or heterogeneity that is of nano-scale dimensionality. The extent of the correlation lengths in glasses were measured and shown to be in the nanometer range order. The clustering and subsequent domain formation assures us that the vibrational properties of the system (phonon modes) now behave in a cooperative sense resulting in excess density of states and allowing for the existence of excess phonons in the system. The observed Raman spectra for the NBS-710, the CAS and the LGG glasses are given in Figs. 1–3. We also have determined the Boson peak locations for SiO2, GeO2 and B2O3 at room temperature and at some selected pressures; in addition the Boson peak frequencies for some halide glasses are given in Table 2. Clearly we see what has been defined as the
Boson peak from the Raman spectra of these glasses. The maximum peak frequency values for the Boson peaks are given in Tables 2 and 3. The peak location changes from 30 cm
1 to 110 cm
1 at atmospheric pressure. The existence of this Boson peak can be reconciled if one considers the Shuker–Gammon model for Raman scattering in glass. The most important factor is the density of states, gb(x), which contributes most of the spectral shape of the Raman lines in a glass since C(x), the light vibration coupling constant seems to be given by C(x) / x at least in the region of the Boson peak. Thus, it is obvious that the occurrence of the
Boson peak, a universal property of all glasses, demands the existence of an excess over the normal Debye values in the vibrational density of states. The excess density of states comes from vibrational excitations of strongly scattered phonons caused by spatial inhomogeneities in the elastic properties, which are in turn caused by the structural disorder in glasses [46,51,39]. If the scattering is sufficiently strong, the mean free path of phonons reaches the Ioffe–Regel limit, i.e., it decreases to their wavelength and the Ioffe–Regel criterion [41,42,57] for phonon localization, kIR(x) Æ lIR(x)  1, is fulfilled. Namely, the phonons are essentially localized by strong scattering. Note that this is analogous to Anderson localization of electrons [58]. In non-crystalline materials there are two possibilities [58]. One is that the mean free path, l, is large, so that kl  1. In this case the wave vector k is then still a good quantum number and since the mean free path is large, the deviation of the density of states from the free-electron form must be small. But if in a liquid or amorphous material the atomic potential (or pseudopotential) is

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strong enough to produce a band gap or any large deviation from the free electron form, then it must give strong scattering and short mean free path, which gives k l 1. Under such conditions the k-selection rule breaks down in optical transitions. If we now assume that the Boson peak frequency be such that xBP  xIR then the Boson peak will allow us to derive information about the spatial extents or correlation lengths of inhomogeneties in glasses. Quitmann et al. [39] show that the spatial range of elastic inhomogeneities in glasses can be given by: lc = S(vs/xBP) with S 6 1 being a weakly varying factor with the composition of the glass, vs is the sound velocity and lc is the spatial extent of the inhomogeneity. This formulation allows the calculation of typical correlation lengths. Thus, the position of the Boson peak maximum which depends on chemical composition, pressure (density), and thermal history of the sample [59,60] leads to a measure of the intermediate range order in glasses. We may state that finding the correlation length gives the structure of a continuous network which is made up of ordered micro-regions (nano-scale inhomogeneities) of the size lc. The Boson peak is attributed to the vibrations of small layered clusters (domains with ill-definedpsurfaces) and the vibrational frequencies (i.e. ffiffiffiffiffiffiffiffiffiffi x ¼ k=M Þ occur at much lower frequencies than typical vibrational frequencies due to longitudinal and transverse optical phonons. This is due to the large effective mass caused by the cooperative vibrations of individual molecules that exists in the intermediate range order expressed in terms of nano-scale inhomogeneities. Hence, a universal property of all glasses, the Boson peak and the hysteresis effect in temperature dependent Rayleigh scattering [19] gives us the universality of the extensive correlation function in glasses and thus, strong evidence for the existence of intermediate range order in glasses. We have found that the broad spectral features of the Boson peaks are the same for different glasses and are mostly independent of their chemical composition. However, the peak position does change substantially with the type of glasses or density (pressure) (see Tables 2 and 3). The size of ordered micro-regions (nano-scale inhomogeneity or domains), or the structural correlation range can be estimated from the position of the low frequency Boson peak and the sound velocity for each sample determined from the Brillouin shift measurements [27,49,61,62]. Thus, the occurrence of the Boson peak in Raman scattering from glasses permits one to determine the structural correlation length and it is a typical sign of intermediate-range order in glasses [27,49]. These values are given in Tables 2–4. In addition to the oxide glass (NBS-710), three halide glasses, three single component glasses (SiO2, GeO2, and B2O3) and the CAS and LGG glasses were studied by Raman scattering at room temperature and all glasses

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revealed a definite Boson peak and corresponding maximum frequency. The position of the Boson peak maximum represented in terms of xmax BP leads to a measure of the magnitude of the intermediate range order in all the glasses that we measured. The correlation length depends on the chemical composition of the glass and also the pressure (density). For SiO2, ltc is 2.55 nm a maximum value, while for the other glasses ltc is between 1.28 nm and 2.0 nm at atmospheric pressure. Here we used ltc  vts =xmax BP to obtain the correlation lengths. Increased pressure increases the maximum Boson peak frequency and results in shortening of the correlation length. Thus, the correlation length which is mostly independent of temperature up to the glass transition region can be modified with pressure [63–66]. As shown above, the Boson peak measurements ascertain that a certain characteristic correlation length exists in the glass with typical sizes in the nanometer range. Values for SiO2 indicate that this is on the order of 8–10 molecular diameters. Analysis of the low frequency Raman spectra of glass-forming liquids and of rapidly quenched glasses shows [67] that the correlation length for vitreous structure is formed in the supercooled melt and does not depend on the cooling rate. Thus the correlation length is essentially a thermodynamic equilibrium property. This fact is supported from our results. From room temperature through the glass transition region both the Brillouin shift (consequently sound velocity) and the Boson peak frequency slowly decrease with increasing temperature. Also for heating and cooling the Brillouin shift as well as the Boson peak exhibits the same behavior with little difference. This means that xmax BP varies with temperature approximately with the variation of sound velocity, vs, i.e., xmax BP =vs  constant, where T 6 Tg.

6. Conclusions The low frequency broad peak of the Raman spectra, the so-called Boson peak, is observed in glasses. From the Boson peak in Raman scattering, we determine a structural correlation length and, we assume, it is a measure of intermediate range order (IRO) in glasses. These results give an indication that intermediate range order (IRO) exists in glasses and that its typical size extent is about 10–20 molecular diameters or in the nanometer size regime. Non-exponential structural relaxation could also be explained by these nano-scale inhomogeneties. Thus, the Rayleigh scattering [19] points to an intermediate range order of similar to nanometer size scale in glasses and this seems to be a universal property. The homogeneous correlated regions give a mechanism where all phonons in this region vibrate in a cooperative fashion and thus provide the

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excess phonon modes and a possible source of local excitations. Raman scattering results further re-enforce the concept of phonon localization and thereby, with the aid of the Ioffe–Regel criteria allowed us to calculate a typical correlation length from the measured Boson peak energy data. The results indicate correlated regions of nanometer size which agree in magnitude with the Rayleigh scattering findings. For the Raman–Boson peak analysis a number of models were examined, and a cluster model was chosen as a plausible explanation of this domain formation We believe from the results and analysis of our measurements that the existence of intermediate range order has been well established as a ubiquitous entity to exist in all glasses. The basic excitations that are necessary for the excess phonon modes at low temperature can also be identified on the basis of the nano-scale inhomogeneities and the resulting phonon localization. The inherent effect of the phonon localization as an entity to give an identity to the basic excitations, and thus to the tunneling states for glasses, has now been put on a more solid foundation. The agreement between the correlation lengths obtained from our Raman studies on various glass formers when compared to earlier measurements employing Rayleigh–Brillouin scattering on some of the same glasses and the resulting cube roots of their correlation volumes complement each other in confirming the existence of
intermediate range order in glasses.

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Acknowledgment Supported in part by Rensselaer FY 03 Revitalization Seed Funds and one of us, W.W., wishes to thank the Meiners Fellowship Fund for summer support.

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