Vibrational spectra, related properties, and structure of inorganic glasses

Journal of Non-Crystalline Solids 253 (1999) 95±118

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Vibrational spectra, related properties, and structure of inorganic glasses Andrei M. E®mov * Vavilov State Optical Institute, 36-1 Babushkina Street, 193171 St. Petersburg, Russia

Abstract Methods for the quantitative analysis of the IR and Raman spectra of various inorganic glasses to determine physically meaningful optical functions and individual band parameters are discussed. Available approaches to the problem of how to describe the mechanism of formation of the vibrational spectra of glasses (such as the quasi-molecular model, central force model and its recent re®nements, and the model of phonon localization regions) are critically discussed. Recent data on band intensities and/or frequencies obtained with these methods for phosphate, borate, and germanate glasses are considered. Trends in the IR and Raman band assignments deduced from the current state of vibrational spectroscopy of glasses are analyzed and structural information obtained by di€erent authors for binary phosphate, borate, and germanate glasses is compared. It is shown that, based on data on individual band parameters obtained, some related optical and dielectric properties of glasses in their transparency range can be calculated and/or modeled. Particular directions that are promising for further development of studies into vibrational spectroscopy of glasses are speci®ed. Ó 1999 Elsevier Science B.V. All rights reserved.

1. Spectral data treatment 1.1. Qualitative and quantitative approaches to the experimental spectra of glasses Current approaches to the problem of interconnection between the vibrational spectra and structures of glasses which are restricted to the consideration of as-recorded IR or Raman spectra are qualitative approaches. The locations, amplitudes, and shapes of spectral features in such spectra are known (see, for example, Refs. [1±3]) to di€er from actual vibrational frequencies, band intensities, and shapes. This discrepancy is due to

* Tel.: +7-812 535 4221; fax: +7-812 560 1022; e-mail: e®[email protected]

(i) the optical functions which can be recorded experimentally and (ii) the fact that as-recorded spectra are determined by not only the material under study but also measurement conditions. Qualitative IR and Raman studies are conducted mostly with respect to new glass types [4±8] for which even approximate estimates of locations and relative amplitudes of principal features can be of interest. For traditional inorganic glass types, on the contrary, characteristics such as the total number of smaller bands within a given range, accurate band frequencies and intensities, and analytic band shapes are mainly of interest now [9,10]. These properties can be obtained only in the context of quantitative analyses (see, for example, Refs. [3,10,11] and literature therein). It is these analyses and data that will be discussed here.

0022-3093/99/$ ± see front matter Ó 1999 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 9 9 ) 0 0 4 0 9 - 3

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As speci®ed in Refs. [10,11], the quantitative analyses involve, as the necessary ®rst stage, the data treatment intended for converting the as-recorded spectra into reduced spectra whose parameters have a physical meaning. Moreover, the vibrational spectra of inorganic glasses (not only oxide but also chalcogenide and ¯uoride glasses) are, as a rule, multiband (see, for example, Ref. [12]), so that band overlapping is always present. Therefore, in a standard case, the complete procedure of the quantitative data treatment should include the deconvolution of multiband spectrum into individual bands. A general scheme for such data treatment and a scheme in more detail related to the IR re¯ection spectra are shown in Figs. 1 and 2, respectively. 1.2. First stage of the quantitative data treatment 1.2.1. IR spectra The formalism of IR spectroscopy is reviewed in many sources [1,3,13±16]. The physically meaningful quantities to be reconstructed from asrecorded spectra are a pair of the optical constants, i.e., either the refractive index, n(x), and absorption index, j(x), or the real, e0 …x† ˆ n2 (x) ÿ j2 (x), and imaginary, e00 …x† ˆ 2n(x) á j(x), parts of the complex dielectric constant, ^e …x† (x being

Fig. 1. A general scheme for the quantitative treatment of experimental vibrational spectra.

Fig. 2. A scheme for the IR re¯ection data treatment in more detail.

the frequency expressed usually in wavenumbers). The absorption index, j(x), is connected to the absorption coecient, a(x), by j(x) ˆ a(x) á k/4p, k being the wavelength. The re¯ected beam is described by the complex amplitude of re¯ection, r^…x† ˆ r…x† exp ‰i/…x†Š, where /(x) is the phase angle. Experimentally, the re¯ectivity, R(x) ˆ [r(x)]2 , can be measured only, so information on the phase angle turns out to be lost. The connection between R(x), /(x), and the optical constants, depending on the incidence angle and direction of incident beam polarization, follows from well-known Fresnel equations described elsewhere (see, for example, Refs. [1,3,13]. For calculating the optical constants from the re¯ectivity spectrum, it is necessary either (i) to ®nd a way for reconstructing, in addition to R(x), also data on the phase angle of re¯ected beam or (ii) to use independent additional information (see below).

A.M. E®mov / Journal of Non-Crystalline Solids 253 (1999) 95±118

When treating the re¯ectivity spectra in the ranges of the fundamental excitations, two methods such as the Kramers±Kronig transform and dispersion analysis are known (see, for example, Refs. [3,10]) to be the most appropriate ones. Two possible ways for conducting data treatment which correspond to the use of these two methods are shown in Fig. 2. Kramers±Kronig transform. The procedure of Kramers±Kronig transform uses the Kramers± Kronig relation for /(x) [3,14,15], Z 1 x ln R …x † dx ; …1† /…x† ˆ ÿ P p x2 ÿ x2 0 where P indicates the Cauchy principal value of the integral and R(x ) is the re¯ectivity. With data on /(x) obtained with Eq. (1), n(x) and j(x) can be calculated through Fresnel equations [1,3,13]. Because Eq. (1) assumes that the integral is over the frequency range from zero to in®nity, some assumptions are necessary about the R(x) function in the ranges for which experimental data are unavailable. Various algorithms di€ering in the presentation of R(x ) at x ® 0 and x ® 1 are known [16±18] to be used (see also Ref. [10] and literature therein). Recent calculations of the spectra of the optical constants with the Kramers±Kronig transform were conducted mostly for borate glasses [17,19,20] and various AgI-containing glasses [21± 23] and also for germanate [24] and ¯uorozirconate glasses [25]. Dispersion analysis: The dispersion analysis, unlike the Kramers±Kronig transform, is based on an analytical model for ^e…x† which serves as additional information. As known [3,10,11,26,27], the procedure of dispersion analysis consists in computer simulation of R…x† spectrum by means of ^e…x† model chosen and Fresnel equations. When the best ®t is obtained, the parameters of the model thus found are assumed to be similar to the actual parameters of spectral bands. Three versions of the dispersion analysis involving di€erent models for ^e…x† are known [10,11] to be used for treating the IR spectra of glasses. In versions ®rst put into practice for crystals by Spitzer and Kleinman [26] and Berreman and

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Unterwald [28], the ^e…x† model from the classical dispersion theory and so-called factorized ^e…x† model, respectively, were used. These models assume Lorentzian band shapes and also are known [10,11] to assume implicitly the occurrence of a translational symmetry lacking in glass structure. Therefore, such models can describe bandwidths only in terms of phonon damping, whereas the greater bandwidths in the spectra of glasses compared to those in the spectra of crystals are mostly considered, either qualitatively [9,16] or quantitatively (see below) to be due to disorder. So, the corresponding versions of the dispersion analysis are not completely valid for glasses. However, there are studies [29±37] in which the IR spectra of (i) chalcogenide, heavy-metal-¯uoride, and silicate glasses and (ii) silicate and aluminosilicate glasses were treated with the classical dispersion analysis [29±35] and the dispersion analysis based on the factorized model [36,37], respectively. As seen from Refs. [10,30±32], the best ®ts obtained with the classical dispersion analysis usually had the less steep band wings than those in the experimental spectra, thus indicating errors in the optical constants. The ^e…x† model ®rst suggested in Refs. [38,39] and discussed in more detail in Refs. [10,11,40±44] involves the convolution of the Lorentzian and Gaussian functions as follows: Z ‡1 J X Sj p ^e …x† ˆ e1 ‡ 2p rj ÿ1 jˆ1 h ÿ
2 . 2 i exp ÿ x ÿ xj 2rj dx: …2†  2 2 x ÿ x ÿ icj x Here x is the variable oscillator frequency, xj is the central frequency for the jth oscillator distribution, and rj is the standard deviation for this distribution. The rest are parameters retaining the same meaning as in the classical model [1,3,10,11,26]. For brevity, I refer to this model as the convolution model. The convolution model is an extension of the classical ^e…x† model to the case of Gaussian band broadening by which an account of the e€ect of structural disorder is taken. The validation of the convolution model, from the viewpoint of its

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physical background, is given in Refs. [10,11, 40,44] and the analysis of errors is given in Refs. [10,43]. The total absolute errors in n…x† and j…x†, for the ranges where j…x† is around unity or greater, were shown to be no more than ‹0.02 and the total relative errors in band frequencies and intensities to be no more than ‹0.8% and ‹10% (for intense bands) or ‹20% (for less intense bands), respectively. In the dispersion analysis, the quality of a ®t is estimated in terms of some error function. The error function used, in particular, in Refs. [10,11,43±46] is as follows: s Z b  2 1 Rmod …x† ÿ R exp …x† dx; …3† Qˆ bÿa a where a and b are the limits of frequency range studied and Rmod …x† and Rexp …x† are the computed and experimental re¯ectivities, respectively. In Refs. [10,39,40], the quality of the best ®ts to the IR re¯ection spectra obtainable with the dispersion analyses using the convolution and classical ^e…x† models was compared. The minimum Qs obtained with the dispersion analysis using the convolution model were shown to be less than the experimental re¯ectivity error. On the contrary, the minimum Qs obtained with the classical dispersion analysis were shown [10,39,40] to be greater than the experimental error by several times because of appreciable deviations of ®ts from the experimental spectra around the re¯ectivity maxima and minima. Thus, the dispersion analysis using the convolution model has, for the spectra of glasses, an unequivocal advantage over the classical dispersion analysis not only in the physical relevance of the band parameters (see below) but also in accuracy. Mathematically, the convolution model is more ¯exible than the classical model, which is due to the use of four free (adjustable) parameters per oscillator instead of three such parameters per oscillator in the classical model [10]. This greater ¯exibility, together with the greater physical relevance, contributes to the above advantage of the convolution model in accuracy over the classical model. However, there is no way to separate reliably the roles of these two factors.

Data obtained with the dispersion analysis using the convolution model were published by (i) me for silicate [10], borate [10,11,44], phosphate [43±45], germanate [10,11,46], and tellurite [10] glasses, (ii) Naiman et al. [47,48] for pure and nitrided SiO2 ®lms, (iii) Kucirkova and Navratil [49] for SiO2 ®lms, (iv) Brendel and Bormann [41] for amorphous Si, SiO, SiN, and aluminum monoxide ®lms, (v) Hobert and Dunken [50] for sodalime and lithium silicate glasses and (vi) Hobert and Seltmann [51] for sol±gel-prepared SiO2 /TiO2 / CdO layers on a soda-lime glass substrate. From my viewpoint, these data show the convolution model to be the most appropriate for analyzing glass spectra. Potentialities of these methods with respect to the ®rst stage of data treatment: When con®ning spectral data treatment to this stage only, the Kramers±Kronig transform can have some advantage over the dispersion analysis [10]. First, as seen from the above, the Kramers±Kronig transform uses more simple computational procedure than that of the dispersion analysis. Second, the underlying Kramers±Kronig relation (1) involves no microscopic physical model for the response of a material to electromagnetic ®elds. As a result, the Kramers±Kronig transform is not connected to any particular band shape, which guarantees the applicability of this transform to any optical spectra irrespective of frequency range and properties of a material under study. Therefore, the Kramers±Kronig transform is an acknowledged method [15±25] for reconstructing the optical constants of solids, including glasses. On the other hand, the dispersion analysis is known [10,40] to have an advantage of insensitivity to some kinds of errors in experimental data that are crucial for the Kramers±Kronig transform. The above results of Refs. [10,40,50] show that the Kramers±Kronig transform and dispersion analysis using the convolution ^e…x† model (Eq. (2)) can provide practically coinciding optical constants. This coincidence is illustrated, for a sodium germanate glass from Ref. [10], by Fig. 3: an agreement between the spectra of the optical constants compared is seen. Other examples are given in Ref. [10] for vitreous silica using the data of di€erent authors and also for a barium borate glass. So, the

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1.3. Second stage of quantitative data treatment

Fig. 3. n…x† and j…x† spectra for 32.5Na2 O á 67.5GeO2 glass obtained with the dispersion analysis based on Eq. (2) and Kramers±Kronig transform. From Ref. [10].

Kramers±Kronig transform and dispersion analysis can be considered, with respect to the ®rst stage of data treatment, to be methods which complement each other. 1.2.2. Raman spectra The main reason for distortion of band shapes in as-recorded Raman intensity spectrum, I…x†, is known [2,52] to be the e€ect of thermal population of the excited states. This e€ect can be eliminated through the calculation of the reduced spectrum, Ired …x†, as follows [52]: Ired …x† ˆ I…x†=‰1 ‡ N …x†Š:

…4†

Here N…x† is the Bose thermal population factor given by N…x† ˆ [exp(hcx/kB T) ÿ 1]ÿ1 , where x is the Raman shift, T is the absolute temperature, c is the light velocity, and h and kB are the Planck and Boltzmann constants, respectively. Changes in the Raman spectra due to thermal population of the excited states are larger at low wavenumbers only and become negligible at wavenumbers more than 400 cmÿ1 . According to Refs. [2,52], the as-recorded Raman intensity is connected to the vibrational density of states through not only the thermal population factor (as in Eq. (4)) but also the 1/x factor, the e€ect of the latter being smaller than that of the former.

This stage consists in calculating the parameters of individual bands in the reduced spectra. When calculating the IR band parameters using, at the ®rst stage, the Kramers±Kronig transform and also the Raman band parameters, the second stage is conducted as depicted in the left-hand side of Fig. 2, i.e., through (i) the deconvolution, with some standard method, of the e00 …x†, j…x†, a…x†, or Ired (x) Raman spectrum into bands, and (ii) the estimation of parameters of bands thus resolved. The necessity to involve an independent method for the deconvolution of multiband e00 …x†, j…x†, or a…x† spectrum reconstructed with the Kramers± Kronig transform is due to the facts that (i) the underlying Kramers±Kronig relation (Eq. (1)) does not assume individual spectral bands and (ii) band overlapping in the multiband spectra of glasses makes it impossible to estimate the parameters of each band directly from the plot of e00 …x†, j…x†, or a…x† versus frequency. Quite the opposite, the dispersion analysis is known [3,10,11,26,40] to combine together the reconstruction of the optical constants and the calculation of band parameters directly from asrecorded re¯ection spectrum. Therefore, when calculating the IR band parameters, two stages of data treatment are reduced, as depicted in the right-hand side of Fig. 2, to a single procedure. 1.3.1. Crucial role of band shapes Any deconvolution method uses a particular band contour (Lorentzian, Gaussian, intermediate, etc. [53]). Because a knowledge of band contour is required, all such methods and, hence, the procedure combining the Kramers±Kronig transform and deconvolution method lack universality. In other words, the applicability of this multi-stage procedure to a spectrum of a material depends on whether the band contour used in the deconvolution method is valid for this material. The relevance of a particular version of the dispersion analysis with respect to a spectrum of a material also depends on whether a particular band contour assumed by the ^e…x† model chosen is valid for this material. Therefore, standard methods for the deconvolution of e00 …x† spectrum and the

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dispersion analysis are equally non-universal and, in this respect, have much in common. 1 Thus, the reliability of data on band parameters, irrespective of whether a standard deconvolution method or the dispersion analysis is used, crucially depends on assumptions about band shapes. Symmetry and asymmetry of maxima for di€erent optical functions: Because di€erent optical functions such as e00 …x†, j…x†, and a…x† are interconnected through the frequency dependent refractive index, n…x†, and/or current frequency, x, it is of importance which optical function is meant when assuming a particular band shape. Quantities that determine the IR and Raman band shapes are known (see Ref. [52]) to be (i) the vibrational density of states, g…x†, and (ii) the matrix element of vibrational transitions, M…x†. Neglecting the frequency dependence of M…x†, Galeener et al. [52] have shown the optical functions that mimic g…x† to be the optical conductivity, xe00 …x†, and the Raman intensity reduced according to Eq. (8) from Ref. [2] (i.e., in terms of Eq. (4), xIred …x†). Therefore, strictly speaking, actual band shapes should be speci®ed as the band shapes in the spectra of xe00 …x† and xIred …x†. The optical conductivity is known [1] to be the only IR optical function whose maximum, according to the classical ^e…x† model, has the strictly symmetric Lorentzian shape. In the mid-IR, bandwidths in glass spectra are small compared to band frequencies. So, when passing from xe00 …x† to e00 …x†, the in¯uence of dividing by x on band shapes is insigni®cant and the e00 …x† maxima are approximately symmetric also. Because the convolution model is an extension of the classical model and Gaussian function entering into Eq. (2) is symmetric, the strict symmetry of xe00 …x† maxima and approximate symmetry of e00 …x† maxima are the case for the convolution model. The quality

1 In particular, when applied to the absorption spectrum only, the dispersion analysis turns out to be similar to other deconvolution methods. In this case, an advantage of the dispersion analysis over standard deconvolution methods is as follows: due to the use of an underlying ^e…x† model, the physical meaning of band parameters is clear (which is not the case for methods using empirical band contours for voluntarily chosen optical functions).

of best ®ts calculated with this model and an agreement of the optical constants thus obtained with those found with the Kramers±Kronig transform (see above) indicate that symmetric e00 …x† maxima are valid for the mid-IR spectra of glasses (the shapes of e00 …x† components being presented in Refs. [10,43,44]). Data on the Raman Ired …x† spectra of glasses (see, for example, Refs. [9,54]) also show that well-resolved higher frequency peaks are symmetric. 2 Thus, band shapes in the xe00 …x† and xIred …x† spectra and also, as the ®rst approximation, in the e00 (x) and Ired …x† spectra should be assumed to be symmetric. At the same time, j…x† and a…x† are interconnected to e00 …x† through n…x† having a great frequency dispersion around band frequencies, which may impose an asymmetry on a…x† and j…x† maxima. Shortcomings of Gaussian band shapes: Among standard deconvolution methods applied in vibrational spectroscopy of glasses, those assuming Gaussian band shapes for the IR [21,23]) and Raman spectra [55±58] are mostly in use. However, the use of Gaussian function results in some problems. The band shape in the xe00 …x† or e00 …x† spectrum una€ected by disorder is known [1,3] to have a Lorentzian contour resulting from the classical [26] and factorized [28] models for ^e…x†. The Lorentzian halfwidths are determined by the damping coecients, cj , which are the reciprocals of the phonon lifetimes in a material [3]. When considering band shapes theoretically, the e€ect of phonon damping on the states of maxima in the density-of-states spectra is often neglected (which is the case for Refs. [2,52]). The use of Gaussian band shapes also implies neglecting the phonon damping because the halfwidths of Gaussian contours are not determined by the phonon lifetimes. In other words, ®tting the xe00 …x† or e00 …x† spectrum with Gaussian band shapes is equivalent to the use of the convolution model under an assumption that cj /rj  1, thus considering the e€ect of band broadening other than the phonon dam-

2 A few greatly asymmetric bands found in the Raman spectra of glasses (see, for example, Ref. [55]) are commonly interpreted as envelopes formed by overlapping of several closely located individual bands with symmetric contours.

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ping to dominate. However, data presented in Refs. [10,39,40] show that, for the mid-IR spectra of glasses, this assumption certainly is not the case: this ratio varies, usually, from about 1:2 to 2:1. The wings of Gaussian contour are steeper than those of Lorentzian [3] and convolution [10] contours with similar halfwidths. Therefore, the 1:2 to 2:1 limits found in Refs. [10,39,40] for cj /rj ratio in experimental IR spectra of glasses indicate that Gaussian band shapes have steeper wings than the wings of bands in the xe00 …x† or e00 …x† spectra of glasses. As to using Gaussian functions for ®tting the a…x† or j…x† spectra, such practice is contradictory to the asymmetry of a…x† or j…x† maxima due to the e€ect of frequency dispersion of the refractive index (see above). Thus, ®tting the a…x† or j…x† spectrum with Gaussian band shapes is not equivalent to the use of the convolution model even under the assumption of cj /rj ratio <1. Thus, accurate models for band shapes should take into account the e€ect of phonon damping on the density of states. Because there is no physically justi®ed function for describing the phonon damping other than the Lorentzian function [10], accurate models for band shapes should include the Lorentzian contribution, which is the case for the convolution model given by Eq. (2). Unlike deconvolution methods involving Gaussian contour only, the Bruker FIT program used in Refs. [17,19,22,24] involves Gaussian or Lorentzian contours and also their linear combination. However, data presented in Ref. [17] for 55Li2 O á 45B2 O3 glass report the fraction of Gaussian function in this linear combination to vary, for the mid-IR, from 82% to 100%, thus showing the role of the Lorentzian function to be negligible. So, these data contradict the above data of Refs. [10,39,40] reporting the comparable e€ects of Gaussian and Lorentzian functions on bandwidths in glass spectra. In general, reasons for this contradiction are not obvious. One of possible reasons can be the fact that the linear combination of Gaussian and Lorentzian functions used in the Bruker FIT program is by no means equivalent to the convolution of these functions in Eq. (2). The problem of estimating the total number of bands in a spectrum: As follows from the above,

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errors of best ®ts in band shapes and intensities can arise when using Gaussian functions for the deconvolution of the IR and Raman glass spectra. Often, it is possible to decrease such errors by introduction of additional Gaussian ÔbandsÕ which are not actual bands in a spectrum. There are studies (such as Ref. [56]) in which the number of Gaussian bands to be resolved from a spectrum is chosen only from considerations of minimizing the error of the best ®t. Therefore, some smaller bands, thus resolved, are artifacts. Fig. 4 illustrates the above with a numerical experiment. The parameters of the 1235 cmÿ1 band in the IR spectrum of 8Na2 O á 92B2 O3 glass [44] obtained with the dispersion analysis (the c/r ratio being close to 2:1) were taken to be the parameters of a single model band (Table 1). Then, three best ®ts to this band were obtained using (i) a single Lorentzian, (ii) a single almost purely Gaussian (negligible ®xed cj of 0.1 cmÿ1 being taken), and (iii) three almost purely Gaussian functions. Fig. 4 curves 2 and 3 show, in the a…x† versus x coordinates, a poor quality of best ®ts obtainable with single Lorentzian and Gaussian functions. As seen, the error function for Gaussian ®t (curve 3) is even greater than that for Lorentzian ®t (curve 2) obtained with the classical e00 …x† model not completely relevant for glasses (see above). The quality of Gaussian ®t can be improved (Fig. 4 curve 5) by

Fig. 4. Results of model experiment showing the inadequacy of Gaussian contours with respect to the mixed a…x† band shapes. For explanations, see Table 1.

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Table 1 Data illustrating the numerical experiment shown in Fig. 4 Band shape under consideration

Band shape characteristics

Curve nos. in Fig. 4

Description

x (cmÿ1 )

De0

c (cmÿ1 )

r (cmÿ1 )

Error function, Q, of Eq. (3) type divided by average a…x† value

1 and 4

mas BO3 absorption band (as ®tted with Eq. (2)) Best ®t with a single Lorentzian function Best ®t with a single Gaussian function Best ®t with three Gaussian functions Gaussian components of the above ®t

1235.0

0.2810

45.3

19.5

±

1235.6

0.3034

65.4

0

11.0 ´ 10ÿ2

1233.2

0.2343

0.1

37.0

19.6 ´ 10ÿ2

2 3 5 6±8

5.4 ´ 10ÿ2 1215.1

0.0652

0.1

40.2

1238.1 1256.7

0.1114 0.0774

0.1 0.1

25.4 67.1

adding two more Gaussian functions, the frequencies of side contours thus formed di€ering from that of the central contour by 21 cmÿ1 (Fig. 4 curves 6 and 8). For such a ®t, the relative magnitude of error function given by Eq. (3) is less, as compared to that for an initial ®t with a single Gaussian function, almost by 4 times (Table 1). However, the side contours have no physical meaning because the band being ®tted is a single band. Unfortunately, there is no independent way for estimating the total number of bands, including the smaller ones, in a spectrum to be analyzed. As follows from above, this shortcoming equally affects the deconvolution of a spectrum by a standard deconvolution method and by the dispersion analysis. As mentioned above, the total number of bands in a spectrum can be overestimated when choosing this number only from considerations of minimizing the error of the best ®t. On the other hand, it is impractical to equate the total number of bands in a spectrum to the minimum number of distinct features such as resolved maxima and welldeveloped shoulders. A reasonable way between these Scylla and Charybdis is the practice of adding extra bands over the total number of distinct features only in cases obeying simultaneously the criteria as follows [10,11,40]: (i) there is independent information (such as the spectra of crystalline phases of the same system) that there should be a band in a certain frequency range and (ii) the ad-

dition of such a band improves the ®t in this range with no measurable decrease in the ®t in other ranges. The validity of these conditions was justi®ed by the studies in Refs. [10,11,40,43±46] conducted with the dispersion analysis; therefore, these conditions should be equally useful for treatment of Raman data. 1.3.2. Potentialities of two methods of the IR data treatment with respect to the calculation of band parameters from the re¯ection spectrum As argued in Ref. [10], the multi-stage procedure for the calculation of IR band parameters depicted in the left-hand side of Fig. 2 has a shortcoming that consists in the accumulation of errors originated from (i) the Kramers±Kronig transform as such and (ii) the methods used for the deconvolution of the total spectra into individual bands. On the contrary, because the dispersion analysis calculates individual band parameters directly from the as-recorded re¯ection spectrum (see the right-hand side of Fig. 2), there is a single kind of error (namely, that inherent in the procedure of the dispersion analysis), so that no accumulation of errors from di€erent sources occurs. This advantage is substantial from the viewpoint of reliability of data on band parameters. Therefore, when aiming to the calculation of band parameters from the re¯ection spectrum, the use of the dispersion analysis (rather than the above multistage procedure involving the

A.M. E®mov / Journal of Non-Crystalline Solids 253 (1999) 95±118

Kramers±Kronig transform and a standard deconvolution method) is preferable [10,11]. Naturally, this conclusion assumes that the accuracy of modeling the experimental band shapes with the band contour used in the dispersion analysis is not worse than that with the band contour used in standard deconvolution method. 2. Appendix. Speci®c features of the IR and Raman spectroscopies of glasses The preparation of glass samples with optical densities in the mid-IR (maximum absorption coecients being 104 cmÿ1 ) measurable with current spectrophotometers causes changes in glass compositions, which makes standard methods of absorption spectroscopy impractical. So, only the methods of the re¯ection spectroscopy are capable of providing reliable quantitative data in this range. There are requirements with regard to the quality of a surface from which the re¯ected beam is recorded. The polishing of samples in a standard way, i.e., using water, can change the re¯ectivity [10,59,60] due to the hydrolysis of the surface layer. To avoid this e€ect, the methods of sample preparation by using water-free organic liquids when polishing [10,11,43,45] and by pressing a liquid of a sample between two copper blocks [17,19,24] are recommended. For the latter method, a change in the surface layer composition due to the volatility of components is possible. Raman spectra of glasses are studied more frequently now than IR spectra, which is due to the experimental advantages of the Raman spectroscopy (such as the generation of a signal in the interior of a sample rather than at its surface). However, a preference given to the Raman spectra makes data on band frequencies incomplete because the Raman and IR spectra are complementary with respect to each other [61]. Also, the Raman spectra do not contain information on n…x† and j…x†; oscillator strengths can be calculated from Raman spectra only in an indirect way (through the splitting of longitudinal and transverse optical modes [62]). Moreover, di€erences in the amplitudes between the largest band and other bands are usually much greater for the Raman

103

spectra of glasses than is the case for their IR spectra. This di€erence is especially so when the largest Raman band is due to the ÔbreathingÕ mode of a ring grouping (such as those in superstructural units [9,63]). Thus, the visual pattern of the Raman spectra often appears simple as compared to that of the IR spectra; as a result, changes in subtle Raman features can easily remain unnoticed. For example, changes in the Raman spectra of Na2 O± GeO2 [24], Na2 O±P2 O5 [64], and K2 O±B2 O3 [65] glasses with composition are observed only for Me2 O content >8 mol%. 3. Vibrational band parameters and glass structures Approaches available [4,66±70] to a correlation between the structures and vibrational spectra of glasses mostly consider bands in the spectra to be due to the vibrations of certain microscopic structural fragments. In general, the expedience of such consideration was justi®ed, as the ®rst approximation [9], by data on spectrum-structure correlations. However, a network may be separated into di€erent types of fragments such as (i) particular types of polyhedra (so-called Qn species), (ii) R±X±R bridges (R being Si, P, etc. and X being O, F, S, etc.) and (RXm )nÿ terminal groups, (iii) superstructural units for example. The problem of reasonable choice of fragments to be separated is interconnected to the problem of changes in a spectrum due to passing from an isolated molecule-like monomer to a polymeric structure or network (which was comprehensively discussed in the spectroscopy of crystals [61,71±73] and organic polymers [74]). Atomic displacements in a structural group entering into a polymeric crystal lattice or glass network necessarily cause atomic displacements in neighboring groups. As a result, the eigenvectors (i.e., the vectors indicating the directions and amplitudes of atomic displacements occurring in the course of the normal vibrations) and selection rules for a polymeric lattice or network di€er from those for the normal modes of an isolated group. In view of these factors, it was understood soon enough (see, for example, Ref. [75]) that no band in the spectrum of a polymeric structure or network can be considered literally to

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be the manifestation of the normal mode of any structural group. Therefore the assignments of the IR and Raman bands to particular groups have sense for the purposes of approximate structural interpretation only [71,72]. Thus, the division of vibrations in a glass network into the vibrations of structural groups is somewhat conventional, so that it is dicult to specify, from general considerations, the most preferable way for separating a network into groups responsible for the origin of particular bands. Due to the lack of translational symmetry in a glass network, it is even more dicult to specify (i) the degree of interaction of vibrations in a network with each other and (ii) the actual dimensions of a region which determines a band. As a result, current approaches to the formation of the vibrational spectra of glasses assume di€erent answers to these questions. Therefore it is necessary to discriminate between these approaches based on whether a particular approach can consistently explain the number, locations, and relative intensities of bands in a glass spectrum. 3.1. Speci®c features of the vibrational spectra of glasses and glass structures The IR and Raman band shapes in glass spectra have substantially greater widths compared to those of crystals (which is common knowledge). The results of quantitative data treatment mentioned above allow for concluding that other principal features of the IR and Raman spectra of typical inorganic glasses are: 1. In spite of greater widths, the shapes of individual bands in the e00 …x† and Raman Ired …x† spectra of glasses remain approximately symmetric (that is, they can be ®tted with symmetric functions ± see above). 2. In frequency ranges of symmetric, ms , and asymmetric, mas , stretching modes of structural groups, the number of the IR bands per each type of structural group can vary from one to three [10,12,43±45] or even more [76] depending on a particular group and glass composition. 3. In the above ranges, the IR spectra of glasses and of corresponding crystals can have an approximate similarity in the locations of princi-

pal bands and in their relative intensities (this by no means excluding the possibility of di€erences). These features are meaningful from the viewpoint of glass structure. According to the dynamical theory of solids [77,78], there is a selection rule by translational symmetry for the vibrational states in an in®nite regular lattice of an ideal crystal. Due to this selection rule, it is only the vibrational states corresponding to the near-zerowavevector phonons that are active in the IR and Raman spectra of such a lattice; hence sharp and, in most cases, resolved maxima in the e00 …x† or I…x† spectra of crystals. If a size of a fragment is limited by discontinuity or if the structure of a material is disordered, the breakdown of the above selection rule occurs, thus potentially activating the entire spectrum of vibrational states including those with non-zero wavevectors [2,74,78]. The transition probabilities for the non-zero-wavevector states decrease with wavevector, the rate of decrease being great at small wavevectors and diminishing at greater wavevectors. These changes in the densityof-states spectrum a€ect on the tops of the e00 …x† or Raman Ired …x† maxima insigni®cantly but cause an asymmetric broadening of one wing of each maximum [74,78]. On the contrary, according to the above ®rst feature of the experimental glass spectra, the wings of broadened maxima in the e00 …x† or Ired …x† spectra are approximately symmetric. Thus, based on this feature, we can consider any models, which assume the crystal-like intermediate-range order in glass structure, to be incapable of explaining the origin of glass spectra. The second and third speci®c features of glass spectra are even more meaningful. Any reasonable approach to the formation of the vibrational spectra of glasses should consistently explain these features. However, as shown below, most approaches available do not meet this requirement. 3.2. Available approaches to the formation of the vibrational spectra of glasses 3.2.1. Approaches involving the normal modes of isolated molecule-like structural groups These approaches were used widely in sixties and seventies (see, for example, Refs. [66,79,80]).

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In particular, Refs. [66,79] described the IR spectrum of vitreous SiO2 and alkali silicate glasses in terms of the normal modes of isolated SiO…4 ÿ m† (Oÿ )m tetrahedra. In Ref. [80], the approach of the normal modes of isolated moleculelike groups was applied to the IR spectra of alkali borate glasses. Lucovsky and Martin [67] assumed that, for chalcogenide glasses such as As2 S3 and As2 Se3 , the ÔintermolecularÕ coupling is weak. Therefore, they interpreted the IR and Raman spectra of arsenic chalcogenides in terms of independent normal modes of the AsS3 or AsSe3 pyramidal ÔmoleculesÕ and also, in addition, of the As±S±As or As±Se±As bridges (which distinguished their paper from the other papers mentioned). In many cases, such approaches allow for approximately interpreting the most principal features of vibrational spectra (such as the locations of the strongest bands) but fail to explain subtle features [9,52,75]. In particular, actual vibrations in the lattices of SiO2 , GeO2 , P2 O5 , and complex silicates, germanates, and phosphates were shown [68,71,72,75] to be poorly traceable to the normal vibrations of monomers such as the RO…4 ÿ m† (Oÿ )m tetrahedra. Due to kinematical reasons and differences in force constants, vibrations predominantly localized at the R±O±R bridges and R±Oÿ , R±(Oÿ )2 , or R±(Oÿ )3 fragments considered to be the di-, tri-, or tetra-atomic terminal groups were shown [71,72] to be better separated at the frequency scale than those of the RO…4 ÿ m† (Oÿ )m monomers. Therefore, it was found preferable to assign bands in the spectra of the above and similar materials to the vibrations of such fragments. For both crystal spectra [71,72,81,82] and glass spectra [4±7,10±12,25,42±46,54,57,58,64,65], the classi®cation of bands in terms of vibrations of the R±O±R bridges and R±(Oÿ )m terminal groups is used widely, whereas the classi®cation in terms of the normal modes of isolated RO…4 ÿ m† (Oÿ )m tetrahedra, from my viewpoint gradually goes out of use. At the same time, the mid-IR spectra of borate glasses remain to be discussed mostly in terms of vibrations of BO3 and/or BO2 Oÿ triangles and (BO4 )ÿ tetrahedra [10,17,19,44,65]. This limitation is due to the fact that the separation of mixed B(III)±O±B(IV) bridge is impractical because of a

105

di€erence in force constants of constituent B(III)± O and B(IV)±O chemical bonds (the boron coordination state being indicated in brackets). However, the assignment of a particular band to the vibration of a BO…3 ÿ n† (Oÿ )n triangle or (BO4 )ÿ tetrahedron should not necessarily assume, in view of these factors, direct analogies with the normal modes of these groups in the isolated state. Notably, the approaches of the normal modes of molecule-like groups assume the increased number of bands in a spectrum to result from the splitting of the doubly or triply degenerated normal modes of these groups due to the lowered symmetry of glass structure (see, for example, Refs. [79,80]). Such assumption, however, does not help in cases (i) when considering the modes of the low-symmetry fragments (such as R±X±R bridges and R±(Xÿ )m terminal groups when m is 1 or 2) which cannot have degenerated modes and (ii) when the number of bands observed is greater than that of the degenerated normal modes (see below). 3.2.2. Central force model Initial version of the model: The initial version of this model [52,68,69] describes the vibrational spectrum of a glass in terms of atomic motions in the R±X±R bridge, calculations being conducted for a Bethe-lattice-type glass network. For the R± X±R bond angle above some limit, an increase in mode coupling and a resultant transition from molecular to phonon-like modes was shown. A single Raman active mode of RX2 glasses was assumed to be the bending mode involving no displacement of the R atom. With a non-central force constant taken into account, three IR active modes of RX2 glasses were assumed to be (i) the rocking mode involving the X atom motion out of the R± X±R plane, (ii) another bending mode involving some displacement of the R atom, and (iii) the mas mode of the R±X±R bridge. When applying this model, Galeener et al. [52,69] and their successors considered the above bending modes to be the ms modes (which, on condition that the consideration is limited by a single R±X±R bridge, made no di€erence). Later, when applying this model to the IR and Raman spectra of heavy-metal-¯uoride glasses [4,5], another Raman active mas mode of the R±X±R bridge involving no displacement of the R

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atom was assumed to occur and also the (RXm )nÿ terminal groups formed by non-bridging X atoms were taken into consideration. In terms of notations frequently used after Almeida [4,5], the Raman active ms and mas modes were denoted by SS and AS, whereas the IR active ms and mas modes were denoted by SS(C) and AS(C), respectively. Thus, in general, the initial version of this model assumed the occurrence of only a single ms and a single mas modes per each kind of structural group. However, such simple assumptions are insucient even for spectra as simple as those of vitreous SiO2 and GeO2 [52,69]. Therefore, the debate went on and other views continued to be developed [10,11,44,70,83,84]. Further re®nements of the model: Lucovsky et al. [70] have re®ned calculations with the central force model and assumed the occurrence of two transverse IR active mas modes of the Ge±O±Ge bridge, thus avoiding the idea of Galeener [52,69] on the IR activity of the longitudinal modes. They concluded that ``the dominant IR band combines outof phase O-atom S motion and Ge motion'' (S standing for ÔstretchingÕ), whereas the second weaker mas mode was interpreted ``in terms of outof phase S motion of near-neighbor O-atoms with little Ge-atom participation''. The same idea was assumed when interpreting the spectrum of vitreous SiO2 . Kirk [83] reinterpreted these considerations in terms of an idea that there are ``(1) an AS1 mode in which adjacent O atoms execute the AS motion in phase with each other and (2) an AS2 mode in which adjacent O atoms execute the AS motion 180o out-of-phase with each other''. However, as was stressed in Ref. [44], there are cases in which the number of bands in the range of the mas modes is three rather than two (or even more [76]). So, the modi®ed versions of the central force model [70,83] also fail to explain the multiband patterns of glass spectra. 3.2.3. Approach of the phonon localization regions This approach was developed, when validating Eq. (2), in Ref. [10] and then discussed in more detail in Ref. [44]. A starting point for this approach is the well-known model calculations by Bell and Dean [85,86] which showed the vibrational excitations in disordered networks to be

phonon-like and substantially localized (the latter conclusion being used also in Ref. [2]). The approach of the phonon localization regions can be speci®ed as follows. (1) The vibrational excitations in disordered networks are phonon-like modes spatially restricted by the boundaries of localization regions for phonons. The dimensions of these regions, for phonons corresponding to the stretching modes, are typically not greater than the correlation radii. Hence there are consequences as follows: (a) phonon localization regions respond to the IR quanta as approximately ordered structural regions (i.e., those within which the e€ect of Ôaccumulation of disorderÕ (see, for example, Ref. [87]) is not yet sucient for making mutual locations of interatomic bonds completely uncorrelated); (b) symmetry conditions within the localization regions determine the regularities of spectrum formation. Therefore, symmetry e€ects known for crystal spectra may a€ect the formation of vibrational spectra of glasses also; (c) as a result, the number of bands observable in the ranges of the ms and mas modes for a particular structural fragment is a€ected by (i) the number of the fragments in the regions of phonon localization and (ii) di€erences in the site symmetries of neighboring fragments in these regions. In Ref. [44], the total e€ect of these factors is discussed in detail, this e€ect being denoted in Refs. [43±46], for brevity, as the multi-site e€ect. (2) Due to small dimensions of localization regions for phonons, there are random variations in the bond angles and/or torsion angles from one virtual localization region to another. This variation results in the frequency distribution of vibrational states with near-zero-wavevectors, which occurs parallel to the known [2,74,78] activation of vibrational states with non-zero wavevectors. Changes in the density-of-states spectrum due to the frequency distribution of states with near-zerowavevectors are much greater than those due to the activation of states with non-zero wavevectors. (3) The frequency distribution of states with near-zero-wavevectors causes the inhomogeneouslike broadening of the IR and Raman bands. As a

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®rst approximation, this inhomogeneous-like broadening can be described with the Gaussian distribution of the near-zero-wavevector phonon intensities, whereas band broadening due to the activation of states with non-zero wavevectors is known [74,78] to be asymmetric. It follows that the inhomogeneous-like broadening of the IR and Raman bands dominates substantially over their asymmetric broadening. Therefore, e00 …x† and Raman Ired …x† band shapes can be approximated with the convolution model for ^e …x† given by Eq. (2). (4) The central frequencies for the distributions of phonon intensities correspond to the phonon frequencies of some regular lattice, the bond and/ or torsion angles in this lattice corresponding to the centers of the bond and/or torsion angle distributions in a glass. If this lattice is close to a lattice of some actual crystal, the central frequencies and relative intensities of individual bands in a glass spectrum turn out to be similar to band frequencies and relative band intensities in a crystal spectrum. Therefore, valid band assignments for the spectra of glasses can be derived, in many cases, from correlations with the spectra of corresponding crystals. This approach imposes no upper limit on the possible total number of the IR and Raman bands in the range of the mas or ms modes. This lack of an upper limit makes it unnecessary to seek for a special reason for the multiband pattern of glass spectra in these ranges and promotes reasonable explanations for the origin of spectral features which are poorly interpretable in terms of other models. It should be noted, at the same time, that this approach, unlike the versions of the central force model, is yet a qualitative idea rather than a strict model and therefore does not allow one to conduct the calculation of the vibrational spectra. 3.2.4. Errors in band assignments: potential sources As follows from the above, the ®rst potential source of dubious band assignments can be the inclusion of unnecessary spectral components for improving the quality of spectrum ®tting with purely or dominantly Gaussian band contours. For example, there is only a negligible di€erence in

107

the shapes of the 700±1000 cmÿ1 j…x† envelopes in the mid-IR spectra of 10Na2 O á 90GeO2 and 10Rb2 O á 90GeO2 glasses (Fig. 5 curves 1 and 4). However, the envelope in the 10Na2 O á 90GeO2 glass spectrum is deconvoluted in Refs. [10,46] into two bands with the convolution contours (Fig. 5 curves 2 and 3), whereas the envelope in the 10Rb2 Oá90GeO2 glass spectrum is deconvoluted in Ref. [24] into three predominantly Gaussian bands (Fig. 5 curves 5 to 7). Therefore, it is not evident whether a negligible di€erence between envelopes 1 and 4 actually requires the addition, to the model spectrum of 10Rb2 O á 90GeO2 glass, of the third band as intense as that (Fig. 5 curve 7) or the third band is an artifact which was added for diminishing the error of Gaussian ®t to the actual (i. e., mixed) band shape. Also, the 1170 cmÿ1 band resolved in Ref. [58] at the high-frequency wing of the principal band in the spectra of low-titania 37.5CaO á xTiO2 á (62.5 ÿ x)SiO2 glasses has no reasonable assignment and may also turn out to be an artifact arising from an attempt to improve a Gaussian ®t. The use, for deconvolution, of Eq. (2) rather than Gaussian function can help to avoid errors of this kind.

Fig. 5. Deconvolution of the 700±1000 cmÿ1 envelopes in the mid-IR j…x† spectra of 10Na2 O á 90GeO2 (1±3) and 10Rb2 O á 90GeO2 (4±7) glasses according to Ref. [10,46] and [24], respectively. (1) and (4) are the total spectra; (2) to (3) and (5)±(7) are individual bands.

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Another potential source of dubious band assignments, as follows from the above, is an intention of an author to ®nd a speci®c reason for the occurrence of a number of the IR or Raman bands that seems to be too great from the viewpoint of a simpli®ed approach used. In this respect, the consideration of spectral data obtained in terms of the approach of the phonon localization regions can allow one to avoid structural conclusions that are not too convincing. Examples of reasonable assignments made based on the approach of the phonon localization regions will be given below. 4. IR and Raman band assignments for some oxide glasses For the last decade, the vibrational spectroscopic studies that included the calculation of the IR and/or Raman band parameters were conducted mostly for studying the structures of phosphate [22,43±46,57], borate [10,17,19,21± 23,44], and, to a lesser extent, germanate glasses [10,11,24,46]. Also, far-IR spectra corresponding to the vibrations of Me‡ cations [17,19,21±24] were investigated. It is impossible, in the framework of a single paper, to consider the results of all the above studies. Therefore, only a few results of maximum interest are reviewed below. 4.1. Mid-IR spectra of phosphate glasses Data on the mid-IR spectra of vitreous metaphosphates [22,45] show the occurrence of three individual IR bands (Fig. 6 and Table 2) under the 990±1190 and 1190±1340 cmÿ1 envelopes each (these envelopes being commonly assigned [81] to the ranges of the ms and mas modes of the (PO2 )ÿ group, respectively). Fig. 6 illustrates the deconvolution of these envelopes for vitreous CaO á P2 O5 [45]. Table 2 presents the IR band frequencies for vitreous CaO á P2 O5 and BaO á P2 O5 and compares these data to those for BaO á P2 O5 crystal [72,88] and vitreous Ag2 OáP2 O5 [22]. The structures of CaO á P2 O5 and BaO á P2 O5 crystals are known [72,89] to contain quasi-in®nite metaphosphate chains lacking symmetry elements. Similar chains were shown (see Refs. [90,91] and

Fig. 6. Deconvolution of the 970±1190 and 1190±1400 cmÿ1 envelopes in the mid-IR e00 …x† spectrum of vitreous CaO á P2 O5 and individual band assignments according to the data of Ref. [45].

literature therein) to occur in the spectra of corresponding glasses also. From such chains, two structural groups only (such as the (PO2 )ÿ groups and P±O±P bridges having low symmetry) can be separated. Therefore, the occurrence of two or even three bands in the ranges of both the mas and ms modes of each group cannot be explained in terms of simple models. So, when trying to interpret the IR spectra of these glasses with the approach of normal modes of molecule-like structural groups or with the central force model, one should admit that, in addition to (PO2 )ÿ groups and P±O±P bridges, some other groups occur in glass structures. In particular, when interpreting the IR spectrum of Ag2 O á P2 O5 glass, Kamitsos et al. [22] assigned the 934 and 996 cmÿ1 bands (similar to those resolved from the spectra of vitreous CaO á P2 O5 and BaO á P2 O5 ) to the mas mode speci®c for the P±O±P bridge in the (P6 O18 )6ÿ and (P3 O9 )3ÿ rings, respectively (Table 2). From the viewpoint of the approach of the phonon localization regions, on the contrary, no additional assumptions are required because the occurrence of multiband pattern shown in Fig. 6 and Table 2 can be non-contradictorily explained in terms of the multi-site e€ect [44]. Proofs for discriminating between these opposite views can be found from the data of Refs.

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109

Table 2 Comparison of band frequencies and assignments for various metaphosphate glasses BaO á P2 O5 and CaO á P2 O5 glasses [45] and BaO á P2 O5 crystal [72,88] ÿ1

xj (cm )

Ag2 O á P2 O5 [22] Assignments

xj (cmÿ1 )

Assignments

ms P±O±P ms P±O±P in rings mas P±O±P mas P±O±P in the (P6 O18 )6ÿ ring mas P±O±P in the (P3 O9 )3ÿ ring mas (PO3 )2ÿ ms (PO2 )ÿ mas (PO2 )ÿ

BaO á P2 O5 glassa

BaO á P2 O5 crystal

CaO á P2 O5 glass

693 ‹ 6 760 ‹ 6 891 ‹ 7 970 ‹ 8

685 775 870 1025

703 ‹ 6 768 ‹ 6 912 ‹ 7 967 ‹ 8

m0 s P±O±P m0 0 s P±O±P m0 as P±O±P m0 0 as P±O±P

670 775 894 934

1021 ‹ 8

1088

1023 ‹ 8

m0 s (PO2 )ÿ

996

1084 ‹ 9 1137 ‹ 9 1236 ‹ 10 1256 ‹ 10 1302 ‹ 10

1095 1158 1260 1298 1308

1089 ‹ 9 1151 ‹ 9 1220 ‹ 10 1268 ‹ 10 1316 ‹ 10

m0 0 s (PO2 )ÿ m0 00 s (PO2 )ÿ m0 as (PO2 )ÿ m0 0 as (PO2 )ÿ m0 00 as (PO2 )ÿ

1060 1122 1235

a Total and random relative errors in band frequencies found with the dispersion analysis are estimated [43] to be 0.8% and 0.5%, respectively (limits indicated in the Table corresponding to the total error).

[72,88]. According to Ref. [72], the experimental IR spectrum of BaO á P2 O5 crystal also contains three IR bands in the 990 to 1190 cmÿ1 range and three such bands in the 1190±1340 cmÿ1 range (Table 2). The calculations of eigenvectors made in Ref. [88] unambiguously assigned these ranges to the ms P±(O- )2 and mas P±(Oÿ )2 vibrations, respectively. The occurrence of two bands in each of the ranges was shown to be due to the above complicated structure of the unit cell and the third band was assumed to be due to Davydov splitting. For vitreous BaO á P2 O5 and CaO á P2 O5 , as seen from Table 2, the numbers of the IR bands in each of the above ranges (and in the ranges of the P±O± P mas and ms modes either) coincide with those for BaO á P2 O5 crystal and the locations of these bands are close to those of the bands in the spectrum of BaO á P2 O5 crystal. 3 Such similarity indicates a similar origin of the IR bands for these glasses and BaO á P2 O5 crystal, thus con®rming the applicability of the approach of the phonon localization regions. By analogy, the same origin of the IR

3 As to the Raman spectra of MOáP2 O5 glasses [92], these spectra are too poor in details for comparison. The Raman spectrum of BaOáP2 O5 crystal (which is not a€ected by Davydov splitting) reveals [72], as should be expected, two bands in the 1060±1160 and 1260±1310 cmÿ1 ranges each.

bands should be expected for Ag2 O á P2 O5 glass, thus making the assumption of mas modes speci®c for the (P6 O18 )6ÿ and (P3 O9 )3ÿ rings [22] unnecessary. Naturally, such ring groupings can occur, in small amounts, depending on the kind of ionic oxide and glass thermal history. However, for validating the presence of these rings in the structures of metaphosphate glasses, proofs other than the occurrence of the 930±970 or 990±1030 cmÿ1 bands should be found because, as shown above, these bands can be due to other reason (namely, to the multi-site e€ect). The structures of vitreous zinc pyrophosphate and mixed alkali zinc pyrophosphate glasses were studied with the Raman [64,93] and IR [43] spectroscopies. A decrease in the intensities of satellites of the principal Raman band with the substitution of Na2 O for ZnO was observed in the spectra of 2ZnO á P2 O5 ±2Na2 O á P2 O5 glasses [93]. Both the Raman [64,93] and IR [43] data con®rm the occurrence of equilibrium between di€erent kinds of phosphate anions. A decrease, with the substitution of Na2 O for ZnO, in the intensities of the IR and Raman bands corresponding to the products of the pyrophosphate anion disproportionation (such as the (PO4 )3ÿ orthophosphate anion and (PO2 )ÿ terminal group) reveals a decrease in the amounts of these products, thus indicating the shift of the equilibrium toward the pyrophosphate

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anion. Moreover, an intensity redistribution was shown [43,44] to occur in a narrow composition range 27±45 mol% Na2 O (the oscillator strength ratio falling from 1.6:1.0 to 1:1) between the 1100 and 1165 cmÿ1 IR bands resolved with the dispersion analysis (these bands being assigned to two components of mas mode of (PO3 )2ÿ terminal group). This intensity redistribution is assumed [44] to indicate variations in the symmetry of the phonon localization regions, which can be due to changes in (i) the number of (PO3 )2ÿ groups in a localization region or (ii) the bond and/or torsion angles.

due to the splitting of the doubly degenerated normal mode of the BO3 triangle. The deconvolution of two bands from the 1360 cmÿ1 envelope makes the total number of bands in the range of the mas modes of the boron±oxygen triangle to be three (Table 3), which has no explanation in terms of simple models. Hence it follows that results presented in Table 3 can be interpreted in terms of the approach of the phonon localization regions only. Namely, three IR bands in the 1200±1450 and 800±1150 cmÿ1 ranges each can be assigned to

4.2. Mid-IR spectra of borate glasses For alkali borate glasses, these spectra were shown [17,44,94] to contain three bands in the ranges of the mas modes of the BO3 triangle and (BO4 )ÿ tetrahedron each (these ranges being known [10,80,95] to be 1200±1450 and 800±1150 cmÿ1 , respectively). Fig. 7 shows the deconvolution of corresponding envelopes for two Na2 O± B2 O3 glasses [44,94]. Table 3 compares these data to the data of Kamitsos et al. [17] for 55Li2 O á 45B2 O3 glass. Individual IR bands in the 800±1150 and 1200± 1450 cmÿ1 ranges each are poorly resolved in asrecorded spectra, so that the envelope around 1360 cmÿ1 , in most cases, was assumed to be a single band. As a result, the band around 1240 and 1360 cmÿ1 envelope were assumed [80,95], to be

Fig. 7. Deconvolution of the 800±1150 and 1150±1500 cmÿ1 envelopes in the mid-IR e00 …x† spectra of 8Na2 O á 92B2 O3 and 35Na2 O á 65B2 O3 glasses [44,94]. (1) and (8) are the total spectra; (2) to (7) and (9) to (14) are individual bands.

Table 3 Comparison of band frequencies for Li2 O±B2 O3 and Na2 O±B2 O3 glasses 55Li2 O á 45B2 O3 glass [17] ÿ1

8Na2 O á 92B2 O3 [44] and 35Na2 O á 65B2 O3 [94] glassesa

xj (cm )

Assignments

xj (cmÿ1 ), for 8Na2 O á 92B2 O3

xj (cmÿ1 ), for 35Na2 O á 65B2 O3

Assignments

871 965 1060 1132 1230 1300 1405 1475

Various (BO4 )ÿ -containing groups Pyroborate group Metaborate chain Orthoborate unit Unspeci®ed Unspeci®ed

855 ‹ 7 939 ‹ 7 1068 ‹ 9 ± 1235 ‹ 10 1365 ‹ 11 1311 ‹ 10 1478 ‹ 12

859 ‹ 7 953 ‹ 7 1057 ‹ 9 1143 ‹ 9 1241 ‹ 10 1316 ‹ 10 1389 ‹ 11 1461 ‹ 12

m0 as (BO4 )ÿ m0 0 as (BO4 )ÿ m0 00 as (BO4 )ÿ ? m0 as BO3 m0 0 as BO3 m0 00 as BO3 Combination mode

a Total and random relative errors in band frequencies found with the dispersion analysis are estimated [43] to be 0.8% and 0.5%, respectively (limits indicated in the Table corresponding to the total error).

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three components of the mas modes of BO3 triangle and of (BO4 )ÿ tetrahedron, respectively. The concentration dependence of the 1200± 1250 cmÿ1 spectral range is complicated, which causes the band assignments available to be diverse. The 1240 cmÿ1 band disappears at the diborate composition [10,80,95], which is why this band is often ascribed to the occurrence of the boroxol ring. With further increase in Me2 O or MeO content of glasses above 37 mol%, a band at 1230 cmÿ1 appears again [10,44,95]. In Ref. [17], the 1230 cmÿ1 band found in the IR spectrum of 55Li2 Oá45B2 O3 glass was assigned to the metaborate chain. In an earlier paper [96], Kamitsos et al. tried to distinguish between the 1250 cmÿ1 band (assigned to boroxol ring, tri-, tetra-, and penta-borate groups) and another band in the 1225±1270 cmÿ1 range (assigned to pyroand ortho-borate units). Thus, a band at practically the same frequency (a decrease in frequency by 10±20 cmÿ1 with alkali content being possible but falling within limits of estimation error) is observed for glasses in a range of compositions; correspondingly, this band is interpreted in literature di€erently depending on glass composition. Therefore, the assignments of a band in the 1230± 1250 cmÿ1 range to the boroxol ring, tri- to pentaborate groups, or meta-borate chain should be understood only as an indication that this band is due to a vibration of either BO3 or BO2 Oÿ triangle. Such stability of the 1240 cmÿ1 band frequency indicates that the corresponding vibration is practically insensitive to a nonbridging oxygen in a boron±oxygen triangle. In other words, it is reasonable to consider the 1240 cmÿ1 band as having a similar origin in the IR spectra of any borate glasses (i.e., as the band due to the lowfrequency component of the mas vibration of the boron±oxygen triangles) irrespective of whether the BO3 or BO2 Oÿ triangle is involved. The Raman spectra of boron oxide and borate glasses contain intense bands, which are known [9,63,97,98] to be due to the totally symmetric ÔbreathingÕ modes of the boroxol ring and other rings forming superstructural units (such rings being the only kind of extended structural motifs detectable with vibrational spectroscopy). In particular, the Raman bands around 806 cmÿ1 (for

111

the boroxol ring ) and 780 cmÿ1 (for rings including the (BO4 )ÿ tetrahedra ) indicate the presence of these rings in borate glass structures. Notably, the ring modes in the Raman spectra cannot be described in terms of the central force model or other simple models but are quite natural in terms of the approach of the phonon localization regions. 4.3. Mid-IR spectra of alkali germanate glasses For the last decade, the IR spectra of alkali germanate glasses were investigated in more detail than in earlier studies. With respect to low-alkali germanate glasses, the mid-IR spectra were shown to be rather insensitive to the kind of alkali, some di€erences in band shapes appearing with an increase in Me2 O content. Fig. 5 compares the 650± 1050 cmÿ1 sections of the total a…x† spectra for two Na2 O±GeO2 [10,46] and Rb2 O±GeO2 [24] glasses derived with di€erent techniques. The band frequency variations with glass composition in Rb2 O±GeO2 and Na2 O±GeO2 systems according to data of the above sources are compared in Fig. 8.

Fig. 8. xj versus glass composition plots for bands corresponding to two kinds of structural groups in Na2 O±GeO2 (1,2) and Rb2 O±GeO2 (3, 4) glasses according to the data of Ref. [10,46] and [24], respectively. (1) and (3) are the bands assigned to the mas mode of the Ge±O±Ge bridge and (2) and (4) are the bands assigned to the stretching mode of the Ge±Oÿ terminal group.

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The speci®c feature of the structures of alkali germanate glasses is known [10,24,99,100] to be the formation of a limited amount (no more than 25% of the total number) of sixfold-coordinated Ge atoms. However, early IR spectroscopic studies of alkali germanate glasses [101,102] revealed no bands due to the vibrations involving the (GeO6 )2ÿ octahedra. For the last decade, it was con®rmed [10,24,46] that neither the Ge(IV)±O±Ge(VI) nor Ge(VI)±O±Ge(VI) bridges manifest themselves in distinct individual bands. Therefore, the spectra of germanate glasses were interpreted, in most cases, in terms of vibrations of two structural fragments only such as (i) the Ge±O±Ge bridge and (ii) the Ge-Oÿ terminal group. In Refs. [24,99], the vibrations of the Ge(IV)-O±Ge(VI) and Ge(VI)±O± Ge(VI) bridges were assumed to contribute to some smaller bands. As shown in Fig. 5, the 650±1050 cmÿ1 sections of the total a…x† spectra for 10Na2 O á 90GeO2 [10,46] and 10Rb2 O á 90GeO2 [24] glasses (curves 1 and 4, respectively) agree. Moreover, in spite of some di€erences in band shapes which appear with an increase in Me2 O [46], the band frequency variations with glass composition in Na2 O±GeO2 and Rb2 O±GeO2 systems shown in Fig. 8 are unmistakably similar. Thus, Figs. 5 and 8con®rm that the conclusions made in Refs. [10,11,46] on the spectrum-structure correlations for Na2 O± GeO2 glasses are valid for Rb2 O±GeO2 glasses also. These conclusions discussed in more detail in Ref. [46] assume that: (i) there is a strong IR band corresponding to the same mas mode of the Ge±O±Ge bridge whose frequency decreases, with an increase in modi®er content, from about 900 to 750 cmÿ1 for Na2 O±GeO2 or from 910 to 770 cmÿ1 for Rb2 O±GeO2 (curves 1 and 2 in Fig. 8), thus obeying the single-mode concentration behavior in both systems; (ii) the IR band in the spectra of high-alkali glasses at 820±830 (for Na2 O±GeO2 ) or 830± 855 cmÿ1 (for Rb2 O±GeO2 ) is due to the stretching mode of the Ge±Oÿ terminal group. However, conclusions made by Kamitsos et al. [24] are entirely unlike: the origin of bands around 770 and 840 cmÿ1 in the IR spectra of high-alkali Rb2 O±GeO2 glasses is considered to be opposite to

the above. This assumption is due to the fact that Kamitsos et al. [24] assumed the Ge±Oÿ group to exhibit both the ms Raman active mode and mas IR active mode di€ering in frequencies (850±870 and 760±790 cmÿ1 , respectively). In terms of classi®cation based on the vibrations of bridges and terminal groups (which seems to be most probable in Ref. [24]), groups of the Ge±Oÿ type are considered as diatomic (see above). So, no ms and mas mode pair can be ascribed to this group; a single stretching mode of the group should be considered to be both IR and Raman active [103], thus resulting in similar IR and Raman band frequencies. Data presented in Ref. [24] con®rm that the 850± 870 cmÿ1 band in the Raman spectra of high-alkali Rb2 O±GeO2 glasses is due to the stretching mode of the Ge±Oÿ group. So, this mode should also manifest itself in the 830±855 cmÿ1 IR band rather than the 760±790 cmÿ1 band. Thus, the 760±790 cmÿ1 IR band should be due to the mas mode of the Ge±O±Ge bridge similar to the case of Na2 O± GeO2 glasses. In terms of classi®cation based on the normal modes of Qn species (which seems to be less probable in Ref. [24] though possible), abbreviations such as Ôms (Ge±Oÿ ) vibrationÕ and Ômas (Ge± Oÿ ) vibrationÕ should correspond to the m1 (A1 type) symmetric mode of the GeO3 Oÿ tetrahedron and to the A1 type vibration due to splitting the asymmetric m3 (F2 type) mode of initial GeO4 tetrahedron, respectively. A trend in mode frequencies assumed in Ref. [24] contradicts known data [104]. As a rule, the trend m1 < m3 is held for XY4 tetrahedra (which is the case for Si(Oÿ )4 ion [71,104]). Under the reduction of tetrahedron symmetry from Td to C3v , the tendency of the smaller m1 mode frequency compared to the frequencies of modes due to splitting the m3 mode does not alter [104]. Therefore, the assumption that the frequencies of the ms (Ge±Oÿ ) and mas (Ge± Oÿ ) modes of the GeO3 Oÿ tetrahedron can be 860 and 770 cmÿ1 , respectively, is improbable. Thus, the assignments of the 830±855 and 760±790 cmÿ1 bands made in Ref. [24] are inappropriate irrespective of which classi®cation of bands is used. Based on a similarity between the IR spectra of alkali-rich silicate and germanate glasses, it was

A.M. E®mov / Journal of Non-Crystalline Solids 253 (1999) 95±118

113

Table 4 Experimental and calculated IR band frequencies for Na2 O á 2SiO2 and BaO á 2SiO2 crystals according to the data of Ref. [82] Na2 O á 2SiO2 Calculated for the [Si4 O10 ]1 layera

Experiment

a

Raman band

IR re¯ection

location (cmÿ1 )

Beam polarization

xTO (cmÿ1 )

De0

± 1079 1018 ± 970 771 746

x z x x z z x

1123 1084 1012 973 968 773 740

0.0006 0.004 0.44 0.05 0.50 0.06 0.09

Symmetry

xTO (cmÿ1 )

Assignments

B1 A1 B1 B1 A1 A1 B1

1128 1084 1035 969 964 754 720

mas Si±O±Si m Si±Oÿ m Si±Oÿ mas Si±O±Si mas Si±O±Si ms Si±O±Si ms Si±O±Si

Calculated data for bands that are too weak for being revealed experimentally are omitted.

concluded in Refs. [10,11,42] that a similar trend in vibrational frequencies, mas Si±O±Si < m Si±Oÿ , should hold for Na2 O±SiO2 glasses also. This conclusion (which rejects a popular view that the trend is opposite, m Si±Oÿ
spectra of oxide glasses remained to be studied [19,21±24]. The band frequencies range from 400 cmÿ1 for Li2 O-containing glasses to 85 cmÿ1 for Cs2 O-containing glasses. In Refs. [19,21± 24], two to three (or even, in the presence of AgI, four [23]) individual bands under the far-IR envelope were resolved. Such far-IR vibrations are known [105] to be the cage-like vibrations of cations in their oxygen (or like) cages, thus being the analogues of external lattice vibrations in the IR spectra of crystalline complex oxides [71,72]. This similarity was con®rmed (see, for example, Ref. [19]) by the observations of proportionality between the band frequencies and inverse square roots of ionic masses. According to Refs. [19,21± 24], individual bands resolvable under the far-IR envelope are due either to di€erent coordination states of a cation or, for the case of AgI-containing glasses [22], to di€erences in the chemical properties of ligands (iodine and oxygen). Now, the debate goes on concerning particular coordination states of a cation corresponding to particular bands.

4.4. Far-IR vibrations in glasses

5.1. Calculations of other optical and dielectric properties

Far-IR active vibrations were shown [105] to result in bands in the spectra of binary or more complex oxide glasses. For the last decade, far-IR

5. Applications of data on band parameters

Data on the IR band parameters allow for calculating optical and dielectric properties of

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glasses in frequency ranges other than those investigated spectroscopically. Using data on the IR band parameters of glasses, the real dielectric constant in the ultra radio frequency (UHF) range 10 000 MHZ, and some optical properties in the visible were estimated [10,11]. The ÔstaticÕ dielectric constant, e0 (i.e., the contribution, to the dielectric constant at zero frequency, from elastic displacements of particles in a material), was calculated using the frequencies and intensities of all the IR fundamentals. This calculation was conducted with Eq. (2) through letting the current frequency go to zero and neglecting the oscillator distributions. Table 5 compares the e0 s thus determined for some glasses [10,11] with their dielectric constants measured experimentally in the UHF range. The errors of e0 calculation cannot be accurately analyzed but can be estimated from known error estimates for the dispersion analysis [10,43]. The errors of the best ®ts to a multiband spectrum are by an order less than the random errors in the oscillator strengths for particular bands. This difference is due to the fact that, in the course of the dispersion analysis, errors in the parameters of neighboring bands in a spectrum tend to compensate each other [10]. Because the calculation of e0 involves the summation of contributions of all bands in a spectrum, the compensation e€ect should occur also, thus causing errors of e0 to be approximately of the same order as those of the best ®ts. Data presented in Table 5 con®rm that e0 magnitudes thus calculated are reliable estimates for the real dielectric constants of glasses in the UHF range.

Data on the vacuum ultraviolet (VUV) and IR band parameters were used for calculating the refractive indices of silica glass in the visible accurate to within ‹1 ´ 10ÿ3 [10,11]. The contribution from the IR fundamentals to the real dielectric constant of a material in the visible is known [3] to be small compared to that from the VUV transitions. However, the e€ect of the IR fundamentals on the frequency dispersion of real dielectric constant in the visible is by no means small [10] because, for xj in the IR, the factor 1/ (xj 2 ÿ x2 ) is much greater than that for xj in the VUV. Therefore, the use of the IR band parameters is necessary for reaching the above accuracy of refractive index calculations. There are also other possibilities of using the IR band parameters for calculating glass properties. In particular, the numerical modeling of interrelation between the relative partial dispersions versus Abbe numbers for optical glasses was conducted [10,106,107] based on experimental data on the IR band parameters, thus providing a reasonable interpretation of regularities observed. The in¯uence of impurity ions and radiation-induced structural defects on the refractive index of a glass in the transparency range can be of practical importance for optical systems. The contribution of impurity absorption or induced absorption to the refractive index of a glass can be calculated either through the Kramers±Kronig relation (as made by Volchek et al. [108] for radiation-induced centers) or directly with Eq. (2) using data on the frequencies and intensities of bands due to impurity or defect centers. Water-related absorption bands are

Table 5 Data on the dielectric constant in the UHF range calculated through the IR band parameters. From Ref. [11], with permission Glass composition, mol% by analysis Na2 O

SiO2

± 19.3 23.6

100 80.7 76.4

BaO 36.0

B2 O3 64.0

Na2 O 30

TeO2 70

Calculated `static' dielectric constant

Experimental data for the UHF Range Measured dielectric constant

Frequency (Hz) ´ 1010

3.73 5.84 6.28

3.75 ‹ 0.05 5.6 ‹ 0.1 6.3 ‹ 0.1

0.94±1.0 1.0 1.0

7.53

7.8 ‹ 0.2

3.56

17.9 ‹ 0.4

3.56

17.4

A.M. E®mov / Journal of Non-Crystalline Solids 253 (1999) 95±118

shown [94] to be capable of causing variations in the refractive index of optical glasses in the sixth decimal ®gure. 5.2. Prospects for further development of the vibrational spectroscopic studies into glass structure Subtle di€erences between the structures of the phonon localization regions in glasses and unit cells of corresponding crystals: A resemblance between the vibrational spectra of glasses and corresponding crystals was shown in many studies [10,11,16,84,109]. This resemblance can be combined with (i) the lack, in a glass spectrum, of a counterpart for some less intense band in a crystal spectrum or (ii) the occurrence of an extra band in a glass spectrum. Such di€erences can be due to reasons as follows [10,11,84]: (i) subtle di€erences in symmetry between the phonon localization regions in glasses and unit cells in corresponding crystals and (ii) the occurrence of the low-frequency limit outside which the phonon localization regions become greater than the correlation radii. A convenient way for revealing such spectral di€erences is shown [10,84] to be the numerical modeling of arti®cial IR spectrum of a randomly disordered crystal and comparison of the model spectrum to the IR spectrum of actual glass. Until now, such modeling was conducted for two glass/ crystal pairs such as vitreous SiO2 /a-quartz [10,84] and vitreous/crystalline lithium disilicate [10,11,84]. The promising pairs to be studied are vitreous GeO2 /hexagonal GeO2 crystal, vitreous/ crystalline BaO á 2SiO2 , and AlPO4 glass (if possible)/AlPO4 crystal. The main problem to be solved is how to obtain better quality IR spectra for glass/ crystal pairs of interest. Rapid variations in the structures of the phonon localization regions with glass composition: Large variations in the relative intensities of the IR bands due to the components of the same stretching mode that occur with small changes in glass composition were assumed [44] to be indicative of variations in the symmetry of the phonon localization regions. One example is the above intensity redistribution, in a narrow composition range, between two IR bands in the 2ZnO á P2 O5 ±

115

2Na2 O á P2 O5 glass spectra [43]. Another case [44] is a di€erence in the concentration dependence of the 1240 cmÿ1 component of the mas BO3 mode in the IR spectra of alkali borate glasses and of its 800 cmÿ1 counterpart in the IR spectra of alkali sul®de thioborate glasses (these glasses being found [7] to be the structural analogues of each other). The problem to be solved is to ®nd a way for correlating changes in the intensities of components of the mode under consideration with particular changes in the symmetry of the phonon localization regions. Prospects for glass spectrum calculation: In Ref. [11], a new way to the semi-empirical calculation of the vibrational spectrum of a glass was suggested. This way should start with generating a series of structural and dynamical models corresponding to various random realizations of structures for virtual regions of phonon localization. Then, the calculations of phonon frequencies and intensities should be conducted for all models thus generated using a reliable calculation method (such as CRYME program [110]). Finally, based on the results of these calculations, the band intensity distributions over frequency should be constructed. Such a way is the more promising than methods tested for glass spectra until now. However, as seen from the above, this way is quite work-consuming, which is why the calculations were not yet put into practice.

6. Conclusions (1) In the methodological aspect, the current state of vibrational spectroscopy of inorganic glasses is characterized by speci®c features as follows. (a) There is a distinct tendency towards a transition from qualitative considerations of spectral data to the methods of quantitative data treatment. (b) For the IR re¯ection spectra, the Kramers± Kronig transform and dispersion analysis are equally good for calculating the optical constants and should be considered to complement each other.

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(c) For obtaining band parameters from multiband IR re¯ection spectra, the dispersion analysis has an advantage in accuracy over the combination of the Kramers±Kronig transform and standard method for the absorption spectrum deconvolution. (2) The key problem for calculating individual IR and Raman band parameters is the choice of an appropriate vibrational band shape. The band shape given by the convolution model for the complex dielectric constant is preferable for the IR spectra from the viewpoint of physical relevance. The preferability of this band shape is supported also by experimental evidence for some spectra considered. A procedure for the deconvolution of the Raman spectra using the convolution band shape is not yet developed. (3) The principal speci®c features of the experimental IR and Raman spectra of typical inorganic glasses are as follows: (a) Symmetric band shapes with widths greater than those in the spectra of crystals of similar compositions. (b) Several overlapping bands in the range of the stretching modes. (c) Resemblance between the spectra of glasses and corresponding crystals in the locations of principal bands and in their relative intensities. (4) There are di€erent approaches and conclusions concerning the spectrum-structure correlations and underlying mechanism of spectrum formation. The reasons for this variability are as follows: (a) Approaches which were in use until recently are insucient with respect to the above speci®c features and their potentialities are exhausted. (b) The newer approach of the phonon localization regions intended for overcoming the insuf®ciency of the former approaches is developed only in a qualitative form and is not yet justi®ed by theoretical calculations of glass spectra. The quantitative IR spectroscopy of glasses can be used as an appropriate tool for calculating some related optical and dielectric properties. On the whole, vibrational spectroscopy of glasses has promising potentialities for further development awaiting appropriate e€orts to be put into practice.

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