X-ray diffraction and EXAFS studies of silicate glasses containing Mg, Ca and Ba atoms

JOURNAL OF

ELSEVIER

Journal of Non-Crystalline Solids 211 (1997) 56-63

X-ray diffraction and EXAFS studies of silicate glasses containing Mg, Ca and Ba atoms T. Taniguchi *, M. Okuno, T. Matsumoto Department of Earth Sciences, Faculty of Science, Kanazawa Uniuersity Kakuma-machi, Kanazawa, lshikawa 920-11, Japan Received 24 April 1995; revised 19 September 1996

Abstract Local structures around Si, Mg, Ca and Ba atoms in MgSiO 3, CaSiO 3, and CaBaSi206 glasses were investigated by X-ray diffraction and EXAFS (extended X-ray absorption fine structure) measurements. From these results, the dependence of the size of modifier ions on glass structures is discussed. The Si-O interatomic distances of these three glasses are in the range of 0.162 to 0.164 nm and the coordination numbers of Si atoms are estimated to be about 4. The Ca-O interatomic distances of CaSiO 3 and CaBaSi206 glasses are in the range of 0.242 to 0.249 nm and their coordination numbers of Ca atoms are estimated to be about 6. The M g - O interatomic distances in MgSiO 3 glass are 0.204 nm and almost equal to those (0.206 nm) in CaMgSi206 glass. The Ba-O interatomic distances in CaBaSi206 glass are 0.27 nm. The Si-, M g - and Ca-O interatomic distances and the coordination numbers of Si, Mg and Ca atoms in alkaline earth silicate glasses have almost the same values. However, the Debye-Waller factors for Ca-O pairs and the S i - O - S i angles become small with increasing size of modifier ions. Comparing the local structures of these glasses with those of the corresponding crystals, the M g - O and Ba-O interatomic distances are shorter than those of the corresponding crystals while the Ca-O distances and the coordination numbers of Ca atoms in the glasses are close to those of wollastonite and parawollastonite crystals. These differences of local structures are caused by the difference of space occupied by modifier ions in glasses and crystals. Namely, as compared with Mg and Ba atoms, Ca atoms probably have suitable atomic radius for the occupation of the space of SiO 4 network of M S i O 3 (M: Mg, Ca and Ba) composition glasses.

I. Introduction The knowledge of glass structures is helpful to understanding crystallization and melting processes, because the structures of silicate glasses are similar

* Corresponding author. Present address: National Institute for Research in Inorganic Materials, 1-1, Namiki, Tsukuba, Ibaraki, 305, Japan. Tel.: +81-298 513 351; fax: +81-298 549 060; e-mail: tanig @nirim.go.jp.

in the liquid state [1]. A structural study of glasses with the composition of alkaline earth oxide and silica in the ratio 1:1 may clarify the effect o f alkaline earth metals on glass structure. A change of modifier ions in the crystal with equivalent mixtures of alkaline earth oxide and silica is sensitive to crystal structure. For example, MgSiO 3 (enstatite, c l i n o e n s t a t i t e and p r o t o e n s t a t i t e ) [ 2 - 4 ] and CaMgSi206 (diopside) crystals [5] have chain silicate structures while CaSiO 3 (wollastonite, parawol-

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T. Taniguchi et al. / Journal of Non-Cr).'stalline Solids 211 (1997) 56-63

lastonite and pseudowollastonite [6-8]), Ca 2 BaSi 309 (walstromite [9]) and BaSiO 3 crystals [10] have bending chain and ring silicate structures. Therefore, the variation of these glass structures with different alkaline earth metal is important to understand the effects of these elements on SiO 4 networks. In our previous study [11], we reported the structures of CaMgSi206, CaCoSi206 and CaNiSi206 glasses and found that these glasses have almost the same structures. Moreover, the coordination numbers of modifier ions are small and the interatomic distances of modifier ion-oxygen are small in the glasses as compared with those in the corresponding crystals. These glasses contain similar sized modifier ions (Mg 2+ = 0.080 nm, Co 2 + = 0.083 nm and Ni 2+ = 0.077 nm [12]). Therefore, it is interesting to reveal the effect of alkaline earth modifier ions on glass structure and the structural difference between glasses and crystals with a change in the modifier ionic radii. The accomplishment of these purposes needs the knowledge of the local structures of Mg, Ca and Ba in silicate glasses and the structural differences between these glasses and the corresponding crystals. Previous reports on structural studies of silicate glasses containing alkaline earth metals: the cation-oxygen interatomic distances and coordination numbers of cation in the MgSiO 3 glass and the glasses of the system M g O - S i O 2 were reported by Waseda and Toguri [13], Yin et al. [14] and Harada et al. [15], those in the CaSiO 3 glass by Yin et al. [16] and Eckersley et al. [17,18] and those in the BaSi205 glass by Hasegawa and Yasui [19]. These results were obtained by X-ray diffraction measurements. In the last ten years, the technique of measurement and analysis for EXAFS (extended Xray absorption fine structure) has been developed thanks to the availability of intense X-ray source at synchrotron facilities [20]. For the local structural study of multicomponent glasses, EXAFS measurements are more useful than X-ray diffraction measurements. In this study, we investigate the structures of MgSiO 3, CaSiO 3 and CaBaSi206 glasses by the X-ray diffraction and EXAFS measurements to determine the dependence of the size of their modifier ions on the structures of these glasses and the difference between glass and crystal structures, referenced to the structure of CaMgSi206 glass [11].

2. Experimental method 2.1. Sample preparation

The MgSiO 3, CaSiO 3 and CaBaSi206 glass samples were prepared by melting stoichiometric mixtures of analytic grade reagents of SiO 2, MgO, CaCO 3 and BaCO 3 purchased from Kanto Chemical Company. The mixtures were fused in Pt crucibles with an electric furnace at 1973 K for 1 h and then quenched in water. We tried the synthesis of BaSiO 3 glass but could not make it. The densities of the glasses measured by pycnometry method are shown in Table 1, together with those of CaMgSi206 glass [11] and MgSiO 3 [2-4], CaMgSi206 [5] and CaSiO 3 [6-8] crystals. It is found that the densities of these glasses increase in the order of MgSiO 3, CaMgSi206, CaSiO 3 and CaBaSi206. In other words, the glasses with heavy alkaline earth atoms have greater densities. 2.2. X-ray diffraction measurements

The X-ray scattering intensity data for the glasses were collected with monochromatized Mo K cr radiation using an X-ray powder diffractometer equipped with bent graphite monochromator. The intensity measurements were carried out by the 0 / 2 0 step Table l Densities of glasses and crystals. The values in parentheses are standard deviations Composition Glass(g/crn 3) Crystal(g/cm 3) MgSiO3

2.75(1)

orthoenstatite clinoenstatite protoenstatite

3.21 b 3.22 c 3.01 a

CaMgSi206

2.85(1)a

diopside

3.28 ~

CaSiO3

2.89( 1)

wollastonite 2.91 t parawollastonite 2.92 g pseudowollastonite 2.89 h

CaBaSi206

3.72(1)

Taniguchi et al. [11]. b Sasaki and Takeuchi [2]. c Ohashi and Finger [3]. d Murakami et al. [4]. e Levien and Prewitt [5]. f Buerger and Prewitt [6]. g Hesse [7]. h Yamanaka and Mori [8].

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T. Taniguchi et al. / Journal of Non-Crystalline Solids 211 (1997) 56-63

scanning method with a stepping angle of 0.5 ° in the range 5-150 ° in 2 0. The X-ray source was operating at electric current of 20 mA and accelerated electric voltage of 40 kV. The total counts accumulated at each measured point were not less than 10,000. The experimental intensity data were corrected for polarization factor, absorption factor and Compton scattering by the conventional method [21] and were normalized with Krogh-Moe and Norman's method [22,23]. The Compton scattering factors reported by Hajdu [24] and PNink~s [25] and the atomic scattering factors listed in the International Tables for X-ray Crystallography, Vol. 4 [26] were used for the above calculations.

2.3. EXAFS measurements The X-ray absorption spectra of glass samples were measured in the ranges of 3.70-5.32 keV (Ca K absorption edge = 4.0381 keV) at 300 K. These measurements were carded out by a transmission mode using synchrotron radiation on the EXAFS spectrometer with a Si(111) two-crystal monochromator at BL6B station of Photon Factory of the National Laboratory for High Energy Physics, Japan [27]. This synchrotron source was operating an injected positron current of about 250 mA and an energy of 2.5 GeV. Higher harmonics were rejected by detuning, a little, the parallel alignment of the monochromator. Ion chambers filled with N 2 gas were used. Energy resolutions at Ca K edge were approximately 0.3 eV. In order to consider the absorption of X-ray radiation by air for EXAFS spectra of Ca K absorption edge, measured time took five times as long as usual measured time. The energy calibration was set as 8.98 k eV at the Cu 3d feature from Cu foil 3d near edge feature. The powdered samples were deposited on tape. The values of A/xt for samples are about 2 and those o f / z t are less than 3.8. The spectrum of diopside (CaMgSi206) was also measured for references.

in Fig. 1, together with those for CaMgSi206 glass [11]. On S. i(S) curves, the amplitudes of first peaks increase and second peaks decrease in the order of MgSiO 3, CaMgSi206, CaSiO 3 and CaBaSi206 samples. The radial distribution functions (RDF), D(r) curves were obtained by Fourier transform of S. i(S) over the range of wave number vector, S < 170.5 nm -~, and are shown in Fig. 2. On the D(r) curve of MgSiO 3 glass, the peaks at r = 0.162 nm and 0.204 nm correspond to the S i - O and M g - O pairs, respectively, which were estimated from the sums of the atomic radii [12]. On the D(r) curve of CaSiO 3 glass, the peaks at r = 0 . 1 6 2 nm and 0.244 nm correspond to the S i - O and C a - O pairs, respectively. The peaks at r = 0.162, 0.242 and 0.272 nm on the D(r) curve of CaBaSi206 glass correspond to the Si-O, C a - O and B a - O pairs, respectively. Moreover, the large wave appeared in CaBaSi206 glass over 0.35 nm is due to the superposed B a - B a and B a - C a pair distributions. Other small peaks over 0.4 nm are probably caused by the effects of Fourier transform over wide range because the accuracy of the experiment is low at large S (S > 140 nm-J). The coordination numbers of Si atoms of these 2-

f

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3. Results .

3.1. X-ray diffraction analyses

CaBaSi206 glass

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20 40

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4

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60 80 100 120 140 160 S (nm-')

The normalized X-ray intensities, S. i(S), for MgSiO 3, CaSiO 3 and CaBaSi206 samples are shown

Fig. 1. The S. i(S) curves of MgSiO 3, CaMgSi206 *, CaSiO 3 and CaBaSi206 glasses ( * after Taniguchi et al. (1995) [11]).

T. Taniguchi et al./ Journal of Non-Cr3'stalline Solids 211 (1997) 56-63

59

j

/

/-\

CaMgSi206 glas

~

i ,I f ' ~ i ~ /\

-fl

x

~

f-~

CaSi03 glass

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glass,

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CaBaSi206 glass._-"

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.

.

.

.

.

.

40

50

60

70

80

90

100

.

k ( nrn ') 0.0

0.1

0.2

0.3

0.4

0.5

0.6

r(nm) Fig. 2. The radial distribution functions, D(r), of MgSiO 3, CaMgSi206 *, CaSiO 3 and CaBaSi206 glasses. Dashed curves represent the radial distributions of structureless media with the same average electron density (" after Taniguchi et al. (1995) [ll]).

glasses are estimated to be about 4 from the calculation of the area of the first peak on the D(r) curves.

Fig. 3. The x(k) spectra of Ca K edge for CaSiO 3 and CaBaSi206 glasses with those for CaMgSi206 glass* and diopside* ( * after Taniguchi et al. (1995) [11]).

IF(r)] curves are shifted by about - 0 . 0 6 nm from the real position due to the phase shift. The IF(r)l curves of the glass samples show one predominant peak at r = 0.18 nm. However, these curves have no significant peaks at r > 0.25 nm. We suggest that

5j i

3.2. EXAFSanalyses The normalized X-ray absorption spectra, g ( k ) , of Ca K are shown in Fig. 3. These spectra were obtained by subtraction of the background level using the Victreen fitting technique and normalization by smoothing technique from EXAFS spectra. The x(k) spectra of Ca K for CaMgSi206, CaSiO 3 and CaBaSi206 samples are similar to one another. Since the x(k) spectrum of Ca K for diopside is almost the same shape as those reported in previous studies [28,29], the accuracy of the spectra for our samples is probably satisfactory. The function kZx(k) was Fourier transformed over the ranges 31.5-100.0 k / n m -~ to obtain the magnitude of the Fourier transform, IF(r)l. The IF(r)l curves of Ca K EXAFS for CaSiO 3 and CaBaSi206 samples are shown in Fig. 4, with those for CaMgSi206 glass and diopside (Taniguchi et al. [11 ]). In this figure, the peak positions of these

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r(nm)

Fig. 4. The IF(r)[ curves of Ca K EXAFS for CaSiO3 and CaBaSi206 glasses with those for CaMgSi206 glass* and diopside* (* after Taniguchi et al. (1995) [11]).

60

T. Taniguchi et a l . / Journal of Non-Crystalline Solids 211 (1997) 56-63

only the first-neighbor shells are resolved in these glass samples. These first peaks of IF(r)l curves for the three glasses and diopside samples correspond to the C a - O pairs. The second prominent peak of diopside corresponds to the Ca-Si pairs. The C a - O distances and the coordination numbers of Ca atoms for CaSiO 3 and CaBaSi206 glass samples are determined by the parameter fitting method [30,31 ]. The procedures have been described in our previous paper [11]. The first peaks in the IF(r)l curves were back transformed to produce k2Xca_o(k) curves. A model k2gc,_o(k) function, using parameters derived from diopside crystal data, was fitted to these back transformed k2Xca_o(k) curves. The mean C a - O interatomic distances, coordination numbers of Ca atoms and Debye-Waller factors, o-2 , were varied to get the best fit between the calculated and back transformed k2Xca_o(k). These values are 0.2492 5- 0.0001 nm, 5.6 ___0.1 and 0.00057 + 0.00002 nm 2 for the CaSiO 3 glass sample and 0.2491 + 0.0002 nm, 6.3 + 0.2 and 0.00044 50.00003 nm 2 for the CaBaSi206 glass sample, respectively.

4. Discussion

4.1. Analysis of glass structure The structural data for MgSiO 3, CaSiO 3 and CaBaSi206 samples obtained by X-ray diffraction and EXAFS measurements are summarized in Table 2, together with those of the CaMgSi206 sample. The data for MgSiO 3, CaMgSi206, and CaSiO 3 crystal samples under ambient pressure are also shown. The results of X-ray diffraction and EXAFS measurements of MgSiO 3, CaSiO 3 and CaBaSi206 samples are described as follows. The S i - O interatomic distances of three glasses are in the range of 0.162 to 0.164 nm and the coordination numbers of Si atoms are about 4 in all glasses. These values resemble those in the corresponding crystals and show that the structures of these glasses are based o n S i O 4 tetrahedra. The C a - O interatomic distances and coordination numbers of Ca atoms in CaMgSi206, CaSiO 3 and CaBaSi206 samples are in the range of 0.242 to

0.249 nm and about 6, respectively. In previous studies, the C a - O interatomic distances and the coordination numbers of Ca are about 0.24 nm and about 6 in CaSiO 3 glass [16-18]. These values are similar to those of our data. The M g - O interatomic distances are 0.204 nm in the MgSiO 3 sample. These obtained interatomic distances of M g - O pairs are almost the same as those of previous studies [11,13,14]. In our previous study [11], moreover, we indicated that the M g - O interatomic distances in CaMgSi206 glass are similar to the C o - O and N i - O interatomic distances in CaCoSizO 6 and CaNiSi206 glasses and the coordination numbers of Mg atoms are about 4 as well as those of Co and Ni atoms. Therefore, we suggest that the similar M g - O interatomic distances in MgSiO 3 and CaMgSi206 samples indicate coordination numbers of Mg atoms in MgSiO 3 glass of about 4. The 4-fold coordinated Mg is also supported by previous investigations [ 13-15]. The B a - O interatomic distances (0.272 nm) in CaBaSi206 sample obtained by RDF analysis are approximately equal to the sums (0.271 nm) of atomic radii of Ba (0.144 nm (VI) [12]) and O (0.127 nm (II) [12]). Therefore, the coordination numbers of Ba atoms in the sample are probably close to 6. The B a - O interatomic distances are also reported 0.275 nm in BaSi205 glass [19]. From our results, the Si-, M g - and C a - O pairs in alkaline earth silicate glasses have almost the same distances. Furthermore, the coordination numbers of the Si, Mg and Ca atoms are also almost the same. Namely, on these silicate glasses, Mg and Ca atoms occupy the interstitial space surrounded by four and six oxygen atoms in SiO 4 network, respectively. However, the peaks near 0.31 nm shift to the left side in the order of MgSiO 3, CaMgSi206, CaSiO 3 and CaBaSi206 samples (0.317, 0.316, 0.315 and 0.310 nm). These peaks are probably contributed by the Si-Si pairs mainly, though these peaks are influenced by broad Mg-Mg, Si-Mg and S i - C a pair distributions. Taking account of these Si-Si and S i - O peaks, the S i - O - S i angles are assumed to become narrow in the order of MgSiO 3, CaMgSi206, CaSiO 3 and CaBaSi206 samples. This decrease of linkage angle between SiO 4 tetrahedron is also observed in chain silicate crystals with increasing size

T. Taniguchi et aL / Journal of Non-C~stalline Solids 211 (1997) 56-63

61

Table 2 Comparison of structural parameters for MgSiO3, CaMgSi206, CaSiO3and CaBaSi206 glasses obtained by X-ray diffraction and EXAFS measurements with those for the corresponding crystals Si-O MgSiO3

CaMgSi206

r (nm)

4.1

0.204

crystal ortho a clino b proto ~

0.163 0.163 0.163

4 4 4

0.211 0.211 0.216

0.162

4.1

crystal wo f para g pseudo h CaBaSi 206

Ba-O

0.162

glass X-ray EXAFS

glass X-ray EXAFS

CN

r (nm)

glass X-ray

glass d X-ray EXAFS

r (nm)

Mg-O

CN

crystal diop e CaSiO3

Ca-O

r (nm)

0.245 0.2492(1)

6.1(1)

0.206

0.164

4

0.250

8

0.163

4.2

0.244 0.2492(1 )

5.6(1)

0.163 0.163 0.163

4 4 4

0.239 0.244 0.254

6 6 8

0.164

4.4

0.242 0.2491(2)

6.3(2)

0.208

0.272

The values in parentheses are standard deviations. The name of the mineral is abbreviated as follows: orthoenstatite: ortho; clinoenstatite: clino; protoenstatite: proto; diop: diopside; wollastonite: wo; para: parawollastonite and pseudowollastonite: pseudo. Sasaki and Takeuchi [2]. b Ohashi and Finger [3]. c Murakami et al. [4]. d Taniguchi et al. [11]. e Levien and Prewitt [5]. f Buerger and Prewitt [6]. Hesse [7]. h Yamanaka and Mori [8].

o f alkali and alkaline earth metal atoms [32]. This tendency o f S i - O - S i angles implies that the interstitial space in SiO 4 n e t w o r k b e c o m e smaller with increasing the size o f m o d i f i e r ions. M o r e o v e r , on E X A F S data, the D e b y e - W a l l e r factor for C a - O pairs, which is related to the vibration o f C a - O atomic pairs, has the same t e n d e n c y as the case of Si-O-Si angles. Namely, those values of C a M g S i 2 0 6 [11], C a S i O 3 and C a B a S i 2 0 6 samples are 0.00064, 0.00057 and 0.00044 n m 2, respectively. This reduction o f D e b y e - W a l l e r factor for C a - O

pairs is also due to the s a m e reason as well as the decrease o f S i - O - S i angles with increasing the size o f m o d i f i e r ions. 4.2. C o m p a r i s o n o f glass structures with the corresponding crystal structures C o m p a r i n g local structures o f M g S i O 3, C a S i O 3 and C a B a S i 2 0 6 samples with those o f the corresponding crystals, the interatomic distances differ: the M g - O interatomic distances (0.204 nm) in the

62

T. Taniguchi et aL / Journal of Non-Crystalline Solids 211 (1997) 56-63

MgSiO 3 sample are less than those (0.208-0.216 nm) in the corresponding crystals [2-5]. The short M g - O distances in the sample compared with those in the corresponding crystal are also observed in the CaMgSi206 composition [11]. Moreover, the Ba-O interatomic distances (0.27 nm) in CaBaSi206 glass are also less than those in walstromite (Ca2BaSi309, 0.284 nm [9]) and BaSiO 3 crystal (0.288 nm [10]). On the other hand, though the Ca-O distances in the glasses obtained by RDF and EXAFS analyses have small differences, the values are close to the intermediate value of those in the crystals (Table 2). The coordination numbers ( N = 6) of Ca atoms are also equal to those in wollastonite (CaSiO 3 [6]) and parawollastonite (CaSiO 3 [7]). However, these values are less than those ( N = 8) in diopside (CaMgSi206 [5]), pseudowollastonite (CaSiO 3, [8]) and walstromite [9]. The pseudowollastonite and walstromite are not chain silicate. Therefore, the Ca coordinations in diopside, pseudowollastonite and walstromite differ from those in the samples because of the difference of framework structure and the influence of Mg and Ba atoms. On the other hand, the similarity of local structures around Ca atoms in the glasses and the wollastonite and parawollastonite crystals indicates that SiO4 network provides almost the same space for Ca in both these samples and in CaSiO 3 crystals with SiO 4 chain. In the case of the short M g - O distances in the samples compared with those in the crystals, we explain these distances by the location of Mg atoms apart from the center of space in SiO 4 tetrahedra networks because small Mg atoms are probably unstable in the large space in the glass structures than in the corresponding crystals [2-5]. Namely, the difference of local structures around modifier ions between glasses and crystals is assumed to be caused by the difference of the space around modifier ions formed by SiO4 tetrahedra. A similar tendency was reported on the structures of SrGa2Si208 and BaGa2Si208 feldspar crystals [33]. This work indicated that Ba atoms with large atomic radius is more stable for size than Sr atoms in the Ga2Si208 frameworks. The Sr atoms in SrGa2SizO 8 feldspar locate farther apart from the center of the space in the framework than the Ba atoms in BaGa2Si208 feldspar, so that Sr-O interatomic distances are shorter than the B a - O distances. This case for the

local structures of Sr and Ba in feldspar crystals corresponds to that of Mg and Ca in silicate glasses in this study. However, the shorter B a - O distances in CaBaSi206 glass cannot be explained by the same argument. In the case of crystal, the frame structure of Ba containing silicate crystals differ from those of Mg and Ca containing crystals. Namely, Ca 2BaSi309 crystal (walstromite) consists of three-membered SiO4 rings [9] while MgSiO 3, CaMgSi206 and CaSiO 3 crystals consist of SiO4 chain except pseudowollastonite (CaSiO3). Though BaSiO 3 crystal [10] consists of SiO4 chain, this composition is out of the region of the glass formation of SiO2-BaO join [34]. Therefore, the framework of CaBaSi206 sample is assumed to differ from the other three samples. For understanding this structure in detail, we expect to perform other systematical X-ray studies and MD calculations based on the newly calculated potential energy parameters [35] on the glass structures of CaSiO3-BaSiO 3 join.

5. Conclusions

The main conclusions of this study are: (1) The EXAFS spectra of Ca atoms were obtained reliably for CaSiO 3 and CaBaSi206 glasses. (2) The Si-, Mg- and Ca-O interatomic pairs in alkaline earth silicate glasses are almost the same. The coordination numbers of Si, Mg and Ca atoms are also similar. (3) The S i - O - S i angles decrease and the Debye-Waller factors for Ca-O pairs decrease in the order of MgSiO 3, CaMgSi2Ot, CaSiO 3 and CaBaSi206 glasses. These two trends suggest that the interstitial space in SiO 4 network becomes smaller with increasing the size of modifier ions. (4) Comparing local structures of glasses with those of the corresponding crystals, M g - O and B a - O interatomic distances are less than those of the crystals. However, the local coordination environment of Ca atoms in the glasses is similar to those in the woilastonite and parawollastonite crystals (CaSiO3). This fact means that S i O 4 network provides almost the same space for Ca in both glasses and CaSiO 3 crystals.

T. Taniguchi et al. / Journal of Non-Cr3'stalline Solids 211 (1997) 56-63

Acknowledgements W e are grateful for the experimental opportunity at the Photon Factory, National Laboratory for H i g h E n e r g y Physics, K E K (proposal No. 89-022). Thanks due to Dr. M. N o m u r a and Dr. A. K o y a m a of K E K for a d v i c e on the data collection at the Photon Factory. W e also wish to thank Dr. A. Nukui o f the National Institute for R e s e a r c h in Inorganic Materials for the critical reading o f this manuscript. The c o m p u t a t i o n s were carried out at the C o m p u t e r Center o f K a n a z a w a U n i v e r s i t y ( F A C O M M - 7 6 0 / 2 0 ) . This w o r k was partly supported by G r a n t - i n - A i d for Scientific Research (No. 02453050 and No. 04302023) f r o m the Ministry o f Education, Science and Culture o f Japan.

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