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Effects of composition on the vibrational properties of sodium silicate glasses
Journal of Non-Crystalline Solids 287 (2001) 231±236
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Eects of composition on the vibrational properties of sodium silicate glasses N. Zotov * Bayerisches Geoinstitut, D-95440 Bayreuth, Germany
Abstract The vibrational modes and the calculated Raman spectra of computer generated models of sodium silicate glasses are presented and discussed. The localization of the vibrational modes on non-bridging oxygens (NBOs) increases linearly with increasing NBO/Si in all spectral ranges. Strongest localization is observed in the low-frequency range near the 225 and 320 cm 1 vibrational bands. The shape of the partial Raman spectra of SiO4 tetrahedra with dierent number of NBOs (Qn species) is complex and depends both on the glass composition and the connectivity of the Qn species. The estimated Qn scattering cross-sections increase with increasing the number of bridging oxygens. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction The structure and the physical properties of the glass-forming system, Na2 OASiO2 , are of basic importance for understanding many commercial as well as natural multicomponent silicate glasses. Correspondingly, Na2 OASiO2 glasses have been studied by various diraction and spectroscopic methods [1,2]. Among them, Raman and infrared (IR) spectroscopy are sensitive probes of the local structure of silicate glasses and many experimental vibrational spectra have been published [3±5]. However, the understanding of the vibrational spectra is still qualitative due to the lack of long-range order in glasses. The most common approach for the interpretation of the vibrational spectra of glasses is the calculation * Present address: Mineralogisches Institut, D-53115 Bonn, Germany. Fax: +49-228 732 770. E-mail address: [email protected] (N. Zotov).
of the vibrational frequencies of characteristic molecular-like units and comparison with experiment [3,6±9]. Calculations of the vibrational density of states (VDOS), the IR and the Raman spectra of large-scale (103 atoms) computer models of silicate glasses have been performed up to now only for amorphous SiO2 (a-SiO2 [10±16]. Accordingly, we have made ®rst calculations of the vibrational properties and the polarized Raman spectra of sodium tetrasilicate glass Na2 O 4SiO2 , denoted NS4) using computer generated models with periodic boundary conditions [17]. In the present paper, we extend this approach to sodium disilicate Na2 O 2SiO2 , denoted NS2) and sodim metasilicate Na2 O SiO2 , denoted NS1) glasses. The results obtained are compared with experimental Raman spectra and previous calculations for NS4 and a-SiO2 [17] to examine the eects of Na2 O content on the vibrational properties over a larger compositional range.
0022-3093/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 1 ) 0 0 5 7 7 - 4
232
N. Zotov / Journal of Non-Crystalline Solids 287 (2001) 231±236
2. Calculation procedures One computer model of NS1 glass and two models of NS2 glass are analysed in the present paper. They have been generated previously by molecular dynamics (MD) [18,19]. The NS1 model (599 atoms) and the ®rst NS2 model (denoted NS1_1), containing 602 atoms, were constructed using a modi®ed Born±Mayer±Huggins potential with a weak Stillinger±Weber three-body term [18]. The second NS2 model (denoted NS2_2), containing 1080 atoms, was constructed using a Vessal-type potential [19]. The MD models are ®rst relaxed with a Kirkwood-type potential (see for details [17]). The force constants are similar to those used for the NS4 model [17]. The atomic vibrations of the relaxed models are then calculated by direct diagonalization of the full dynamical matrix. Each mode in the VDOS is additionally broadened by 38 cm 1 to account, empirically, for the ®nite size of the models, anharmonic eects and disorder not present in the generated models.
The properties of the vibrational modes are investigated by calculating the participation ratio, pc , the correlation length, Lc , and the atomic participation ratios, APR. De®nition of these parameters and discussion of their properties is given elsewhere [17]. The ®rst-order Raman spectra of the structural models is calculated in the bond polarizabiliy ap-
(a)
(b)
(c)
Fig. 1. VDOS for the NS1 (full line), NS2_1 (dotted line) and the a-SiO2 (dashed line) models.
Fig. 2. Participation ratio pc x: (a) NS1; (b) NS2_1; (c) a-SiO2 models.
N. Zotov / Journal of Non-Crystalline Solids 287 (2001) 231±236
proximation [12,17]. The bond polarizability parameters used for NS4 [17] are employed in the present study as well. Each Raman line is additionally broadened by 20 cm 1 to reproduce, approximately, the full-width at half-maximum of the high-frequency band at about 1100 cm 1 in the experimentally measured spectra. The partial Raman spectra of short-range order clusters are calculated by projecting the contribution of the corresponding atoms out of the dynamical matrix.
233
O and Si±non-bridging oxygen (NBO) contributions. The participation ratio, pc x, for the NS1, NS2_1 and the a-SiO2 models is shown in Fig. 2. Most of the localized modes (those for which pc x ! 0 ) are grouped around the band edges. Fig. 3, in which data for a-SiO2 and two NS4 models from [17] are also incorporated, shows that
3. Results The calculated normalized VDOS for the NS1 and the NS2_1 models are compared with that for a-SiO2 [17] in Fig. 1. The calculated VDOS for aSiO2 is in relatively good agreement with experimental VDOS measured by inelastic neutron scattering [20]. Unfortunately, there are no, to the best of our knowledge, published VDOS of sodium silicate glasses. The new vibrational states at about 50, 225, 320 cm 1 and in the range 850±1050 cm 1 as well as the decrease of the intensity of the 800 and 450 cm 1 bands are due to increasing Na±
(a)
(b)
Fig. 3. Dependence of the average participation ratio on the number of NBO/Si: band 200±500 cm 1 (full circles); band 850± 1200 cm 1 (full triangles). The dashed lines are linear ®t through the data points.
(c) Fig. 4. Correlation length Lc x: (a) NS1; (b) NS2_1; (c) a-SiO2 models.
234
N. Zotov / Journal of Non-Crystalline Solids 287 (2001) 231±236
the average participation ratio of the main vibrational bands in the ranges 200±500 cm 1 and 850± 1200 cm 1 decreases linearly with increasing NBO/ Si. The NBO/Si parameter is calculated as NBO=Si R 4 nxn , where xn is the concentration of SiO4 tetrahedra with n bridging oxygens (so-called Qn species). Another measure of mode localization is the correlation length, Lc x: Lc is approximately equal to L/2 for fully delocalized modes, where L is the box edge and approaches zero for the most localized modes. As can be seen from Fig. 4, the number of such localized modes (modes with Lc < 0:5 nm) increases with increasing Na2 O content. The calculated and experimentally measured Raman spectra of the NS2 glass are shown in Fig. 5. The calculated VV polarized spectra reproduce all features in the experimental one. The agreemenent between the calculated and previously published [7] experimental Raman spectra of sodium metasilicate glass is worse, especially in the high-frequency range where only one peak larger than the others at about 985 cm 1 with shoulders at 1060 and 855 cm 1 is experimentally observed compared with four peaks at about 850, 910, 975 and 1060 cm 1 in the calculated spectrum. The discrepancies between calculations and experiment are due to two reasons. The ®rst one is related to the bond polarizability parameters used. The better agreement obtained in the case of a-
SiO2 and NS4 [17], compared to the NS2 and NS1 models, indicates that the bond polarizability parameters of the Si±BO and Si±NBO bonds probably depend on the composition. The second kind is associated with the topology of the computer models and will be analysed in the next section. 4. Discussion The NBO/Si parameters and the Q-species distribution of the investigated models are compared in Table 1 with the results from 29 Si MAS NMR spectroscopy [21]. The concentration of the dominant Q-species (Q2 in the NS1 glass and Q3 in the NS2 glass) is much less in the glass models. The partial Raman spectra (PRS) of the most abundant Qn species in the models (Fig. 6) show that the presence of 100% more Q1 and Q3 species in the NS1 model, as well as Q2 and Q4 species in the NS2 model is the main reason for the discrepancy between the calculated and experimental Raman spectra. The PRS reveal also two other important features of the vibrational dynamics of binary sodium silicate glasses. The Q-species distributions derived from the integral PRS intensities in the range 850± 1200 cm 1 , assuming equal scattering cross-sections, do not agree with the actual Q-species distributions (see Table 1). In other words, the bond polarizability parameters of the dierent Qn spe-
Fig. 5. Comparision between the reduced experimental (full line) and calculated polarized (VV and VH) Raman spectra of the NS2 models: NS2_1 (dashed line); NS2_2 (dashed±dotted line).
N. Zotov / Journal of Non-Crystalline Solids 287 (2001) 231±236
235
Table 1 Q-species distributions (in %) and NBO/Si parameters of the investigated sodium silicate glasses Glass model
Q species a
NMR
Model analysis
Partial Raman spectrab
Estimated cross-sections
NS1
Q0 Q1 Q2 Q3 Q4 NBO/Si
1 13 70 16 ± 1.99
4 28 36 24 8 1.96
n.d.c 40.6 39.2 16.6 3.4 2.168
± 0.69 0.92 1.44 2.35
NS2_1
Q0 Q1 Q2 Q3 Q4 NBO/Si
± ± 10 79 11 0.99
± 2.2 21.6 44.1 32.1 0.94
± n.d.c 40.4 46.8 12.8 1.276
± ± 0.53 0.94 2.50
NS2_2
Q0 Q1 Q2 Q3 Q4 NBO/Si
± ± 10 79 11 0.99
0.8 4.6 22.9 50.4 21.3 1.13
n.d.c 9.6 36.1 46.0 8.3 1.47
± 0.48 0.63 1.10 2.57 ±
a
From [21]. Assuming equal cross-sections. c Not determined due to the very small number of corresponding Q-species. b
cies probably dier. Therefore, the data in Table 1 were used to estimate the corresponding scattering cross-sections (Table 1). Second, the shape of the high-frequency part of the PRS is complex and cannot be described by a
(a)
single Gaussian function as usually assumed [5]. The PRS for a given Qn vary with composition and even from one model to another for a given composition. Calculations of the next-nearest neighbours distributions around a given Qn species
(b)
Fig. 6. Calculated VV polarized partial Raman spectra of the NS1 (a) and NS2_1 (b) models for: Q1 species (dashed-dotted line); Q2 species (full line); Q3 species (dashed line).
236
N. Zotov / Journal of Non-Crystalline Solids 287 (2001) 231±236
indicate that these spectral variations are due to dierences in the connectivity of the Qn species. 5. Conclusions The addition of Na2 O to a-SiO2 leads to breaking of some of the Si±O±Si linkages and formation of Si±NBO bonds. We predict that this disruption causes changes in the vibrational density of states of sodium silicate glasses. Dierent quantitative measures of the degree of localization of the vibrational modes were calculated and show that increasing the number of NBO/Si increases linearly the localization of the vibrational modes on NBO in the 200±500 and 850±1100 cm 1 ranges. The calculated polarized Raman spectra were analyzed in terms of the dierent Q-species present. The partial Raman spectra have complex shapes and are composition dependent. This dependence is mainly due to changes in the connectivity of the Q-species with increasing Na2 O content and I suggest that the corresponding scattering cross-sections are also a function of composition. Acknowledgements We thank Dr W. Smith and Dr D. Timpel for the coordinates of the NS2 and NS1 models.
References [1] S.R. Elliot, Physics of Amorphous Materials, Longman, Harlow, 1990. [2] H. Bach, D. Krause (Eds.), Analysis of the Composition and the Structure of Glass and Glass Ceramic, Springer, Berlin, 2000. [3] T. Furukawa, K.E. Fox, W.B. White, J. Chem. Phys. 75 (1981) 3226. [4] D.W. Matson, S.K. Sharma, J.A. Philpotts, J. Non-Cryst. Solids 58 (1983) 323. [5] B.O. Mysen, J.D. Frantz, Chem. Geology 96 (1992) 321. [6] P.H. Gaskell, Phys. Chem. Glasses 8 (1967) 69. [7] S.A. Brawer, Phys. Rev. B 11 (1975) 3173. [8] N. Zotov, M. Marinov, L. Konstantinov, J. Non-Cryst. Solids 197 (1996) 179. [9] T. Uchino, T. Yoko, J. Phys. Chem. B 102 (1998) 8372. [10] R.J. Bell, N.F. Bird, P. Dean, J. Phys. C 1 (1968) 299. [11] R.J. Bell, D.C. Hibbins-Butler, J. Phys. C 9 (1976) 1171. [12] R.J. Bell, D.C. Hibbins-Butler, J. Phys. C 9 (1976) 2955. [13] R.A. Murray, W.Y. Ching, Phys. Rev. B 39 (1989) 1320. [14] S.N. Taraskin, S.R. Elliot, Phys. Rev. B 56 (1997) 8605. [15] J. Sarnthein, A. Pasquarello, R. Car, Science 275 (1997) 1975. [16] A. Pasquarello, R. Car, Phys. Rev. Lett. 79 (1997) 1766. [17] N. Zotov, I. Ebbsj o, D. Timpel, H. Keppler, Phys. Rev. B 60 (1999) 6383. [18] D. Timpel, K. Scheerschmidt, S.H. Garofalini, J. NonCryst. Solids 221 (1997) 187. [19] W. Smith, G.N. Greaves, M.J. Gillan, J. Non-Cryst. Solids 192&193 (1995) 267. [20] D.L. Price, J.M. Carpenter, J. Non-Cryst. Solids 92 (1987) 153. [21] H. Maekawa, T. Maekawa, K. Kawamura, T. Yokokawa, J. Non-Cryst. Solids 127 (1991) 53.
www.elsevier.com/locate/jnoncrysol
Eects of composition on the vibrational properties of sodium silicate glasses N. Zotov * Bayerisches Geoinstitut, D-95440 Bayreuth, Germany
Abstract The vibrational modes and the calculated Raman spectra of computer generated models of sodium silicate glasses are presented and discussed. The localization of the vibrational modes on non-bridging oxygens (NBOs) increases linearly with increasing NBO/Si in all spectral ranges. Strongest localization is observed in the low-frequency range near the 225 and 320 cm 1 vibrational bands. The shape of the partial Raman spectra of SiO4 tetrahedra with dierent number of NBOs (Qn species) is complex and depends both on the glass composition and the connectivity of the Qn species. The estimated Qn scattering cross-sections increase with increasing the number of bridging oxygens. Ó 2001 Elsevier Science B.V. All rights reserved.
1. Introduction The structure and the physical properties of the glass-forming system, Na2 OASiO2 , are of basic importance for understanding many commercial as well as natural multicomponent silicate glasses. Correspondingly, Na2 OASiO2 glasses have been studied by various diraction and spectroscopic methods [1,2]. Among them, Raman and infrared (IR) spectroscopy are sensitive probes of the local structure of silicate glasses and many experimental vibrational spectra have been published [3±5]. However, the understanding of the vibrational spectra is still qualitative due to the lack of long-range order in glasses. The most common approach for the interpretation of the vibrational spectra of glasses is the calculation * Present address: Mineralogisches Institut, D-53115 Bonn, Germany. Fax: +49-228 732 770. E-mail address: [email protected] (N. Zotov).
of the vibrational frequencies of characteristic molecular-like units and comparison with experiment [3,6±9]. Calculations of the vibrational density of states (VDOS), the IR and the Raman spectra of large-scale (103 atoms) computer models of silicate glasses have been performed up to now only for amorphous SiO2 (a-SiO2 [10±16]. Accordingly, we have made ®rst calculations of the vibrational properties and the polarized Raman spectra of sodium tetrasilicate glass Na2 O 4SiO2 , denoted NS4) using computer generated models with periodic boundary conditions [17]. In the present paper, we extend this approach to sodium disilicate Na2 O 2SiO2 , denoted NS2) and sodim metasilicate Na2 O SiO2 , denoted NS1) glasses. The results obtained are compared with experimental Raman spectra and previous calculations for NS4 and a-SiO2 [17] to examine the eects of Na2 O content on the vibrational properties over a larger compositional range.
0022-3093/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 3 0 9 3 ( 0 1 ) 0 0 5 7 7 - 4
232
N. Zotov / Journal of Non-Crystalline Solids 287 (2001) 231±236
2. Calculation procedures One computer model of NS1 glass and two models of NS2 glass are analysed in the present paper. They have been generated previously by molecular dynamics (MD) [18,19]. The NS1 model (599 atoms) and the ®rst NS2 model (denoted NS1_1), containing 602 atoms, were constructed using a modi®ed Born±Mayer±Huggins potential with a weak Stillinger±Weber three-body term [18]. The second NS2 model (denoted NS2_2), containing 1080 atoms, was constructed using a Vessal-type potential [19]. The MD models are ®rst relaxed with a Kirkwood-type potential (see for details [17]). The force constants are similar to those used for the NS4 model [17]. The atomic vibrations of the relaxed models are then calculated by direct diagonalization of the full dynamical matrix. Each mode in the VDOS is additionally broadened by 38 cm 1 to account, empirically, for the ®nite size of the models, anharmonic eects and disorder not present in the generated models.
The properties of the vibrational modes are investigated by calculating the participation ratio, pc , the correlation length, Lc , and the atomic participation ratios, APR. De®nition of these parameters and discussion of their properties is given elsewhere [17]. The ®rst-order Raman spectra of the structural models is calculated in the bond polarizabiliy ap-
(a)
(b)
(c)
Fig. 1. VDOS for the NS1 (full line), NS2_1 (dotted line) and the a-SiO2 (dashed line) models.
Fig. 2. Participation ratio pc x: (a) NS1; (b) NS2_1; (c) a-SiO2 models.
N. Zotov / Journal of Non-Crystalline Solids 287 (2001) 231±236
proximation [12,17]. The bond polarizability parameters used for NS4 [17] are employed in the present study as well. Each Raman line is additionally broadened by 20 cm 1 to reproduce, approximately, the full-width at half-maximum of the high-frequency band at about 1100 cm 1 in the experimentally measured spectra. The partial Raman spectra of short-range order clusters are calculated by projecting the contribution of the corresponding atoms out of the dynamical matrix.
233
O and Si±non-bridging oxygen (NBO) contributions. The participation ratio, pc x, for the NS1, NS2_1 and the a-SiO2 models is shown in Fig. 2. Most of the localized modes (those for which pc x ! 0 ) are grouped around the band edges. Fig. 3, in which data for a-SiO2 and two NS4 models from [17] are also incorporated, shows that
3. Results The calculated normalized VDOS for the NS1 and the NS2_1 models are compared with that for a-SiO2 [17] in Fig. 1. The calculated VDOS for aSiO2 is in relatively good agreement with experimental VDOS measured by inelastic neutron scattering [20]. Unfortunately, there are no, to the best of our knowledge, published VDOS of sodium silicate glasses. The new vibrational states at about 50, 225, 320 cm 1 and in the range 850±1050 cm 1 as well as the decrease of the intensity of the 800 and 450 cm 1 bands are due to increasing Na±
(a)
(b)
Fig. 3. Dependence of the average participation ratio on the number of NBO/Si: band 200±500 cm 1 (full circles); band 850± 1200 cm 1 (full triangles). The dashed lines are linear ®t through the data points.
(c) Fig. 4. Correlation length Lc x: (a) NS1; (b) NS2_1; (c) a-SiO2 models.
234
N. Zotov / Journal of Non-Crystalline Solids 287 (2001) 231±236
the average participation ratio of the main vibrational bands in the ranges 200±500 cm 1 and 850± 1200 cm 1 decreases linearly with increasing NBO/ Si. The NBO/Si parameter is calculated as NBO=Si R 4 nxn , where xn is the concentration of SiO4 tetrahedra with n bridging oxygens (so-called Qn species). Another measure of mode localization is the correlation length, Lc x: Lc is approximately equal to L/2 for fully delocalized modes, where L is the box edge and approaches zero for the most localized modes. As can be seen from Fig. 4, the number of such localized modes (modes with Lc < 0:5 nm) increases with increasing Na2 O content. The calculated and experimentally measured Raman spectra of the NS2 glass are shown in Fig. 5. The calculated VV polarized spectra reproduce all features in the experimental one. The agreemenent between the calculated and previously published [7] experimental Raman spectra of sodium metasilicate glass is worse, especially in the high-frequency range where only one peak larger than the others at about 985 cm 1 with shoulders at 1060 and 855 cm 1 is experimentally observed compared with four peaks at about 850, 910, 975 and 1060 cm 1 in the calculated spectrum. The discrepancies between calculations and experiment are due to two reasons. The ®rst one is related to the bond polarizability parameters used. The better agreement obtained in the case of a-
SiO2 and NS4 [17], compared to the NS2 and NS1 models, indicates that the bond polarizability parameters of the Si±BO and Si±NBO bonds probably depend on the composition. The second kind is associated with the topology of the computer models and will be analysed in the next section. 4. Discussion The NBO/Si parameters and the Q-species distribution of the investigated models are compared in Table 1 with the results from 29 Si MAS NMR spectroscopy [21]. The concentration of the dominant Q-species (Q2 in the NS1 glass and Q3 in the NS2 glass) is much less in the glass models. The partial Raman spectra (PRS) of the most abundant Qn species in the models (Fig. 6) show that the presence of 100% more Q1 and Q3 species in the NS1 model, as well as Q2 and Q4 species in the NS2 model is the main reason for the discrepancy between the calculated and experimental Raman spectra. The PRS reveal also two other important features of the vibrational dynamics of binary sodium silicate glasses. The Q-species distributions derived from the integral PRS intensities in the range 850± 1200 cm 1 , assuming equal scattering cross-sections, do not agree with the actual Q-species distributions (see Table 1). In other words, the bond polarizability parameters of the dierent Qn spe-
Fig. 5. Comparision between the reduced experimental (full line) and calculated polarized (VV and VH) Raman spectra of the NS2 models: NS2_1 (dashed line); NS2_2 (dashed±dotted line).
N. Zotov / Journal of Non-Crystalline Solids 287 (2001) 231±236
235
Table 1 Q-species distributions (in %) and NBO/Si parameters of the investigated sodium silicate glasses Glass model
Q species a
NMR
Model analysis
Partial Raman spectrab
Estimated cross-sections
NS1
Q0 Q1 Q2 Q3 Q4 NBO/Si
1 13 70 16 ± 1.99
4 28 36 24 8 1.96
n.d.c 40.6 39.2 16.6 3.4 2.168
± 0.69 0.92 1.44 2.35
NS2_1
Q0 Q1 Q2 Q3 Q4 NBO/Si
± ± 10 79 11 0.99
± 2.2 21.6 44.1 32.1 0.94
± n.d.c 40.4 46.8 12.8 1.276
± ± 0.53 0.94 2.50
NS2_2
Q0 Q1 Q2 Q3 Q4 NBO/Si
± ± 10 79 11 0.99
0.8 4.6 22.9 50.4 21.3 1.13
n.d.c 9.6 36.1 46.0 8.3 1.47
± 0.48 0.63 1.10 2.57 ±
a
From [21]. Assuming equal cross-sections. c Not determined due to the very small number of corresponding Q-species. b
cies probably dier. Therefore, the data in Table 1 were used to estimate the corresponding scattering cross-sections (Table 1). Second, the shape of the high-frequency part of the PRS is complex and cannot be described by a
(a)
single Gaussian function as usually assumed [5]. The PRS for a given Qn vary with composition and even from one model to another for a given composition. Calculations of the next-nearest neighbours distributions around a given Qn species
(b)
Fig. 6. Calculated VV polarized partial Raman spectra of the NS1 (a) and NS2_1 (b) models for: Q1 species (dashed-dotted line); Q2 species (full line); Q3 species (dashed line).
236
N. Zotov / Journal of Non-Crystalline Solids 287 (2001) 231±236
indicate that these spectral variations are due to dierences in the connectivity of the Qn species. 5. Conclusions The addition of Na2 O to a-SiO2 leads to breaking of some of the Si±O±Si linkages and formation of Si±NBO bonds. We predict that this disruption causes changes in the vibrational density of states of sodium silicate glasses. Dierent quantitative measures of the degree of localization of the vibrational modes were calculated and show that increasing the number of NBO/Si increases linearly the localization of the vibrational modes on NBO in the 200±500 and 850±1100 cm 1 ranges. The calculated polarized Raman spectra were analyzed in terms of the dierent Q-species present. The partial Raman spectra have complex shapes and are composition dependent. This dependence is mainly due to changes in the connectivity of the Q-species with increasing Na2 O content and I suggest that the corresponding scattering cross-sections are also a function of composition. Acknowledgements We thank Dr W. Smith and Dr D. Timpel for the coordinates of the NS2 and NS1 models.
References [1] S.R. Elliot, Physics of Amorphous Materials, Longman, Harlow, 1990. [2] H. Bach, D. Krause (Eds.), Analysis of the Composition and the Structure of Glass and Glass Ceramic, Springer, Berlin, 2000. [3] T. Furukawa, K.E. Fox, W.B. White, J. Chem. Phys. 75 (1981) 3226. [4] D.W. Matson, S.K. Sharma, J.A. Philpotts, J. Non-Cryst. Solids 58 (1983) 323. [5] B.O. Mysen, J.D. Frantz, Chem. Geology 96 (1992) 321. [6] P.H. Gaskell, Phys. Chem. Glasses 8 (1967) 69. [7] S.A. Brawer, Phys. Rev. B 11 (1975) 3173. [8] N. Zotov, M. Marinov, L. Konstantinov, J. Non-Cryst. Solids 197 (1996) 179. [9] T. Uchino, T. Yoko, J. Phys. Chem. B 102 (1998) 8372. [10] R.J. Bell, N.F. Bird, P. Dean, J. Phys. C 1 (1968) 299. [11] R.J. Bell, D.C. Hibbins-Butler, J. Phys. C 9 (1976) 1171. [12] R.J. Bell, D.C. Hibbins-Butler, J. Phys. C 9 (1976) 2955. [13] R.A. Murray, W.Y. Ching, Phys. Rev. B 39 (1989) 1320. [14] S.N. Taraskin, S.R. Elliot, Phys. Rev. B 56 (1997) 8605. [15] J. Sarnthein, A. Pasquarello, R. Car, Science 275 (1997) 1975. [16] A. Pasquarello, R. Car, Phys. Rev. Lett. 79 (1997) 1766. [17] N. Zotov, I. Ebbsj o, D. Timpel, H. Keppler, Phys. Rev. B 60 (1999) 6383. [18] D. Timpel, K. Scheerschmidt, S.H. Garofalini, J. NonCryst. Solids 221 (1997) 187. [19] W. Smith, G.N. Greaves, M.J. Gillan, J. Non-Cryst. Solids 192&193 (1995) 267. [20] D.L. Price, J.M. Carpenter, J. Non-Cryst. Solids 92 (1987) 153. [21] H. Maekawa, T. Maekawa, K. Kawamura, T. Yokokawa, J. Non-Cryst. Solids 127 (1991) 53.