Pressure and temperature dependence of elastic properties of lanthanum gallogermanate glasses

Pressure and temperature dependence of elastic properties of lanthanum gallogermanate glasses

Materials Research Bulletin 37 (2002) 1293±1306 Pressure and temperature dependence of elastic properties of lanthanum gallogermanate glasses L.G. Hw...

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Materials Research Bulletin 37 (2002) 1293±1306

Pressure and temperature dependence of elastic properties of lanthanum gallogermanate glasses L.G. Hwaa,*, W.C. Chaoa, S.P. Szub a

Department of Physics, Fu-Jen Catholic University, Taipei 24205, Taiwan, ROC b Department of Physics, Chung-Hsing University, Taichung, Taiwan, ROC (Refereed) Received 19 December 2001; accepted 25 March 2002

Abstract The hydrostatic and the uniaxial pressure, and the temperature dependence of elastic properties of a series of lanthanum gallogermanate glasses were determined by ultrasonic pulse-echo techniques. The experimental results are used to obtain the complete set of secondand third-order elastic constants of these glasses. The normal behavior of negative temperature dependence and positive pressure dependence of the ultrasonic velocities were observed in these glasses. The pressure derivative of shear and bulk moduli, and the longitudinal and shear acoustic mode Gruneisen parameters are all positive for these glasses. # 2002 Elsevier Science Ltd. All rights reserved. Keywords: Glasses; Optical materials; Ultrasonic measurements; Elastic properties

1. Introduction Glasses containing heavy metal oxides are of interest for infrared (IR) transmission application due to their long IR cut-off wavelength relative to phosphate, borate and silicate glasses. Dumbaugh found that the addition of Ga2O3 into PbO±Bi2O3 and CdO±Bi2O3 dramatically improves glass stability and these glasses have high optical transmission up to about 7 mm [1]. Kokubo et al. found that alkali- and alkaline-earth Ta2O±Ga2O3 or Nb2O±Ga2O3 glasses also have high optical transmission to about 7 mm [2,3]. A study of barium gallogermanate glasses has demonstrated their *

Corresponding author. Tel.: ‡886-2-29031111, ext. 2436; fax: ‡886-2-29021038. E-mail address: [email protected] (L.G. Hwa). 0025-5408/02/$ ± see front matter # 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 5 - 5 4 0 8 ( 0 2 ) 0 0 7 6 0 - 2

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potential for both IR optical wave-guide and bulk optic components at 3±5 mm [4]. Lindquist and Shelby studied the properties of rare earth gallogermanate glasses, which have an IR cut-off at 6.05 mm and show excellent chemical durability in water [5]. These glasses also have very high Verdet constants due to their high rare earth ion concentration making them suitable for use in Faraday rotation devices. We have conducted several investigations on the structural, physical, elastic and optical properties of lanthanum gallogermanate glasses [6±9] and concluded that (i) these glasses are transparent over frequencies ranging from the near UV to the mid-IR (8 mm) and have ionic bond properties in their structure [6]; (ii) the Raman and IR spectra indicate that the majority of Ge and Ga are four-coordinated in the glass network. The role of La3‡ ion acts either as charge-compensator for GaO4 or as glass modi®er for non-bridging oxygen [7,8]; (iii) these glasses exhibit high values of Young's modulus, shear modulus and bulk modulus due to high packing density of each individual atom [9]. Since the strength of materials increase with their elastic moduli, it is easier to assess strength indirectly from their elastic properties. Studies of the elastic constants of the glassy materials give considerable information about the structure of noncrystalline solids since they are directly related to the interatomic forces and potentials [10±18]. In this paper, the pressure and temperature dependence of the elastic properties of a series of the lanthanum gallogermanate glasses (0.20La2O3± GeO2±yGa2O3 with y ˆ 0:20, 0.33, and 0.50) is studied. The experimental results provide the complete set of non-vanishing components of the second- and third-order elastic stiffness tensors. The second-order elastic constants (SOECs) give the slope of dispersion curves at long wavelength limit and the understanding of the vibration properties of a glass [19]. The third-order elastic constants (TOEC) characterize the anharmonic properties or non-linearity of the atomic force with respect to atomic displacements of a glass [19]. The determination of TOECs of a solid is of great interest since they are the measure of ®rst-order anharmonic terms of the inter-atomic potential or of the intrinsic non-linearity of the solid, such as thermal expansion. The knowledge of TOEC allows the evaluation of the generalized Gruneisen parameters, which measures the coupling of the stress wave with the thermal phonons in Debye approximation [20]. The measurement of pressure and temperature dependence of ultrasonic sound wave is a good way to study this effect. Recently, another method by a stressinduced Brillouin scattering measurement was carried out to study TOEC of soda± lime±silica glass [21]. 2. Experimental aspect Three samples with the formula: 0.20La2O3±GeO2±yGa2O3, where y ˆ Ga2 O3 mol%/GeO2 mol% were prepared by melting appropriate amounts of La2O3 (Aldrich 99.9%), GeO2 (Aldrich 99.9%), Ga2O3 (Aldrich 99.9%) in an open atmosphere at temperatures between 1550 and 16508C, for 1±1.5 h in a platinum crucible. The melt

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was quenched with the bottom of the crucible in water. The glassy samples were identi®ed by X-ray diffraction. In order to produce samples large enough for the experimental requirements, samples were batched to obtain 10 g. High temperature ultrasonic measurements were performed by a pulse-echo method with a Panametrics model 5800 pulser/receiver instrument and with two specially designed high temperature PZT transducers from Etalon Inc., for temperature up to 3508C at 20 MHz frequency. X-cut transducers were employed for longitudinal modes and Y-cut for shear modes. The pulse transit time was measured with a Hewlett-Packard model 54502A oscilloscope. The high temperature couplant used in the experiment was purchased from Sonotech Inc. High-pressure PZT transducers were designed by us and fabricated by a Canadian company. The piezoelectric crystals in the transducers were protected by high strength alloy steel and by hydraulic component. These transducers were designed to be able to stand the force till 200,000 N. X- and Y-cut transducers have the delay time 8.4 and 15.5 ms, respectively. The detailed description of the pressure systems are discussed elsewhere [22]. The pressure systems (both uniaxial and hydrostatic) were calibrated by a well-known data of a SiO2 glass sample. Pressure-induced changes in the sample dimensions were accounted for by using the natural velocity technique [20,23,24]. Several effects, such as multiple internal re¯ections within the transducer, sample thickness, and the acoustic impedance mismatch between the glass sample and the transducer, in¯uence the accuracy of ultrasonic velocity measurements. The uncertainty of the measurements is estimated to be about 1%. The density was determined by an Archimedes technique for which n-hexadecane was used as the working ¯uid. The accuracy of the measurement was about 0.001 g/cm3. The glass samples were polished on both sides before the measurements. The size of the glass sample is about 5 mm  5 mm  2 mm. The glass transition temperature Tg and the crystallization temperature Tx of our glasses were determined by DTA with a heating rate of 58C/min. The values of both temperatures (Tg, Tx) in 8C are (751, 819), (732, 861), and (728, 854) for 0.20La2O3±GeO2±0.2Ga2O3, 0.20La2O3± GeO2±0.33Ga2O3, and 0.20La2O3±GeO2±0.50Ga2O3 glasses, respectively. 3. Results For isotropic materials (such as glass), two independent second-order elastic constants (C11 and C44) exist. The SOEC can be derived directly from the measurement of sound velocities. For pure longitudinal waves: C11 ˆ rVL2 , and for pure transverse waves: C44 ˆ rVT2 , where VL and VT are the longitudinal and transverse velocities, respectively. The sound velocities also allow the determination of Young's modulus, E, bulk modulus, B, and Poisson's ratio, s, by the following equations: E ˆ rVT2

3VL2 VL2

4VT2 VT2

(1)

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Table 1 The density, sound velocities (both longitudinal and transverse), the calculated elastic constants (C11 and C44), Young's modulus E, bulk modulus B, Poisson ratio s and fractal bond connectivity ``d ˆ 4C44/B'' from experimental data for lanthanum gallogermanate glasses and GeO2 glass Sample

VL r (g/cm3) (m/s)

VT C11 C44 E B s (m/s) (GPa) (GPa) (GPa) (GPa)

4C44/B

0.20La2O3±GeO2±0.20Ga2O3 0.20La2O3±GeO2±0.33Ga2O3 0.20La2O3±GeO2±0.50Ga2O3 GeO2

5.16 5.18 5.21 3.63

2701 2718 2790 2233

1.75 1.71 1.71 3.03

5144 5207 5346 3636

136.6 140.4 148.8 48.0

37.7 38.3 40.5 18.1

98.7 100.5 106.4 43.3

86.4 89.4 94.7 23.9

S ˆ rVT2 ˆ C44 Bˆr sˆ

3VL2

3

0.31 0.31 0.31 0.19

(2)

4VT2

(3)

VL2 2VT2 2…VL2 VT2 †

(4)

The density, longitudinal and transverse sound velocities, the calculated secondorder elastic constants (C11, and C44), Young's modulus, bulk modulus and the Poisson's ratio for the present glasses, and the pure GeO2 glass are given in Table 1. The overall uncertainty for the above calculated quantities is estimated about 2%. Bergman and coworkers suggested an expression: d ˆ 4C44 =B, which was derived for an inhomogeneous random mixture of ¯uid and a solid backbone near the percolation limit [25±27]. This new parameter ``d'', Bogue and Sladek [26] called the fractal bond connectivity, can give information on effective dimensionality of the materials: d ˆ 3 for 3D tetrahedral coordination polyhedra; d ˆ 2 for 2D layer structures; and d ˆ 1 for 1D chains. The calculated d-value is also given in Table 1. The temperature dependence of both longitudinal and transverse sound velocities of three lanthanum gallogermanate glasses is given in Fig. 1. The results indicate that the ultrasonic velocities, both longitudinal and transverse, decrease slowly and monotonically with increasing temperature in the range of 25±3508C. The variation of sound velocities has a small negative temperature coef®cient. Similar negative temperature dependence is observed for shear modulus (@S/@T) and bulk modulus (@B/@T). Figs. 2±4 show the hydrostatic pressure dependence of both longitudinal and transverse sound velocities of 0.20La2O3±GeO2±0.20Ga2O3, 0.20La2O3±GeO2± 0.33Ga2O3, and 0.20La2O3±GeO2±0.50Ga2O3 glasses up to around 1200 bar, respectively. Fig. 5 gives the uniaxial pressure dependence of transverse sound velocity of three lanthanum gallogermanate glasses up to around 100 bar. An isotropic material has six TOECs, and three of those are independent. If we take C123 Bn1 , C144 Bn2 , and C456 Bn3 , then the others are given by the linear combinations of these three C112 ˆ n1 ‡ 2n2 ;

C155 ˆ n2 ‡ 2n3 ;

C111 ˆ n1 ‡ 6n2 ‡ 8n3

(5)

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Fig. 1. The temperature dependence of both longitudinal and transverse sound velocities for a series of lanthanum gallogermanate glasses.

Fig. 2. The hydrostatic pressure dependence of both longitudinal and transverse sound velocities for a 0.20La2O3±GeO2±0.20Ga2O3 glass.

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Fig. 3. The hydrostatic pressure dependence of both longitudinal and transverse sound velocities for a 0.20La2O3±GeO2±0.33Ga2O3 glass.

Fig. 4. The hydrostatic pressure dependence of both longitudinal and transverse sound velocities for a 0.20La2O3±GeO2±0.50Ga2O3 glass.

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Fig. 5. The uniaxial pressure dependence of transverse sound velocity for a series of lanthanum gallogermanate glasses.

Fig. 6. The hydrostatic pressure dependence of relative natural velocity changes (DW/W0)L of longitudinal sound wave for a series of lanthanum gallogermanate glasses.

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Fig. 7. The hydrostatic pressure dependence of relative natural velocity changes (DW/W0)T of transverse sound wave for a series of lanthanum gallogermanate glasses.

Fig. 8. The uniaxial pressure dependence of relative natural velocity changes (DW/W0)T of transverse sound wave for a series of lanthanum gallogermanate glasses.

Hydrostatic

Arbitrary

L

ˆ to N

1

Hydrostatic

Arbitrary

T

? to N

1

Uniaxial compression

? to stress

T

? to stress

1 S …2C11 ‡ 3n1 ‡ 10n2 ‡ 8n3 † 3BT 1 …2C44 ‡ 3n2 ‡ 4n3 † 3BT

1 T ‰s …2C44 ‡ 4n3 † ‡ n2 …2sT ET

A: 0.20La2O3±GeO2±0.20Ga2O3, B: 0.20La2O3±GeO2±0.33Ga2O3, C: 0.20La2O3±GeO2±0.50Ga2O3.

1†Š

B

C

8.86

9.36

11.24

1.66

1.80

1.95

7.15

7.46

8.01

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Table 2 Thurston±Brugger relations for isotropic media [20] and the measured ‰d…r0 W 2 †=dPŠPˆ0 for lanthanum gallogermanate glasses   d DW A Type of Propagation Mode Displacement 2…r0 W02 † dP W0 Pˆ0 stress direction (N) direction

1301

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The three independent TOECs are obtained from the measured hydrostatic and uniaxial pressure derivative of the ultrasonic natural velocities ‰d…r0 W 2 †=dPŠPˆ0 using the following expressions as [20,23,24]   d DW 1 S ˆ 1 …2C11 ‡ 3n1 ‡ 10n2 ‡ 8n3 † (6) 2…r0 W02 †L dP W0 L Pˆ0 3BT   DW 1 2 d ˆ 1 …2C44 ‡ 3n2 ‡ 4n3 † (7) 2…r0 W0 †T dP W0 T Pˆ0 3BT   DW 1 2 d 2…r0 W0 †T ˆ T ‰sT …2C44 ‡ 4n3 † ‡ n2 …2sT 1†Š (8) dP W0 T Pˆ0 E where isothermal Young's modulus ET, bulk modulus BT, and Poisson's ratio sT are obtained from the calculation of Eqs. (1), (3) and (4). The data in Figs. 6±8 provide the necessary values of ‰d…r0 W 2 †=dPŠPˆ0 for longitudinal and transverse waves in lanthanum gallogermanate glasses under hydrostatic pressure and for transverse wave propagating perpendicularly to an applied uniaxial stress. The measured pressure derivatives of three glasses are given in Table 2. 4. Discussion The calculated longitudinal and transverse elastic constants (C11 and C44) of the lanthanum gallogermanate glasses are about three times and two times larger than that of pure GeO2 glass, respectively. This implies that lanthanum gallogermanate glasses have a very rigid lattice structure. Poisson's ratio for the lanthanum gallogermanate glasses is about 0.31, where GeO2 shows this ratio to be 0.19. The higher Poisson's ratio may be because most of the bonds are ionic, in comparison with GeO2, which is predominantly covalent [14,28±30]. Bridge et al. [14] suggested a close correlation between Poisson's ratio and the crosslink density (de®ned as the number of bridging bonds per cation) of the glass structure. The crosslink density of two, one and zero are related to the value of Poisson's ratio of 0.15, 0.30 and 0.40, respectively. Consequently, Poisson's ratio for the lanthanum gallogermanate glasses is about 0.31, which implies a 2D layer structure. Furthermore, the fractal bond connectivity of the glasses being studied gives a connectivity d  1:75 (as seen in Table 1), where the pure GeO2 glass shows this ratio to be 3.03. The values of n1,n2 and n3 was calculated from Eqs. (6)±(8) using the pressure data from Table 2. The other three TOECs are obtained with the further help of Eq. (5). All the TOECs for the glasses studied are given in Table 3, and the values of SiO2 glass are also given for comparison. Most of the TOECs of the lanthanum gallogermanate glasses are negative, in marked contrast to those of SiO2 glass, which are anomalously positive, exception being (C123, C456) and C456 for lanthanum gallogermanate and SiO2, respectively. The reason for this discrepancy may be due to lanthanum

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Table 3 The third-order elastic constants of lanthanum gallogermanate glasses and SiO2 glass Elastic constant n1 n2 n3 C111 C112

C123 C144 C155 C456

Glass 0.20La2O3±GeO2±0.20Ga2O3 0.20La2O3±GeO2±0.33Ga2O3 0.20La2O3±GeO2±0.50Ga2O3 SiO2 502 702 335 1026 901 502 31 702 335

533 750 355 1124 968 533 39 750 355

484 845 405 1347 1206 484 35 845 405

50 90 10 530 240 50 70 90 10

gallogermanate glasses falling into the category of highly coordinated solids with respect to the tetrahedrally coordinated SiO2 glass. Although previous studies suggested that the majority of Ge and Ga are four-fold coordinated tetrahedral [7,8], however, the elastic results didn't rule out the possibility of the existence of high coordinated octahedral of GeO6 and/or GaO6 in the glass network. Further experiments using X-ray absorption spectroscopy (synchrotron source in Taiwan and Japan) are underway to test this suggestion. The calculated values of all TOECs for our glasses show that C111 has the largest magnitude among all the TOECs. A qualitative physical reason was given by Brassington et al. [31] to account for the negative sign of the TOEC of a-As2S3. They argued that the shorter range and larger exponent of the repulsive potential require the negative values of TOEC. Since C111 ˆ @ 3 [email protected] is dominated by the short-range repulsive force between nearest-neighbor atoms (where f is the interatomic potential, and Z is generalized coordinate) while C112 and C123 are not, therefore C111 should have the largest magnitude of the TOEC, as it does for a-As, aAs2S3, and many glass systems [28,31,32]. The Cauchy relation for SOEC (2C44 ˆ C11 C12 ) allows one to determine C12. The ratio of C44/C12 is an indicator of the character of the force ®eld, central forces if C44 =C12 ˆ 1 and non-central forces if C44 =C12 6ˆ 1. For lanthanum gallogermanate glasses, the ratio is about 0.6, which is due to the non-central force ®elds (the anharmonicity of the vibrational states or the non-linearity of atomic force) resulting from ion-overlapping and covalent bonding [33]. With new TOEC available, it gives us the opportunity to test the nature of a non-central force ®eld, further. The Cauchy relation for TOEC (C123 ˆ C144 ˆ C456 ) is not obeyed. The large deviations imply the existence of the large component of anharmonic forces between the atoms. The 0.20La2O3±GeO2±0.50Ga2O3 glass shows the largest deviation among three glasses being studied and hence the highest non-linear effect. Table 4 gives the temperature and pressure dependence of both shear modulus (@S/@T and @S/@P) and bulk modulus (@B/@T and @B/@P) for a series of lanthanum gallogermanate glasses, the values of GeO2 and SiO2 glasses is also given for

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Table 4 The temperature and the pressure derivatives of elastic moduli and Gruneisen parameters of lanthanum gallogermanate glasses, GeO2 and SiO2 glasses Glass

Property (@S/@T)  10 3 GPa/K

0.20La2O3±GeO2±0.20Ga2O3 0.20La2O3±GeO2±0.33Ga2O3 0.20La2O3±GeO2±0.50Ga2O3 GeO2 SiO2

(@B/@T)  10 3 GPa/K

5.5 5.8 2.9 0.69 4.6

9.8 12.8 13.8 2.32 12.3

@S/@P 1.8 2.0 2.1 1.1 4.1

@B/@P 7.0 7.3 9.0 0.07 6.3

g11 2.8 3.0 3.6 2.0 2.8

g44 1.9 2.1 2.3 2.3 2.4

comparison. The normal behavior of negative temperature dependence of our glasses agrees with most oxide glasses, ceramic and metallic materials; and in sharp contrast with most tetrahedral bonded glasses, such as SiO2, BeF2, and GeO2, which show the anomalous positive temperature dependence of sound velocities. TOECs measurements allow the evaluation of the generalized Gruneisen parameters. Gruneisen mode gammas express the volume dependence of the normal mode frequency oi as the following equation: gi ˆ

d ln oi : d ln V

(9)

They are the measure of anharmonic effect of the glass and to some extent, exhibit the degree of the vibrational phonon energy shift with respect to the volume change under applied compressions. For an isotropic solid there are two contributions to the acoustic Gruneisen gammas (g11 and g44), and they are associated with longitudinal and transverse modes, respectively. Two Gruneisen parameters were calculated using the following expression [23]: g11 ˆ g44 ˆ

…6C11 † 1 …3BT ‡ 2C11 ‡ k11 † 1

…6C44 † …3BT ‡ 2C44 ‡ k44 †

(10) (11)

where k11 ˆ C111 ‡ 2C112 and k44 ˆ 1=2(C111 C123 ). The positive pressure derivatives of the longitudinal and transverse wave velocities ensure that the calculation of both g11 and g44 are positive for lanthanum gallogermanate glassess (Table 4). The Gruneisen parameters of three lanthanum gallogermanate glasses are given in Table 4 with the values of GeO2 and SiO2 glasses for comparison. The positive values of both g11 and g44 for our glasses implies that the acoustic modes stiffen under pressure (more closely packed structure) at the long wavelength limit. However, the covalently bonded and tetrahedrally coordinated SiO2 glass shows an anomalous negative value of the Gruneisen parameters, which is due to the very open lattice of SiO2 glass and the density increments can be accommodated primarily by bond bending alone. For GeO2 glass, g11 and g44 are 2.0 and 2.3, respectively.

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This implies the longitudinal modes stiffen in the usual way and shear modes soften under pressure. For our glasses, elastic results suggest that the addition of La2O3 and Ga2O3 into GeO2 glass network give a normal pressure dependence behavior. Furthermore, the Gruneisen parameters increase with increasing Ga2O3 content for a series of 0.20La2O3±GeO2±xGa2O3 glasses. Again the 0.20La2O3±GeO2± 0.50Ga2O3 glass shows the largest positive values of the Gruneisen parameters among three glasses and hence the highest anharmonic effect. 5. Conclusions We have successfully performed the measurements of both pressure and temperature dependence of ultrasonic velocities of a series of lanthanum gallogermanate glasses. The experimental results give the complete set of the components of the second- and third-order elastic stiffness tensors of these glasses. The non-central force character or non-linearity of atomic force of our glasses is further understood through the availability of Gruneisen parameters. The positive values of these parameters imply that long wavelength acoustic mode frequencies (or the vibrational phonon energies) increase with increasing applied strains or compressions. Acknowledgments This work was supported in part by the National Science Council of the Republic of China under Grant number NSC-89-2112-M-030-007 and by the Sapintia Education Foundation. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

W.H. Dumbaugh, Phys. Chem. Glasses 27 (1986) 119. T. Kokubo, Y. Inaka, S. Sakka, J. Non-Cryst. Solids 80 (1986) 518. T. Kokubo, Y. Inaka, S. Sakka, J. Non-Cryst. Solids 81 (1987) 337. P.L. Higby, I.D. Aggarwal, J. Non-Cryst. Solids 163 (1993) 303. B.H. Lindquist, J.E. Shelby, Phys. Chem. Glasses 35 (1994) 1. L.G. Hwa, Y.R. Chang, S.P. Szu, J. Non-Cryst. Solids 231 (1998) 222. S.P. Szu, C.P. Shu, L.G. Hwa, J. Non-Cryst. Solids 240 (1998) 22. L.G. Hwa, J.G. Shiau, S.P. Szu, J. Non-Cryst. Solids 249 (1999) 55. L.G. Hwa, H.Y. Lay, S.P. Szu, J. Mater. Sci. 34 (1999) 5999. H.T. Smyth, J. Am. Ceram. Soc. 42 (1959) 277. A. Makishima, J.D. Mackenzie, J. Non-Cryst. Solids 12 (1973) 35. A. Makishima, J.D. Mackenzie, J. Non-Cryst. Solids 17 (1975) 147. A. Makishima, Y. Tamura, T. Sakaino, J. Am. Ceram. Soc. 61 (1978) 247. B. Bridge, N.D. Patel, D.N. Waters, Phys. Stat. Sol. (a) 77 (1983) 655. J. Rocherulle, C. Ecolivet, M. Poulain, P. Verdier, Y. Laurent, J. Non-Cryst. Solids 108 (1989) 187. [16] A. Elsha®e, Mater. Chem. Phys. 51 (1997) 182.

1306 [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33]

L.G. Hwa et al. / Materials Research Bulletin 37 (2002) 1293±1306 A. Abd El-Moneim, I.M. Yousssof, M.M. Shoaib, Mater. Chem. Phys. 52 (1998) 258. R. El-Mallawany, Mater. Chem. Phys. 53 (1998) 93. M. Born, K. Huang, Dynamical Theory of Crystal Lattices, Oxford, London, 1954. R.N. Thurston, K. Brugger, Phys. Rev. 133 (1964) A1604. D. Cavaille, C. Levelut, R. Vialla, R. Vacher, E. Le Bourhis, J. Non-Cryst. Solids 260 (1999) 235. L.G. Hwa, Y.J. Wu, W.C. Chao, C.H. Chen, Mater. Chem. Phys. 74 (2002) 160. M.P. Brassington, W.A. Lambson, A.J. Miller, G.A. Saunders, Y.K. Yogurtcu, Phil. Mag. B 42 (1980) 127. E.F. Lambson, G.A. Saunders, B. Bridge, R.A. El-Mallawany, J. Non-Cryst. Solids 69 (1984) 117. D.J. Bergman, Y. Kantor, Phys. Rev. Lett. 53 (1984) 511. R. Bogue, R.J. Sladek, Phys. Rev. B 42 (1990) 5280. G.A. Saunders, T. Brennan, M. Acet, M. Cankurtaran, H.B. Senin, H.A.A. Sidek, M. Federico, J. Non-Cryst. Solids 282 (2001) 291. P. Brassington, T. Halling, A.J. Miller, G.A. Saunders, Mater. Res. Bull. 16 (1981) 613. R. Ota, N. Soga, J. Non-Cryst. Solids 56 (1983) 105. A.N. Streeram, A.K. Varshneya, D.R. Swiler, J. Non-Cryst. Solids 128 (1991) 294. M.P. Brassington, A.J. Miller, G.A. Saunders, Phil. Mag. B 43 (1981) 1049. J.C. Thompson, K.E. Bailey, J. Non-Cryst. Solids 27 (1978) 161. C.C. Chen, Y.J. Wu, L.G. Hwa, Mater. Chem. Phys. 65 (2000) 306.