CHAPTER 6. ULTRASONIC STUDIES ON Ca0-B203-A1203-Na20 AND Ca0-B203-AI2O3-Fe203 GLASS SYSTEMS

Transcription

1 CHAPTER 6 ULTRASONIC STUDIES ON Ca0-B203-A1203-Na20 AND Ca0-B203-AI2O3-Fe203 GLASS SYSTEMS

2 CHAPTER VI ULTRASONIC STUDIES ON Ca0-B203-A1203-Na20 and CaO-B 0 -A1203-Fe203 GLASS SYSTEMS Introduction There is an ever increasing interest in the measurement of elastic properties of solids using ultrasonic methods, due to their non-destructive nature. Elastic and acoustical properties of glasses are si9nificant from the point of view of their application in special devicesli]. The main reason for extensive ultrasonic investigations of solids is the need for elastic properties of materials like crystals, alloys, plastics, ceramics, glasses and so on in a variety of applications. The development of electronic circuits has resulted in a variety of techniques, ranging in precision from a per cent to a hundredth of per cent under various conditions of temperature and pressure. The older static and dynamic methods of measuring elastic constants of large samples have gained wide acceptance due to their simplicity. Among the various newer techniques pulse echo methods are useful where measurements of highest precision are needed.

3 An ultrasonic investigation of solids will help to understand various solid state phenomena such as grain and domain boundary effects in metals, ferromagnetic and ferroelectric materials, the diffusion of atoms, molecules and vacancies through a solid, the motion of imperfection such as dislocations as well as the interaction between the lattice sound vibrations and free electrons in metals at low temperatures. All these effects are studied by measuring elastic properties, internal friction properties and their change with temperature, frequency and applied electric field[2-41. The measurement of elastic constants of solids is of considerable interest and significance to both science and technology. This measurement yields valuable information reqarding the forces operative between the atoms or ions in a solid. Since the elastic properties describe the mechanical behaviour of materials, this information is of fundamental importance in interpreting and understanding the nature of bonding in the solid state. When a material is subjected to a stress it will get strained and within the elastic limit stress applied on a material is directly proportional to strain (Hooke's law). The proportionality constant relating the stress and strain is the modulus of elasticity or the elastic constant. three types of elastic constants[5]. Commonly there are They are (i) Young's

4 modulus (Y) (ii) Bulk modulus (K) and (iii) Rigidity or shear modulus (G). The Young's modulus relates a unidirectional stress to the resultant strain. It also represents the resistance to traction along the axis of a thin bar or rod. The Bulk modulus (K) provides a good link between the macroscopic elasticity theory and the atomistic view points such as lattice dynamics. Basically 'lit relates the pressure with volume st,ain. The shear modulus (G) shows the relation between shear stress and shear strain. In addition to the above elastic constants there is a longitudinal modulus (L ) determined from the velocity of propagation of longitudinal waves through a solid. The kinds and number of elastic constants for non-isotropic solids like crystals have been discussed by various workers like Huntinqton[6], Nye[7It Bhagavantam[B], Hearmon[9,10], Federov[ll], Musgrave[l2] and others, and the use of physical acoustics to study the properties of solids has been discussed by Mason[ Amorphous materials like glasses exhibit some unique properties which are not usually found in other engineering materials. These materials lack the longranye periodicity in the arrangement of atoms. The study of the propagation and attenuation of waves in c,lasses[16,17] is of special and vital significance due to the observation of anomalous specific heat[l8] and thermal

5 conductivity at low temperature[l9]. The ever increasing study of glasses is also due to their anomalous physical properties apart from practical applications[2]. Inspite of the immense use of ultrasonic techniques in understanding the structure and properties of glasses only a limited number of reports have appeared on such studies. Ultrasonic studies on binary alkali oxide and other oxide glasses have been reported. The studies of ternary glasses are sparse, while on quarternary glasses are almost totally lacking. A brief review of the latest ultrasonic studies in binary and ternary glasses is given in the following section Ultrasonic Investigations in Oxide Glasses - A Brief Review The ever increasing interest in the investigation of 5lasses is motivated by their widespread practical application and the fact that they exhibit a number of anomalous physical properties, which suggest specific structural singularities that differentiate the glassy state of matter from the crystalline as well as the ordinary amourphous state[21. So far, however, a unified theory of the glassy state of matter has failed to emerge, and so the specific structure of glasses continue to be less than fully understood. These specific attributes

6 are extremely pronounced, in particular, in the acoustical properties of glasses, primarily in the composition and temperature dependence of the velocity and absorption of ultrasonic waves[2,20]. For that reason a great many publications have been devoted to the investigation of glasses by ultrasonic methods. A brief review of the earlier works on ultrasonic studies of inorganic glasses is given in this section. Reports on ultrasonics investigations on glasses up to 1976 have been reviewed by Kul'bitskaya et a1.[20] In 1985 Kodama[21] reported the elastic properties of barium borate glasses. By making use of the ultrasonic pulse echo overlap method, ultrasonic velocities in barium borate glasses were measured at 298 K over the single phase composition range. The results of the elastic constant measurements of the glasses as a function of composition were discussed with the help of the relation wv2 2 2 = ( a Vm/ and ) which was derived from the Sm finite elastic strain theory, where M is the molar mass, V the velocity of the longitudinal or transverse wave, Urn per the internal energy unit mole, nh the Lagranqean strain component specifying the sound wave, and Sm the molar entropy. Based on this relation, elastic internal energies per unit mole of the glasses are determined as

7 functions of composition in relation to the behaviour of the fraction of boron atoms in tetrahedral N4r coordination. Elastic constants and structure of the glass system Co 0 -P 0 had been determined by Higazy et a1.[21] by the ultrasonic techniques at 15 MHz. They found that Young's, bulk, shear and longitudinal moduli and the Poisson ratio are sensitive to the composition of the glass. From the ultrasonic data obtained, it was found that the glass system could be divided into three "compositional regions". This behaviour had been qualitatively interpreted in terms of the cobalt coordination, crosslink densities, interatomic force constants and atomic ring sizes. They also presented a full discussion of effects of annealing on elastic properties of the cobalt phoskhab glasses. Ultrasonic sound velocities behaviour in silver borate glasses were investigated by Carimi et a1.[23]. They studied the sound velocity of 5 MHz longitudinal and transverse waves in silver iodide - silver borate glasses and observed in the K temperature range the presence of dispersive effects, whose contribution increased with the AgI content. These effects arise from + the thermally activated jumps of Ag ions, between nearly

8 equivalent positions available in the glassy network. The whole behaviour was explained by the overlap of two different mechanisms, the relaxational one and the one coming out from the anharmonicity of the system. This last effect implies, in a quasi-harmonic approximation, a linear temperature dependence of the elastic constants in all the explored ranges. The velocity and absorption coefficient o f longitudinal ultrasonic waves of frequency 5 and 10 MHz in molten glassy Na 0-SiO K 0-SiO and PbO-SiO and molten 2 2' 2 2 2' Na 0-B 0 and PbO-B 0 were measured by means of the pulse-echo method at 300 to 1600 K by Kazuhira Nagata et a1.[24]. They observed that the velocity of sound decrease with increasing ternprature and decreased rapidly near the transition temperature of the glass system. The mean free path of phonons was also estimated from the velocity of ultrasonic sound, thermal conductivity, and specific heat capacity. The temperature dependence of 15 MHz ultrasonic bulk wave velocity in the range 4 to 600 K in Moo3-P205 glass system was reported by Bridge et a1.[25] in They concluded that a complete understanding of temperature gradients of elastic moduli in glasses generally requires

9 the measurement of both acoustic wave velocity and wave absorption as a function of temperature, so that the relaxational contribution to the gradients can be computed and substrated from the experimental gradients. Damodaran et al.[26] reported the elastic properties of lead containing MOO -P 0 glasses using ultrasonic velocity measurements at 10 MHz. They observed that the composition dependence of elastic moduli, Poisson's ratio and the Debye temperature were consistant with a structural model proposed by Selvaraj et a According to this model lead acts both as a network former and as a network modifier in different composition regimes. They suggested that the incorporation of lead into the network is accompanied by the conversion of three-connected tetrahedra into four-connected tetrahedra in the network. Longitudinal and shear velocities were found to decrease gradually as the concentration of PbO increased. The results were interpreted with the help of the structural model proposed by Selvaraj et a1.[27]. Ultrasonic studies and calculation of elastic and thermodynamic properties of alkaline earth containing silicate glasses were investigated by Batti et a1.[28].

10 They made an effort to test the model proposed by Makishima and Mackenzie[29,30] for the direct calculation of the Young's modulus of silicate glasses of different compositions. Batti et a1.[31] also studied the softening temperature and Debye temperature for alkaline earth silicate glasses. the They also reported[32] the attenuation and velocity measurements of ultrasonic waves in strontium borate glasses and their elastic properties. They observed a variation of velocity, attenuation, longitudinal modulus and coefficient of thermal expansion of the glasses with the frequency of the ultrasonic waves. Ultrasonic velocities in Vanadium-barium-borate glasses were measured at 298 K by making use of the ultrasonic pulse-echo technique at three frequencies by Anand Pal Singh et a1.[33]. They calculated the molecular weight, packing density, mean atomic volume and effective number of atoms in these glass samples. They also calculated the longitudinal modulus of elasticity, internal friction and thermal expansion coefficient with the help of the ultrasonic propagation velocity. They observed that the values of ultrasonic velocity and the dynamic modulus of elasticity exhibit considerable variation at each frequency due to variation in structure

11 and composition of the glass. Values of longitudinal modulus were found to increase with the B203 content and with the frequency of the ultrasonic waves. The results of ultrasonic, X-ray and infrared measurements on xbao- (0.9-x)B Fe203 glasses have been reported recently by Anand Pal Singh et a1.[34]. They have concluded that introduction of Fe203 in the matrix of BaO-B 0 softens 2 3 the material and that Fe203 do not enter the boron-oxygen network but, after dissociation into Fe3+ and 02-, sit in cavities inside the structure. Ultrasonic studies in sodium borate glasses were reported by Sidkey et a1.[35] in They observed that ultrasonic velocity increased as the sodium oxide concentration was increased upto 27.2 mol%. A similar trend was observed in the case of Young's, bulk and shear moduli. The increase in velocity was attributed to the increase in packing density due to a decrease of B203, and therefore an increase in the B04 groups and consequently occupation of the intersticies by the alkali ions. They compared the experimental results with those calculated theoretically from equation derived by Makishima and Mackenzie[29,30]. They also studied the boron anomaly and the results showed that this anomaly should appear at concentrations of sodium oxide above 28 mol%.

12 Padake et a1.[36] investigated ultrasonic velocity, and absorption in ZnO - B203 glasses at 2 MHz frequency for different temperatures. They observed a peak in the value of attenuation for all glasses and the velocity was found to be decreasing with increase of temperature. Experimental results were explained on the basis of tunneling defect atom and the structural mechanism which is totally responsible for the strong absorption in glasses. Ultrasonic studies in binary zinc borate glasses xzn0-(1-x) B203 were also reported by Singh et al. in 1992[371. They had calculated the elastic moduli of the glasses and compared the results with those predicted by P,lakishima-Machenzie mode1[29,30]. Temperature dependence of velocity of longitudinal and transverse ultrasonic waves in V 0 -P 0 glass system was investigated by Mukherjee et a1.[381. The experimental results showed that unlike most of the glasses having tetrahedrally coordinated structures, '2'5- '2'5 glasses which contain both tetrahedral and octahedral structures[39] do not indicate any minimum in the variation of sound velocity with temperature but instead show a steady decrease of velocity with a small negative temperature coefficient.

13 Recently Kodama[40] reported ultrasonic velocity in potassium borate glasses as a function of concentration of K20. They observed a strong dependence of the ultrasonic velocity on the concentration of K20. The elastic properties of these glasses were analysed in terms of the three structural units, on the assumption that these structural units have their respective elastic constants. They have shown that the elastic constants of these structural units are defined on the basis of the elastic internal energy due to deformation. Ultrasonic velocity and elastic properties of the ternary glass system Sr0-Ba0-B203 were reported very recently by Anand Pal Singh et a1.[41]. They observed that ultrasonic velocity and acoustic impedance in these glasses increased with the concentration of strontium oxide. The role of SrO and BaO (modifier) was shown to be diametrically opposite to their role in silicate glasses. The elastic moduli of these glasses were obtained making use of Makishima and Mackenzie mode1[29,30].

14 6.3. Theory The ultrasonic velocity in solids yields the appropriate elastic modulus of the mode being propagated. The relation can be expressed as Where P is the density of the solid and M is the apgropriate combination of the elastic moduli of the solid. The combination depends on the mode of propagation, and the mode in turn depends on the interaction of the wave with the boundaries of the solid. Since solids can sustain shearing strains elastically, they will support the propagation of waves with transverse as well as longitudinal particle motion. The moduli of materials are influenced by many physical phenomena which may in turn be studied by measuring the ultrasonic wave velocities. Within the elastic limit, majority of solids obey Hooke's law which states that stress is directly proportional to strain. Then,

15 Where p is the normal (tensile) stress and is the strain. E is the moduli of elasticity. Similarly the shear stress 1 is directly proportional to the shear strain. where G is the modulus of elasticity in shear. When a sample is extended in tension, there is an accompanying decrease in thickness; the ratio of the thickness decrease to the length increase in the Poisson's ratio 6 where Ad and ~l are the change in thickness and length, and d and 1 are original thickness and length respectively. Poisson's ratio relates the Young's modulus and shear modulus by the following equation. This relationship is only applicable to an isotropic body in which there is only one value for the elastic

16 constant independent of direction. Generally this equation is a good approximation for glasses and for most polycrystalline ceramic materials. Under conditions of isotropic pressure the applied pressure P is equivalent to a stress of -P in each principal directions. In each principal direction, we have a relative strain. The relative volume change is given by The Bulk modulus K defined as the isotropic pressure divided by the relative volume change is given by The elastic constants of the solids are calculated from the measured densities and the velocities of longitudinal (VL) and transverse (Vs) ultrasonic waves 5, using the following expressions[$2].

17 Longitudinal modulus L = 2 "L... (6.9) -~ Shear modulus G = PVs 2... (6.10) Bulk modulus K = L - (4/3) G... (6.11) 1-2 (VS/VL) 2 Poisson's ratio 6 = (6.12) 2-2 (VS/VL) Young's modulus E = (1 +6 ) 2G... (6.13) 6.4. Work Undertaken in the Present Study In the present study two systems of quarternary glasses CaO-B 0 - A1203-~a 0 and CaO-E 0 -A1 0 -Fe containing different concentrations of Ma20 and Fe203 respectively were prepared. Longitudinal and transverse ultrasonic velocity in these glasses were determined using ultrasonic pulse echo overlap technique. The elastic moduli and Poisson's ratio with concentration of Na20 and Fe 0 are discussed Experimental Details Two systems of ylass samples 10Ca0-(75-x) B A1 0 -xna 0, x varying from 15 to 24 mol% and CaO-(70-y) B 0-10 A y Fe203, y varying from 2 to mol% were prepared as described in Section 3620f Chapter 3. Glass samples of thickness about 10 mm and with smooth and parallel end faces were obtained.

18 Velocity of longitudinal and transverse ultrasonic waves in the glass samples were determined using Matec 7700 ultrasonic velocity system and using respectively x cut and y cut quartz transducers each of frequency 3 MHz. block diagram of the experimental set up (figure 2.5) The and the procedure for the measurement of the ultrasonic velocity are described in detail in Section 2.5 of Chapter 2. The path length of the ultrasonic waves in the glass samples were determined by measuring the thickness of the glass samples using a micrometer. Longitudinal and transverse ultrasonic velocity in the glass samples containing different concentrations of Na 0 and Fe203 were 2 determined. The density of the glass samples were measured making use of Archimede's principle and using water as the immersion liquid Results and Discussions Longitudinal (VL) and transverse (V ) velocities of T * ultrasonic waves of frequency 3 MHz in quarteanary glass systems CaO-B 0 -A1203-Ba20 and 2 3 CaO-B 0 -A1 0 -Fe containing different concentrations of Na20 and Fe203, respectively, are given in table 6.1. The density of the glass samples was found to increase with increase in the concentration of Na 0 and Fe203. It is seen from 2 figure 6.1 and 6.2 that both VL and VT increase almost

19 Table 6.1 Variation of ultrasonic velocities, Poisson's ratio and elastic moduli in CaO-B 0 -Al 0 -Na 0 (SS) with varying concentration of Na 0 and in CaO-B 0 -Al 0 -Fe 0?F$) with varying concentration of P$ Sample Longitudinal Transverse Dens'ty Poisson's Longitudinal Shear Bulk Young's f Name velocity velocity kg/m ratio modulus modulus modulus modulus m/sec m/sc K bar K bar K bar K bar

20 Figure 6.1 Variation of longitudinal and transverse velocities in CaO-B 0 -A1 0 -Na 0 with varying concentrations20? N~;O! 2

21 Figure 6.2 Variation of longitudinal and transverse velocities in CaO-B 0 -A1 0 -Fe203 with varying concentration$ df ~ 6 ~ d ~.

22 regularly with the concentration of Na20 or Fe203. But the rate of increase of V L is greater than that of VT for both the glass systems investigated. The values of the three elastic constants and the ~oisson's ratio evalu ated usins expressions 6.9 to 6.13 are given in tables 6.1. It is seen that for both the glass systems the modulii of elasticity show almost a regular increase over the entire variation of concentration of Na 0 and Fe 0 [figure and 6.4). (figure 6.5 and 6.6). But the Poisson's ratio exhibit a reverse trend From X-ray diffraction studies by Biscoe and Warren[43] had pointed out that as an alkali oxide is added to B203, the coordination of boron which is 3 in B203 changes to 4. It is known for some time that the physical properties of binary borate - glasses display unusual trends with change in their composition. behaviour known as "boron oxide anomaly", has This been investigated by many workers. Internal friction studies of sodium borate glasses[44,45] showed that this anomaly occurs at 15 mo18 of alkali oxide, while Abe screening theoryl461 suggested the saturation of BO to occur at 4 16 mol%. But ultrasonic studies of Gladkov and Tarasov[47] showed this anomaly to occur at 35 mol% Na20.

23 Figure 6.3 Variation of elastic constants in CaO iil 2 o 3 -Na20 with varying concentrations2 at Na20.

24 Figure 6.4 Variation of elastic constants in CaO-B 0 - A1203-Fe203 with varying concentrations2 af Fe 0 2 3'

25 Figure 6.5 Variation of Poisson's ratio in CaO-B 0 - A Na20, with varying concentrations2 af Na203. Figure 6.6 Variation of Poisson's ratio in CaO-B 0 - A1203-Fe 0 with varying concentrations2 df Fe

26 Recent ultrasonic studies by Sidkey et a on sodium borate glasses have showed that both longitudinal and transverse ultrasonic velocity in sodium borate glasses and the elastic constants increased with concentration of Na20 upto 27 mol%. The increase in ultrasonic velocity has been attributed to an increase in packing density due to the transformation of coordination of boron from 3 to 4 and consequent occupation of the intersticies by the alkali ions. But once BO groups get saturated, non- 4 bridging oxygens start appearing producing a loose structure. This phenomenon was not observed by Sidkey et a1.[35] upto a concentration of 28 mol% of Na20. In these studies Poisson's ratio was found to increase with increase in Na 0 concentration. They pointed out 2 that addition of Na 0 changes the coordination of boron 2 from three to four making the glass strong and rendering maximum rigidity. KodamaI401 have measured the elastic properties of potassium borate glasses as a function of concentration of K 0 and analysed the elastic properties 2 in terms of the three structural units represented by Bg3, + X+B g26 and K B o4, where P) represents a bridging - oxysen and 0 a non-bridging oxygen, on the assumption that the three structural units have their respective elastic constants. It was shown numerically that the structural unit BB4 increases the rigidity of the glass + whereas the unit K ~(3~0- decreases it.

27 In the present ultrasonic investigations both the longitudinal and transverse ultrasonic velocities were found to increase with concentration of Na 0 and Fe Also the elastic constants showed almost a regular increase with concentration of Na20 or Fe203. These results may be explained by making use of the results of ultrasonic investigations on binary borate glasses reported in the literature[35,40] as is done in the case of laser Raman spectra where results from the Raman studies of binary glasses are made use of in the interpretation of spectra of ternary and quarternary glasses[48,49]. Results of laser Raman studies (Chapter 5 of this thesis) of the ternary glass CaO-B 0 -A1203 showed 2 3 that the structure of the glass consists of mainly boroxol rings containing only bridging oxygen. When Na20 is added to this glass system (so that the resultant glass is CaO- B 0 -A1203- Na20) the structure was found to consist 2 3 mainly of tetraborate groups and at high concentration of Na 0 pentaborate groups were formed (Chapter 5 of this 2 thesis). A few percentage of diborate-pentaborate and other groups having bridging oxygens were also detected in this structure. The main structural units in quarternary glass CaO-B 0 -A1 0 -Fe203 were found to be boroxol rings for low concentration of Fe 0 where as at 2 3 high concentration boroxol rings transform into other 9roups all having only bridging oxygens (Chapter 5).

28 In these transformations boron undergo a change from three coordinated to four it is reported that the presence of Ba4 increases the rigidity of the glass[40]. It has also been reported that in the case of binary sodium borate glasses as the concentration of Na20 is increased the packing density increased due a transformation of coordination of boron from 3 to 4 and consequent occupation of the intensities by the alkali ions[35]. The increase in ultrasonic velocity and the elastic moduli in the present study may also be attributed to the increase in packing density and rigidity of the c,lass samples as the concentration of Na20 or Fe20j is increased. The laser Raman spectra indicated the presence of a few loose diborate and loose B04 groups. Their concentration should be small since a large concentration of these groups should Lead to a decrease in the rigidity of the glass resulting in the decrease of ultrasonic velocity and the elastic constants, whereas an increase in these quantities were observed. Poisson's ratio had been reported to be increasing with alkali oxide concentration in binary oxide glasses[35], while it had been observed to decrease upto a certain concentration of ZnO and then increase in the case of zinc oxide glasses[37]. In the present study, Poisson's ratio showed a regular decrease with increase in concentration of both Na 0 and Fe203. 2

29 The regular variation of ultrasonic vel(-~ities and the elastic constants of the two systems of quarternary glasses investigated in the present study show that the transformation of the structural groups in these glasses to other groups is systematic and does not cause a disruption of the structure which is also supported by the Raman scattering results (Chapter 5 of thls thesis) that the Ranan peak characteristic of a continuous random network was prominently present in the spectra of all the $lass samples investigated. The ultrasonic velocity or the elastic constants do not show a decreasing trend in any of the glasses. This may be attributed to the reason that within the variation in Na 0 and Fe 0 studied, B groups do not get saturated and show a trend for the fornation of nonbridging oxygens leading to a loose structure Conclusion Ultrasonic velocity of longitudinal and transverse waves of frequency 3 ElHz has been determined in two quarternary glass systems. The elastic constants and Poisson's ratio have been evaluated. The increase in the values of ultrasonic velocity and elastic constants has been attributed to an increase in the packing density and

30 rigidity of the glass samples as a result of a transformation of the coordination of boron from 3 to 4 when the concentration of Na 0 and Fe 0 respectively in the two systems of glasses is increased. It is also concluded that the transformation of the groups constituting the structure of the glass into other groups on increasing the concentration of Na 0 or Fe 0 does not affect the rigidity of the glass so that the random continuous network of glass is maintained.

31 References 1. True11 R., Elbaurin C. and Chick B.B., Ultrasonic Eiethods in Sold State Physics, Academic Press, N.Y. (1965). ~opkins I.L. and Kukijian C.R., in Physical Acoustics, ed., Mason W.P., Vol. 2B, Academic Press, p.91 (1965). Heydernanna P.L.M., Rev. Sci. Instr., 42, 983 (1971). Kul'bitskaya I.l.N., Romanov V.P. and Shutilov V.A., Sov. Phys. Acoust., 19, 399 (1974). Bhatia A.B., Ultrasonic Absorption, Clarendon Press, Oxford (1967). Muntigton H.B., The Elastic Constants of Crystals, ed., Seitz F. and Turnbul D., "Solid State Physics", Advances in Research and Applications, Vo1.7, p.213, Academic Press, N.Y. (1958). Nye J.F. "Physical Properties of Crystals", Oxford University Press, N.Y. (1960). Bhayavantan S., "Crystal Symmetry and Physical Properties", Academic Press, New York (1966). Mearrnon R.F.S., Advan. Phy., 5, 323 (1956). 10. Hearrnon.R.F.S., "Introduction to Applied Anisotropic Elasticity", Oxford Press (1961). 11. Federov E.S., "Synrnetry of Crystals", Polycrystal Book Service (1971). 12. Kusgrave N.J.P., "Crystal Acoustics", Holden-day, San Francisco (1970). 13. 'lason W.P., "Physical Acoustics and Properties of Solids" Van Nostrand. Reinhold, Princeton, New Jersey (1950). 14. llason W.P., "Piezoelectric Crystals and their Application to Ultrasonics", Van Nostrand-Reinhold, Princeton, New Jersey (1950). 15. Ilason M.P., "Crystal Physics of Interaction Processes", Academic Press, Inc., N.Y. (1966).

32 16. Anderson P.W., Malperin B.I. and Varma C.M., Phil. Mag., 25, 1 (1972). 17. Phillip W.A., J. Low Temp. Phys., 7, 351 (1972). Ng. D and Sladek R.J., Phys. Rev., B11, 4017 (1975). Claytor T.N. and Sladek R.J., Phys. Rev. B18, 10, 5842 (19781, Kul'bitskaya Acous t., 451 (1976). M.N. and Shytilor V.A., J. Sov. Phys. Kodana M., Phys. & Chern. Glasses, 26, 105 (1985). Higazy A.A. and Bridge B., J. Non-Cryst. Solids, 81 (1985). Carini G., Cutroni M., Federico M. and Tripodo G., Solid St. Ion., 18 & 19, 415 (1986). - Kazuhira Uasata, Katsurni Ohira, Hisao Yarnada and Kazuhiro S. Goto, Metall. Trans., 18B, 549 (1987). Bridge B. and pate1 N.D., J. Mat. Sci., 22, 781 (1987). Danodaran K.V., Selvaraj V. and Rao K.J., Mater. Res. Bul., 23, 1151 (1988). Selvaraj V. and Rao K.J., J. Non-Cryst. Solids, (1988). Bhatti S.S. and Santokh Singh, Acustica, 65, 2 61 (1988). Makishima A. and Mackenzie J.D., J. Non-Cryst. Solids, 12, 35 (1975). i.lakishima A. and Mackenzie J.D., J. Non-Cryst. Solids, 17, 147 (1975). Bhatti S.S. and Anand Pal Singh, Acustica, 68, 181 (1989). Bhatti S.S. Santokh Singh, J. Pure and Appl. Ultrason, 8, 101 (1986). Anand Pal Singh, Gurjit and Bhatti S.S., 11, 49 (1989).

33 Anand Pal Singh, Kanwar Jit Singh and Bhatti S.S., J. Pure and Appl. Ultrason., 12, 70 (1990). Sidkey M.A., Abd-Ei Fattah, Abd-Ei Latif and Nakhla R.I., J. Pure and Appl. Ultrason, 12, 93 (1990). Padake S.V., Yawale S.P. and Adgaokar C.S., Proc. Int. Cong. Ultrason., C-1 (1990). Kanbrar Jit Singh, Singh D.P. and Bhatti S.S., J. Pure and Appl. Ultrason., 14 (1992). Mukherjee S., Basu C. and Ghosh U.S., J. Acoust. Soc. Ind., 3 &4, 57 (1990). Janakirama Rao BH. V., J. Am. Ceram. Soc., 18, 311 (1965). Kodama M., J. Non-Cryst. Solids, 127, 65 (1991). Anand Pal Singh and Bhatti S.S., J. Pure and Appl. Ultrason., 14, 36 (1992). Borjesson L. and Torell L.M., Phys. Lett., 107 A, 190 (1985). Biscoe J. and Warren B.E., J. Am. Ceram. Soc., 21, 287 (1938). Karsch K.M. and Jenckel E., Glastech Ber., 34, 397 (1961). Coenen PI.,, 2. Elekrochen., 65, 903 (1961). Abe, J. Am. Ceram. Soc., 35, 284 (1952). Gladkov A.V. and Tarasov V.V.,.The Structure of Glass, Vol.11, Consultants Bureau, N.Y. (1960). Konijnendijk W.L. and Stevels J.M., J. Non-Cryst. Solids, 18, 307 (1975). Pieera B.N. and Rarnakrishna J., J. Non-Cryst. Solids. 159, 1 (1993).