chapter 6 ultrasonic studies on ca0-b203-a1203-na20...
TRANSCRIPT
CHAPTER 6
ULTRASONIC STUDIES ON Ca0-B203-A1203-Na20
AND Ca0-B203-AI2O3-Fe203 GLASS SYSTEMS
CHAPTER VI
ULTRASONIC STUDIES ON Ca0-B203-A1203-Na20 and
CaO-B 0 -A1203-Fe203 GLASS SYSTEMS 2 3
6.1. 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.
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. Commonly there are
three types of elastic constants[5]. They are (i) Young's
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
'l- it 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[13-151.
Amorphous materials like glasses exhibit some unique
properties which are not usually found in other
engineering materials. These materials lack the long-
ranye 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
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.
6.2. 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
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
functions of composition in relation to the behaviour of
N4r the fraction of boron atoms in tetrahedral
coordination.
Elastic constants and structure of the glass system
Co 0 -P 0 had been determined by Higazy et a1.[21] by the 3 4 2 5
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 77-430 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
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 2 2 3 2 3
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 1987. They
concluded that a complete understanding of temperature
gradients of elastic moduli in glasses generally requires
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 3 2 5
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 a1.1271.
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].
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 the
alkaline earth silicate glasses.
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
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)B203-0.10Fe203 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 1990. 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%.
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 2 5 2 5
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.
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].
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,
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 A d 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
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].
Longitudinal modulus L = 2 "L ..... (6.9)
-~
Shear modulus G = P V s 2 ..... (6.10) Bulk modulus K = L - (4/3) G ..... (6.11)
1-2 (VS/VL) 2 Poisson's ratio 6 = -------------
2 ..... (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 0 2 3 2 2 3 2 3 2 3
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. 2 3
6.5. Experimental Details
Two systems of ylass samples 10Ca0-(75-x) B 0 -15 2 3
A1 0 -xNa 0, x varying from 15 to 24 mol% and 20 2 3 2
CaO-(70-y) B 0 - 10 A1203 - y Fe203, y varying from 2 to 2 3
8 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.
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. The
block diagram of the experimental set up (figure 2 . 5 ) 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.
6.6. Results and Discussions
Longitudinal ( V L ) 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 -Fe203 2 3 2 3
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. 2 It is seen from
figure 6.1 and 6.2 that both VL and VT increase almost
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$)
2 3 2 2
with varying concentration of P$ 3 2 3 2 3 2 3
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
Figure 6.1 Variation of longitudinal and transverse velocities in CaO-B 0 -A1 0 -Na 0 with varying concentrations20? N~;O! 2
Figure 6.2 Variation of longitudinal and transverse velocities in CaO-B 0 -A1 0 -Fe203 with varying concentration$ df ~ 6 ~ d ~ .
regularly with the concentration of Na20 or Fe203. But
the rate of increase of V is greater than that of VT for L
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 6.3 2 2 3
and 6.4). But the Poisson's ratio exhibit a reverse trend
(figure 6.5 and 6.6).
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. This
behaviour known as "boron oxide anomaly", has 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.
Figure 6.3 Variation of elastic constants in CaO-8 0 - iil 2 o 3 -Na20 with varying concentrations2 a t Na20.
Figure 6 . 4 Variation of elastic constants in CaO-B 0 - A1203-Fe203 with varying concentrations2 af Fe 0
2 3'
Figure 6.5 Variation of Poisson's ratio in CaO-B 0 - A 1 2 0 3 -Na20, with varying concentrations2 a f Na203.
Figure 6.6 Variation of Poisson's ratio in CaO-B 0 - A1203-Fe 0 with varying concentrations2 df Fe203. 2 3
Recent ultrasonic studies by Sidkey et a1.1351 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 B B 4 increases the rigidity of the glass
+ whereas the unit K ~(3~0- decreases it.
In the present ultrasonic investigations both the
longitudinal and transverse ultrasonic velocities were
found to increase with concentration of Na 0 and Fe203. 2
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 2 3 2 3
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).
In these transformations boron undergo a change from three
coordinated to four coordinate @"and it is reported that
the presence of B a 4 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
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 0 4 2 2 3
groups do not get saturated and show a trend for the
fornation of nonbridging oxygens leading to a loose
structure.
6.7. 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
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 2 2 3
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 2 2 3
affect the rigidity of the glass so that the random
continuous network of glass is maintained.
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).
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 M.N. and Shytilor V.A., J. Sov. Phys. Acous t . , 451 (1976).
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 6 1 (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).
Anand P a l S i n g h , Kanwar J i t S i n g h a n d B h a t t i S.S., J. P u r e and Appl . U l t r a s o n . , 1 2 , 70 ( 1 9 9 0 ) .
S i d k e y M . A . , Abd-Ei F a t t a h , Abd-Ei L a t i f a n d N a k h l a R . I . , J . P u r e a n d A p p l . U l t r a s o n , 1 2 , 9 3 ( 1 9 9 0 ) .
P a d a k e S.V., Yawale S .P . a n d Adgaokar C.S. , P r o c . I n t . Cong. U l t r a s o n . , C - 1 ( 1 9 9 0 ) .
Kanbrar J i t S i n g h , S i n g h D.P. a n d B h a t t i S .S . , J . P u r e a n d Appl . U l t r a s o n . , 1 4 ( 1 9 9 2 ) .
M u k h e r j e e S . , Basu C. a n d Ghosh U.S., J . A c o u s t . S o c . I n d . , 3 & 4 , 57 ( 1 9 9 0 ) .
J a n a k i r a m a Rao BH. V . , J. Am. Ceram. Soc . , 1 8 , 311 ( 1 9 6 5 ) .
Kodama M . , J . Non-Crys t . S o l i d s , 1 2 7 , 6 5 ( 1 9 9 1 ) .
Anand P a l S i n g h a n d B h a t t i S . S . , J . P u r e a n d A p p l . U l t r a s o n . , 1 4 , 3 6 ( 1 9 9 2 ) .
B o r j e s s o n L. a n d T o r e l l L .M. , P h y s . L e t t . , 1 0 7 A , 1 9 0 ( 1 9 8 5 ) .
B i s c o e J. a n d W a r r e n B.E., J . Am. Ceram. S o c . , 2 1 , 287 ( 1 9 3 8 ) .
K a r s c h K.M. a n d J e n c k e l E . , G l a s t e c h B e r . , 3 4 , 397 ( 1 9 6 1 ) .
Coenen PI. ,, 2 . E l e k r o c h e n . , 6 5 , 903 ( 1 9 6 1 ) .
Abe, J . Am. Ceram. S o c . , 3 5 , 284 ( 1 9 5 2 ) .
G l a d k o v A.V. a n d T a r a s o v V . V . , . T h e S t r u c t u r e o f Glass , V o l . 1 1 , C o n s u l t a n t s B u r e a u , N . Y . ( 1 9 6 0 ) .
K o n i j n e n d i j k W.L. a n d S t e v e l s J . M . , J . Non-Crys t . S o l i d s , 1 8 , 307 ( 1 9 7 5 ) .
Pieera B.N. a n d R a r n a k r i s h n a J . , J. Non-Crys t . S o l i d s . 1 5 9 , 1 ( 1 9 9 3 ) .