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CChhaapptteerr 11
A BRIEF REVIEW ON THE TRIGONAL CRYSTALS
CaCO3, Bi AND LiNbO3 AND AN INTRODUCTION
TO THE FINITE STRAIN ELASTICITY THEORY
1.1 Introduction
The elastic properties are basic cohesive properties related to
the anharmonicity of crystal lattices. Elastic constants also provide
insight into the nature of binding forces between atoms, since they are
represented by the derivatives of internal energy. A complete set of
higher order elastic constants of materials is essential to estimate
physical properties such as thermal expansion, specific heat, Debye
temperature, compressibility and acoustic anisotropy. Third-order elastic
constants play an important role in the analysis of the non-linear effects in
finite amplitude acoustic waves such as second harmonic generation,
acoustical mixing and parametric oscillation.
The present objective is to study the vibrational anharmonicity of
long wavelength acoustic modes of the trigonal crystals; Calcite
(CaCO3), Bismuth (Bi) and Lithium Niobate (LiNbO3). Also here we
make an attempt to calculate the complete set of second and third-
order elastic constants. The trigonal class of crystals has seven second-
order elastic constants and fourteen third-order elastic constants.
2
Pressure derivatives of the second-order elastic constants and
generalized Gruneisen parameters of elastic waves are determined.
Low temperature lattice thermal expansions of these trigonal crystals
are also obtained. In this chapter, we present an overview of the
physical properties exhibited by the above trigonal crystals along with
the elasticity studies done so far.
1.2 Calcite
CaCO3 is one of the most abundant minerals in the environment
and of fundamental importance. Calcite, which gets its name from
"chalix" the Greek word for lime, is a most amazing and yet, one of the
most common minerals on the face of the Earth, comprising about 4%
by weight of the Earth's crust and is formed in many different
geological environments. Colour is extremely variable but generally
white or colourless or with light shades of yellow, orange, blue, pink,
red, brown, green, black and grey. Lustre is vitreous to resinous to dull
in massive forms. Crystals are transparent or translucent. Double
refraction, fluorescence, phosphorescence and thermo-luminescence
are some of the important properties of calcite. Although not all
specimens demonstrate these properties, some do quite well and is
diagnostic in some cases. Specific gravity is approximately 2.7 and
refractive indices are 1.49 and 1.66 causing a significant double
refraction. Being the building block of shells and skeletons, it is used
as scaffolding material for growth of a variety of cells and even for
3
bone and cartilage auto-grafts [1-3]. It is used as a carbon isotope
counter in meteorology [4]. Its strong interaction with heavy metals makes
it favourite in environment management [5, 6], energy storage [7] and
industry [8, 9]. It is used in the preparation of organic-inorganic nano-
composites [10]. Its special properties of double refraction and thermo-
luminescence find its effective applications in the fields of optical and
geological device fabrication and fossil dating [11, 12]. The dielectric
behaviour of these crystals is also of much interest [13]. Due to its low
work function, low cost and enhanced chemical stability it is
extensively used in electron field effect displays [14].
There are several compounds that crystallize with calcite
structure at ambient conditions. The trigonal unit cell is defined by the
lattice parameters a = 6.375 and α = 46o
5’. The calcite-structure
carbonates (space group c3R_
) represent a mineral group that is
structurally simple and different from oxides and silicates; the slightly
distorted octahedra are exclusively corner-linked through shared
oxygen anions of CO3 groups [15, 16]. In CaCO3 there are six different
ion-pair types, but the carbon atoms are deeply buried inside oxygen
triads. So the carbon atoms will not be considered explicitly and the
number of different ion pairs is reduced to three: metal ion - metal ion
(M-M), metal ion - oxygen ion (M-O) and oxygen ion - oxygen ion
(O-O) [17].
4
The elastic constants of CaCO3 have been determined by the static
flexture and torsion method by Vogit [18] and from the measurements of
resonant frequencies of oriented plates by Bhimsenachar [19]. Reddy and
Subrahmanyam [20] determined the elastic constants by composite piezo-
electric oscillator method and Peselnick and Robie [21] by the pulse-echo
method. Dandekar [22, 23] also determined the elastic constants by the
pulse-echo method. The values of the elastic constants obtained by these
investigators vary widely. A large variation in the estimated values of elastic
constants of a substance may arise from several sources. These sources
may be conveniently grouped into (a) error due to physical effects, e.g.,
structural imperfections, orientation uncertainty, geometric feature of the
specimen, (b) error in the measurement of the variables like specimen
thickness, density of specimen, temperature fluctuations etc. and (c) error
due to a limitation inherent in a technique, e.g., frequency measurement in
the phase-comparison method, a travel-time measurement in the pulse-
echo method.
Thermal expansion and elastic properties of single crystal
CaCO3 are extensively studied using ultrasonic techniques [22-25],
Brillouin spectroscopy [26] and other techniques [27-30].
1.3 Bismuth
Bi is an interesting semimetal because of its numerous
applications in the different fields of device fabrication, nanotechnology,
5
superconductivity, etc. Bi is a rare and strange semimetal and an
element belonging to the fifth main group of the periodic table, where we
will find nitrogen, phosphorous, arsenic, antimony and bismuth. There is
only twice as much Bi available as gold.
The specific resistance is unusually high. It also shows the
strongest diamagnetism of all metals and the strongest Hall effect. Very
unusual and very useful is the fact that solid Bi has a lower specific
gravity compared to liquid Bi. So solid Bi floats on its own melt like ice
on water and expands on crystallizing. This effect is furthermore known
for water (ice), gallium and germanium. Solid Bi is brittle at room
temperature and occurs in nature as a mineral. Bi is of atomic weight
208.98, melting point 271.4oC, boiling point 1564
oC, specific gravity
9808kg/m3 and specific electrical resistance 1.20µΩ-m. Its thermal
conductivity is the lowest of all metals. Its thermo-electric properties
render it especially valuable for the construction of thermopiles. Bi readily
forms alloys with other metals. The salts of Bi are feebly antiseptic. Its
high magneto-resistance, low carrier density and high purity make it
the most popular material for studies on quantum magnetic field effects
[31]. Because of its high anisotropy in the electronic behaviour, low
conduction band, high electron mobility and potential for inducing a
semiconductor transition, it is widely used in electronics and
nanotechnology [32, 33]. Bi also has potential applications in the effective
6
fabrication of high performance thermoelectric devices due to their
remarkable transport properties [34, 35]. Superconductor-insulator
transitions in ultra thin films of amorphous Bi by the electron field effect
are of special interest in the advanced studies on quantum phase
transitions [36]. It is a rhombohedral (trigonal) crystal (space group
c3R_
), which shows many features typical of layer like crystals [37-39].
The rhombohedral unit cell is defined by the lattice parameters
a= 4.746 and α = 57o 14’.
The second-order and third-order elastic constants of Bi have
been determined from the pressure dependence of ultrasonic wave
velocities by Hailing and Saunders [37]. Eckstein et al. [40] also have
measured the second-order elastic constants. Hydrostatic pressure
derivatives of effective second-order elastic constants of Bi are
determined by the ultrasonic method. Pressure dependence of the
velocities of the longitudinal and transverse ultrasonic waves are
studied by Vornov and Stal’gorova [41]. The Gruneisen parameters
along with the low temperature thermal expansion of Bi are studied by
Bunton and Weintroub [42] and White [43].
1.4 Lithium Niobate
LiNbO3 has received much attention in recent years because of
its extensive applications in the field of device fabrication as well as
material characterization. The stability of the phase over a wide range
7
of temperature and optical anisotropy explore its use as an efficient
ferroelectric material [44, 45]. It is also used in the field of electro and
elasto-optics because of its large electro-mechanical coupling
coefficients [46-48]. An acoustical tone burst in the crystal makes it
special in the field of acoustics and ultrasonics [48-50]. LiNbO3 finds its
application also in the field of holographic imaging, optical waveguides
and modern optical parametric oscillators [51-53]. Its large
spontaneous polarization and non-linear optical activity make it
favourite in the thermal, electrical and optical areas [54-58]. Its
superior piezo-electric performance makes it a potential candidate for
replacing quartz. The knowledge of higher order elastic constants is
essential for the study of anharmonic properties of LiNbO3. LiNbO3
exhibits a perovskite structure having two phases of trigonal symmetry
(space groups R3c and R3c) [58-60]. The unit cell is defined by the
lattice parameters a = 5.492 and α = 55o 53’.
Warner et al. [61] have determined the second-order elastic
constants Cij of LiNbO3 at room temperature using resonance
techniques. Smith and Welsh [62] have reported that the elastic constants
decrease linearly with temperature in the range 273K and 383K. Tomeno
and Matsumura [63] have measured elastic constant C33 as a function of
temperature. C33 decreases linearly with increasing temperature up to
1080K. Its value is 230 GPa at 1000K. Elastic constants C11, C44 and
8
C66 are determined using resonance techniques, while C33, C13 and C14
are obtained from the combination of ultrasonic, piezo-electric and
dielectric measurements. Elastic constants C11, C44 and C66 determined
at room temperature are reported by Nakagawa and Yamanouchi [64].
The lattice properties of LiNbO3 are calculated earlier using a potential
model suggested by Donnerberg et al. [65-67]. This model cannot
interpret and reproduce the available experimental data. A new interatomic
potential based on a fully ionic description of the material is used by
Jackson and Valerio [44] to calculate the lattice properties. They have
studied the lattice properties including the elastic constants and
dielectric constants as well as powder X-ray diffraction patterns of both
phases. All the elastic constants have been determined from
specimens prepared from Z-cut plates only, using a series resonance
method by Damle [68].
Cho and Yamanouchi [69] have determined all the fourteen third-
order elastic constants of the LiNbO3 crystal at room temperature from
measured values of the velocity variation of small amplitude ultrasonic
waves. Nakagawa and Yamanouchi [64] have determined all the fourteen
elastic constants of the congruent crystal at room temperature using the
same method [69] but they have determined the constants without
correction. On the other hand Philip and Breazeale [70] have determined
C111 by measuring the amplitude of the second-harmonic wave of the
longitudinal wave propagating along the X-axis. In the case of
9
determining C111, this method does not need correction because the
longitudinal wave propagating along the X-axis has no electro-mechanical
coupling coefficient. The larger magnitudes of C111 (as well as C222 and
C333) appear to be more valid [64, 69, 70].
Kamal Singh et al. [54] have determined the thermal expansion
coefficient of LiNbO3 single crystal by Newton’s rings experiment.
Takanaga and Kushibiki [71] have determined the acoustical constants of
LiNbO3 using line-focus beam acoustic microscopy. Ogi et al. [72] have
calculated the lattice properties of LiNbO3 including the elastic constants,
piezo-electric coefficients and dielectric coefficients using acoustic
spectroscopy.
1.5 The Theory of Elasticity
Consider an elastic medium where the co-ordinates of any point
can be denoted as 1 2 3(a ,a ,a ) . Choose a set of orthonormal vectors e1, e2
and e3 as the basis vectors for the coordinate system and denote the
kth component of the stress acting on the plane ei = 0 by ik where i
and k are the component indices. Consider the equilibrium of a small
element centred at the point ia and bounded by the plane
. Let
ui denotes the elastic displacement of the point ia of the body and
the density of this point. The equation of motion can be derived by
considering the total force acting on the volume element. If we ignore
the body forces, the equation of motion of an elastic solid can be
10
written as (the convention that repeated indices indicate summation
over the indices, will be followed here)
(1.01)
here the stress tensor
φ
(1.02)
where φ is the crystal potential and ik are the components of the
strain tensor given by
(1.03)
here ik and ik are symmetric tensors of second rank.
According to Hooke’s law
(1.04)
The constants iklmC form a fourth rank tensor with 34 components.
From equations (1.02) and (1.04), we have
φ φ
(1.05)
Hence the elastic constants iklmC are multiple strain derivatives of the
state functions and since the strains lm are symmetric, the elastic
constants possess complete Voigt’s symmetry. Thus,
(1.06)
11
These quantities are symmetric with respect to the interchange of the
subscripts. It will be convenient to abbreviate the double subscript
notation to the single subscript Voigt’s notation running from 1 to 6,
according to the following scheme
; ; ; ; and .
Hence the matrix of elastic constants iklmC would contain a 6 x 6 array
of 36 independent quantities in the most general case. This number is,
however, reduced to 21 by the requirements that the matrices be
symmetric on interchange of double indices. The number of independent
elastic constants will be further reduced by the symmetry of the
respective crystal classes. The three trigonal crystals; CaCO3, Bi and
LiNbO3 belong to the classes which have seven
independent second-order elastic constants [73]. Elastic constant
matrix for this class of compounds is given by
11 12 (C C )
−−−−
−−−−
−−−−
(1.07)
In the equation of motion for an elastic medium, the forces on an
element of volume are given by the divergence of the stress field.
12
Using equations (1.03) and (1.04), the equation (1.01) can be written as
(1.08)
For an elastic plane wave we have
( ) −−−− (1.09)
where kA are the components of the amplitude of vibration, is the
angular frequency and k the wave vector corresponding to π2
=k
. The
resulting equations of motion from equation (1.08) are
( )−−−− (1.10)
substituting ˆ , where n is the unit vector, we get
( ) −−−− (1.11)
where
are the reduced elastic constants and v is the phase
velocity given by
. The components of second rank tensor are
given by
(1.12)
hence equation (1.11) can be written as
( ) (1.13)
13
This shows that u is the eigen vector of tensor where eigen
value is . Hence is the root of the equation
(1.14)
This is the Christoffel equation. The theory of elastic waves
generally reduces to find u and v for all plane waves propagating in an
arbitrary direction for crystals possessing different symmetries. In this
situation, all terms in equation (1.11), which involve differentiation with
respect to the co-ordinates other than that along the propagation
direction, drop out.
A more fundamental significance to the second-order elastic
constants is implied by their appearance as the second derivatives of
elastic energy with respect to strains. It should be noted that the stored
elastic energy is only a part of the complete thermodynamic potential
of the crystal, since it depends on many other variables. Also, one can
introduce elastic constants as a constitutive, local relation between
stress and strain for materials in which long-range atomic forces are
unimportant.
1.6 Finite Strain Theory of Elasticity
In the finite strain elasticity theory [74] the three states of a
crystal are defined as
1. Natural State: when there is no stress upon a crystal, it is said to
be in the natural state.
14
2. Initial State: when a finite stress is applied on a crystal, it is said
to be in the initial state or undeformed state.
3. Final State: when an infinitesimal strain is superimposed on a
finite strain by applying an infinitesimal stress, the crystal is said
to be in the final state or deformed state.
Let the position co-ordinates of a material particle in the unstrained
state be . Let the co-ordinates of a material particle in the
deformed state be ix . Consider two material particles located at ia and
. Let their co-ordinates in the deformed state be ix and .
The elements are related to ida by the equation
( )
(1.15)
The convention that repeated indices indicate summation over the
indices will be followed here. ij is the Kronecker delta and ij are the
deformation parameters. The Jacobian of the transformation
(1.16)
is taken to be positive for all real transformation. If adV is the volume
element in the natural state and xdV its volume after deformation
(1.17)
15
where 0 and are the densities in the natural and strained states
respectively. Let the square of the length of arc from ia to be
20dl in the unstrained state and 2dl in the strained state. Then
− −− −− −− −
= −−−−
= jk j k2 da da (1.18)
where ik are the Lagrangian strain components which are symmetric
with respect to the interchange of the indices j and k . In terms of ik ,
(1.19)
The internal energy function jkU(S, ) for the material is a
function of the entropy S and Lagrangian strain components. U can be
expanded in powers of the strain parameters about the unstrained
state as
……..
or
…….. (1.20)
16
The linear term in strain is absent because the unstrained state is one
where U is minimum. We shall define the elastic constants of different
orders referred to the unstrained state as [75].
(1.21)
and
(1.22)
The subscript ‘o’ means the coefficients have to be evaluated in the
undeformed state and S is the entropy. Here the derivatives are to be
evaluated at equilibrium configuration and constant entropy. ij,klC and
ij,kl,mnC are the adiabatic elastic constants of second and third-orders
respectively. They are tensors of fourth and sixth ranks. The number of
independent second-order and third-order elastic constants for different
crystal classes are tabulated by Bhagavantam [73]. There exist various
theoretical [76-79] as well as experimental methods [80-84] for the
determination of higher order elastic constants of solids.
References:
1. D. Nave, S. Rosenwaks, R. Vago and I. Bar, J. Appl. Phys. 95 (2004)
8309.
2. D. Beruto and M. Giordani, J. Chem. Soc., Faraday Trans. 89 (1993) 2457.
3. K. M. Beck, D. P. Taylor and W. P. Hess, Phys. Rev. B 55 (1997) 13253.
4. C. S. Romanck, E. L. Grossman and J. W. Morse, Geochim. Cosmochim.
Acta 56 (1992) 419.
17
5. S. L. Stipp and M. F. Hochella, Geochim. Cosmochim. Acta 55 (1991)
1723.
6. N. S. Park, M. W. Kim, S. C. Langford and J. T. Dickinson, J. Appl. Phys.
80 (1996) 2680.
7. D. Chakraborty and S. K. Bhatia, Ind. Chem. Res. 35 (1996) 1995.
8. P. M. Dove and M. F. Hochella, Geochim. Cosmochim. Acta 57 (1993) 705.
9. N. H. de Leeuw, S. C. Parker and J. H. Harding, Phys. Rev. B 60 (1999)
13792.
10. C. Xiong, S. Lu, D. Wang, L. Dong, D. D. Jiang and Q. Wang, Nanotechnology
16 (2005) 1787.
11. Z. Shao, Phys. Rev. E 52 (1995) 1043.
12. J. F. de Lima, M. E. G. Valerio and E. Okuno, Phys. Rev. B 64 (2001)
014105.
13. N. Bogris, J. Grammatikakis and A. N. Papathanassiou, Phys. Rev. B 58
(1998) 10319.
14. D. Zhou, A. R. Krauss and D. M. Gruen, J. Appl. Phys. 82 (1997) 4051.
15. N. L. Ross, Am. Mineral. 82 (1997) 682.
16. J. Zhang and R. J. Reeder, Am. Mineral. 84 (1999) 861.
17. P. W. Bridgeman, Am. J. Sci. 10 (1925) 483.
18. W. Voigt, Lehrbuch der Kristallphysik B. G. Teubner, Berlin, (1910) 754.
19. J. Bhimsenachar, Proc. Indian Acad. Sci. 22 (1945) 199.
20. P. J. Reddy and S. V. Subrahmanyam, Acta Cryst. 13 (1960) 493.
21. L. Peselnick and R. A. Robie, J. Appl. Phys. 33 (1962) 2889.
22. D. P. Dandekar, Communications (1968) 2971.
23. D. P. Dandekar, J. Appl. Phys. A 39 (1968) 3694.
24. R. F. S. Hearmon, The Elastic Constants of Crystals and other Anisotropic
Materials, In K. H. Hellwege, and A. M. Hellwege, Eds., Landolt-Bernstein
Tables, III / 11, Springer-Verlag, Berlin, (1979).
18
25. D. Vo. Thanh and A. Lacam, Physics of the Earth and Planetary interiors
34 (1984) 195.
26. C. C. Chen, C. C. Lin, L. G. Liu, S. V. Sinogeikin and J. D. Bass, Am.
Mineral. 86 (2001) 1525.
27. A. Pavese, M. Catti, S. C. Parker and A. Wall, J. Phys. Chem. Minerals 23
(1996) 89.
28. H. Kaga, Phys. Rev. 172 (1968) 900.
29. R. Ramji Rao and A. Padmaja, J. Appl. Phys. 62 (1987) 440.
30. S. Sarkar, T. K. Ballabh, T. R. Middya and A. N. Basu, Phys. Rev. B 54 (1996)
3926.
31. X. Du, S. W. Tsai, D. L. Maslov and A. F. Hebard, Phys. Rev. Lett. 94
(2005) 166601.
32. L. Balan, R. Schneider, D. Billaud, Y. Fort and J. Ghanbaja, Nanotechnology
15 (2004) 940.
33. P. Chiu and I. Shih, Nanotechnology 15 (2004) 1489.
34. T. W. Cornelius, J. Brotz, N. Chtanko, D. Dobrev, G. Miehe, R. Neumann
and M. E. Toimil Molares, Nanotechnology 16 (2005) S426.
35. A. D. Grozav and E. Condrea, J. Phys. Condens. Matter 16 (2004) 6507.
36. K. A. Parendo, K. H. Sarwa, B. Tan, A. Bhattacharya, M. Eblen- Zayas,
N. E. Staley and A. M. Goldman, Phys. Rev. Lett. 94 (2005) 197004.
37. Tu Hailing and G. A. Saunders, Phil. Mag. A 48 (1983) 571.
38. M. Bastea, S. Bastea, J. A. Emig, P. T. Springer and D. B. Reisman, Phys.
Rev. B 71 (2005) 180101.
39. C. R. Ast and H. Hochst, Phys. Rev. B 67 (2003) 113102.
40. Y. Eckstein, A. W. Lawson and D. H. Reneker, J. Appl. Phys. 31 (1960)
1534.
41. F. F. Voronov and O. V. Stal’gorova, Sov. Phys. Solid State 33 (1991) 223.
42. G. V. Bunton and S. Weintroub, J. Phys. C Solid State Phys. 2 (1969) 116.
43. G. K. White, J. Phys. C Solid State Phys. 2 (1968) 575.
19
44. R. A. Jackson and M. E. G. Valerio, J. Phys. Condens. Matter 17 (2005) 837.
45. D. A. Scrymgeour, V. Gopalan, A. Itagi, A. Saxena and P. J. Swart, Phys. Rev.
B 71 (2005) 184110.
46. M. Jazbinsek and M. Zgonik, Appl. Phys. B 74 (2002) 407.
47. S. Kakio and Y. Nakagawa, J. Appl. Phys. 34 (1995) 2917.
48. O. Beyer, D. Maxein, K. Buse, B. Sturman, H. T. Hsieh and D. Psaltis,
Phys. Rev. E 71 (2005) 056603.
49. M. S. McPherson, I. Ostrovskii and M. A. Breazeale, Phys. Rev. Lett. 89
(2002) 115506.
50. J. Kushibiki, M. Arakawa and R. Okabe, IEEE Trans. Ultrason., Ferroelect.,
Freq. Contr. 49 (2002) 827.
51. J. Zhao, J. G. Wang, G. D. Liu, Q. S. He, M. X. Wu and G. F. Jin, Chin.
Phys. Lett. 20 (2003) 377.
52. P. Zhang, D. X. Yang, J. L. Zhao, K. Su, J. B. Zhou, B. L. Li and D. S.
Yang, Chin. Phys. Lett. 21 (2004) 1558.
53. K. A. Tillman, R. R. J. Maier, D. T. Reid and E. D. McNaghten, J. Opt. A:
Pure Appl. Opt. 7 (2005) S 408.
54. K. Singh, P. V. Saodekar and S. S. Bhoga, Mater. Sci. 21 (1998) 469.
55. H. Chaib, T. Otto and M. Eng, Phys. Rev. B 67 (2003) 174109.
56. M. R. Chowdhury, G. E. Peckham, R. T. Ross and D. H. Saunderson, J. Phys.
C: Solid State Phys. 7 (1974) 99.
57. D. Xue, N. Iyi and K. Kitamura, J. Appl. Phys. 92 (2002) 4638.
58. D. Xue and K. Kitamura, Ferroelectrics 297(2003) 19.
59. M. Veithen and P. Ghosez, Phy. Rev. B 65 (2002) 214302.
60. H. Boysen and F. Altorfer, Acta Crystallogr. Sect. B: Struct. Sci. B 50 (1994)
405.
61. A. W. Warner, M. Onoe and G. A. Coquin, J. Acoust. Soc. Amer. (USA) 41
(1967) 1223.
62. R. T. Smith and F. S. Welsh, J. Appl. Phys. (USA) 42 (1971) 2219.
20
63. I. Tomeno and S. Matsumura, J. Phys. Soc. Jpn. (Japan) 56 (1987) 163.
64. Y. Nakagawa, K. Yamanouchi and K. Shibayama, J. Appl. Phys. 44 (1973)
3969.
65. H. Donnerberg, S. M. Tomlinson, C. R. A. Catlow and O. F. Schirmer, Phys.
Rev. B 40 (1989) 11909.
66. S. M. Tomlinson, C. R. A. Catlow, H. Donnerberg and M. Leslie, Mol.
Simul. 4 (1990) 335.
67. S. M. Tomlinson, C. M. Freeman, H. Donnerberg and C. R. A. Catlow, J. Chem.
Soc. Faraday Trans. 85 (1989) 367.
68. R. V. Damle, J. Phys. D 25 (1992) 1091.
69. Y. Cho and K. Yamanouchi, J. Appl. Phys. 61 (1987) 875.
70. J. Philip and M. A. Breazeale, IEEE US Symp. Proc. 2 (1982) 1022.
71. I. Takanaga and J. Kushibiki, IEEE Trans. Ultrason., Ferroelect., Freq.
Contr. 49 (2002) 893.
72. H. Ogi, Y. Kawasaki, M. Hirao and H. Ledbetter, J. Appl. Phys. 92 (2002)
20451.
73. S. Bhagavantam, Crystal Symmetry and Physical Properties. Academic
Press Inc. London Ltd, (1966) 135.
74. F. D. Murnaghan, Finite Deformation of an Elastic Solid, Wiley and Sons
Inc., New York, (1951) 46.
75. K. Brugger, Phys. Rev. A 133 (1964) 1161.
76. R. N. Thurston and K. Brugger, Phys. Rev. A 133 (1964) 1604.
77. R. Srinivasan and R. Ramji Rao, J. Phys. Chem. Solids 32 (1971) 1769.
78. A. Batana and I. Gomez, Phys. Stat. Sol.(b) 166 (1991) K 81.
79. S. Matar, V. Fonnet and G. Demazeau, J. Physique 4 (1994) 335.
80. K. C. Goretta, D. S. Kupperman, S. Majumdar, M. W. Such and Norinutsu
Murayama, Supercond. Sci. Technol. 11 (1998) 1409.
81. M. S. Kala and J. Philip, Indian J. Phys. 71 A (1997) 117.
21
82. A. Migliori, J. L. Sarrao, William M. Visscher, T. M. Bell, Ming Lei, Z. Fisk and
R. G. Leisure, Physica B 183 (1993) 1.
83. Willis, R. G. Leisure and T. Kanashiro, Phys. Rev. B 54 (1996) 9077.
84. G. A. Saunders, C. Fanggao, Li Jiaqiang, Q. Wang, M. Cankurtaran,
E. F. Lambson, P. J. Ford and D. P. Almond, Phys. Rev. B 49 (1994) 9862.