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.

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