Chapter 1

Structural Phase Transitions: An

Introductory Note

1

2

1.1 Introduction

Phase transitions are exhibited by a wide variety of systems from simple metals and

alloys to complex inorganic and organic materials. Phase transition is an area of great

academic interest and has proved to be of technological importance in material science

[1-6]. Various aspects such as critical phenomena, soft modes, mechanisms and changes

in properties at the phase transition have been discussed in several reviews and books [7-

12]. Systems undergoing transitions are routinely discovered and study of their physico-

chemical properties lead to the development of new materials.

1.2 Definition

Any phase is characterized by thermodynamic properties like volume, pressure,

temperature and energy. An isolated phase is stable only when its free energy is a

minimum for the specified thermodynamic condition. The system is said to be in a

metastable phase if it exhibits several local minima along with the global minimum and

settles in one of the local minima [9]. As the temperature, pressure or any other variable

like electric field or magnetic field acting on a system is varied, the free energy of the

system changes continuously. Whenever such variations of free energy are associated

with changes in structure of the phase (i.e. either electronic or spin configuration), a

phase transformation is said to occur [10].

1.3 Thermodynamic Considerations

Changes in quantities like entropy and volume which represent discontinuities in the first

derivative of Gibbs’ free energy follow the Clausius Clayperon equation [10]. These

phase changes are referred to as first order transitions and are generally brought about by

variation in temperature or pressure. In a second order transition, there is a discontinuity

in the second derivative of the free energy such as heat capacity and compressibility.

These include transitions where the heat capacity tends towards infinity at the transition

temperature and are referred to as -transitions. Every phase transition is associated with

a change in symmetry as well as in order. The concept of order parameter was

introduced by Landau [13]. The average value of vanishes above Tc (critical

temperature) in a second order transition. Ubbelohde [14] has classified phase transitions

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broadly into two groups, continuous and discontinuous. In the former, there is no

discontinuity in enthalpy at Tc and the crystal structure changes continuously from one

polymorph to another.

1.4 Kinetic Considerations

Phase transition in many solids occur through the process of nucleation and propagation,

each of these processes being associated with specific activation energy. The formation

of the product phase in the matrix of the parent phase is called nucleation. This requires

higher activation energy than the propagation step [15,16]. The theory of nucleation has

been invoked to satisfactorily to explain the kinetics of phase transition. The

transformation in TiO2 from anatase to the rutile form is an example of a nucleation-

growth mechanism [16]. The empirical relation developed by Avrami [17] is an example

of a kinetic expression to measure the overall rate of transformation.

1.5 Structural Considerations

While the thermodynamic treatment of phase transition is very fundamental and useful, it

does not provide a structural insight into the microscopic changes accompanying a

transition. Thus, an essential part of the study of a phase transition in solids [18,19]

involves a detailed understanding of crystal chemistry in terms of atomic arrangements

and bonding. A new phase obtained after a transition may be related to the parent phase

in more than one way. The transition may be accompanied by a change in the primary

coordination and/or secondary coordination and/or there could be major changes in the

electronic structure and/or bond type. Hence, a detailed study of the structures of the

parent and transformed phases, especially in terms of their orientational relation, becomes

important to understand the transition mechanism.

Buerger [20,21] classified phase transitions on the basis of structural changes involving

primary or higher coordination as follows:

(i) Transformation involving first coordination

(a) Reconstructive (sluggish)

(b) Dilatational (rapid)

(ii) Transformations involving second or higher coordination

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(a) Reconstructive (sluggish)

(b) Displacive (rapid)

Reconstructive transitions involve a major reorganization of the crystal structure, in

which many bonds are broken and new bonds formed. An example is the reversible

transition from graphite to diamond involving a complete change in crystal structure.

Reconstructive transitions usually have high activation energies due to bond breaking and

making and therefore proceed slowly. Transformations involving second coordination are

reconstructive only if the mechanism involves breaking and forming of bonds of first

coordination like in the different polymorphs of silica [10].

In dilatational transitions the extent of bond breaking is much less than for reconstructive

type with no intermediate state of high energy and hence transition rates are rapid.

Buerger proposed a simple mechanism for the transformation of CsCl or NH4Cl from the

CsCl structure to the NaCl structure [22] as shown in figure 1.1.

Figure1.1: Dilatational mechanism for the transformation from CsCl structure to NaCl

structure

Displacive phase transitions involve the distortion of bonds and the structural changes

that occur are usually small. Hence, these transitions take place with zero or very small

activation energies, and a symmetry relationship usually exists between the parent and

the transformed phase. Figure 1.2 illustrates the distinction between reconstructive and

displacive phase transitions [10]. In order to convert structure A into any of the other

structures B, C and D, bond breaking is essential and hence the transition is

reconstructive. On the other hand, interconversions between structures B, C and D

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involve only small rotational movements and no bond breaking. These transitions are

hence displacive in nature.

Figure 1.2: Transformation from structure A to any other structure requires the breaking

of first coordination bonds. Transformation among B, C and D are only distortions.

Structural transitions are also classified into ferrodistortive (representing either no

change in the number of formula units in the unit cell) or antiferrodistortive (change in

the number of formula units in the unit cell) [7]. Ferrodistortive transitions can however

be displacive or of order-disorder type . Such transitions are exhibited only by

ferroelectric materials. Antiferrodistortive displacive transitions are shown by both

ferroelectric and antiferroelectric materials.

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1.5.1 Order-disorder transitions

It is known based on third law of thermodynamics that only at zero Kelvin perfect order

in solids is realized and the extent of order or disorder in a system thus becomes relevant

at any other temperature. These transitions are generally classified as (i) positional

disorder (ii) orientational disorder (iii) disorder of electronic (or nuclear) spin states. The

configurational entropy due to disorder is given by

S =Rln (II/I)

where I and II are the total number of configurations in the product and parent phases

respectively. An example of positional disorder is AgI, which undergoes a transition from

a hexagonal to a cubic phase, resulting in a high ionic conductivity due to the

randomization of Ag+ ions in the higher symmetry system.

1.5.2 Martensitic transitions

Although this type of transformation was originally discovered in steel, they were

considered to provide the mechanism for transitions in a variety of inorganic solids. A

martensitic [23] transition is a structural change caused by atomic displacements

corresponding to a homogeneous deformation which gives rise to an invariant strain

plane. At this plane the parent and product phases are related by a precise orientational

relationship [23].

1.6 Material Properties and Phase transition

The occurrence of a phase transition in a solid alters several material properties.

Measurement of a particular property across the phase transition using sophisticated

techniques provides insights into the nature of phase transition. Indeed, such changes in

properties result in the development of new materials of technological interest. The

following sections discuss various types of phase transitions and their correlation with

material properties such as ferroelectricity and ferroelasticity.

1.6.1 Phase transitions in ferroics

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A ferroic may be defined as a material possessing two or more orientation states or

domains, which can be switched from one to another by the application of appropriate

forces [24,25]. In a ferroelectric, the orientation state of spontaneous electric polarization

can be altered by the application of an electric field and in a ferromagnet, the orientation

state of magnetization in domains can be switched by the application of a magnetic field

while in a ferroelastic, the direction of spontaneous strain in a domain can be switched by

the application of a mechanical stress. Examples for these materials are BaTiO3

(ferroelectric), CrO2 (ferromagnetic) and CaAl2Si2O8 (ferroelastic). The electric

polarization, magnetic polarization and elastic strain are the properties for which

directionality changes in the above examples. Ferroics are classified into primary and

secondary ferroics depending on the nature of the switchable property. Table 1.1 lists the

different types of ferroic effects [7]. In primary ferroics the switchability involves the

property (for example electric polarization) while in secondary ferroics, the switchability

occurs on the derivative of the property (for example dielectric susceptibility).

At hig temperature, ferroelectric materials transform to the paraelectric state,

ferromagnetic to the paramagnetic state and ferroelastic to the paraelastic state. The

transitions are characterized through order parameters [9] which are specific properties

parameterized in such a way that the resulting quantity is unity for the ferroic state, below

Tc and zero in the nonferroic phase beyond Tc. Whenever transitions are governed by the

expected variations of these order parameters they are called proper ferroics or else the

materials are termed as improper ferroics. Polarization, magnetization and strain are the

proper order parameters for the ferroelectric, ferromagnetic and ferroelastic transitions

respectively. Newnham and Cross have proposed a hexagonal representation (figure 1.3)

of proper and improper primary ferroics [25]. The order parameter for proper ferroics

appears on the diagonals of the hexagon, while the sides of the hexagon represent

improper ferroics.

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Table 1.1

Primary and secondary ferroics

Ferroic property Switching field Example

Primary ferroics

Ferroelectric Spontaneous

polarization

Electric field BaTiO3

Ferromagnetic Spontaneous

magnetization

Magnetic field CrO2

Ferroelastic Spontaneous

strain

Mechanical

stress

CaAl2Si2O8

Secondary ferroics

Ferrobielectric Dielectric

susceptibility

Electric field SrTiO3

Ferrobimagnetic Magnetic

susceptibility

Magnetic field NiO

Ferrobielastic Elastic

compliance

Mechanical

stress

-quartz

Ferroelasoelectric Piezoelectric

coefficients

Electric field and

mechanical stress

NH4Cl

Ferromagnetoelastic Piezomagnetic

coefficients

Magnetic field

and mechanical stress

FeCO3

Ferromagnetoelectric Magnetoelectric

coefficients

Magnetic field

and electric field

Cr2O3

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Figure 1.3: Illustration of several types of order parameters in proper and improper

ferroics.

1.6.2 Symmetry aspect in ferroics

Ferroic phase transitions in crystals usually involve a change of the space group

symmetry. It enables the classification of ferroic crystals in terms of their macroscopic

physical properties such as polarization and strain. According to the Newnham’s

principle, any macroscopic physical property of a crystal must include all the symmetry

elements of its point group [25].

Figure 1.4 illustrates the symmetry classification of phase transitions in crystals based on

the work of several groups [26]. The term distortive is used for both displacive and order-

disorder phase transitions [27]. This can be divided into two categories: isomorphous and

nonisomorphous In case of isomorphous phase transition, there is no change in the

space-group symmetry of the crystal. A well studied example is the transition occurring

in Ce at a pressure of 7.7kbar (0.77GPa), involving a volume decrease of about 14%

leaving the space-group symmetry unchanged as Fm3m. The change of space group

symmetry at a nonisomorphous phase transition can be either nonferroic or ferroic. In a

nonferroic phase, transition a change in the translational symmetry is seen without

change in the point-group symmetry [28]. The order-disorder phase transition occurring

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in the alloy Cu3Au where the symmetry changes from Fm3m to Pm3m is a classic

example.

DISTORTIVE PHASE TRANSITIONS

ISOMORPHOUS NONISOMORPHOUS 7.7kbarCe: Fm3m Fm3m

NONFERROIC FERROIC

Cu3Au: Fm3m Pm3m

NONFERROELASTIC FERROELASTIC TGS: 2/m 2 BiVO4: 4/m 2/m SiO2: 622 32 BaTiO3, Mn3O4

Figure 1.4: A classification of phase transitions based on considerations of symmetry

descent from prototype symmetry

Ferroic phase transitions involve a change of the point-group symmetry with or without a

change of the translational symmetry. If there is a change in the point-group symmetry

but no change of the crystal system, the phase transition is nonferroelastic [29]. The

2/m(C2h) to 2(C2) transition (figure 1.4) in triglycine sulfate (TGS) is such an example. In

general, it is seen that a change of crystal system is a necessary and a sufficient condition

for a ferroelastic phase transition [30]. The crystal structure of BiVO4 [26] is an example

of a purely ferroelastic transition while BaTiO3 [26] displays both ferroelastic and

ferroelectric transitions.

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1.7 Major Techniques Employed to Characterize Phase Transitions

Structural phase transitions can be analyzed using several characterization techniques

such as diffraction, dielectric measurements, spectroscopy, thermal analysis and

microscopy. A brief description of these techniques is outlined.

1.7.1 Diffraction techniques

The two important thermodynamic variables in the study of phase transitions are

temperature and pressure. Any study of a phase transition would therefore involve

measurement of properties as a function of temperature or pressure. Variable temperature

X-ray diffraction studies of crystalline substances are useful in the study of phase

transitions, thermal expansion and thermal vibrational amplitudes of atoms in solids. For

temperatures as low as 80K liquid nitrogen cryosystem is used. High temperature studies

are carried out using a graphite heating filament fitted with a thermocouple. High

pressure apparatus employ a diamond anvil cell for the study of phase transitions. All

these apparatus are commercially available.

1.7.2 Single crystal X-ray diffraction

One of the best means of obtaining an accurate and detailed structural analysis of a

crystalline solid is by single crystal X-ray diffraction technique. If a crystal is to be

satisfactory for collecting X-ray diffraction data, two main requirements must be met. It

must possess a uniform internal structure and must be of proper size and shape. This

means that the crystal should not be twinned or composed of microscopic subcrystals. It

should not be grossly fractured, bent, or otherwise physically distorted. It need not,

however, have particularly uniform or well-formed external faces [31]. The morphology

of the crystal can be screened rapidly and conveniently using a polarizing microscope. If

rotated about an axis, normal to the polarizing material, the crystals should either appear

uniformly dark in all positions or be bright and extinguish, i.e., appear uniformly dark,

once every 90º. Crystals which are made up of two or more fragments with different

orientations will often reveal themselves by displaying both dark and light regions at one

time. The ultimate evidence, however, of the internal structure of a crystal is furnished by

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the diffraction pattern itself. The reflections that appear should be single spots, without

‘tails’ or streaks connecting them and should be uniquely indexable.

The size of the crystal is normally of the order of 0.2mm3 for structures with light atoms

whereas even smaller size crystals are suitable for diffraction if the compound has one or

more heavy atoms. The structure determination from the single crystal diffraction

involves the measurement of intensity data from the reciprocal lattice, reduction and

scaling of the data using well established data reduction procedures. Any structure

determination has to be done in two stages because of the well known phase problem in

crystallography. The determination of the phases of each reflection is performed using

standard packages such as SHELXS[97] [32] and SIR97 [33], which allow the use of

either direct methods or the Patterson methods. The starting model thus derived can be

refined using the standard packages that are available for example on a program suite like

WinGX [34] as described in subsequent chapters.

1.7.3 Powder X-ray diffraction

It is not always possible to obtain good quality crystals of suitable size for single crystal

structure determination by X-ray diffraction. In fact traditionally, phase transitions have

been studied mainly via powder X-ray diffraction techniques. With the advent of the

Rietveld method [35], the refinement of the structures has received an enormous impetus.

A brief description of the Rietveld method followed by a discussion on the abinitio

approach is given below.

1.7.3.1 The Rietveld method

In the Rietveld method, the least-squares refinements are carried until the best fit is

obtained between the observed powder diffraction pattern taken as a whole and the

calculated pattern [36]. A powder diffraction pattern of a crystalline material may be

thought of as a collection of individual reflection profiles, each of which has a peak

height, a peak position, a breadth, tails which decay gradually with distance from the

peak position, and an integrated area which is proportional to the Bragg intensity, Ik,

where k stands for the Miller indices, h,k,l. Ik is proportional to the square of the absolute

value of the structure factor.

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The diffraction pattern is recorded in digitized form i.e. as a numerical intensity value, y i,

i representing the step value in 2 [36]. Depending on the method, the increments may be

either in scattering angle 2, or some energy parameter such as velocity (for time of

flight neutron data) or wavelength (for X-ray data collected with an energy dispersive

detector) and an incident beam of ‘white’ X-radiation. Typical step sizes range from 0.01

to 0.050 in 2 for fixed wavelength X-ray data on a conventional powder X-ray

diffractometer. With the advent of synchrotron radiation, it is now possible to record the

pattern at step sizes of ~ 0.0050 in 2. The number of steps in the powder diffraction

pattern is usually in thousands (e.g. Number of data points for a scan range of 3 to 80 in

steps of 0.02º is 3850).

The quantity minimized in the least-squares refinement is the residual, Sy :

Sy = wi (yi-yci)2

Where wi = 1/yi

yi = observed intensity at the ith step

yci = calculated intensity at the ith step

Typically, many Bragg reflections contribute to the intensity, yi; observed at any

arbitrarily chosen point i in the pattern. The calculated intensities yci are determined from

the Fk2 values calculated from the structural model by summing up the calculated

contributions from neighboring (i.e. within a specified range) Bragg reflection and the

background.

yci = s LK |FK|2 (2i - 2K)PKA+ybi

where

s is the scale factor,

K represents the Miller indices, h,k,l for a Bragg reflection,

Lk contains the Lorentz polarization and multiplicity factors,

is the reflection profile function.

Pk is the preferred orientation function,

A is an absorption factor,

Fk is the structure factor for the Kth Bragg reflection,

ybi is the background intensity at the ith step.

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In a number of available computer programs for the Rietveld method, the ratio of the

intensities for the two X-ray K wavelengths (if used) is included in the calculation of

Fk2, so that only a single scale factor is required. The model parameters that may be

refined include not only the atomic parameters like positional, thermal and site-

occupancy but also parameters for the background, lattice, instrumental geometrical -

optical features, specimen aberrations (e.g. specimen displacement and transparency), an

amorphous component and specimen reflection. Although it is in general a much less

severe problem with powders than with single crystals, extinction can be quite important

in some powder specimens. Multiple phases may be refined simultaneously and

comparative analysis of the separate overall scale factors for the phases is probably the

most reliable method for doing quantitative phase analysis.

The Rietveld method is a powerful tool, but it is limited by the same drawback that

affects powder methods in general. There is loss of information that arises from the

compression of the three-dimensional diffraction pattern into a single dimension. It is

also important to mention that the Rietveld method, though an excellent technique for

refining structures, requires a good starting model if it is to converge successfully.

Standard packages such as GSAS [37], FULLPROF [38], DBWS [39] and RIETAN[40]

perform Rietveld refinements for single as well as multiple phase in a routinely and

user-friendly manner.

1.7.3.2 The abinitio approach

There has been a great deal of interest concerning the determination of unknown

structures from powder diffraction data during the last decade and there have been

several reviews on the subject [41-44]. The process may be broken into several steps. An

essential prerequisite for crystal structure determination is that the lattice parameters and

the space group are known. Determination of the lattice parameters from the powder

diffraction pattern requires accurate determination of the peak positions (accurate d-

spacing data), which can normally be achieved using a peak-search process, provided all

systematic errors have been eliminated. Although in some cases the lattice parameters

can be determined from first principles, it is usually necessary to use an ‘autoindexing’

program such as ITO, TREOR or DICVOL [45-47]. In general these programs generate

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several possible sets of lattice parameters that are consistent with a set of measured peak

positions; a variety of figures of merit can be used to rank the proposed sets of lattice

parameters. The space group is assigned by identifying the conditions for systematic

absences in the indexed powder diffraction data. From the unit cell parameters and the

selected space group, the positions of the reflections in the diffraction pattern can be

calculated. The diffraction intensity associated with each reflection can then be

determined by applying a whole-profile fitting technique similar to that used in a Rietveld

refinement but with the intensities of the reflections rather than the structural parameters.

This procedure, known as intensity extraction, can be performed using either a least-

square method (Pawley 1981) [48] or an iterative approach (Le Bail et all.1988) [49].

Even if integrated intensities are not used in subsequent steps, this procedure is still

necessary to establish the appropriate profile parameters for whole-profile applications.

Those reflections that are too close to one another to be considered independently (i.e.

strong overlapping reflections) were earlier ignored or assigned arbitrary (often equal)

contributions to the total intensity of the overlapping set. However, this approach was

clearly unsatisfactory, and more sophisticated approaches for determining reliable

relative intensities from overlapping peaks have been developed [41]. These pattern

decomposition techniques are incorporated in a number of programs including ALLHKL,

WPPF [50], GSAS [37], FULLPROF [38], LSQPROF [51] and EXTRA [52].

As most approaches for structure solution from powder diffraction data depend heavily

on extracting reliable intensity information, pattern decomposition constitutes an

important step dictating the overall success of these approaches. Methods are currently

being developed to allow the relative intensities of such overlapping peaks to be

determined accurately and include the applications of relations between structure factors

derived from direct methods and the Patterson function (DOREES) [53]. This is an

iterative procedure involving the calculation of a squared Patterson map and the

subsequent back-transformation giving a new set of structure factors for the overlapping

reflections. Another program (FIPS) [54] is a method based on entropy maximization of a

Patterson function and a Bayesian fitting procedure. The result of the above procedure is

a pseudo single crystal dataset (i.e. a list of hkl and Ihkl) which can be subjected to (a)

adaptations of single-crystal techniques, (b) direct-space methods that exploit prior

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chemical knowledge and (c) hybrids of the two to arrive at the satisfactory structural

model.

The final and often most time-consuming step in the structure determination maze is the

completion of the structure (eg. finding any missing atoms by Fourier analysis, resolving

disorder problems, etc) and the refinement of the structural parameters using the Rietveld

method. Only when the refinement has been brought to a successful conclusion can the

structure proposed from the structure determination step be considered to be confirmed.

Throughout the whole procedure, chemical information and intuition play an important

role in guiding the user through the maze [41].

1.7.4 Neutron diffraction

Thermal neutrons with a velocity of about 4000ms-1 associated with a wavelength

of~1.0 Å are used in neutron diffraction experiments. Variable temperature neutron

powder diffraction experiments are used to study phase transitions in many materials.

Whereas X-rays are scattered primarily by the electrons in atoms, neutrons are scattered

mainly by the atomic nucleus. Since the neutron-scattering amplitude does not show a

smooth dependence on the atomic number of the atoms, neutron diffraction is

particularly useful in locating light atoms in crystals. Additional scattering of neutrons

can arise owing to the magnetic moment of neutrons. In the absence of an external field,

the magnetic moments of atoms in a paramagnetic crystal are arranged at random, so that

the magnetic scattering of neutrons by such a crystal is also random. It only contributes

diffuse background to the diffraction pattern. In magnetically ordered materials however,

the magnetic moments are regularly aligned. Neutron diffraction provides an

experimental means whereby the magnetic structures can be determined. In addition to

the two scattering effects that are elastic, neutrons can also undergo inelastic scattering by

crystals. This involves energy exchange between the lattice and neutrons. Inelastic

neutron scattering by crystals is used to study quantized vibrational modes and dynamics

in solids [55].

Since neutron beams are much weaker in intensity than X-rays, neutron diffraction

requires large single crystals. However, in recent years, powder neutron diffraction

analysis has also been used to obtain structural information [56]. The Rietveld method

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described in earlier sections is also used for analysis of the powder neutron data [35]. The

profile analysis is particularly suitable for neutron data since the profiles could be

accurately described by a Gaussian function. In case of structures consisting of atoms

which are near neighbors in the periodic table, it is recommended to use neutron

diffraction rather than X-ray diffraction techniques. In general, neutron diffraction is

employed as a surrogate technique along with either single crystal or the powder X-ray

diffraction techniques.

1.7.5 Electron diffraction

Electron beams have small wavelengths than X-rays and carry charge. Hence, electron

diffraction is a valuable technique for structural studies of solids. These are the small

wavelength of electron beams and charge carried by them. Smaller wavelength leads to

smaller Bragg angles in electron diffraction. The radius of the Ewald sphere, 1/, is

therefore much larger for electron diffraction than X-ray diffraction. This makes it

possible to record extensive sections of the reciprocal lattice with a small stationary

crystal. Due to the charge, interaction of electrons with atoms is about 103 times stronger

than that of X-rays and this makes it possible to record electron diffraction patterns

almost instantaneously. Electron diffraction patterns are readily obtained with

commercial electron microscopes. It is possible to investigate defect ordering, the Bravais

lattice type, superstructures and fine particle sample by electron diffraction. However, it

has the disadvantage of having secondary diffraction effects that severely limit its

application as a stand alone structure determination technique. The requirement of very

high vacuum also becomes a serious rate limiting step especially for the study of phase

transition.

1.7.6 Dielectric measurements

The dielectric properties of a material are governed by its response to an applied electric

field at the electronic, atomic, molecular and macroscopic levels. Application of a

potential difference across a dielectric leads to polarization of charge within the material

although long range motion of ions and electrons cannot occur. The polarization

disappears when the voltage is removed. Polarization and dielectric loss in materials are

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phenomena of interest, and thus are generally studied as a function of frequency.

Piezoelectrics, pyroelectrics, ferroelectrics, paraelectrics, ferroelastics all fall under a

broad class of dielectrics since most of the materials exhibiting these properties undergo

transitions with changes in polarization on application of an electric field.

Ferroelectric materials retain a large, residual polarization of charge after the electric

field has been removed. One of the important characteristics of a ferroelectric is that the

dielectric constant obeys the Curie-Weiss law,

e = r – 1 = 3Tc/T-Tc

where e is the susceptibility of the material, r is the relative dielectric constant of the

material and Tc is the critical or Curie temperature. A number of displacive, order-

disorder and hydrogen bonded ferroelectric sulfates have been evaluated for their

pyroelectric behavior. For example, ferroelectricity was observed in A3H(SO4)2

(A=K,NH4,Rb) through dielectric measurements [57,58]. A conspicuous increase in the

Curie temperature and enhancement in the spontaneous polarization by deuterium

substitution was found in (NH4)H3(SO4)2 [12].

1.7.7 Spectroscopic techniques

A number of spectroscopic methods are available for the analysis of structure and

dynamics during phase transition. Most of these techniques yield valuable information

regarding the nature of phase transition. Magnetic transitions in solids can be studied by

Mossbauer spectroscopy and magnetic resonance spectroscopy (NMR, ESR and NQR).

Inelastic neutron scattering provides valuable information on phonons and magnons in

crystals.The slow vibrational motion of H atom in RBHSO4 governed by the reoriented

motion of the sulfate ion through the riding model mechanism due to the isotope effect is

studied by this technique [59]. NMR spectroscopy has been employed to study phase

transitions of solids containing the appropriate nuclei as in the case of NaCN and NaHS

[9]. Studies of hindered rotations of CH3 or NH4+ groups and phase transitions in

hydrogen-bonded ferroelectrics like KH2PO4 are other important applications of NMR

spectroscopy. ESR spectra of solids undergoing transitions have been reported in the

literature [9]. NQR spectroscopy has been employed to study phase transition of halides,

nitrates and nitrites containing nuclei with quadrupole moments [9]. Raman spectroscopy

19

is specially used for investigating soft modes [9]. The changes in electronic structure can

be followed by X-ray and ultraviolet photoelectric spectroscopy. Optical spectroscopy is

used for studying phase transformations, particularly with respect to the movement of

boundaries, growth of nuclei and changes of grain size [9]. Spectroscopic techniques like

mass spectrometry, atomic absorption spectrophotometry and electron microprobe

analysis are used to confirm the purity of the material which is crucial to study phase

transitions [9].

1.7.8 Thermal analysis

Thermal measurements have been widely used to identify and characterize transitions.

Heat capacity measurements provide precise enthalpy changes and indicate

thermodynamic order. The two main thermal analysis techniques are thermogravimetric

analysis (TGA) which automatically records the change in weight of a sample as a

function of either temperature or time and differential thermal analysis (DTA) which

measures the difference in temperature, T, between a sample and an inert reference

material as a function of temperature. A technique that is closely related to DTA is

differential scanning calorimetry (DSC), where, the equipment is designed to allow a

quantitative measure of the enthalpy changes that occur in a sample as a function of

either temperature or time. The H values obtained here are more reliable. A fourth type

of thermal analysis technique is dilatometry, in which the change in linear dimension of a

sample as a function of temperature can be recorded.

Thermal changes can be followed on cooling as well as on heating. It is evident that if a

particular process, on heating, is endothermic, then the reverse process on cooling must

be exothermic. This enables us to study reversibility of any phase transition. Another

phenomenon is the thermal hysteresis that occurs when the exotherm may be displaced to

occur at lower temperatures than the corresponding endotherm. Hence hysteresis is more

in transitions involving breaking of strong bonds.

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1.7.9 Electron microscopy

Electron microscopy is an extremely versatile technique capable of providing structural

information over a wide range of magnification. It gives useful information on

dislocations and structural aspects. High resolution lattice imaging of transforming solids

is an area of great potential. The main mode of operation makes use the fact that when a

sample is placed in the microscope and bombarded with high energy electron, X-rays

characteristic of the elements present in the sample are generated. By scanning either the

wavelength (wavelength dispersive,WD) or the energy(energy dispersive, ED) of the

emitted X-rays it is possible to identify the elements present. By a suitable calibration a

quantitative elemental analysis may be made for elements heavier than sodium. The

scanning electron microscopy (SEM) allows for preliminary verification of the purity of

the phase along with transmission electron microscopy (TEM). Other techniques include

use of electron probe microanalysis (EPMA), electron microscopy with microanalysis

(EMMA) and analytical electron microscopy (AEM) to gain insights into phase

transitions A recent advance is the development of the scanning transmission electron

microscope (STEM). This combines the scanning feature of the SEM with the

intrinsically higher resolution obtainable with TEM. SEM is invaluable for surveying

materials under high magnification and providing information on particle sizes and

shapes. Using thin foils with TEM (transmission electron microscopy), crystal defects

such as dislocations, stacking faults, anti phase boundaries may be seen directly. With

HREM (high resolution electron microscopy), it is now possible to it to view details on

an atomic scale. Variations in local structure such as site occupancies and vacancies can

be observed directly; this is especially useful in studying compositional phase transitions

by doping.

1.8 Foreword

The subsequent chapters report the detailed phase transition analysis mainly using X-ray

diffraction techniques. The materials investigated belong to the family of sulfates and

complex oxides. The following preamble is an account of the existing literature related to

phase transitions in these materials.

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1.8.1 Sulfates

The alkali and alkaline earth metal mixed sulfates have been investigated for exhibiting

properties such as ferroelectricity, pyroelectricity, ferroelasticity and proton ion

conduction. These properties have been characterized by dielectric, optical, spectroscopic

and X-ray diffraction measurements. The compounds which have been subjected to these

studies are A3H(SO4)2 (A= K, Cs, Rb, NH4) [60-70], AHSO4 (A=NH4, K, Cs, Rb),

A4LiH3(SO4)4 (A= NH4, K, Rb, Cs) and AB(SO4)3 (A=Rb,K,NH4,Tl; B= Ca, Cd, Mg, Mn,

Zn), ABSO4(A=Li, K, Na, Cs; B=NH4, Li, Rb) [71-74].

The family of ABSO4 (A=Li, K, Na, Cs; B=NH4, Li, Rb)[74-77] and AHSO4 (A=NH4,

K, Cs, Rb)[78-80] have been shown to exhibit ferroic properties and possess extensive

hydrogen bonding network. For example, the crystal structure of LiNH4SO4 has been

studied in detail and the ferroelastic to ferroelectric phase transitions characterized by

modulated differential scanning calorimeter, capacitance and ac thermal measurements

[81,82].

1.8.1.1 Rubidium hydrogen sulfate

The ferroelectric transition in NH4HSO4 was observed by dielectric measurements and

attributed to the character of the N-H…O bond [83] and hence it was surprising to

discover that RbHSO4 [79], which is isomorphous with NH4HSO4 at room temperature, is

also ferroelectric below -15C. Dielectric and thermal measurements confirmed this phase

transition [84]. Preliminary X-ray diffraction reports showed that the room temperature

paraelectric phase crystallized in a pseudo orthorhombic system (space group B21/a),

while the ferroelectric phase crystallized in a noncentrosymmetric (space group Pc or

P1). The room temperature crystal structure of RbHSO4 was determined by X-ray and

neutron diffraction [85]. There was evidence for disorder of one of the sulphate groups

and the hydrogen atoms were ordered in the paraelectric phase. In another report [86] the

disordered structure of RbHSO4 at room temperature was redetermined by assuming that

one of the two independent sulfate ions is disordered

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.

Figure 1.5. Pressure-temperature phase diagram of RbHSO4.

A high pressure phase of this compound has also been observed by dielectric constant

measurements and the crystal structure at 1GPa has been analyzed [87]. The pressure

temperature phase diagram of RbHSO4 is shown in figure 1.5.

1.8.1.2 Langbeinites

Langbeinites are a class of sulfates having the general formula (A+1)2(B2+)2(SO4)3, where

A+1=K,Rb,NH4, Tl and B2+=Zn,Co,Cd,Mn or Mg [88]. The structural phase transitions in

these compounds were first characterized by Dvorak [89], using Landau theory.

According to Dvorak’s theory, the langbeinite type crystal could transform from the

parent cubic phase (space group P213) to any one of the four space groups P21, P1,

P212121 and R3. Langbeinites are classified into three categories based on their successive

phase transition sequence as shown below where P and F represent paraelectric and

ferroelectric or ferroelastic phases respectively [90]:

Type ICubic (P213) (P) Monoclinic (P21) (F)* Triclinic (P1) (F)* Orthorhombic

(P212121) (F)

* This type has one or two intermediate phases followed by the lowest phase.

Eg. Tl2Cd2(SO4)3, Rb2Cd2(SO4)3, K2Zn2(SO4)3, K2Co2(SO4)3 [91]

Type II Cubic (P213) (P) Orthorhombic (P212121) (F)

Eg. K2Mn2(SO4)3, (NH4)2Cd2(SO4)3

Type III There are no phase transitions eg. K2Ni2(SO4)3

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Several Type II structures have been analyzed by single crystal X-ray diffraction and the

mechanism of phase transition has been attributed to the simultaneous translation and

rotation of the SO4 group and the subsequent rearrangement in the B2+ and A+ cation

positions [88,92].

It is of interest to note that so far there is only one detailed single crystal X-ray diffraction

study followed by a subsequent phase transition analysis in literature for type I

langbeinite Tl2Cd2(SO4)3 [ 93]. Rb2Cd2(SO4)3 also belongs to the typeI langbeinite as

observed from dielectric studies[94]. This compound undergoes successive ferroelectric

phase transition at -144C and -170C [94]. Preliminary studies suggest the presence of

super lattice reflections in both these phases [95]. The lattice parameters were measured

as a function of temperature and the phase transitions were confirmed by piezoelectric

resonance, EPR and dielectric measurements [95-97]. The phase transition pathway

indicated in Tl2Cd2(SO4)3 is not supported by any of these measurements.

1.8.1.3 Tetra rubidium lithium tri hydrogen tetra sulfate (Rb4LiH3(SO4)4)

The crystals of the common formula A4LiH3(SO4)4 were found to be tetragonal at room

temperature by X-ray diffraction and crystal morphology studies [98]. Pyroelectric,

thermal and elastic studies of Rb4LiH3(SO4)4 show the existence of a phase transition at

137K [99].

Figure 1.6: Plot of nc against temperature for Rb4LiH3(SO4)4 crystal

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Linear birefringence measurements (figure 1.6) indicate a linear dependence of the

morphic birefringence n on the order parameter confirming that the transition in

Rb4LiH3(SO4)4 is a proper ferroelastic phase transition [100]. The Curie temperature (Tc)

for this compound and its dueterated analogue are determined to be at 122.44K and

108.43K respectively [100]. The crystal structure of an optically laevorotatory crystal

was determined in the space group P41 [101]. Dielectric and DTA measurements show

anomalies at 458K, 470K and 490K indicating structural phase transitions. The behavior

of the dielectric constant [102] was found to be very similar to that of a superionic

conductor.

1.8.1.4 Tetra potassium lithium tri hydrogen tetra sulfate (K4LiH3(SO4)4)

Low temperature pyroelectric and dielectric behavior of K4LiH3(SO4)4 show an anomaly

at 114K [99]. EPR studies show a change in the line intensity with temperature relating

to the existence of a phase transition at 110K [103].

Figure 1.7. Raman spectra of K4LiH3(SO4)4 crystal at (1) 90K (2) 300K

Polarization Z(XX)Y

Further thermal expansion investigations do not show any anomalous changes suggesting

that either no phase transition related with change of the lattice parameter occurs in

K4LiH3(SO4)4 or the relative length induced by these changes is less than 10-6 [99].

Brillouin scattering technique used to study elastic properties in the temperature range 90

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to 300K confirm the nonferroelastic [104] character of the phase transition in this

compound. Raman investigations unambiguously show the existence of a phase transition

(figure 1.7) near 120K and possible changes related to the dynamics of the proton on the

central H-bond were conjectured [105]. However the microscopic mechanism of the

phase transition governing could not be determined.

1.8.2 Oxides

1.8.2.1 Aurivillius phases

The complex bismuth oxides with layered structures were discovered by Aurivillius more

than 50 years ago [106]. These compounds can be described by the general formula

[Bi2O2] [An-1BnO3n+3] (n=1,2,3,4) where the [Bi2O2]2+ layers are interleaved with n

perovskite-type layers having the composition [An-1 Bn O3n+1] [107]. To date, an intensive

search will yield more than 100 compounds in this group, which have been characterized

by dielectric, single crystal and powder X-ray diffraction, Raman spectroscopy, electron

diffraction and conductivity measurements [108-112]. Representative members are

Bi2WO6 (n=1), Bi3TiNbO9 (n=2), Bi4Ti3O12 (n=3), Bi5Ti3FeO15 (n=4) (figure 1.8) [113].

All the compounds in this family are tetragonal (I4/mmm) and centrosymmetric at high

temperature but transform to different polar groups on cooling to room temperature. The

crystal structure of Bi3TiNbO9 and Bi4Ti3O12 was determined by single crystal X-ray

diffraction methods and it was found that both crystallized in the orthorhombic system

with different space groups [113,114]. Further studies describe the modulated structures

of both these compounds in greater detail and accuracy [115,116]. The structure and

property of the higher members of the series (n=4 and above) has also been a subject of

interest as they exhibit ferroelectric properties in thin films [117].

26

Figure 1.8: Idealized structures of : (a) Bi2WO6 (n=1); (b)Bi3TiNbO9 (n=2) (c) Bi4Ti3O12

(n =3) projected along ‘a’ axis [108]

1.8.2.2 Crystal structures of ABi4Ti4O15 (A=Ba, Sr and Pb) and phase transitions at

high temperature

Ferroelectricity has been observed in thin films of the n=4 Aurivillius oxides [117].

Raman studies on these oxides show under damped soft modes in SrBi4Ti4O15, low

frequency over damped modes in BaBi4Ti4O15 in the vicinity of the ferroelectric to

paraelectric transition [118]. In PbBi4Ti4O15, the softening of the under damped modes

was not observed clearly. The reported Tc for these compounds is 803K for BaBi4Ti4O15,

693K for SrBi4Ti4O15 and 843K for PbBi4Ti4O15. Recently a variable temperature neutron

diffraction study was reported for SrBi4Ti4O15 [119]. It is observed that these oxides

exhibit no spontaneous polarization along the c axis. Thin film studies on BaBi4Ti4O15

27

show that the macroscopic ferroelectric properties of these layered oxides depend on the

crystalline orientation of the films [120].

1.8.2.3 Structural studies on lanthanum doped n=2 Bi3TiNbO9

The lanthanum substituted bismuth titanates have shown a high fatigue free resistance

which find application in ferroelectric random access memories (FRAM) [121]. It has

been observed previously that substitution of the trivalent Bi ions with increasing

concentrations of trivalent La ions in Bi2-xLaxTiNbO9 result in smaller orthorhombic

distortion [122]. At the maximum substitution rate for Bi2-xLaxTiNbO9 corresponding to

x=1, it was observed that the value of the dielectric constant is reduced drastically in the

vicinity of the transition. These results have been correlated to the change of symmetry

with increasing concentration of La. In the analogous Bi2SrTa2O9, where the ‘A’ site is

occupied by the divalent Sr ion, the reduced orthorhombic distortion observed is

attributed to the high BVS at that site [123]. The marked preference for Bi to occupy a

highly distorted coordination site is complicated by the presence of the stereochemically

active lone pair [124].

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