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
3
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
5
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
15
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
17
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
18
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
24
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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