electro ceramics web course (nptel) · 2017. 8. 4. · of technological applications such as...

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Electroceramics Overview

Electro Ceramics Web Course (NPTEL)

Contact information of the course instructor:

Ashish Garg, Associate ProfessorDepartment of Materials science and EngineeringIndian Institute of Technology KanpurKanpur 208016, IndiaTelephone: 0512-259-7904Email: [email protected]: http://home.iitk.ac.in/~ashishg

Introduction

Electro-ceramics or broadly speaking electronic, optical and magnetic ceramics are useful in a varietyof technological applications such as sensors, actuators, transducers, data storage devices etc. Some of the examples are

Dielectric materials such as SiO2 are used as data storage elements in random access

memories or RAMsFerroelectrics such as BaTiO3 and PbTiO3 are used as sensors and actuators

Magnetic oxides such as iron oxides are used for data storage in magnetic headsZnO is used circuit protection materials in devices named as varistorsZrO2 stabilized with other oxides is used in fuel cells and batteries

Hence, to understand these materials better and to engineer them as per our needs, we need tounderstand their science viz. their structure, defects in these materials, phenomenon of conduction,fundamentals of various functional properties. A sound understanding of these would (hopefully)enable us tailor the structure and properties of these material with good degree of control.

Pre-requisites

Basic courses on structure of materials, thermodynamics, and solid state physicsSuited for final year undergraduate students of most disciplines and fresh graduate students.

List of Topics

Module Topics Equivalent Lectures (50-60 meach)

1: Structure of Ceramic Materials 5

2: Defect Chemistry and Equilibria 7

3: Diffusion and Conduction in Ceramics 7

4: Linear Dielectric Ceramics 8

5: Nonlinear Dielectric Ceramics 6

6: Magnetism and Magnetic Ceramics 5

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7: Superconducting Ceramics 1

8: Multiferroic and Magnetoelectric Ceramics 1

9: Synthesis Methods 1

Total number of equivalent lectures 41

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Electroceramics Table of Contents

Table of Contents

1. Structure of Ceramic Materials

1.1 Brief Review of Structure of Materials1.2 A Brief Review of Bonding in Materials1.3 Packing of atoms in metals1.4 Interstices in Structures1.5 Structure of Covalent Ceramics1.6 Ionically Bonded Ceramic Structures1.7 Compounds based on FCC Packing of ions1.8 Other cubic structures1.9 Orthogonal Structures1.10 Structures based on HCP packing of ions1.11 Summary

2. Defect Chemistry and Defect Equilibria

2.1 Point Defects2.2 Kröger–Vink notation in a metal oxide, MO2.3 Defect Reactions2.4 Defect Structures in Stoichiometric Oxides2.5 Defect Structures in Non-stoichiometric Oxides:2.6 Dissolution of foreign cations in an oxide2.7 Concentration of Intrinsic Defects2.8 Intrinsic and Extrinsic Defects2.9 Units for defect Concentration2.10 Defect Equilibria2.11 Defect Equilibria in Stoichiometric Oxides2.12 Defect Equilibria in Non-Stoichiometric Oxides2.13 Defect Structures involving Oxygen vacancies and interstitials:2.14 Defect Equilibrium Diagram2.15 A Simple General Procedure for constructing at Brouwer’s Diagram2.16 Extent of non-stoichiometry2.17 Example: Comparative behaviour of TiO2 and MgO vis-à-vis oxygen pressure

2.18 Electronic Disorder2.19 Examples2.20 Summary

3. Defects, Diffusion and Conduction in Ceramics

3.1 Diffusion 3.2 Diffusion Kinetics 3.3 Examples of Diffusion in Ceramics 3.4 Mobility and Diffusivity

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3.5 Analogue to the electrical properties 3.6 Conduction in Ceramics vis-à-vis metallic conductors: General Information3.7 Ionic Conduction: Basic Facts 3.8 Ionic and Electronic Conductivity3.9 Characteristics of Ionic Conduction 3.10 Theory of Ionic Conduction Conduction in Glasses 3.11 Conduction in Glasses3.12 Fast Ion Conductors3.13 Examples of Ionic Conduction3.14 Electrochemical Potential3.15 Nernst Equation and Application of Ionic Conductors3.16 Examples of Ionic Conductors in Engineering Applications3.17 Summary

4. Dielectric Ceramics: Basic Principles

4.1 Basic Properties: Dielectrics in DC electric field 4.2 Mechanisms of Polarization4.3 Microscopic Approach 4.4 Determination of Local Field 4.5 Analytical treatment of Polarizability 4.6 Effect of alternating field on the behavior of a dielectric material 4.7 Frequency dependence of dielectric properties: Resonance 4.8 Dipolar Relaxation i.e. Debye Relaxation is Polar Solids 4.9 Circuit Representation of a Dielectric and Impedance Analysis 4.10 Impedance Spectroscopy 4.11Dielectric Breakdown 4.12 Summary

5. Nonlinear Dielectrics

5.1 Introduction 5.2 Classification based on Crystal Classes 5.3 Ferroelectric Ceramics 5.3.1 Permanent Dipole Moment and Polarization 5.3.2 Principle of Ferroelectricity: Energetics 5.3.3 Proof of Curie-Weiss Law 5.3.4 Thermodynamic Basis of Ferroelectric Phase Transitions 5.3.5 Case I: Second order Transition 5.3.6 Case – II: First Order Transition 5.3.7 Ferroelectric Domains 5.3.8 Analytical treatment of domain wall energy 5.3.9 Ferroelectric Switching and Domains 5.3.10 Measurement of Hysteresis Loop 5.3.11 Structural change and ferroelectricity in Barium Titanate (BaTiO3)

5.3.12 Applications of Ferroelectrics

5.4 Piezoelectric Ceramics 5.4.1 Direct Piezoelectric Effect 5.4.2 Reverse or Converse Piezoelectric Effect

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5.4.3 Poling of Piezoelectric Materials 5.4.4 Depolarization of Piezoelectrics 5.4.5 Common Piezoelectric Materials 5.4.6 Measurement of Piezoelectric Properties 5.4.7 Applications of Piezoelectric Ceramics

5.5 Pyroelectric Ceramics 5.5.1 Difference between and pyroelectric and ferroelectric material 5.5.2 Theory of Pyroelectric Materials 5.5.3 Measurement of Pyroelectric coefficient 5.5.4 Direct and Indirect effect 5.5.5 Common Pyroelectric Materials 5.5.6 Common Applications

5.6 Summary

6. Magnetic Ceramics

6.1 Magnetic Moments6.2 Macroscopic view of Magnetization 6.3 Classification of Magnetism 6.4 Diamagnetism6.5 Paramagnetism6.6 Ferromagnetism6.7 Antiferromagnetism 6.8 Ferrimagnetism6.9 A Comparison 6.10 Magnetic Losses and Frequency Dependence 6.11 Magnetic Ferrites 6.12 Summary

7. High temperature Superconductors

7.1 Background7.2 Meissner Effect7.3 The critical field, Hc7.4 Theory of Superconductivity7.5 Discovery of high temperature superconductivity7.6 Mechanism of high temperature superconductivity7.7 Applications7.8 Summary

8. Multiferroic and Magnetoelectric Ceramics

8.1 Introduction8.2 Historical Perspective8.3 Requirements of a magnetoelectric and multiferroic material8.4 Magnetoelectric Coupling8.5 Type I Multiferroics8.6 Type II Multiferroics8.7 Two Phase Materials8.8 Summary

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9. Synthesis Methods

9.1 Bulk Preparation Methods9.2 Thin Film Preparation Methods9.3 Thin film deposition: Issues9.4 Summary

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Electroceramics General Bibliography

General Bibliography

The following are the books which can be referred for general reading. More references are providedin each module.

Recommended Reading

1. Physical Ceramics: Principles for Ceramic Science and Engineering, Y.-M. Chiang, D. P.Birnie, and W. D. Kingery, Wiley-VCH

2. Introduction to Ceramics, 2nd Edition, W. D. Kingery, H. K. Bowen, D. R. Uhlmann, Wiley3. Principles of Electronic Ceramics, by L. L. Hench and J. K. West, Wiley4. Electroceramics: Materials, Properties, Applications, by A. J. Moulson and J. M. Herbert, Wiley5. Nonstoichiometry, Diffusion and Electrical Conductivity in Binary Metal Oxides (Science &

Technology of Materials), P.K. Kofstad, John Wiley and Sons Inc.

Supplementary Reading

6. Introduction to Solid State Physics, C. Kittel, Wiley7. Electrical Properties of Materials, L. Solymer and D. Walsh, Oxford University Press8. Introduction of Solid State Physics, N.W. Ashcroft and N.D. Mermin, Brooks Cole9. Solid State Physics, A.J. Dekker, Prentice-Hall

10. Transition Metal Oxides: An Introduction to Their Electronic Structure and Properties, P.A. Cox,Oxford University Press

11. Basic Solid State Chemistry, A.R. West, Wiley12. Non-stoichiometric Oxides, O. Toft Sørensen, Academic Press13. Dielectrics and Waves, A.R. von Hippel, John Wiley and Sons14. Feynman Lectures on Physics, Volume 1-3, R.P. Feynman, Addison Wesley Longman15. Materials Science and Engineering: A first course, V. Raghavan, Prentice Hall of India16. Materials Science And Engineering: An Introduction, W.D. Callister, Wiley

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Module 1: Structure of Ceramic Materials Introduction

In this module, we will first review the structure and bonding in the materials in general followed by abrief discussion on how atoms pack together in the solids and what are the types of intersticespresent in various structures. Then we would briefly delve into the types of bonding with reference tothe nature of materials. Together, this information will form the basis for structures in ceramicmaterials which are typically bonded with a mix of ionic and covalent bonding. Subsequently, wewould discuss the structure of ceramic materials with purely covalent bonding followed by ratherdetailed description of ceramic materials with ionic bonding. These are essentially based on packing ofanions closed packed forms where cations fill the interstices.

The Module contains:

Brief Review of Structure of Materials

A Brief Review of Bonding in Materials

Packing of Atoms in Metals

Interstices in Structures

Ionically Bonded Ceramic Structures

Compounds based on FCC Packing of ions

Other Cubic Structures

Orthogonal Structures

Structures based on HCP packing of ions

Summary

Suggested Reading:

Materials Science and Engineering, W.D. Callister, Jr., Wiley

Physical Ceramics: Principles for Ceramic Science and Engineering, Y.-M.Chiang, D. P. Birnie, and W. D. Kingery, Wiley-VCH

Introduction to Ceramics, W. D. Kingery, H. K. Bowen, D. R. Uhlmann, Wiley

Fundamentals of Ceramics, Michael Barsoum, McGraw Hill

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Module 1: Structure of Ceramics Brief Review of Structure in Materials

1.1 Brief Review of Structure of Materials

In the following sections, we will quickly look at the concepts of lattice, unit-cell and crystalstructures which will be useful to understand the crystal structures of common ceramic compounds.

1.1.1 Point Lattice

In a point lattice, the following characteristics are obeyed:

There is a periodic arrangement of points in space. (Figure 1.1(a))

In addition, each point must have identical neighbourhood.(Figure 1.1(b))

Lack of regular arrangementNo periodicityNon-identical neighbourhood

Regular arrangementPeriodicityIdentical neighbourhood of each point

Figure 1.1(a) Unit-cellrepresentation

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Figure 1.1(b) Schematics of arrangement of points inspace

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1.1.2 Unit Cell

A unit cell is the smallest repeatable unit in a point lattice (Figure 1.2).

Choice of unit cell shape is not unique.

Figure 1.2 Representation of a point lattice and anunit cell

Unit-cell parameters for a 3-D unit cells

axis lengths: a, b and c

angles:a ß and γ

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1.1.3 Motif and Crystal Structure

Crystal structure: a combination of motif and point lattice

Motif is defined as a unit or a pattern. For a crystal, it can be an atom, an ion or a group ofatoms or ions or a formula unit or formula units. Often it is also called as Basis.

When motif replaces points in a periodic point lattice, it gives rise to what is called as acrystal with a defined structure.

Figure 1.3 Formation of a periodic crystal structure

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1.1.4 Types of Lattice

Lattice can further be classfied into two types

Primitive lattice having one formula unit or one lattice point or one unit of motif perunit cell, and Non-primitive lattices having more than one lattice points or more than one unitof motif per unit cell.

Figure 1.4 Primitive and Non-primitivelattices

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1.1.5 Symmetry in Crystals

Symmetry is an operation which brings the object back to its original confiscation.

Symmetry elements underlying a point lattice (see the figure) Reflection: reflection across a mirror plane

Rotation: rotation around a crystallographic axis by an angle θ such as 360°/θ isan integer of value 1, 2, 3, 4 and 6 and is referred to as n -fold rotation.Inversion: a point at x,y,z becomes its equivalent at (–x,-y,-z)Rotation-Inversion: Rotation followed by inversion OR Rotation-Reflection:Rotation followed by reflection

Figure 1.5 Basic symmetry operations in crystals

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1.1.6 Crystal Systems

As you can see now, the choice of unit cell is not unique and we can define any unit cell ofany shape as long it contains one lattice point.

However, as one starts defining various shapes, we come up with seven categories, called ascrystal systems, in which all possible unit cells shapes would fit provided space filling criteriais fulfilled.

Seven crystal systems are shown below.

Crystal system and latticeparametersMinimumsymmetryelements

Cubic

a = b =c,

Four 3-foldrotation axes

Tetragonal One 4-foldrotation (orrotation-inversion)axis

Orthorhombic Threeperpendicular 2-fold rotation (orrotation-inversion)axis

Rhombohedral

a = b = c

One 3-foldrotation (orrotation-inversion)axis

Hexagonal One 6-foldrotation (orrotation-inversion)axis

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MonoclinicOne 2-foldrotation (orrotation-inversion)axis

Triclinic None

Figure 1.6 Seven crystal systems

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1.1.7 Bravais Lattices

Taking seven crystal systems and symmetry elements into account, Bravais came out with thefact that there are a total of 14 Bravais Lattices which are shown below.

Cubica=b=c,

Tetragonal Orthorhombic Rhombohedrala=b=c

Hexagonal Monoclinic Triclinic

Figure 1.7 Fourteen Bravais lattices

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1.1.8 Planes and Directions

Faces and directions joining atoms in crystals can be best described by Miller Indices (in the names of W. H. Miller ) ascribed to determine various planes and directions. While planes are determined little empirically, directions are nothing but vectors.

1.1.8.1 Crystallographic Planes

Identification of various faces seen on the crystal

( h.k.l ) for a plane or {h..k.l} for identical set of planes

A crystallographic plane in a crystal satisfies the following equation

(1.1)

h/a, k/b, and c/l are the intercepts of the plane on x, y, and z axes.

a, b, c are the unit cell lengths

h, k, l are integers called Miller indices and the plane is represented as (h, k, l)

Any negative indices in Miller indices of a plane is written with a bar on top such as .

1.1.8.2 Directions

These are basically atomic directions in the crystal.

Miller indices are [ u , v , w ] for a direction or < u , v , w > for identical set of directionswhere u , v , w are integers

Vector components of the direction resolved along each of the crystal axis reduced to smallestset of integers

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Figure 1.8(a) Planes and Directions in Crystals

Crystal Directions

How to locate a direction:

Example: [231] direction would be

1/3 intercept on cell a-length

1/2 intercept on cell b-length and

1/6 intercept on cell c-length

Directions are always denoted with [uvw ] with square brackets and family

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of directions in the form < uvw >

Figure 1.8(b) Planes and Directions in Crystals

We will not go into too much details of this assuming that you would know about planes anddirections in a crystal. If you are not sure, then refer to any elementary materials science text bookon structure of materials (see bibliography) or else refer to other NPTEL modules.

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Module 1: Structure of Ceramics

A Brief Review of Bonding in Materials

1.2 A Brief Review of Bonding in Materials

Bonding in materials is a very important criterion and determines many of the physical properties ofthe materials. For basics of bonding, you can refer to any elementary materials science book (seebibliography) to get familiar with the fundamental aspects of bonding between atoms i.e. how todetermine the equilibrium distance, bond energy and fundamental properties like young’s modulus.Bonding in materials can be divided in two categories:

Primary bondingSecondary bonding

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Module 1: Structure of Ceramics A Brief Review of Bonding in Materials

1.2.1 Primary Bonding There are three types of primary bonding mechanisms: metallic, covalent and ionic bonding.

Figure 1.9 Metallic Bonding

1.2.1.1 Metallic bonding :

This kind of bonding is characterized by presence of a sea of electrons around atoms in metalgiving rise to flexible bonds, good malleability, high electrical and thermal conductivity. Mostmetals such as Ni, Fe, Cu, Au, Ag etc exhibit this kind of bonding.

1.2.1.2 Covalent Bonding:

In this bonding, atoms share their outer shell unpaired electrons leading to a stronger anddirectional bonding.

Examples of materials showing this bonding are mainly group IV elements and compoundssuch as Si, C, Ge, and SiC and gases like methane.

Figure 1.10 Schematic of covalent bonding

1.2.1.3 Ionic Bonding :

This bonding occurs due to large differences in the electronegativities of two elements, forexample in NaCl, MgO etc.

This type of bonding typically leads to high bond energies, high bond strength, high modulus,brittle nature, generally low thermal and electrical conductivities making them excellentinsulators.

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Figure 1.11 Schematic of ionic bonding

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1.2.2 Secondary Bonding :

It arises from the interaction between charge dipoles.

1.2.2.1 Fluctuating Dipoles:

Observed in gases like hydrogen.

Figure 1.12 Secondary bonding due to fluctuating dipoles

1.2.2.2 Permanent Dipole Moment Induced

Induced to permanent dipoles in the materials

General case

Figure 1.13 Secondary bonding due to permanentdipoles

Examples are materials like polymers.

Figure 1.14 Secondary bonding inpolymers

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1.2.3 Bonding, Bond Energy and General Remarks

Type Bond Energy Comments

Ionic

Large magnitude

(Large Tm, large E and small a

)

Example: MgO – 1000 kJ /

mol, Tm- 2800oC

Non-directional

(Typically Ceramics)

Covalent

Variable for materials such asSi, Ge

Large for Diamond (Carbon)and small for Bismuth

Typically Large Tm and large E

Example: Si - 450 kJ/mol; Tm-

1410oC

Directional

(Typically Semiconductors,Polymers and some Ceramics)

Metallic

Variable

Large: Tungsten (W)

Small: Mercury (Hg)

Moderate: Al: 68 kJ/mol, Tm~

670oC

Non–directional (Metals)

Secondary

Smallest

Characterized by low Tm, low

E and large a

Directional

Inter–chain (Polymer)

Inter–molecular

Symbols: Tm: Melting point, a: Coefficient of thermal expansion, E: Elastic modulus

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Module 1: Structure of Ceramics Packing of Atoms in Metals

1.3 Packing of Atoms in Metals

In solids, we consider atoms as hard incompressible spheres which can be packed in variousforms. First we will see how atoms pack in metals.

Atoms in many metals form closed packed structures either in the form of hexagonal closedpacked structure or face-centered cubic structures. Some metals are a little loosely packed inthe form of body-centered cubic structure. Very rarely atoms pack in metals in the form ofsimple cubic structure.

1.3.1 Simple Cubic Structure

Simplest structure crystallographically but in the entire periodic table only polonium (Po)possesses this structure.

Structure contains only one atom per unit-cell.

Figure 1.15 Simple cubic structure

1.3.2 Body Centered Cubic or BCC Structure

Many metals like W, Fe (room temperature form) possess BCC structure.

Contains 2 atoms per unit-cell

Figure 1.16 BCC Structure

One of the important parameters of interest is packing factor, determining how loosly or densely astructure is packed by atoms.

Packing Factor: Volume of all atoms in one unit cell divided by Volume ofone unit-cell

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If r is the atomic radii in these structures, then

Packing Factor (Simple Cubic) =

Packing Factor (BCC) =

1.3.3 Closed Packed Structures

Each atom has 12 nearest neighbours touching the atom to each other.

Figure 1.17 Closed packing of atoms in FCC/HCPmetals

ABC ABC ABC . . . stacking leads to the formation of cubic closed packed (CCP) or face centered

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cubic (FCC) structure which has higher symmetry than other structures. The closed packed A, B, Cplanes are (111) planes in the structure.

AB AB AB . . . stacking leads to hexagonal closed packed (HCP) structure. The A or B planes areclosed packed c-plane or (001) planes of hexagonal structure.

Figure 1.18 FCC and HCPStructures

Now you can work out yourself that packing factor of both FCC and HCP is 0.74.

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Module 1: Structure of Ceramics Interstices in Structures 1.4 Interstices in Structures

Since the unit cell is not completely packed as packing efficiency in the previous structures is lessthan 100%, there are empty spaces inside which are called as interstices.

These interstices are very useful because there can contain smaller atoms which modify theproperties of materials tremendously, such as Carbon (C) in Iron (Fe) makes steel and makes ironstronger.

1.4.1 Interstices in FCC Structure

Tetrahedral Interstices

2 per atom

Octahedral Interstices

1 per atom

Figure 1.19 Interstices in a FCCstructure

So, by simple geometry, you can also estimate the size of the largest interstitial atom that would fit in theseinterstices without distorting them.

rtet = 0.225* rrtet = 0.225* r

1.4.2 Interstices in BCC Structures

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Figure 1.20 Interstices in a BCCstructure

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Module 1: Structure of Ceramics Structure of Covalent Ceramics

1.5 Structure of Covalent Ceramics

Most ceramic materials are neither purely covalently or ionically bonded materials. In most ionicallybonded materials, there is a significant level of covalency which is decreases as the differencebetween the electronegativities of cations and anions increases. While covalent bonding is prevalentamong the group IV solids such as diamond and many other compound semiconductors, mostceramics such as NaCl, MgO, BaTiO3, Fe3O4 etc are predominantly ionically bonded. Covalent

bonding, as we saw in preceding sections, arises from the sharing of orbitals and as a resultmaterials with this type of bonding are characterized by significant hybridization of orbitals anddirectionality of the bonds which play a crucial role in determining the crystal structure. In contrast,ionically bonded solids are predominantly based on the size difference between the cations and theanions and the formation of structures in them is determined by a set of rules called as Pauling’sRules which we will see later in this module.In this section, we will understand the structures of a few covalently bonded materials with emphasison the Diamond structure.

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1.5.1 Diamond Cubic Structure

Typical and well known purely covalent bonded materials are carbon (Diamond), Si, Ge andSiC.

For example, in diamond, the base lattice is FCC and is built by the C atoms with half of thetetrahedral sites filled by C atoms. Thus, the unit cell of diamond contains a total of 8 atoms.

The structure is typically called as diamond cubic structure.

Orbital hybridization of C atoms (sp3) requires that the atoms are tetrahedrally co-ordinatedand thus the structure has high degree of directionality.

One unit-cell consists of two FCC motifs, one at (0 0 0) and another at (¼ ¼ ¼).

What it means is that there are two FCC unit-cells of C intermingled into each other,with origin of one at (0,0,0) and another at (¼,¼,¼).

In case of compounds, FCC lattice can be formed by one type of atom and remaining atoms,usually from the same group, occupy half of the tetrahedral sites.

Figure 1.21 shows the crystal structure of diamond where one can clearly observe thetetrahedrally co-ordinated C atoms.

Figure 1.21 Diamond cubic structure (a) the unit cell showing all theatoms and (b) (001)-plan view of the structure where positionsmarked show the position in the z-direction only while x- and y-positions are self-explanatory.

You can also work out the packing factor of this unit which is lower than the typical FCC unitcell. This is because the tetrahedral site size in a normal FCC unit cell is 0.225*r while inthis structure, the size of the atom sitting at the site is much larger i.e. same size as thebase lattice atom.(Self Evaluation)

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1.5.2 Structure of Graphite

Other forms of Carbon such as graphite and fullerene are also covalent bonded but thestructures are entirely different.

Graphite has a layered structure where in each layer, carbon atoms are sp2 hybridized andthey make a hexagonal pattern. However, the bonding between individual layers is Van derWalls type of bonding. That is why Graphite is a soft material and is used as a lubricant.

Figure 1.22 Structure ofGraphite

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Module 1: Structure of Ceramics Ionically Bonded Ceramic Structures

1.6 Ionically Bonded Ceramic Structures

Most of the ceramic materials are compounds with anions and cations with differentelectronegativities. Hence, when these ions are brought together, they form a very strong ionicbond.

Typically, since anions are bigger in size than cations, anions tend to form the base latticeand cations fill in the interstices..However, it is not so simple. As there is an involvement of two different types of ions to forma crystal structure, there are certain rules or say guidelines which need to be followed to giverise to a stable crystal structure. These rules are called Pauling’s rules.

Based on these rules, typically ceramic structures are based on anions forming the baselattice and cations occupying the interstices in them. Fortunately, most ceramic compoundsare completely or partially ionically bonded and happen to be based on either of FCC or HCPpacking of anions. As a result, we can categorize the structures of most ceramic materials intofollowing categories

Compounds based on cubic closed packing (CCP or FCC) of ions

Compounds based on hexagonal closed packing (HCP) of ions

Other structures with some deviations from above two.

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Module 1: Structure of Ceramics Ionically Bonded Ceramic Structures

1.6.1 Pauling’s Rules

Anions being the larger ions form the base lattice and lead to the formation of coordinatedpolyhedrons around cations. The co-ordination is determined by the radius ratio of cations (rc)

to anions (ra) i.e. (rc/ra). Also, another point to note is that the ionic radius of each ion is also

dependent on its co-ordination.

Ligancy orCoordination

number

Range ofRadius Ratio

(rc/ra)

Configuration

2 0.0-0.155 Linear

3 0.155-0.225 Triangular

4 0.225-0.424 Tetrahedral

6 0.414-0.732 Octahedral

8 0.732-1.0 Cubic

12 1.0 or above FCC or HCP

The structure will be stable when it preserves the charge neutrality (Electrostatic valencerule).

Corner linking of polyhedrons is preferred over face or edge sharing to ensure largerseparation between cations. This is especially true for solids with smaller cations and cations

with bigger charges e.g. Ti4+ and Zr4+. For example, in SiO2, due to +4 charge on Si atoms,

corner linking of tetrahedrons is preferred.

In a crystal containing different cations, those of high valence and small coordination numbertend not to share the polyhedron elements with one another such as in materials like BaTiO3.

The number of essentially different kinds of constituents in a crystal tends to be small. Therepeating units will tend to be identical because each atom in the structure is most stable in aspecific environment. There may be two or three types of polyhedra, such as tetrahedra oroctahedra, but there will not be many different types (Rule of parsimony).

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Ionically Bonded Ceramic Structures

1.6.2 Bond strength

Bond strength is a useful parameter to determine whether the derived structure is correct ornot, at least whether the charge is neutral and stoichiomteric or not. Bond strength of an ion isdefined as the ratio of the valence of an ion to its co-ordination, i.e.

In a stoichiometric and charge neutral solid, the bond strengths of cations must be equal tothose of anions. Alternatively, you can work out bond strength of one ion and from this, youcan work out the valence of other ion which should match what is needed to maintain thestoichiometry and most cases, the common valence state.

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Compounds based on FCC Packing of Ions

1.7 Compounds based on FCC Packing of ions

In these structures, typically anions form the FCC lattice and cations fill tetrahedral or octahedralsites in most cases, although in some cases, some other co-ordination may be preferred. Here wewill discuss the following structures:

Rocksalt structures with NaCl as parent compound

Fluorite and anti-fluorite structures based on CaF2 and Na2O

Zinc Blende or Sphalerite structure based on ZnS

Spinel structure based of formula AB2O4

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Module 1: Structure of Ceramics Compounds based on FCC Packing of Ions

1.7.1 Rocksalt Structure

MX Type Compounds Based on NaCl or Rocksalt Structure

Anions (X) form the cation sub lattice with FCC structure.

Cations (M) fill the octahedral sites.

100% occupancy of sites according to the stoichiometry since there will be one octahedral siteper anion.

Radius ratio, rc/ra is typically between 0.414 - 0.732 with some exceptions.

Examples of ceramic materials with such structure as NaCl, MgO, NiO, FeO etc.

Figure 1.23 Schematic of structure of Rocksalt structuredcompounds

Lattice type: FCC and motif will be M at 0 0 0 and X at ½ 0 0

Four formula units per unit cell.

Some of the radius values of cations for selected rocksalt structured compounds are givenbelow:

Compound rc (nm) ra (nm) rc/ra

NaCl 0.102 0.181 0.564

MgO 0.072 0.140 0.514

SrO 0.118 0.140 0.842

NiO 0.069 0.140 0.492

FeO 0.078 0.140 0.557

MnO 0.053 0.140 0.378

PbO 0.119 0.140 0.85

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We can verify the bond-strength of ions in NaCl

Bond Strength of cations i.e. Na+ =

Valence of anions

which is the valence of chlorine and hence, proposed packing is appropriate.

The octahedrons shares at the edges. If there was corner sharing of polyhedra, co-ordinationnumber will be 2 for anions, which won’t maintain the stoichiomentry as you can verify usingbond-strength relationship.

Schematic representation of atomic arrangement on (110) plane of rock-salt structurecompounds shows the empty rows of tetrahedral voids along [001]-direction.

Figure 1.24 (110) plane of a Rocksalt structuredcompound

You can see that if both octahedral and tetrahedral voids were filled, this would bring cationsquite close to each other, leading to large electrostatic repulsion as like neighbours do notmake an energetically preferred configuration.

Another way of looking at this structure is to visualize hexagonal arrays of cations and anionsstacked along [111]-direction repeating in an alternative fashion.

The complete structure can be viewed as two FCC lattices, one of Na and another of Cl,interpenetrating into each other.

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1.7.2 Antifluorite (A2X) and Fluorite (AX2) Structures

1.7.2.1 Antifluorite

FCC packing of anions

All tetrahedral sites filled by cations

Coordination : Anions: 8, Cations: 4

Chemical formula: M2X

Example: Li2O, Na2O, K2O

Radius ratio (rc/ra): 0.225-0.414

Examples: r(Li+) : 0.059 nm, r(Na+) : 0.099 nm, r(O-) : 0.14 nm

Figure 1.25 Antifluorite structure

Lattice type: FCC

Motif – X: 0 0 0, M -

Four formula units per unit cell

In this structure in many cases, although rc/ra ratio predict an octahedral co-ordination,tetrahedral coordination is preferred to fulfill the stoichiometry requirements. In turn, anionsare cubic coordinated by cations (CN: 8)

The structure shows corner sharing of tetrahedra.

1.7.2.2 Fluorite Structure (CaF2 Structure)

Slightly bigger cations in comparison to other structures

Example:UO2, ZrO2, CaF2, CeO2

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Typical representation of the structure appears as if cations make a FCC lattice and anionsoccupy the tetrahedral sites.

Figure 1.26 Fluorite Structure

While more appropriate Fluorite structure representation is shown below where eight primitivecubic unit cells made by anions are joined together to make a big cube and cations occupythe centers of four of these small cubes in an ordered fashion.

Figure 1.27 A more appropriate representation of fluoritestructure

Co-ordination number: Cations - 8 ; Anions - 4

Lattice: FCC

Motif: M – 0 0 0; X – ¼ ¼ ¼; ¾ ¾ ¾

Examples of ionic radii of a few ions:

U4+ : 0.1 nm, Zr4+: 0.084 nm, Ce4+: 0.097 nm, O2- : 0.14 nm (observe that cations are quitelarge as compared to oxygen ions)

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The structure as you can also see has a large void in the center of unit cell made by cations.

These empty spaces make such oxides, good ionic conductors which is useful in applicationssuch as energy storage e.g. batteries.

For having some fun with the structure, we can also draw as projection of this material on(110) plane. Here you can see the row of empty octahedral sites along [110]-direction.

Figure 1.28 View of (110) plane of fluorite structure

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1.7.3 Zinc Blende (MX) Structure

MX type compounds, also called as sphalerite structured compounds based on a mineralname of sphalerite.

Mostly oxides and sulphides follow this structure. Examples are ZnO, ZnS, BeO etc.

Some covalently bonded materials and compounds have similar structure such as GaAs, SiC,BN. You can also visualize diamond also having similar structure with both anion and cationbeing of same type.

Typically compounds with tetrahedral co-ordination assume this structure.

In this structure anions form FCC lattice and cations occupy the tetrahedral interstices.

Due to stoichiometry, half of the tetrahedral sites are filled.

Compounds with radius ratio : 0.225-0.414 follow this structure with a few exceptions

where bonding favours a tetrahedral coordination despite unfavourable radius ratio, especiallycovalently bonded compounds.

Examples:Zn2+ - 0.06 nm,Be2+ - 0.027 nm,O2- - 0.14 nm, S2-- 0.184 nm

Figure 1.29 Zinc Blende or Sphaleritestructure

Coordination numbers: M - 4, X - 4

Lattice type: FCC

Motif: M – 0 0 0; X – ¼ ¼ ¼

4 formula units per unit cell.

Tetrahedra are shared at corners.

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1.7.4 Spinel Structure

Formulae – (A2+)(B3+)2O4 or AB2O4 or AO.B2O3

FCC Packing of anions

Partial occupancy of both tetrahedral and octahedral sites i.e.1/8th of tetrahedral and ½ of theoctahedral sites are occupied.

A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration2×2×2, so it is a big structure consisting of 32 oxygen atoms, 8 A atoms and 16 B atoms.

Depending on how cations occupy different interstices, spinel structure can be Normal orInverse.

1.7.4.1 Normal Spinel

Chemical formula: (A2+)(B3+)O4

Examples are many aluminates such as MgAl2O4, FeAl2O4, CoAl2O4 and a few ferrites such

as ZnFe2O4 and CdFe2O4.

In this structure, all the A2+ ions occupy the tetrahedral sites and all the B3+ ions occupy theoctahedral sites.

Apply bond strength rule to verify the stoichiometry

Cations: - A2+ - 2/4 ; B3+ - 3/6

Oxygen valence = (2/4x1)+ (3/6x3) = 2

Figure 1.30 Schematic of spinel structure

1.7.4.2 Inverse Spinel B(AB)O4

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Chemical formula: (A2+)(B3+)2O4 but can be more conveniently written as B(AB)O4.

Most ferrite follow this structure such as Fe3O4 (or FeO.Fe2O3), NiFe2O4, CoFe2O4 etc.

In this structure, ½ of the B3+ ions occupy the tetrahedral sites and remaining ½ B3+ and all

A2+ ions occupy the octahedral sites (now you can hopefully make sense of the formula in theprevious line).

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Module 1: Structure of Ceramics Other Cubic Structures

1.8 Other Cubic Structures

There are a few structures, which appear as if they are based on cubic closed packing ofanions. However the actual structure is rather different and many of these structures are merelybased on the cubic packing of anions. Here, we discuss the perovskite structure based onABO3 structure, CsCl structure and ReO3 structure.

1.8.1 Perovskite (ABO3) Structure

ABO3 type compounds

Examples are many titanates like BaTiO3, SrTiO3, PbTiO3 etc. which happen to be

technologically very useful compounds as we will see in later modules.

In ABO3 structured compounds, A ion is twelve fold coordinated by oxygen (like a

dodecahedra) and B ion is octahedrally coordinated by oxygen ions.

Oxygen atoms form an FCC-like (not FCC) cell with atoms missing from the corners whichare occupied by A atoms.

Bond strength check:

Cation: Ba: 2/12 = 1/6 and Ti: 4/6 = 2/3

Oxygen valence = 1/6 x Coordination number by Ba + 2/3 x coordination number by Ti .

Figure 1.31 Perovskite structure

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Figure 1.32 Polyhedra model of perovskitestructure

Lattice type: Primitive Cubic (NOT FCC!)

Motif: A ion - 0 0 0, B ion – ½ ½ ½, O ion - ½ ½ 0, 0 ½ ½, ½ 0 ½

One Formula unit per unit cell

Coordination

B cation is surrounded by oxygen octahedra which share corners.

A cation is surrounded by oxygen dodecahedra which touch faces of octahedra.

An important parameters about perovskites is the their “Tolerance Factor (t)” which is definedas

This is derived from the geometry of a cube in which the atoms are of such sizes that theytouch each other and hence, the face diagonal of the unit cell would be times the unit-celllength, as result t = 1 for a perfect cubic perovskite

.

However, due to variations in ionic radii of various ions, many perovskites show deviationsfrom t = 1 and may not even have a cubic structure. Deviations from t = 1 signify the level oflattice distortion.

For example, BaTiO3 has cubic structure only above ~120°C while it is tetragonal at room

temperature and further adopts orthorhombic and rhombohedral structure if cooled below RT.

Perovskites can also have various combinations of ionic valence such as

e.g. A2+B4+O4 , BaTiO3, PbTiO3, CaTiO3, SrTiO3 etc.

e.g. A3+B3+O4 , LaAlO3, LaGaO3, BiFeO3 etc.

Mixed Perovskites:

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A2+(B2+1/3B5+2/3)O3 eg. Pb(Mg1/3Nb2/3)O3

A2+(B3+1/2B5+1/2)O3 eg. Pb(Sc1/2Ta1/2)O3

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1.8.2 ReO3 Structure

Stoichiometry : MX3

Lattice type: Primitive cubic

Atomic Positions: M- 0 0 0; X - ½ 0 0, 0 ½ 0, 0 0 ½

Coordination Numbers M CN = 6 Octahedral coordination X CN = 2 Linear coordination

Can be visualized as perovskite ABO3 structure with empty B-sites

Representative Oxides

ReO3, UO3, WO3

Used for gas sensing and electrochromic applications

Figure 1.33 ReO3 structure and polyhedramodel

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1.8.3 CsCl Structure

MX type compounds, parent compound being CsCl.

Examples: Halides such as CSCl, AgI, AgBr etc.

Radius ratio governs cubic co-ordination of both cations and anions.

Lattice type: Primitive cubic lattice.

Motif: Anions (X): 0 0 0, Cations (M): ½ ½ ½

One formula unit per unit cell.

Figure 1.34 (a) CsCl structure (b) Ball-stickmodel

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Module 1: Structure of Ceramics Orthogonal Structures

1.9 Orthogonal Structures

Many superconductors follow the structures which are perovskite based i.e., the structurecontains the perovskite structured units stacked along c-axis or [001]-direction in most cases.The examples are superconductors such as YBa2Cu3O7, ferroelectrics such as Bi4Ti3O12etc. In some other compounds such as La-Sr-Cu-O, the structure is composed of alternatingperovskite and rocksalt structure units. Such a representation makes it easy to understandthem.

Here we will take examples of Y-Ba-Cu-O and La-Sr-Cu-O and discuss them very briefly.

1.9.1 Yttrium Barium Copper Oxide or YBCO (YBa2Cu3O7)

Parent compound is Y3Cu3+3O9 (see Fig. 1.35) which also contains perovskite units.

Doping of Y by Ba leads to structure modification (step 1) as well as reduction of Cu3+ to

Cu2+ state (step 2) and thus resulting in the reduction in the number of required oxygen ionsand hence creates oxygen vacancies in the structure. This gives a transition temperature of~92 K below which the compound has zero electrical resistance i.e. is a superconductor.

Y3Cu3+3O9→YBa2Cu3+3O8→ YBa2Cu2+2Cu3+O7-x

Figure 1.35 Origin of the structure of YBa2Cu3O7-x as a triple-perovskite unit. (D.M. Smyth, PP.1-10 in ceramic superconductors II.Research Update 1988, M.F.Yan, Ed. The American Ceramic Society,1988.)

Here Cu coordination is of interest:

Cu2+ atoms have four-fold coordination along Cu-O chains.

Cu3+ atoms have five-fold coordination in the Cu-O planes.

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Figure 1.36 Atomic coordination inYBCO

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Module 1: Structure of Ceramics Orthogonal Structures

1.9.2 Lanthanum Strontium Copper Oxide La2-xSrxCuO4

Parent compound La2CuO4 is actually a mixture of one Rocksalt structured compound, LaO

and one perovskite structured compound, LaCuO3 and can also be written as LaO.LaCuO3.

The structure shows a layered structure with layers stacked as A4O-AO4-A4O as shown

below where A is La.

Substitution of La by Sr results in the compound La2-x Srx CuO4 turning into a

superconductor with a Tc ~ 35K.

Figure 1.37 (a) Origin of La2-xSrxCuO4 structure, shown in(b) as two perovskite and cells

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Structures based on HCP Packing of Ions

1.10 Structures based on HCP packing of ions

Similar to FCC packing of anions, many ceramic structures are also based on another type of closed packing of anions i.e. hexagonal closed packed (HCP). In this category we will look at the following structures:

Wurzite structured compounds

Corundum structured compounds

Ilmenite structure compounds

Lithium niobate structured compounds and

Rutile structure

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Module 1: Structure of Ceramics Structures based on HCP Packing of Ions

1.10.1 Wurtzite (MX) structured compounds

Compounds with M2+X2- stoichiometry

Examples are the polymorphs of Sphalerite structured compounds such as ZnS ZnO, SiC.

Co-ordination of both anions and cations is 4, as in Sphalerite structured compounds.

Anions form an HCP lattice with ½ of the tetrahedral sites occupied by cations.

The only difference to Sphalerite structure is that here anions pack in the form ofABCABC… stacking.

Figure 1.38 Wurtzite structure and polyhedralmodel

As you can notice, all the tetrahedrons point in one direction i.e. along the c-axis of the unit-cell and they share the corners.

Lattice type: Primitive, HCP

Motif: M: 0 0 0 and ; X: and

The filling of structure can be seen below.

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Figure 1.39 Layer by layer filling in Wurtzite

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1.10.2 Corundum (Al2O3) Structured Compounds

M2X3 type of compounds

- Alumina or Sapphire (Al2O3) is the parent compound.

Other examples are compounds like Cr2O3, Fe2O3

Anions form an HCP lattice

Two-third of octahedral voids are occupied by the cations to maintain the stoichiometry.

Coordination numbers: M: 6, X: 4.

This arrangement preserves the charge neutrality as you can also verify using bond strengthformula.

This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell andstart filling the interstices.

Figure 1.40 Layered filling of Corundum

One unit-cell consists of six layers of oxygen ions.

A side view of the structure on plane can be seen below where you can see columns of

cations along the c-axis with 2/3 rd filling of octahedral sites.

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Figure 1.41 View of {1010} plane ofCorundum

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Module 1: Structure of Ceramics Structures based on HCP Packing of Ions 1.10.3 Ilmenite Structure

The stoichiomteric formula is ABO3 (different to perovskite ABO3)

The parent compound is FeTiO3.

Other compounds which follow this structure are CdTiO3, CoTiO3, CrRhO3, FeRhO3, FeVO3, LiNbO3,

MgGeO3, MgTiO3.

This structure is very similar to Corundum or a - Al2O3.

Imagine the Corundum structure and replace Al atoms in the octahedral sites in one (0001)-layer i.e. half ofthe total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sitesand continue this order of substitution along the c-axis of the unit-cell.

Hence, the atomic arrangement is similar to Al2O3 except with alternate layers of Fe and Ti in place of Al

.

Coordination numbers: both Fe and Ti remain octahedrally coordinated while O is coordinated by 4 cations i.e. 2 Fe and 2 Ti.

Bond strength rule gives correct oxygen valence:

+ =2=Oxygen

valence

Figure 1.42 Layered filling of Ilmenite

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One unit-cell consists of six layers of oxygen ions.

A side view of the structure on {10-10} plane, as shown below, shows the columns of cations along the c-

axis with 2/3rd filling of octahedral sites which are alternately filled by Fe and Ti ions and then followed by a

vacant site.

Figure 1.43 Ilmenite structure on {10-10}plane

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1.10.4 Lithium Niobate Structure

Structure is similar to Al2O3 except that Al sub-lattice is substituted in an ordered manner by

Li and Nb ions in the same layer unlike in alternating layer in Fe2O3.

The parent compound LiNbO3 is ferroelectric in nature and hence, is technologically

important.

LiNbO3 also has highly anisotropic refractive index and it shows birefringence which is

changeable by electric field.

Such materials are used in electro-optic devices.

Figure 1.44 Atomic arrangement of a layer inLiNbO3 structure

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Figure 1.45 Structure on {10-10} plane in LiNbO3

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1.10.5 Rutile Structure

Polymorph of titanium di-oxide or TiO2

Other forms are Anatase and Brookite.

It is formed by quasi-HCP packing of anions.

Half of the octahedral sites are filled by cations.

The resulting structure has a tetragonal crystal structure due to a slight distortion in the lattice.

Anisotropic diffusion properties of cations are found in TiO2.

Materials shows large and anisotropic refractive index and high birefringence.

TiO2 is often used as pigments and is non-toxic.

Figure 1.46 Structure of a layer of oxygen and Titanium inRutile

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Figure 1.47 Unit-cell of Rutile

Figure 1.48 Polyhedral model of Rutile

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Module 1: Structure of Ceramics Summary

Summary

In ionic solids, anions typically form the base lattice.

Interstices can be completely or partially filled depending on the size of cations andstoichiometry.

Pauling’s rules play an important role in structure determination and deviations lead tostructural distortions.

Most ceramic compounds follow three types of common structures based on packing ofanions i.e.

Structures based on FCC packing of anions

Structures based on HCP packing of anions

Primitive cubic or other structures

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