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Electroceramics Overview
Electro Ceramics Web Course (NPTEL)
Contact information of the course instructor: Ashish Garg, Associate Professor
Department of Materials science and Engineering
Indian Institute of Technology Kanpur
Kanpur 208016, India Telephone: 0512-259-7904 Email: [email protected] Web: http://home.iitk.ac.in/~ashishg Introduction
Electro-ceramics or broadly speaking electronic, optical and magnetic ceramics are useful in a variety of 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 RAMs Ferroelectrics such as BaTiO3 and PbTiO3 are used as sensors and actuators Magnetic oxides such as iron oxides are used for data storage in magnetic heads ZnO is used circuit protection materials in devices named as varistors ZrO2 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 to understand 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 physics Suited for final year undergraduate students of most disciplines and fresh graduate students.
List of Topics
Module Topics Equivalent Lectures (50-60 m each)
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 Materials 1.2 A Brief Review of Bonding in Materials
1.3 Packing of atoms in metals
1.4 Interstices in Structures
1.5 Structure of Covalent Ceramics 1.6 Ionically Bonded Ceramic Structures 1.7 Compounds based on FCC Packing of ions 1.8 Other cubic structures 1.9 Orthogonal Structures 1.10 Structures based on HCP packing of ions 1.11 Summary
2. Defect Chemistry and Defect Equilibria
2.1 Point Defects 2.2 Kröger–Vink notation in a metal oxide, MO 2.3 Defect Reactions 2.4 Defect Structures in Stoichiometric Oxides 2.5 Defect Structures in Non-stoichiometric Oxides: 2.6 Dissolution of foreign cations in an oxide 2.7 Concentration of Intrinsic Defects 2.8 Intrinsic and Extrinsic Defects 2.9 Units for defect Concentration 2.10 Defect Equilibria 2.11 Defect Equilibria in Stoichiometric Oxides 2.12 Defect Equilibria in Non-Stoichiometric Oxides 2.13 Defect Structures involving Oxygen vacancies and interstitials: 2.14 Defect Equilibrium Diagram 2.15 A Simple General Procedure for constructing at Brouwer’s Diagram 2.16 Extent of non-stoichiometry 2.17 Example: Comparative behaviour of TiO2 and MgO vis-à-vis oxygen pressure 2.18 Electronic Disorder 2.19 Examples 2.20 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 Information 3.7 Ionic Conduction: Basic Facts 3.8 Ionic and Electronic Conductivity 3.9 Characteristics of Ionic Conduction 3.10 Theory of Ionic Conduction Conduction in Glasses 3.11 Conduction in Glasses 3.12 Fast Ion Conductors 3.13 Examples of Ionic Conduction 3.14 Electrochemical Potential 3.15 Nernst Equation and Application of Ionic Conductors 3.16 Examples of Ionic Conductors in Engineering Applications 3.17 Summary
4. Dielectric Ceramics: Basic Principles
4.1 Basic Properties: Dielectrics in DC electric field 4.2 Mechanisms of Polarization 4.3 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 Moments 6.2 Macroscopic view of Magnetization 6.3 Classification of Magnetism 6.4 Diamagnetism 6.5 Paramagnetism 6.6 Ferromagnetism 6.7 Antiferromagnetism 6.8 Ferrimagnetism 6.9 A Comparison 6.10 Magnetic Losses and Frequency Dependence 6.11 Magnetic Ferrites 6.12 Summary
7. High temperature Superconductors
7.1 Background 7.2 Meissner Effect 7.3 The critical field, Hc 7.4 Theory of Superconductivity 7.5 Discovery of high temperature superconductivity 7.6 Mechanism of high temperature superconductivity 7.7 Applications 7.8 Summary
8. Multiferroic and Magnetoelectric Ceramics
8.1 Introduction 8.2 Historical Perspective 8.3 Requirements of a magnetoelectric and multiferroic material 8.4 Magnetoelectric Coupling 8.5 Type I Multiferroics 8.6 Type II Multiferroics 8.7 Two Phase Materials 8.8 Summary
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9. Synthesis Methods
9.1 Bulk Preparation Methods 9.2 Thin Film Preparation Methods 9.3 Thin film deposition: Issues 9.4 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 provided in 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, Wiley
3. Principles of Electronic Ceramics, by L. L. Hench and J. K. West, Wiley 4. Electroceramics: Materials, Properties, Applications, by A. J. Moulson and J. M. Herbert, Wiley 5. 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, Wiley 7. Electrical Properties of Materials, L. Solymer and D. Walsh, Oxford University Press 8. Introduction of Solid State Physics, N.W. Ashcroft and N.D. Mermin, Brooks Cole 9. 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, Wiley 12. Non-stoichiometric Oxides, O. Toft Sørensen, Academic Press 13. Dielectrics and Waves, A.R. von Hippel, John Wiley and Sons 14. Feynman Lectures on Physics, Volume 1-3, R.P. Feynman, Addison Wesley Longman 15. Materials Science and Engineering: A first course, V. Raghavan, Prentice Hall of India 16. 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 a brief discussion on how atoms pack together in the solids and what are the types of interstices present in various structures. Then we would briefly delve into the types of bonding with reference to the nature of materials. Together, this information will form the basis for structures in ceramic materials which are typically bonded with a mix of ionic and covalent bonding. Subsequently, we would discuss the structure of ceramic materials with purely covalent bonding followed by rather detailed description of ceramic materials with ionic bonding. These are essentially based on packing of anions 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 crystal
structures 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 arrangement Regular arrangement No periodicity Periodicity Non-identical neighbourhood Identical neighbourhood of each point
Figure 1.1(a) Unit-cell representation
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Figure 1.1(b) Schematics of arrangement of points in space
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Module 1: Structure of Ceramics Brief Review of Structure in Materials
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 an unit 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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Module 1: Structure of Ceramics Brief Review of Structure in Materials
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 of atoms 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 a
crystal with a defined structure.
Figure 1.3 Formation of a periodic crystal structure
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Module 1: Structure of Ceramics Brief Review of Structure in Materials
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 per unit cell, and Non-primitive lattices having more than one lattice points or more than one unit of motif per unit cell.
Figure 1.4 Primitive and Non-primitive lattices
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Module 1: Structure of Ceramics Brief Review of Structure in Materials
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 is an 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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Module 1: Structure of Ceramics Brief Review of Structure in Materials
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 of any shape as long it contains one lattice point.
However, as one starts defining various shapes, we come up with seven categories, called as crystal systems, in which all possible unit cells shapes would fit provided space filling criteria is fulfilled.
Seven crystal systems are shown below.
Minimum Crystal system and lattice symmetry parameters elements Cubic Four 3-fold a = b = rotation axes c,
Tetragonal One 4-fold rotation (or rotation-inversion) axis
Orthorhombic Three perpendicular 2- fold rotation (or rotation-inversion) axis
Rhombohedral One 3-fold rotation (or a = b = c rotation-inversion) axis
Hexagonal One 6-fold rotation (or rotation-inversion) axis
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One 2-fold Monoclinic rotation (or rotation-inversion) axis
Triclinic None
Figure 1.6 Seven crystal systems
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Module 1: Structure of Ceramics Brief Review of Structure in Materials
1.1.7 Bravais Lattices
Taking seven crystal systems and symmetry elements into account, Bravais came out with the fact that there are a total of 14 Bravais Lattices which are shown below.
Cubic Tetragonal Orthorhombic Rhombohedral Hexagonal Monoclinic Triclinic a=b=c, a=b=c
Figure 1.7 Fourteen Bravais lattices
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Module 1: Structure of Ceramics Brief Review of Structure in Materials
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 directions where u , v , w are integers
Vector components of the direction resolved along each of the crystal axis reduced to smallest set 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 and directions in a crystal. If you are not sure, then refer to any elementary materials science text book on 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 of the materials. For basics of bonding, you can refer to any elementary materials science book (see bibliography) to get familiar with the fundamental aspects of bonding between atoms i.e. how to determine the equilibrium distance, bond energy and fundamental properties like young’s modulus. Bonding in materials can be divided in two categories:
Primary bonding Secondary 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 metal giving rise to flexible bonds, good malleability, high electrical and thermal conductivity. Most metals 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 and directional bonding.
Examples of materials showing this bonding are mainly group IV elements and compounds such 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, for example 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 excellent insulators.
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Figure 1.11 Schematic of ionic bonding
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Module 1: Structure of Ceramics A Brief Review of Bonding in Materials
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 permanent dipoles
Examples are materials like polymers.
Figure 1.14 Secondary bonding in polymers
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Module 1: Structure of Ceramics A Brief Review of Bonding in Materials
1.2.3 Bonding, Bond Energy and General Remarks
Type Bond Energy Comments Large magnitude
(Large T , large E and small a m Non-directional
Ionic ) (Typically Ceramics) Example: MgO – 1000 kJ / o mol, Tm- 2800 C
Variable for materials such as Si, Ge
Large for Diamond (Carbon) Directional and small for Bismuth Covalent (Typically Semiconductors, Typically Large Tm and large E Polymers and some Ceramics)
Example: Si - 450 kJ/mol; Tm- 1410oC
Variable Large: Tungsten (W)
Metallic Small: Mercury (Hg) Non–directional (Metals)
Moderate: Al: 68 kJ/mol, Tm~ 670oC
Smallest Directional
Secondary Inter–chain (Polymer) Characterized by low Tm, low E and large a 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 various forms. First we will see how atoms pack in metals.
Atoms in many metals form closed packed structures either in the form of hexagonal closed packed structure or face-centered cubic structures. Some metals are a little loosely packed in the form of body-centered cubic structure. Very rarely atoms pack in metals in the form of simple 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 a structure is packed by atoms.
Packing Factor: Volume of all atoms in one unit cell divided by Volume of one 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/HCP metals
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, C planes are (111) planes in the structure.
AB AB AB . . . stacking leads to hexagonal closed packed (HCP) structure. The A or B planes are closed packed c-plane or (001) planes of hexagonal structure.
Figure 1.18 FCC and HCP Structures
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 less than 100%, there are empty spaces inside which are called as interstices.
These interstices are very useful because there can contain smaller atoms which modify the properties of materials tremendously, such as Carbon (C) in Iron (Fe) makes steel and makes iron stronger.
1.4.1 Interstices in FCC Structure
Tetrahedral Interstices
2 per atom
Octahedral Interstices
1 per atom
Figure 1.19 Interstices in a FCC structure
So, by simple geometry, you can also estimate the size of the largest interstitial atom that would fit in these interstices without distorting them.
rtet = 0.225* r
rtet = 0.225* r
1.4.2 Interstices in BCC Structures
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Figure 1.20 Interstices in a BCC structure
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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 ionically
bonded materials, there is a significant level of covalency which is decreases as the difference
between the electronegativities of cations and anions increases. While covalent bonding is prevalent among the group IV solids such as diamond and many other compound semiconductors, most ceramics 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 result materials with this type of bonding are characterized by significant hybridization of orbitals and directionality 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 the anions and the formation of structures in them is determined by a set of rules called as Pauling’s Rules which we will see later in this module.
In this section, we will understand the structures of a few covalently bonded materials with emphasis on the Diamond structure.
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Module 1: Structure of Ceramics Structure of Covalent Ceramics
1.5.1 Diamond Cubic Structure
Typical and well known purely covalent bonded materials are carbon (Diamond), Si, Ge and SiC.
For example, in diamond, the base lattice is FCC and is built by the C atoms with half of the tetrahedral 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-ordinated and 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 the tetrahedrally co-ordinated C atoms.
Figure 1.21 Diamond cubic structure (a) the unit cell showing all the atoms and (b) (001)-plan view of the structure where positions marked 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 unit cell. This is because the tetrahedral site size in a normal FCC unit cell is 0.225*r while in this structure, the size of the atom sitting at the site is much larger i.e. same size as the base lattice atom.(Self Evaluation)
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Module 1: Structure of Ceramics Structure of Covalent Ceramics
1.5.2 Structure of Graphite
Other forms of Carbon such as graphite and fullerene are also covalent bonded but the structures are entirely different.
Graphite has a layered structure where in each layer, carbon atoms are sp2 hybridized and they make a hexagonal pattern. However, the bonding between individual layers is Van der
Walls type of bonding. That is why Graphite is a soft material and is used as a lubricant.
Figure 1.22 Structure of Graphite
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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 different electronegativities. Hence, when these ions are brought together, they form a very strong ionic bond.
Typically, since anions are bigger in size than cations, anions tend to form the base lattice and cations fill in the interstices. . However, it is not so simple. As there is an involvement of two different types of ions to form a crystal structure, there are certain rules or say guidelines which need to be followed to give
rise 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 base lattice and cations occupying the interstices in them. Fortunately, most ceramic compounds are completely or partially ionically bonded and happen to be based on either of FCC or HCP packing of anions. As a result, we can categorize the structures of most ceramic materials into following 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 coordinated polyhedrons 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 or Range of Configuration Coordination Radius Ratio number (r /r ) c a 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 valence rule).
Corner linking of polyhedrons is preferred over face or edge sharing to ensure larger separation between cations. This is especially true for solids with smaller cations and cations 4+ 4+ with bigger charges e.g. Ti and Zr . 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 number tend 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. The repeating units will tend to be identical because each atom in the structure is most stable in a specific environment. There may be two or three types of polyhedra, such as tetrahedra or octahedra, but there will not be many different types (Rule of parsimony).
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Module 1: Structure of Ceramics
Ionically Bonded Ceramic Structures
1.6.2 Bond strength
Bond strength is a useful parameter to determine whether the derived structure is correct or not, at least whether the charge is neutral and stoichiomteric or not. Bond strength of an ion is defined 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 to those of anions. Alternatively, you can work out bond strength of one ion and from this, you can work out the valence of other ion which should match what is needed to maintain the stoichiometry and most cases, the common valence state.
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Module 1: Structure of Ceramics
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 octahedral sites in most cases, although in some cases, some other co-ordination may be preferred. Here we will 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 site per 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 structured compounds
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 given below:
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-ordination number will be 2 for anions, which won’t maintain the stoichiomentry as you can verify using bond-strength relationship.
Schematic representation of atomic arrangement on (110) plane of rock-salt structure compounds shows the empty rows of tetrahedral voids along [001]-direction.
Figure 1.24 (110) plane of a Rocksalt structured compound
You can see that if both octahedral and tetrahedral voids were filled, this would bring cations quite close to each other, leading to large electrostatic repulsion as like neighbours do not make an energetically preferred configuration.
Another way of looking at this structure is to visualize hexagonal arrays of cations and anions stacked 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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Module 1: Structure of Ceramics Compounds based on FCC Packing of Ions
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, anions are 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 anions occupy the tetrahedral sites.
Figure 1.26 Fluorite Structure
While more appropriate Fluorite structure representation is shown below where eight primitive cubic unit cells made by anions are joined together to make a big cube and cations occupy the centers of four of these small cubes in an ordered fashion.
Figure 1.27 A more appropriate representation of fluorite structure
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:
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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 applications such 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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Module 1: Structure of Ceramics Compounds based on FCC Packing of Ions
1.7.3 Zinc Blende (MX) Structure
MX type compounds, also called as sphalerite structured compounds based on a mineral name 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 cation being 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, especially covalently 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 Sphalerite structure
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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Module 1: Structure of Ceramics Compounds based on FCC Packing of Ions
1.7.4 Spinel Structure
2+ 3+ Formulae – (A )(B )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 the octahedral sites are occupied.
A spinel unit-cell is made up of eight FCC cells made by oxygen ions in the configuration 2×2×2, 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 or Inverse.
1.7.4.1 Normal Spinel
2+ 3+ Chemical formula: (A )(B )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 the octahedral sites.
Apply bond strength rule to verify the stoichiometry
2+ 2 3+ 3 Cations: - A - /4 ; B - /6 2 3 Oxygen valence = ( /4x1)+ ( /6x3) = 2
Figure 1.30 Schematic of spinel structure
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2+ 3+ Chemical formula: (A )(B )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 the previous 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 of anions. However the actual structure is rather different and many of these structures are merely based on the cubic packing of anions. Here, we discuss the perovskite structure based on ABO3 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 which are occupied by A atoms.
Bond strength check: 2 1 4 2 Cation: Ba: /12 = /6 and Ti: /6 = /3 1 2 Oxygen valence = /6 x Coordination number by Ba + /3 x coordination number by Ti .
Figure 1.31 Perovskite structure
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Figure 1.32 Polyhedra model of perovskite structure
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 defined as
This is derived from the geometry of a cube in which the atoms are of such sizes that they touch each other and hence, the face diagonal of the unit cell would be times the unit-cell length, as result t = 1 for a perfect cubic perovskite
.
However, due to variations in ionic radii of various ions, many perovskites show deviations from t = 1 and may not even have a cubic structure. Deviations from t = 1 signify the level of lattice 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
2+ 4+ e.g. A B O4 , BaTiO3, PbTiO3, CaTiO3, SrTiO3 etc.
3+ 3+ e.g. A B O4 , LaAlO3, LaGaO3, BiFeO3 etc.
Mixed Perovskites:
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2+ 2+ 5+ A (B 1/3B 2/3)O3 eg. Pb(Mg1/3Nb2/3)O3
2+ 3+ 5+ A (B 1/2B 1/2)O3 eg. Pb(Sc1/2Ta1/2)O3
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Module 1: Structure of Ceramics Other Cubic Structures
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 polyhedra model
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Module 1: Structure of Ceramics Other Cubic Structures
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-stick model
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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 structure contains the perovskite structured units stacked along c-axis or [001]-direction in most cases. The examples are superconductors such as YBa2Cu3O7, ferroelectrics such as Bi4Ti3O12 etc. In some other compounds such as La-Sr-Cu-O, the structure is composed of alternating perovskite and rocksalt structure units. Such a representation makes it easy to understand them.
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)
3+ Parent compound is Y3Cu 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 ions and 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.
3+ 3+ 2+ 3+ Y3Cu 3O9→YBa2Cu 3O8→ YBa2Cu 2Cu 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 in YBCO
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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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Module 1: Structure of Ceramics
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 of ABCABC… stacking.
Figure 1.38 Wurtzite structure and polyhedral model
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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Module 1: Structure of Ceramics Structures based on HCP Packing of Ions
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 strength formula.
This can be best viewed when we look at the basal plane of (0001)-plane of the unit-cell and start 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 2 rd cations along the c-axis with /3 filling of octahedral sites.
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Figure 1.41 View of {1010} plane of Corundum
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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 of the total aluminum atoms by Fe and the remaining half in the next layer by Ti atoms in the octahedral sites and 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- 2 rd axis with /3 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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Module 1: Structure of Ceramics Structures based on HCP Packing of Ions
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 in LiNbO3 structure
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Figure 1.45 Structure on {10-10} plane in LiNbO3
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Module 1: Structure of Ceramics Structures based on HCP Packing of Ions
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 in Rutile
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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 and stoichiometry.
Pauling’s rules play an important role in structure determination and deviations lead to
structural distortions. Most ceramic compounds follow three types of common structures based on packing of
anions 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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