Crystallographic Texture and Group Representations

Foreword
Crystallographic Texture and Group Representations
Contents
Preface
Introduction
Comments on Contents
PART I. RUDIMENTS OF CLASSICAL TEXTURE ANALYSIS
1 Parametrization of Rotations
1.1 Preliminaries in Vector Algebra
1.1.1 Linear Transformations
1.1.2 Tensor Product of Two Vectors
1.1.3 Change of Orthonormal Basis
1.2 Affine Coordinate Systems
1.2.1 Basis and Reciprocal Basis
1.2.2 Change of Basis
1.3 Space of Linear Transformations
1.4 Exponential and Logarithmic Function of aMatrix
1.4.1 Basic Definitions
1.4.2 Properties of the Matrix Exponential
1.5 Active Versus Passive View of Rotations
1.5.1 Product of Rotations
1.6 Euler’s Theorem
1.7 Parametrization of Rotations by Euler Angles
1.8 Comparison with Other Conventions
1.8.1 Convention Adopted by Roe and by Matthies
1.8.2 Conventions Adopted by Bunge and Gel’fand et al.
1.9 Description of Rotations by Axis-Angle Parameters
1.10 Misorientation and Distance Between Two Rotations
2 Ideal Crystals and the Crystallographic Groups
2.1 Preliminaries
2.2 The Euclidean Group
2.3 Ideal Crystals and Crystallographic Groups
2.3.1 Lattice Groups and Lattices
2.3.2 Primitive Unit Cells
2.3.3 Crystallographic Point Groups
2.3.4 Characterization of Elements of Space Group
2.3.5 Change of Coordinate System
2.3.6 Space-Group Types and International Tables A
2.3.7 Point-Group Types and Conjugacy Classes of Subgroups of O(3)
2.4 Finite Subgroups of the Rotation Group
2.4.1 Poles, Stabilizers, and Rotations About the Same Axis
2.4.2 Enumeration of Finite Rotational Groups
2.5 The Crystallographic Point Groups
2.5.1 The Crystallographic Restriction
2.5.2 The Proper Crystallographic Point Groups
2.5.3 The Improper Crystallographic Point Groups
2.6 Geometric Crystal Classes and Laue Classes
2.6.1 The 32 Crystallographic Point-Group Types
2.6.2 The 11 Laue Classes
2.7 Holohedries
2.8 The Bravais Lattices
2.8.1 Lattice Types
2.8.2 Conventional Lattice Basis and Unit Cell
2.8.3 The Hexagonal, Tetragonal, and Rhombohedral Lattice Systems
2.8.4 The Orthorhombic and Cubic Lattice Systems
2.8.5 The Monoclinic and Triclinic Lattice Systems
2.8.6 Summary. Metric Specialization
2.9 The Seven Crystal Systems
2.10 Some Crystal Structures and their Space Groups
2.10.1 Close-Packed Structures
2.10.2 Hexagonal Close-Packed Structure
2.10.3 Interlude: Symmorphic Types of Space Groups
2.10.4 Face-Centered Cubic Structure
2.10.5 Body-Centered Cubic Structure
2.11 Complete Symmetry Group of Single Crystal inMacroscopic Physics
3 The Invariant Integral on SO(3)
3.1 Introductory Remarks on OrientationMeasures
3.1.1 Crystallite Orientation at a Point in a Polycrystalline Sample
3.1.2 Orientation Measures
3.1.3 Ensemble Average and Volume Average
3.2 The Haar Integral on SO(3)
3.2.1 Uniqueness
3.3 Left-Invariant Integral in Euler Angles
3.4 Invariant Integral in Axis-Angle Parameters
3.5 Further Properties of the Invariant Integral
3.5.1 Right-Invariance
3.5.2 Inverse-Invariance
3.6 Integrals with Complex-Valued Integrands
4 Orientation Distribution Function
4.1 Definition of the ODF for Aggregates of Triclinic Crystallites
4.1.1 ODF Under the Active and Passive View of Rotations
4.2 TheWigner D-Functions
4.2.1 Symmetry Properties of dlmn
4.2.2 Symmetry Properties ofDlmn
4.3 Series Expansion and Texture Coefficients
4.4 Alternate Expressions for the Wigner D-functions
4.4.1 Generalized Spherical Functions of Gel’fand and Šapiro
4.5 Alternate Formulations of the Series Expansion
4.5.1 Roe’s Generalized Spherical Harmonics and Texture Coefficients
4.5.2 Viglin’s Generalized Spherical Functions and Texture Coefficients
4.5.3 Bunge’s Generalized Spherical Harmonics and Texture Coefficients
4.5.4 The D-Functions, ODF Expansion, and Texture Coefficients of Matthies
5 Texture and Crystallite Symmetries
5.1 Transformation Formulas
5.1.1 Rotation of Polycrystal
5.1.2 Rotation of Reference Ideal Crystal
5.2 Restrictions on Texture Coefficients Imposed by Sample and CrystalliteSymmetries
5.3 Examples of Low Symmetry
5.3.1 Orthorhombic Texture Symmetry
5.3.2 Orthorhombic Crystallite Symmetry
5.3.3 Orthorhombic Aggregates of Orthorhombic Crystallites
5.3.4 Trigonal, Tetragonal, and Hexagonal Crystallite Symmetries
5.4 Some Cases Frequently Encountered in Applications
5.4.1 Fiber Textures
5.4.2 Orthorhombic Aggregates of Hexagonal Crystallites
5.4.3 Cubic Crystallite Symmetry
5.4.4 Orthorhombic Aggregates of Cubic Crystallites
6 Orientation Space for Polycrystals with Crystallite Symmetry
6.1 Introduction
6.1.1 Introductory Remarks on Orientation Space
6.2 Polycrystals with Non-trivial Crystallite Symmetry
6.2.1 Fundamental Domains
6.2.2 Quotient Measure, Orientation Density Function, and Roe’s ODF
6.2.3 Orientation Averaging and Texture Coefficients
6.3 The Roe Approach
6.4 Disorientation Angle as Distance Function on Orientations
6.5 Polycrystals with Non-trivial Sample Symmetry
6.5.1 Sample Symmetry Revisited
6.5.2 Fundamental Domains
6.5.3 Quotient Measure and Texture Coefficients
6.6 Examples of Explicit Fundamental Domains
6.6.1 Triclinic Aggregates of Crystallites with a Dihedral-Group Symmetry
6.6.2 Triclinic Aggregates of Cubic Crystallites
6.7 Spaces of Symmetrized Functions
6.8 The Bunge Approach: Symmetric Generalized Spherical Functions
7 Reciprocal Space and Reciprocal Lattice
7.1 Dual Space
7.2 Reciprocal Space
7.3 Reciprocal Lattice
7.4 Families of Parallel Lattice Planes
7.4.1 Distance Between Two Adjacent Lattice Planes
7.4.2 Fictitious Lattice Planes and Higher-Order Bragg Reflections
7.5 Metric Tensors
7.6 Description of Orientations of Lattice Basis byMiller Indices
7.6.1 Cubic Crystallites in a Sheet Metal
8 Texture Approximation by Individual Orientation Measurements
8.1 Introduction
8.1.1 EBSD and “Ghost Correction”
8.2 Mathematical Preliminaries
8.3 Texture Coefficients of Discrete Orientations
8.3.1 Texture Coefficients of a Single Crystal
8.3.2 Texture Coefficients of Ideal Orientations
8.4 Model Functions and Texture Components
8.4.1 Texture Components Defined by Central Functions
8.4.2 Bunge’s “Gaussian” Components
8.5 The Bunge–Haessner Method and ItsModification
8.5.1 Mathematical Basis
8.5.2 Modification of the Bunge–Haessner Method
8.5.3 Ideal Orientations Versus Gaussian Components
8.6 Further Issues Concerning the Bunge–Haessner Method in Practice
8.6.1 “Grains” and Independence in Orientation Measurements by EBSD
8.6.2 Minimum Number of Independent Single-Orientation Measurements Required
9 Determination of Texture Coefficients via X-Ray Diffraction
9.1 Representation of Orientations in Pole Figures
9.1.1 Stereographic Projection and Pole Figures
9.1.2 Pole Figure of a Cubic Crystallite in a Sheet Metal
9.1.3 Effects of Crystal Symmetry
9.1.4 Effects of Texture Symmetry
9.2 Rotation of Sample in Pole-Figure Measurements
9.3 Pole Figures and the Orientation Distribution Function
9.4 Pole Figures of Sheet Metals
9.5 Inversion of Pole Figures for l-Even Part of ODF
9.5.1 Inversion of Complete Pole Figures
9.5.2 Inversion of Incomplete Pole Figures
9.6 Ghost Correction (I): Generalized PositivityMethod
9.6.1 The Positivity Method
9.6.2 Inclusion of an Isotropic Component
9.6.3 An Improved Algorithm
9.7 Ghost Correction (II):Method of Texture Components
PART II. MATHEMATICAL FOUNDATIONS AND EXTENSIONS
10 SO(3) and O(3) as Riemannian Manifolds
10.1 SO(3) and O(3) as Riemannian Submanifolds of M(3)
10.1.1 Smooth Structure on SO(3) and on O(3)
10.1.2 Bi-invariant Metric on O(3) and SO(3)
10.2 SO(3) as Metric Space
10.3 Riemannian Metric on SO(3) in Euler Angles
10.4 Riemannian Metric on Orientation Space
10.5 Riemannian Metric on O(3)
10.5.1 O(3) as Metric Space
10.6 Invariant Integration
10.6.1 On SO(3)
11 Rotations Revisited
11.1 Euler-Rodrigues Parameters
11.1.1 The Quaternions
11.1.2 Rotations and the Symplectic Group Sp(1)
11.2 More on Sp(1)
11.2.1 Sp(1) as a Differentiable Manifold and a Lie Group
11.2.2 Sp(1) as a Riemannian Manifold with a Bi-invariant Metric
11.3 The Sp(1)?SO(3) Double Covering
11.3.1 Volume Element on SO(3) in Axis-Angle Parameters
11.4 SU(2)
11.4.1 Identification with Sp(1)
11.4.2 Convention in Physics
12 Texture Analysis Based on the Orthogonal Group
12.1 Orientation Distribution Functions Defined on O(3)
12.1.1 Series Expansions and Texture Coefficients
12.2 Discrete Probability Measures on O(3)
12.3 Transformations and Symmetries
12.3.1 Inversion of Reference Placement
12.3.2 Polycrystalline Aggregates of Type II Crystallites
12.3.3 Texture and Crystallite Symmetries
12.4 The Reduced ODF
12.4.1 Polycrystalline Aggregates of Type III Crystallites
PART III. GROUP REPRESENTATIONS
13 Group Representations
13.1 Preliminaries: Complex Inner-Product Spaces
13.1.1 Adjoint, Hermitian, Unitary, and Normal Transformations
13.2 Basic Definitions and Theorems
13.2.1 Finite-Dimensional Representations of Groups
13.2.2 Equivalence of Representations
13.2.3 Irreducible Representations, Schur’s Lemma
13.2.4 Unitary Representations, Unitary Equivalence
13.2.5 Complete Reducibility
13.3 The Space L2(G). The Regular Representations
13.4 Orthogonality Relations
13.5 Completeness Theorem for Finite Groups
13.6 Characters of Group Representations
13.6.1 Basic Properties
13.6.2 Completeness Theorem on Characters of Finite Groups
13.6.3 Example
13.7 Tensor Product of Representations
13.8 Unitary Representations on Spaces of Symmetric Tensors
13.8.1 Symmetric Tensors
13.8.2 Characters of Representations on Symmetric Tensors
13.9 Irreducible Representations of Direct Product of Groups
13.10 Irreducible Representations of Improper Crystallographic Point Groups
13.10.1 Crystallographic Point Groups of Type II
13.10.2 Crystallographic Point Groups of Type III
14 Irreducible Representations of SU(2), SO(3), and O(3)
14.1 Irreducible Representations of SU(2)
14.1.1 Construction of a Set of Continuous Unitary Representations
14.1.2 Characters and Irreducibility
14.1.3 Completeness
14.1.4 A Simple Criterion for Irreducibility
14.2 The Wigner D-Functions
14.2.1 Wigner D-Functions in Euler Angles
14.2.2 Wigner D-Functions in Euler-Rodrigues Parameters
14.3 Irreducible Unitary Representations of SO(3)
14.4 Irreducible Representations of O(3)
15 The Peter-Weyl Theorem
15.1 Preliminaries
15.2 Proof of the Peter-Weyl Theorem
15.3 The Right-Regular Representation
16 Tensor and Pseudotensor Representations of SO(3), O(3), and TheirFinite Subgroups
16.1 Mathematical Preliminaries
16.1.1 Tensor Algebra
16.2 Material Tensors and Pseudotensors
16.3 Decomposition of Representations on Tensor and Pseudotensor Spaces intoIrreducible Parts
16.3.1 Method of Characters
16.4 Decomposition of Tensor Representations of SO(3)
16.4.1 Characters of Irreducible Representations of SO(3)
16.4.2 Examples
16.5 Decomposition of Tensor and Pseudotensor Representations of O(3)
16.5.1 Characters of Irreducible Representations of O(3)
16.5.2 Decomposition Theorem
16.6 Point-Group Symmetry of Tensors and Pseudotensors
16.6.1 Proper Point Groups (Type I)
16.6.2 Improper Point Groups (Type II)
16.6.3 Improper Point Groups (Type III)
17 Harmonic Tensors and Tensorial Texture Coefficients
17.1 Symmetric Tensors and Homogeneous Polynomials
17.2 Homogeneous Harmonic Polynomials
17.2.1 Trace of a Symmetric Tensor
17.2.2 Laplace Equation and Homogeneous Harmonic Polynomials
17.3 Spaces of Harmonic Tensors
17.4 Harmonic Decomposition of Symmetric Tensors
17.5 Irreducible Tensor Basis in Space of Harmonic Tensors
17.5.1 Cartan Decomposition
17.6 Tensorial Fourier Expansion of the ODF
17.6.1 Harmonic Tensor Basis and Tensorial Fourier Expansion
17.6.2 Classical ODF Expansion as Tensorial Fourier Series
17.6.3 Texture and Crystallite Symmetries
APPENDICES
Appendix A
Appendix B
Appendix C
Appendix D
Bibliography
Index
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