Crystallographic Texture and Group Representations

Description

This book starts with an introduction to quantitative texture analysis (QTA), which adopts conventions (active rotations, definition of Euler angles, Wigner D-functions) that conform to those of the present-day mathematics and physics literature. Basic concepts (e.g., orientation; orientation distribution function (ODF), orientation density function, and their relationship) are made precise through their mathematical definition. Parts II and III delve deeper into the mathematical foundations of QTA, where the important role played by group representations is emphasized. Part II includes one chapter on generalized QTA based on the orthogonal group, and Part III one on tensorial Fourier expansion of the ODF and tensorial texture coefficients. 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 Blank Page