Texture Analysis in Materials Science

ContentsList of Symbols Used 1. Introduction 12. Orientation of Individual Crystallites 2.1. Various Representations of a Rotation 2.1.1. Eulebian Angles 2.1.2. Rotation Axis and Rotation Angle 2.1.3. Crystal Direction and Angle 2.1.4. Sample Direction and Angle 2.1.5. Representation of the Orientation in the Pole Figure 2.1.6. Representation of the Orientation in the Inverse Pole Figure 2.1.7. Representation by Miller Indices 2.1.8. Matrix Representation 2.1.9. Relations between Different Orientation Parameters 2.1.10. The Invariant Measure 2.2. Some Properties of Rotations 2.3. Ambiguity of Rotation as a Consequence of Crystal and Specimen Symmetry 2.4. Orientation Distance 2.5. Orientation for Rotational Symmetry 3. Orientation Distributions 4. Expansion of Orientation Distribution Functions in Series of Generalized Spherical Harmonics (Three-dimensional Textures) 4.1. Determination of the Coefficients Cµvl 4.1.1. Individual Orientation Measurements 4.1.2. Interpolation of the Function f(g) 4.2. The General Axis Distribution Functions A(h,y) 4.2.1. Determination of the Coefficients Cµvl by Interpolation of the General Axis Distribution Function 4.2.2. Pole Figures Ph(y) 4.2.3. Inverse Pole Figures Ry(h) 4.2.4. Comparison of the Representations of a Texture by Pole Figures and Inverse Pole Figures 4.3. The Angular Distribution Function Why(Θ) 4.3.1. Integral Relation between Pole Figures and Inverse Pole Figures 4.4. Determination of the Coefficients Cµvl by the Method of Least Squares 4.5. Measures of Accuracy 4.5.1. A Special Accuracy Measure for Pole Figures of Materials with Cubic Symmetry 4.5.2. A Method for the Adaption of Back-reflection and Transmission Range 4.6. Truncation Error 4.6.1. Decrease of the Truncation Error by Smearing 4.7. Determination of the Coefficients Cf from Incompletely Measured Pole Figures 4.8. Texture Index 4.9. Ambiguity of the Solution 4.9.1. Non-random Textures with Random Pole Figures 4.9.2. The Refinement Procedure of KBIGBAUM 4.9.3. The Extremum Method of TAVARD 4.10. Comparison with ROE'S Terminology 4.11. The Role of the Centre of Inversion 4.11.1. Right-and Left-handed Crystals 4.11.2. Centrosymmetric Sample Symmetries 4.11.3. Centrosymmetric Crystal Symmetries 4.11.4. Friedel's Law 4.11.5. Black—White Sample Symmetries 4.11.6. Determination of the Odd Part of the Texture Function5. Fiber Textures 5.1. Determination of the Coefficients Cµvl 5.1.1. Individual Orientation Measurements 5.1.2. Interpolation of the Function R(h) 5.2. The General Axis Distribution Function A(h Φ) 5.2.1. Pole Figures Ρh(Φ) 5.2.2. Inverse Pole Figures RΦ(h) 5.3. Determination of the Coefficients Cµvl According to the Least Squares Method 5.4. Measures of Accuracy 130 5.4.1. A Special Measure of Accuracy for Pole Figures of Materials with Cubic Symmetry 5.5. Truncation Error 5.5.1. Decrease of the Truncation Error by Smearing 5.6. Determination of the Coefficients Cf from Incompletely Measured Pole Figures 5.7. Texture Index 5.8. The Approximation Condition for Fibre Textures 5.9. Calculation of the Function Β(Φ, β) for Various Crystal Symmetries 5.9.1. Orthorhombic Symmetry 5.9.2. Cubic Symmetry 5.10. The Role of the Centre of Inversion 5.10.1. Right- and Left-handed Crystals 5.10.2. Centrosymmetric Sample Symmetries 5.10.3. Centrosymmetric Crystal Symmetries 5.10.4. Friedel's Law 5.10.5. Black—White Sample Symmetries 5.10.6. Determination of the Odd Part of the Texture Function 6. Methods not Based on the Series Expansion 6.1. The Method of Perlwitz, LÜCKE and Pitsch 6.2. The Method of Jetter, Mchargue and Williams 6.3. The Method of Ruer and Baro 6.4. The Method of IMHOF 7. Special Distribution Functions 7.1. Ideal Orientations 7.2. Cone and Ring Fibre Textures 7.3. 'Spherical' Textures 7.4. Fibre Axes 7.5. Line and Surface Textures (Dimension of a Texture) 7.6. Zero Regions 7.7. Gaussian Distributions 7.8. Polynomial Approximation (Angular Resolving Power) 8. Texture Transformation 9. A System of Programs for the Texture Analysis of Sheets of Cubic Materials 9.1. The Subroutines 9.2. The Mainline Programs 9.3. The Library Program 9.4. Calculation Times and Storage Requirements 9.5. Supplementary Programs 9.6. A Numerical Example 9.7. Listings of the ODF and Library Programs 10. Estimation of the Errors 10.1. A Reliability Criterion for Pole Figures of Materials with Cubic Symmetry 10.2. The Error Curve ΔFvl 10.3. The Error Curve ΔCµvl 10.4. Error Estimation According to the HARRIS Relation 10.5. Comparison of Experimental and Recalculated Pole Figures 10.6. Negative Values 10.7. Estimation of the Truncation Error by Extrapolation 10.8. The Integration Error 10.9. The Statistical Error 10.10. The Indetermination Error for Incomplete Pole Figures11. Some Results of Texture Analysis 11.1. Three-dimensional Orientation Distribution Functions (ODF) 11.1.1. Determination of the Coefficients Cµvl from Individual Orientation Measurements 11.1.2. The Rolling Textures of Face-centred Cubic Metals and Alloys 11.1.3. The Theoretical Rolling Texture for {111} Slip 11.1.4. The Rolling Textures of Body-centred Cubic Metals 11.1.5. Textures of Tubes 11.1.6. Orthorhombic Crystal Symmetry 11.1.7. Hexagonal Crystal Symmetry 11.1.8. Trigonal Crystal Symmetry (Separation of Real Coincidences) 11.1.9. Transformation Textures 11.1.10. Cubic-triclinic Symmetry 11.1.11. Representation of the Orientation Distribution Function by Rotation Axis and Rotation Angle 11.2. Fibre Textures 11.2.1. The Drawing Texture of Aluminium Wires 11.2.2. Transformation Texture in Au—Ge TAYLOR Wires 11.2.3. Hexagonal Crystal Symmetry (Titanium) 11.2.4. Orthorhombic Crystal Symmetry (Separation of Partial Coincidences) 11.2.5. Triclinic Crystal Symmetry (Application of the Refinement Procedure) 11.2.6. Orientation Distribution of the Number of Crystallites and the Mean Grain Size 11.2.7. Shape of the Spread about Preferred Orientations12. Orientation Distribution Functions of Other Structural Elements 12.1. Orientation Distribution Functions of the Grain Surfaces 12.1.1. Orientation Distribution of the Crystallographic Planes in the Outer Surface of an Arbitrary Section 12.2. Orientation Distribution Functions of the Grain Boundaries 12.2.1. The Distribution Function f(Δg) of the Orientation Differences 282 12.2.2. The Distribution Function ϕ(y) of the Grain Boundaries in the Sample Fixed Coordinate System 12.2.3. The Distribution Function ϕ(h) of the Grain Boundaries in the Crystal Fixed Coordinate System 12.3. Orientation Distribution Functions of the Grain Edges 12.3.1. The Distribution Function ϕ(y) of the Grain Edges in the Sample Fixed Coordinate System 12.3.2. The Distribution Function ϕ(h) of the Grain Edges in the Crystal Fixed Coordinate System 13. Physical Properties of Polycrystalline Materials 13.1. Physical Properties of Single Crystals 13.1.1. Representation by Tensors 13.1.2. Representation by Surfaces 13.1.3. Representation by Functions of the Orientation g 13.2. The Problem of Averaging 13.3. The Calculation of the Simple Mean Valued Ē 13.3.1. Tensor Representation 13.3.2. Surface Representation 13.3.3. Representation by Orientation Functions 13.4. Average Values of Special Properties 13.4.1. Magnetization Energy in a Homogeneous Magnetic Field 13.4.2. The Remanence in Ferromagnetic Materials 13.4.3. Tensor Properties of Second Order 13.4.4. Elastic Properties 13.4.5. Plastic Anisotropy 13.4.6. The Reflectivity of Crystallites for X-rays 13.5. Determination of the Texture Coefficients from Anisotropic Polycrystal Properties 13.6. Determination of Single Crystal Properties from Polycrystal Measurements 13.7. Textures With Equal Physical Properties 13.7.1. Fibre Textures of Ferromagnetic Cubic Materials 13.7.2. Magnetic Anisotropy of an Fe—Si Sheet 13.7.3. Tensor Properties of Second Rank for Fibre Textures 13.8. Physical Meaning of the Coefficients Cµvl14. Mathematical Aids 14.1. Generalized Spherical Harmonics 14.2. Spherical Surface Harmonics 14.3. FOUBIER Expansion of the Ρmnl(Φ) 14.4. CLEBSCH—GORDAN Coefficients 14.5. Symmetric Generalized Spherical Harmonics 14.5.1. Transformation of the Coefficients Anvl 14.5.2. The Fundamental Integral 14.5.3. Convolution Integrals 14.6. Symmetric Spherical Surface Harmonics 14.7. The Symmetric Functions of the Various Symmetry Groups 14.7.1. 'Lower' Symmetry Groups (Non-cubic) 14.7.2. 'Higher' Symmetry Groups (Cubic) 14.7.3. Subgroups 14.7.4. Explicit Representation of Symmetric Generalized Spherical Harmonics 14.7.5. Representation of the Cubic Spherical Surface Harmonics by Products of Powers of Cubic Polynomials 14.7.6. Space Groups in the EULER Space 14.7.7. Cubic Symmetry 14.8. CLEBSCH—GORDAN Coefficients for Symmetric Functions 15. Numerical Tables References Appendix 1 Tables 9.2-9.14 Appendix 2 Listings of the ODF and Library ProgramsAppendix 3 Tables for Chapter 15 15.1. Fourier Coefficients 15.1.1. Qmnl 15.1.2. a'mnsl 15.1.3. a'mnsl 15.2. Symmetry Coefficients Bmμl 15.2.1. Cubic, Fourfold Axis 15.2.2. Cubic, Threefold Axis 15.2.3. Tetragonal, Orthogonal to Cubic 15.2.4. Cubic, ROE'S Notation 15.3. Generalized Legendre Functions Pmnl(Φ) 15.4. Cubic Surface Harmonics kml(Φβ) 15.4.1. In Steps of Φ and β 15.4.2. For Low-index Directions 15.5. Cubic Generalized Spherical Harmonics 15.5.1. Cubic-orthorhombic Τmnl(φ1Φφ2) 15.5.2. Cubic-Cubic Tµµ'l(φ1Φφ2) Appendix 4 Graphic Representations A4.1. The Generalized Legendre Functions Pmnl(Φ) A4.2. Cubic Spherical Harmonics kml(Φβ) A4.3. Cubic Generalized Spherical Harmonics A4.3.1. Cubic-orthorhombic Generalized Spherical Harmonics A4.3.2. Cubic-cubic Generalized Spherical Harmonics Subject Index