# Texture Analysis in Materials Science

### Mathematical Methods

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Texture Analysis in Materials Science Mathematical Methods focuses on the methodologies, processes, techniques, and mathematical aids in the orientation distribution of crystallites. The manuscript first offers information on the orientation of individual crystallites and orientation distributions. Topics include properties and representations of rotations, orientation distance, and ambiguity of rotation as a consequence of crystal and specimen symmetry. The book also takes a look at expansion of orientation distribution functions in series of generalized spherical harmonics, fiber textures, and methods not based on the series expansion. The publication reviews special distribution functions, texture transformation, and system of programs for the texture analysis of sheets of cubic materials. The text also ponders on the estimation of errors, texture analysis, and physical properties of polycrystalline materials. Topics include comparison of experimental and recalculated pole figures; indetermination error for incomplete pole figures; and determination of the texture coefficients from anisotropie polycrystal properties. The manuscript is a dependable reference for readers interested in the use of mathematical aids in the orientation distribution of crystallites.

## Table of Contents

Contents

List of Symbols Used

1. Introduction 1

2. 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 Function

5. 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 Figures

11. 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} <110> 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 Orientations

12. 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 Cµvl

14. 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 Programs

Appendix 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

## Product details

- No. of pages: 614
- Language: English
- Copyright: © Butterworth-Heinemann 1982
- Published: December 15, 1982
- Imprint: Butterworth-Heinemann
- eBook ISBN: 9781483278391

## About the Author

### H.-J. Bunge

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