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.

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(T)


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 F)


5.2.1. Pole Figures h(F)


5.2.2. Inverse Pole Figures RF(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 (F, ß) 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 Fvl


10.3. The Error Curve Cµvl


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 Rol…