Defect and Microstructure Analysis by Diffraction

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1 Defect and Microstructure Analysis by Diffraction ROBERT L. SNYDER Deparnnent of Materials Science and Engineering, The Ohio State University, Columbus, Ohio, USA JAROSLAV FIALA Department of Metallurgy, Central Research Institute Skoda, Czech Republic and HANS J. BUNGE Department of Physical Metallurgy, Technical University of Clausthal, Germany INTERNATIONAL UNION OF CRYSTALLOGRAPHY OXFORD UNIVERSITY PRESS

2 Contents List of contributors Abbreviations 1. Introduction to defect and microstructure analysis or the analysis of real structure 1 Jaroslav Fiala and Robert L. Snyder 1.1 Introduction What is real structure? X-ray diffraction analysis of real structure Restoring physical patterns from observed values Conclusions References 10 Part I. Fundamentals of defect analysis by diffraction Some applications of the kinematical theory of X-ray diffraction 19 Hans Bradaczek 2.1 Introduction The basis of the kinematical theory Bragg's law Direct methods of phase determination Particle size effect Convolution square root Influence of the periodic elements (atoms, clusters) General epilogue References Profile fitting and analytical functions 29 Stefano Enzo and Liliana Schiffini 3.1 Historical presentation of the theoretical framework The selection and the mathematical properties of the line shape function The use of line shape parameters in practical cases Conclusions References Effects of instrument function, crystallite size, and strain on reflection profiles 41 V. Honkimäki and P. Suortti 4.1 Introduction Total pattern 42

3 CONTENTS 4.3 Instrument function Modeling of background Crystallite size and strain Anisotropic strain and particle size Summary References 56 Appendix 1. Spherical harmonics Use of pattern decomposition or simulation to study microstructure: theoretical considerations 59 J. Jan Langford 5.1 I ntroduction Diffraction effects due to crystallite or domain size Diffraction effects due to structural `mistakes' Interpretation of apparent strain Characterization of microstructure References 76 Appendix 1. Analytical functions commonly used to model powder diffraction line profiles 77 Appendix 2. Apparent and `true' size of crystallites or domains from the integral breadth Classical treatment of line profiles influenced by strain, small size, and stacking faults 82 C.R. Houska and R. Kufel 6.1 Introduction Classical treatment Discussion References Voigt function model in diffraction-line broadening analysis 94 Davor Balzar List of symbols Introduction Diffraction-line broadening Size strain analysis Concluding remarks References X-ray analysis of precipitation-related crystals with dislocation substructure 127 R.I. Barabash 8.1 References 140

4 CONTENTS xi 9. Analytic functions describing line profiles influenced by size distribution, strain, and stacking faults 141 C. R. Houska and R. Kufel 9.1 Introduction Correlated dislocations Analytic profiles General analytic method Discussion References 160 Appendix Appendix 2. Krivoglaz expansion of strain 163 Appendix 3. Column distances, conversions, units The dislocation-based model of strain broadening in X-ray line profile analysis 165 T. Ungar 10.1 Introduction The theory of strain broadening Experimental aspects Single-profile measurements and results Two-profile analysis of the dislocation structure The analysis of small grain specimens Conclusions References Diffraction-line broadening analysis of dislocation configurations 200 A.C. Vermeiden, R. Delher, Th.H. de Keijser, and E.J. Mittemeijer 11.1 Introduction Integral breadths and Fourier coefficients: previous work Theory of diffraction-line broadening by dislocations How to deal with eqn (8) [111] Fibre-textured f.c.c metal layers plastically deformed by thermal stresses Practical example: aluminium layers Summary References Diffraction-line broadening analysis of strain fields in crystalline solids 214 J.G.M. van Berkum, R. Th.H. de Keifser, and E.J. Mittemeijer 12.1 Introduction Basic considerations and scope of the chapter Line profile description Line profile decomposition: resolving size and strain 218

5 xii CONTENTS 12.5 Line profile synthesis and matching: refining microstructure parameters Order dependence of line broadening: evaluation of line profile decomposition methods Line profile syntheses and matching: practical example Summary References Paracrystallinity 234 Hans Bradaczek 13.1 Introduction The basic idea The real paracrystal Computer simulation of paracrystals Summary References The model of the paracrystal and its application to polymers 247 W. Wilke 14.1 Introduction Ideal paracrystal Scattering intensity of ideal paracrystals Examples of application of the method to polymers Some remarks about paracrystal models for the interpretation of small-angle scattering References Effect of planar defects in crystals on the position and profile of powder diffraction lines 264 A.I. Ustinov 15.1 Introduction Equation for intensity calculation and common law of X-ray scattering crystals with planar defects Planar defects in body-centered cubic (body-centered tetragonal and body-centered rhombohedral) lattices Planar defects in crystals with face-centered cubic lattices Planar defects in hexagonal close-packed crystals One-dimensionally disordered crystals with long periods Conclusions References Effect of stacking disorder on the profile of the powder diffraction line 318 Z. Weiss and P. e'apkovä 16.1 Introduction 318

6 CONTENTS xiii 16.2 Translation stacking faults Diffraction patterns Results of calculations References 328 Part II. Experimental techniques Crystallite statistics and accuracy in powder diffraction intensity measurements 333 Deane K. Smith 17.1 Introduction Intensity measurement Crystallite statistics Crystallite statistics in diffractometer geometry Crystallite statistics for other instruments Summary References Reciprocal space mapping and ultra-high resolution diffraction of polycrystalline materials 346 Paul F. Fewster and Norman L. Andrew 18.1 Introduction Why high resolution? The diffraction equipment Conventional methods Three-dimensional diffraction in practice Practical examples of three-dimensional diffraction Summary References 363 Part UI. Macrostress X-ray analysis of the inhomogenous stress state 367 I. Kraus and N. Ganev 19.1 Introduction Basic elements of X-ray stress measurement The actual residual state of stress in products is inhomogenous Stress gradient Non-uniform stress problems solved by X-ray at Czech TU in Prague Conclusions References 401

7 xiv CONTENTS Part IV. Texture Texture and structure of polycrystals 405 Hans J. Bunge List of symbols Introduction Structure of polycrystalline materials Correspondence of structure in reciprocal space Problems to be considered in texture analysis Definition of the texture Symmetries of the texture functions Statistical relevance of the texture functions The texture degree Calculation of the ODF from experimental data Compatibility of pole figures (measures of error) Series expansion Special distribution functions Experimental methods of texture analysis Generalized textural quantities Applications of texture analysis References Texture effects in powder diffraction and their correction by simple empirical functions 520 V. Valvoda 21.1 Introduction Simple texture characterization and correction Theory Experimental Summary References 530 Part V. Whole-pattern fitting Accounting for size and microstrain in whole-powder pattern fitting 535 A. Le Ball List of symbols Introduction Theory WPPF with direct quantitative account for S/M effects WPPF with qualitative accounts for S/M effects Two experimental illustrations which qualitatively account for S/M in WPPF Conclusions References 552

8 CONTENTS xv 23. Modelling of texture in whole-pattern fitting 556 Maul Järvinen 23.1 Introduction Orientation distribution of grains Integrated intensity The harmonic model Symmetrized harmonics Examples Further simplifications Rietveld method Discussion References A new whole-powder pattern-fitting approach 570 P. Scardi 24.1 Introduction Methodology Applications Conclusions References The role of whole-pattern databases in materials science 597 Deane K. Smith 25.1 Introduction The nature of the diffraction pattern Def.-ming pattern parameters for a whole-pattern databases The role of calculated and simulated patterns in a whole-pat tern database A database for the analysis of clay minerals Uses of the Clay Minerals Database Future of whole-pattern databases Summary References 609 Part VI. Restoring physical patterns from the observed variables Restoration and preprocessing of physical profiles from measured data Marian üerrianskj, 26.1 Introduction 26.2 Preprocessing of data 26.3 Methods of restoration 26.4 Concluding remarks 26.5 References

9 xvi CONTENTS 27. Towards higher resolution: a mathematical approach 652 Derk Reefinan 27.1 Introduction to the problem The instrument function Deconvolution Comparison of deconvolution methods Applications Conclusions References 669 Part VII. Applications Use of pattern decomposition to study microstructure: practical aspects and applications 673 Daniel LOLA' 28.1 Introduction The problem of line overlap in powder diffraction Pattern decomposition Instrumental correction Examples Discussion References X-ray diffraction broadening effects in materials characterization 698 Giora Kin und and David Da_van 29.1 Introduction Techniques and methodology Metals and alloys Ceramic materials Summary and discussion References Crystal size and distortion parameters in fibres using Wide-Angle X-ray Scattering (WAXS) 728 R. Soinashekar 30.1 Introduction Paracrystallinity Warren- Averbach method Single order method Experimental method for some polymer fibres Matching of simulated and experimental intensity profiles Conclusions References 753

10 CONTENTS xvii 31. Pressure-induced profile change of energy-dispersive diffraction using synchrotron radiation 755 Takamitsu Yamanaka 31A I ntrod uction High pressure powder diffraetion Profile change with pressure Discussion Rderenees 765 Index 767

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