Indexing of Crystal Diffraction Patterns: From Crystallography Basics to Methods of Automatic Indexing

Description

This book provides a detailed, self-contained description of automatic indexing of crystal diffraction patterns, considering both ab initio indexing and indexing of patterns originating from known structures. Introductory chapters equip the reader with the necessary basic knowledge of geometric crystallography, as well as kinematic and dynamic theories of crystal diffraction. Subsequent chapters delve and describe ab initio indexing of single crystal diffraction patterns and indexing of patterns for orientation determination. The book also reviews methods of indexing powder diffraction and electron spot-type patterns, as well the subject of multigrain indexing. Later chapters are devoted to diffraction by helical structures and quasicrystals, as well as some aspects of lattice parameter refinement and strain determination. The book is intended equally for materials scientists curious about ‘nuts and bolts’ of diffraction pattern indexing and orientation mapping systems, as well as interdisciplinary researchers from physics, chemistry, and biology involved in crystallographic computing. It provides a rigorous, yet accessible, treatment of the subject matter for graduate students interested in understanding the functioning of diffraction pattern indexing engines. Preface Contents Preliminaries Points and Vectors in Space Index Notation List of Selected Symbols NIST Values of Physical Constants 1 Elements of Geometric Crystallography 1.1 Linear Oblique Coordinate Systems 1.1.1 Component-free Tensor Notation 1.1.2 Frames—Overcomplete Sets of Vectors 1.2 Lattices 1.2.1 Lagrange-Gauss Reduction 1.2.2 Buerger- and Niggli-Reduced Bases 1.2.3 Delaunay Reduction 1.2.4 Sublattices and Superlattices 1.2.5 Centerings and Non-Primitive Lattice Cells 1.3 Crystal Symmetry Groups 1.3.1 Euclidean Group 1.3.2 Finite Point Groups 1.3.3 Crystallographic Point Groups 1.3.4 Space Groups 1.3.5 Crystal Systems 1.3.6 Bravais Types 1.3.7 Symmetry of the Reciprocal Lattice 1.3.8 Bravais Type from Niggli Character or Delaunay Sort 1.4 Conventional Crystallographic Settings 1.5 Indices of Directions and Planes 1.5.1 Direction and Miller Indices 1.5.2 Generalized Indices of Directions and Planes 1.6 Families of Equivalent Stacks of Planes 1.7 Comparison of Lattices and Bravais-class Determination 1.7.1 Lattice Symmetry from Distribution of Two-fold Axes 1.7.2 Method Based on Metric Tensor 1.8 Crystal Orientation 1.9 Homogeneous Strain 1.9.1 Change of Lattice Metric 1.9.2 Effect of Lattice Transformation on Its Reciprocal Lattice 1.9.3 Strain Tensor in the Crystal Reference System 1.9.4 Strain Tensor in Cartesian Reference System 1.10 Lattice and Fourier Transformation 1.11 Appendix: Fourier Transformation 1.11.1 Fourier Series and Fourier Transformation 1.11.2 Distributions 1.11.3 Convolution 1.11.4 Fourier Transform of Dirac Comb 1.11.5 Projection-Slice Theorem References 2 Basic Aspects of Crystal Diffraction 2.1 Scattering of Waves in Solids 2.1.1 Coherence 2.1.2 Diffraction Theories 2.2 Geometry of Crystal Diffraction 2.2.1 Laue Equation 2.2.2 Ewald Construction 2.2.3 Bragg's Law 2.3 Geometries of Selected Diffraction Techniques 2.3.1 X-ray Diffractometry 2.3.2 Planar Detector 2.3.3 Geometry of K-lines 2.3.4 Electron Spot Patterns 2.3.5 Geometry of Laue Patterns 2.4 Structure Factor 2.4.1 Introduction 2.4.2 X-ray Form Factors 2.4.3 Electron Atomic Scattering Factors 2.5 Formal Approach to Crystal Diffraction 2.5.1 Fourier Transform of the Transfer Function of an Unbounded Crystal 2.5.2 Crystal of Finite Dimensions 2.6 Intensities of Reflections 2.6.1 Systematic Absences 2.6.2 Friedel's Law 2.7 Other Factors Affecting Intensities 2.7.1 Absorption 2.7.2 Occupancy and Thermal Vibrations 2.8 Appendix: A Note on the Diffraction of Light 2.8.1 Pattern at the Focal Plane of a Converging Lens References 3 Diffraction of High Energy Electrons 3.1 Introduction to Dynamical Diffraction 3.1.1 Bloch Waves 3.2 Wave equation for a Single Electron in an Electrostatic Potential 3.2.1 Solutions for an Unbounded Crystal 3.2.2 Two-Beam Centro-Symmetric Case 3.3 Bloch Waves in Semi-Infinite and Plate-Like Crystals 3.4 Intensities on TEM Diffraction Patterns References 4 Cartesian Reference Frames in Diffractometry 4.1 X-ray Diffractometer 4.2 Crystal Orientation in Transmission Electron Microscope 4.2.1 Tilt Angles and Specimen Orientation 4.2.2 Crystal Orientation with Respect the Microscope Axis 4.2.3 Tilting a Crystal to a Given Zone Axis 4.2.4 Determination of `Magnetic' Rotation Angle 4.3 Orientation in Scanning Microscope References 5 Ab Initio Indexing of Single-Crystal Diffraction Patterns 5.1 Indexing in General 5.2 Ab Initio Indexing for Structure Determination 5.3 Experimental Single-Crystal Techniques 5.4 The Problem of Indexing Single-Crystal Data 5.4.1 Basics 5.4.2 Indexing Error-Free Data 5.4.3 Impact of Errors 5.4.4 Some Objective Functions 5.5 Real-Space Indexing 5.5.1 Obtaining Test Vectors 5.5.2 Interpretations of t- .4 cdoth- .4 n 5.6 Period Detection 5.6.1 Domains 5.6.2 Test Periods 5.6.3 Period Determination Without Binning the Data 5.6.4 Folding 5.6.5 Correlations with Other Functions 5.6.6 One-Dimensional Fourier Transformation 5.6.7 Rayleigh Test 5.6.8 Lomb-Scargle Periodogram 5.6.9 Combining Various Techniques 5.7 Difference Vectors 5.8 Indexing via Three-Dimensional Fourier Transformation 5.9 Clustering in Reciprocal Space 5.10 Directions of Zone Axes from Difference Vectors 5.11 Constructing a Three-Dimensional Lattice 5.12 An Example Indexing Program Ind_X 5.12.1 Method 5.13 A Bird's Eye View on Ab Initio Indexing 5.14 Appendix: Auxiliary Tools 5.14.1 Obtaining the Scattering Vector from a Kossel Line 5.14.2 Linear Optimization Problem 5.14.3 Generation of Integer Triplets References 6 Ab-Inito Indexing of Laue Patterns 6.1 Geometry of Laue Patterns 6.1.1 Experimentally Accessible Part of the Reciprocal Space 6.2 Gnomonic Projection of Reciprocal Lattice Nodes 6.3 Gnomonic Projection of a Cell 6.4 Laue Indexing 6.4.1 Indexing Software 6.4.2 An Approach Referring to Direct Space 6.4.3 Getting Zone Axes via Integral Transforms 6.4.4 Fitting a Consistent Mesh 6.4.5 Indexing Limited to Reciprocal Space 6.4.6 Using Sextuplets of Points 6.4.7 Testing Superlattices 6.4.8 Indices of an Individual Reflection 6.4.9 Quality of Solution—Figure of Merit 6.5 Indexing of Pink-Beam Diffraction Patterns 6.5.1 Algorithm for Fitting the Scaling Factor and Orders of Reflections References 7 Indexing of Powder Diffraction Patterns 7.1 Link Between Peaks Positions and Reflection Indices 7.2 Ambiguities 7.3 Figures of Merit 7.4 Indexing Procedures 7.4.1 Search in the Continuous Parameter Space 7.4.2 Search in the Discrete Index Space 7.4.3 Relationships Between Line Positions 7.4.4 Metric in Conventional Crystallographic Setting 7.4.5 Indexing Based on Complete Pattern 7.5 Integrated Software Packages References 8 Indexing for Crystal Orientation Determination 8.1 Orientation Mapping 8.2 Orientation via Pattern Indexing 8.2.1 Scattering Vectors and Reciprocal Lattice Vectors 8.2.2 Vector Magnitudes and Reflection Intensities 8.3 Formal Aspects of End-Indexing 8.3.1 Basic Relationships 8.3.2 Related Solvable Problems 8.3.3 Rotations Versus Proper Rotations 8.3.4 Computational Context 8.4 Spurious Scattering Vectors 8.4.1 Accumulation 8.5 Accumulation in Discrete Space 8.5.1 Triplet Voting 8.5.2 Example Implementation 8.6 Accumulation in Rotation Space 8.6.1 Accumulation at Points of the Rotation Space 8.6.2 Accumulation Along Curves in the Space of Rotations 8.6.3 Maxima in Rotation Space 8.6.4 Other Orientation-Based Algorithms 8.7 Testing of Indexing Algorithms 8.8 Figures of Merit and Other Issues 8.8.1 Three Remarks 8.9 Orientation Determination via Direct Pattern Matching 8.9.1 Direct Matching Limited by a Detected Reflection References 9 Indexing of Electron Spot-Type Diffraction Patterns 9.1 Conventional Indexing of Zone Axis Patterns 9.1.1 180°-Ambiguity 9.1.2 Computer-Assisted Conventional Indexing 9.2 Automatic Orientation Determination 9.2.1 Precession Electron Diffraction 9.3 Three-Dimensional Ab Initio Indexing 9.3.1 Automatic Recording of Tilt Series 9.4 Note on Other TEM-Based Patterns References 10 Example Complications in Indexing 10.1 Pseudosymmetry 10.2 Indexing of `Multi-lattice' Diffraction Patterns 10.2.1 Twins 10.2.2 Types of Twins 10.2.3 Diffraction Patterns Originating From Twins 10.3 Ambiguities in Crystal Orientation Determination 10.4 Indexing of Satellite Reflections 10.4.1 Sinusoidally Commensurately Modulated One-Dimensional `Crystals' 10.4.2 Modulation Propagation Vector 10.4.3 Indexing 10.4.4 Incommensurately Modulated Structures 10.5 Non-Conventional Structure Determination Methods 10.5.1 Indexing Grazing-Incidence X-ray Diffraction Data 10.5.2 Serial Crystallography References 11 Multigrain Indexing 11.1 Three-Dimensional X-ray Diffraction 11.2 X-ray Diffraction Contrast Tomography 11.3 Processing of Diffraction Data 11.3.1 Location of a Diffraction Spot as a Function of Grain Position 11.3.2 Algebraic Reconstruction Technique 11.3.3 Friedel Pairs 11.3.4 Indexing and Reconstruction 11.4 Other Methods of Three-Dimensional Mapping 11.4.1 Laboratory X-ray Diffraction Contrast Tomography 11.4.2 Differential Aperture X-ray Microscopy 11.4.3 Three-Dimensional Orientation Mapping in TEM 11.4.4 Three-Dimensional Mapping Using Neutron Diffraction References 12 An Excursion Beyond Diffraction by Periodic Crystals 12.1 Debye Scattering Formula 12.2 Single-Particle Diffraction Imaging 12.2.1 Phase Problem 12.2.2 Iterative Phase Retrieval Algorithms 12.2.3 Single-Particle Imaging With XFEL 12.3 Indexing of Diffraction Patterns of Helical Structures 12.3.1 Helix 12.3.2 Helical Structure 12.3.3 Structure Factor 12.3.4 Selection Rule 12.3.5 Single-Wall Tubes 12.3.6 Intensities in Layer Lines 12.3.7 Indices of Helical Reflections 12.3.8 Indices (l, n, m) and a Frame 12.3.9 Helical Structures Spanning a Range of Radial Values 12.3.10 Procedures for Indexing Helical Diffraction Patterns References 13 Indexing of Quasicrystal Diffraction Patterns 13.1 Example One-Dimensional Quasicrystal 13.1.1 Fourier Transform 13.1.2 Fibonacci Chain 13.1.3 Four-Segment Quasicrystal 13.1.4 The Strip Projection in One-Dimensional Cases 13.1.5 Indexing 13.2 The Strip Projection Method 13.3 Two-Dimensional Pentagonal Case 13.3.1 Diffraction by Pentagonal Primitive Quasicrystal 13.3.2 Ambiguity Due to Rational Linear Dependence of Vectors a-.4µ 13.3.3 Ambiguity in Assignment of Indices Due to Scaling `Symmetry' 13.4 Frame-Based Tilings 13.4.1 Grid Method of De Bruijn 13.5 Indices of Reflections 13.5.1 Indices of Symmetrically Equivalent Reflections 13.5.2 Transformation of Indices Between Frames 13.5.3 Zone Law 13.5.4 Indices of Peaks in Powder Diffraction Diagrams 13.6 The Decagonal and Other Axial Quasicrystals 13.6.1 Other Frames 13.6.2 Other Axial Quasicrystals 13.7 The Icosahedral Quasicrystal 13.7.1 Alternative Icosahedral Indexing Scheme 13.8 Practical Aspects of Indexing References 14 Refinement of Lattice Parameters and Determination of Local Elastic Strains 14.1 Methods of Local Strain Determination 14.2 CBED-Based Determination of Micro-Strains 14.2.1 K-Line Equation Based Scheme 14.2.2 Fitting Distances Between Line Intersections 14.2.3 Ambiguities 14.2.4 Software for CBED-Based Refinement of Lattice Parameters 14.3 Kossel Micro-Diffraction 14.3.1 KSLStrain 14.4 Appendix: Intersections of K-Lines References Appendix Index Index