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

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