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Indexing of crystal diffraction patterns : from crystallography basics to methods of automatic indexing / Adam Morawiec.
Author
Morawiec, Adam
[Browse]
Format
Book
Language
English
Published/Created
Cham, Switzerland : Springer, [2022]
©2022
Description
1 online resource (427 pages)
Details
Subject(s)
Crystallography
[Browse]
Molecular structure
[Browse]
Series
Springer Series in Materials Science
[More in this series]
Springer Series in Materials Science ; v.326
[More in this series]
Notes
Includes index.
Source of description
Description based on print version record.
Contents
Intro
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
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
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
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
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
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
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
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
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
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
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.
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ISBN
9783031110771 ((electronic bk.))
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