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)
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.
ISBN
9783031110771 ((electronic bk.))
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