Indexing of crystal diffraction patterns : from crystallography basics to methods of automatic indexing / / Adam Morawiec

• Record Nr.: UNISA996490350603316
• Autore: Morawiec Adam
• Pubbl/distr/stampa: Cham, Switzerland : , : Springer, , [2022]; ©2022
• ISBN: 9783031110771; 9783031110764
• Descrizione fisica: 1 online resource (427 pages)
• Collana: Springer Series in Materials Science ; ; v.326
• Disciplina: 548
• Soggetti: Crystallography; Molecular structure
• Lingua di pubblicazione: Inglese
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Note generali: Includes index.
• Nota di contenuto: 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 -- 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.