Understanding single-crystal x-ray crystallography

Foreword

Preface

Acknowledgements

1. Crystal Lattices

1.1. The Solid State

1.2. The Crystal Lattice

1.2.1. Two-dimensional Lattices

1.2.2. Three-dimensional Lattices

1.3. Vectors in Crystallography

1.3.1. Geometric Vector Addition and Multiplication

1.3.2. Basis Vectors and Coordinates

1.3.3. Orthonormal Bases

1.3.4. The Scalar Product in an Orthonormal Coordinate System

1.3.5. The Vector Product in an Orthonormal Coordinate System

1.4. Matrices in Crystallography

1.4.1. Matrix Definitions

1.4.2. Matrix Operations

1.4.3. Matrix Transformations

1.4.4. The Determinant of a Matrix

1.4.5. The Inverse of a Matrix

1.4.6. The Rules of Matrix Algebra

1.4.7. The Eigenvectors and Eigenvalues of a Matrix

1.5. Coordinate Systems in Crystallography

1.5.1. Change of Basis

1.5.2. Transformation from the Unit Cell Basis to an Orthonormal Basis

1.5.3. Determining Distances and Angles In the Unit Cell

1.5.4. Determining the Volume of the Unit Cell

1.5.5. Important Identities

2. Crystal Symmetry

2.1. Symmetry

2.2. Symmetry Group Theory

2.2.1. Sets of Symmetry Operations

2.2.2. Symmetry Groups

2.3. Point Groups

2.3.1. Molecular Point Symmetry

2.3.2. Matrix Representations of Groups

2.3.3. Character Tables

2.3.4. Lattice and Crystal Point Symmetry. The Hermann-Mauguin Notation

2.3.5. Stereographic Projections: Crystallographic Point Symmetry Elements

2.3.6. The 32 Crystallographic Point Groups

2.3.7. The Symmetry Classification of Crystal Systems

2.4. Space Groups

2.4.1. Translational Symmetry

2.4.2. Crystal Space Symmetry

2.4.3. The Triclinic Space Groups

2.4.4. The Monoclinic Space Groups

2.4.5. The Orthorhombic Space Groups

2.4.6. The Trigonal Space Groups

2.4.7. The Tetragonal Space Groups

2.4.8. The Hexagonal Space Groups

2.4.9. The Cubic Space Groups

2.4.10. General Considerations

3. Crystal Diffraction: Theory

3.1. Electromagnetic Radiation

3.1.1. The Electric Field

3.1.2. Waves

3.1.3. Particles

3.1.4. Interference

3.2. Diffraction

3.2.1. The Reciprocal Lattice

3.2.2. X-ray Diffraction: The Diffraction Equation

3.2.3. X-ray Diffraction: The Electron Density Equation

3.2.4. X-ray Diffraction: The Spherical Atom Approximation

3.2.5. Calculating Structure Factors and Electron Density

4. Crystal Diffraction: Experiment

4.1. The Sphere of Reflection

4.2. Recording the Diffraction Pattern: Film Methods

4.3. Recording the Diffraction Pattern: Counter Methods

4.3.1. Serial Detectors

4.3.2. Area Detectors

4.4. Determining the Orientation Matrix and Unit Cell

4.5. Refining the Orientation Matrix and Unit Cell

4.6. Determining the Bravais Lattice

4.6.1. Reduction of the Unit Cell

4.6.2. Searching for Higher Lattice Symmetry

4.6.3. Symmetry of the Reciprocal Lattice

4.7. The Measurement of Integrated Intensities

4.7.1. Reflections

4.7.2. The Integrated Intensity

4.7.3. Intensities From Serial Detectors

4.7.4. Intensities From Area Detectors

4.7.5. Limits to the Collection of Intensity Data

5. Crystal Diffraction: Data

5.1. Experimental Error

5.1.1. Random Error

5.1.2. Systematic Error

5.2. Scaling the Intensity Data

5.3. Determining the Space Group

5.3.1. Systematic Absences

5.3.2. Intensity Statistics

6. Crystal Structure Solution: Experimental

6.1. The Patterson Function

6.1.1. Patterson Solution: Structures Without Heavy Atoms

6.1.2. Patterson Solution: Structures With Heavy Atoms

6.1.3. Patterson Solution: Search Methods

6.2. Other Experimental Methods

6.2.1. The Isomorphous Replacement Method

6.2.2. The Anomalous Dispersion Method

6.3. Completion of the Structural Solution: Fourier Methods

6.3.1. Electron Density Synthesis

6.3.2. Difference Electron Density Synthesis

6.3.3. The Completion of Macromolecular Structures

7. Crystal Structure Solution: Statistical

7.1. Direct Methods

7.1.1. Probability Methods: Structure Invariants

7.1.2. Probability Methods: Initial Phases

7.1.3. Probability Methods: Solving the Structure

7.2. Other Direct Methods

7.2.1. Dual-Space Iteration

7.2.2. Maximum Entropy

7.3. Completion of the Structural Solution: Probability Methods

7.3.1. Phases from Fourier Structure Factors

7.3.2. Phases from Difference Fourier Structure Factors

8. Crystal Structure Refinement

8.1. Linear Least Squares

8.1.1. Weighted Least Squares

8.1.2. Estimation of Parameter Errors

8.1.3. Constrained Least Squares

8.1.4. Restrained Least Squares

8.2. Non-linear Least Squares: Structure Refinement

8.2.1. Weights in Refinement

8.2.2. Estimation of Parameter Errors in Refinement

8.2.3. Refinement Figures of Merit

8.2.4. Constrained Refinement

8.2.5. Restrained Refinement

8.2.6. The Refinement of Twinned Structures

8.2.7. The Refinement of Chiral Structures

8.3. Macromolecular Refinement

8.3.1. Heavy Atom Solutions

8.3.2. Molecular Replacement Solutions

8.3.3. Completion of the Model

8.3.4. Refinement of the Model

Appendix

A. A Geometric Derivation of Bragg's Law

B. The Fourier Transform: Electron Density & The Structure Factor

C. Determination of the Phase Parameter in the Amplitude Reflectivity Ratio

D. Reflection From a Single Plane

E. A Discussion of Kinematical Models for Extinction

F. Probability Integrals: The Modified Bessel Function

G. Monte Carlo Optimization A Simple Example

H. Constrained Optimization

I. Taylor Series

Bibliography

Index