Structure Determination by X-Ray Crystallography

1 Crystal Geometry. I.- 1.1 Introduction.- 1.2 The Crystalline State.- 1.2.1 Reference Axes.- 1.2.2 Equation of a Plane.- 1.2.3 Miller Indices.- 1.2.4 Axial Ratios.- 1.2.5 Zones.- 1.3 Stereographic Projection.- 1.4 External Symmetry of Crystals.- 1.4.1 Two-Dimensional Point Groups.- 1.4.2 Three-Dimensional Point Groups.- Problems.- 2 Crystal Geometry.II.- 2.1 Introduction.- 2.2 Lattices.- 2.2.1 Two-Dimensional Lattices.- 2.2.2 Choice of Unit Cell.- 2.2.3 Three-Dimensional Lattices.- 2.3 Families of Planes and Interplanar Spacings.- 2.4 Reciprocal Lattice.- 2.5 Rotational Symmetries of Lattices.- 2.6 Space Groups.- 2.6.1 Two-Dimensional Space Groups.- 2.6.2 Limiting Conditions Governing X-Ray Reflection.- 2.6.3 Three-Dimensional Space Groups.- 2.6.4 Screw Axes.- 2.6.5 Glide Planes.- 2.6.6 Analysis of the Space-Group Symbol.- Problems.- 3 Preliminary Examination of Crystals by Optical and X-Ray Methods.- 3.1 Introduction.- 3.2 Polarized Light.- 3.3 Optical Classification of Crystals.- 3.3.1 Uniaxial Crystals.- 3.3.2 Birefringence.- 3.3.3 Biaxial Crystals.- 3.3.4 Interference Figures.- 3.4 Direction of Scattering of X-Rays by Crystals.- 3.4.1 Laue Equations for X-Ray Scattering.- 3.4.2 Bragg’s Treatment of X-Ray Diffraction.- 3.4.3 Equivalence of Laue and Bragg Treatments of X-Ray Diffraction.- 3.5 X-Ray Techniques.- 3.5.1 Laue Method.- 3.5.2 Oscillation Method.- 3.5.3 Ewald’s Construction.- 3.5.4 Weissenberg Method.- 3.5.5 Precession Method.- 3.6 Recognition of Crystal System.- Problems.- 4 Intensity of Scattering of X-Rays by Crystals.- 4.1 Introduction.- 4.2 Path Difference.- 4.3 Mathematical Representation of a Wave: Amplitude and Phase.- 4.4 Combination of Two Waves.- 4.5 Argand Diagram.- 4.6 Combination of N Waves.- 4.7 Combined Scattering of X-Rays from the Contents of the Unit Cell.- 4.7.1 Phase Difference.- 4.7.2 Scattering by Atoms.- 4.8 Structure Factor.- 4.9 Intensity Expressions.- 4.10 Phase Problem in Structure Analysis.- 4.11 Applications of the Structure Factor Equation.- 4.11.1 Friedel’s Law.- 4.11.2 Structure Factor for a Centrosymmetric Crystal.- 4.11.3 Limiting Conditions and Systematic Absences.- 4.11.4 Determination of Unit-Cell Type.- 4.11.5 Structure Factors and Symmetry Elements.- 4.11.6 Limiting Conditions from Screw-Axis Symmetry.- 4.11.7 Centrosymmetric Zones.- 4.11.8 Limiting Conditions from Glide-Plane Symmetry.- 4.12 Preliminary Structure Analysis.- 4.12.1 Practical Determination of Space Groups.- Problems.- 5 Methods in X-Ray Structure Analysis. I.- 5.1 Introduction.- 5.2 Analysis of the Unit-Cell Contents.- 5.2.1 Papaverine Hydrochloride,C20H21NO4 HCl.- 5.2.2 Naphthalene, C10H8.- 5.2.3 Molecular Symmetry.- 5.2.4 Special Positions.- 5.2.5 Nickel Tungstate, NiWO4.- 5.3 Two Early Structure Analyses Revisited.- 5.3.1 Sodium Chloride, NaCl.- 5.3.2 Pyrite, FeS2.- Problems.- 6 Methods in X-Ray Structure Analysis. II.- 6.1 Introduction.- 6.2 Fourier Series.- 6.2.1 Computation of ?(x) for a Square Wave.- 6.2.2 Exponential Form of Fourier Expressions.- 6.3 Representation of Crystal Structures by Fourier Series.- 6.3.1 Electron Density and Structure Factors.- 6.3.2 Electron Density Equations.- 6.3.3 Interpretation of Electron Density Distributions.- 6.4 Methods of Solving the Phase Problem.- 6.4.1 Number of Reflections in the Data Set.- 6.4.2 The Patterson Function.- 6.4.3 Examples of the Use of the Patterson Function in Solving the Phase Problem.- 6.4.4 Absolute Scale of |Fo| and Overall Temperature Factor.- 6.4.5 Heavy-Atom Method and Partial Fourier Synthesis.- 6.4.6 Difference-Fourier Synthesis.- 6.4.7 Limitations of the Heavy-Atom Method.- 6.4.8 Isomorphous Replacement.- 6.4.9 Further Details of the Isomorphous Replacement Phasing Procedure.- 6.4.10 Anomalous Scattering.- Problems.- 7 Direct Methods and Refinement.- 7.1 Introduction.- 7.2 Direct Methods of Phase Determination.- 7.2.1 Normalized Structure Factors.- 7.2.2 Structure Invariants and Origin-Fixing Reflections.- 7.2.3 Sign Determination—Centrosymmetric Crystals.- 7.2.4 Amplitude Symmetry and Phase Symmetry.- 7.2.5 ?2 Listing.- 7.2.6 Symbolic-Addition Procedure.- 7.2.7 Calculation of E Maps.- 7.2.8 Phase Determination—Noncentrosymmetric Crystals.- 7.2.9 Phase Determination in Space Group P21.- 7.2.10 Advantages and Disadvantages of Symbolic Addition.- 7.2.11 Multisolution Philosophy and Brief Description of the Program multan.- 7.2.12 Figures of Merit.- 7.2.13 Example of the Use of multan.- 7.2.14 Some Experiences.- 7.3 Patterson Search Methods.- 7.3.1 Orientation Search.- 7.3.2 Translation Search.- 7.3.3 Example of a Structure Solution by Vector Verification.- 7.4 Least-Squares Refinement.- 7.4.1 Unit-Cell Dimensions.- 7.4.2 Atomic Parameters.- 7.5 Molecular Geometry.- 7.5.1 Bond Lengths and Angles.- 7.5.2 Torsion Angles.- 7.5.3 Conformational Analysis.- 7.5.4 Mean Planes.- 7.6 Precision.- 7.7 Correctness of a Structure Analysis.- 7.8 Limitations of X-Ray Structure Analysis.- 7.9 Disorder in Single Crystals.- Problems.- 8 Examples of Crystal Structure Analysis.- 8.1 Introduction.- 8.2 Crystal Structure of 2-Bromobenzo[b]indeno[1,2-e]pyran (BBIP).- 8.2.1 Preliminary Physical and X-Ray Measurements.- 8.2.2 Intensity Measurement and Correction.- 8.2.3 Structure Analysis in the (010) Projection.- 8.2.4 Three-Dimensional Structure Determination.- 8.2.5 Refinement.- 8.2.6 Molecular Geometry.- 8.3 Crystal Structure of Potassium 2-Hydroxy-3,4-dioxocyclobut-1-ene-1-olate Monohydrate (KHSQ).- 8.3.1 Preliminary X-Ray and Physical Measurements.- 8.3.2 Intensity Measurement and Correction.- 8.3.3 ?2 Listing.- 8.3.4 Specifying the Origin.- 8.3.5 Sign Determination.- 8.3.6 The E Map.- 8.3.7 Completion and Refinement of the Structure.- 8.4 Concluding Remarks.- Problems.- A.1 Stereoviews and Crystal Models.- A1.1 Stereoviews.- A1.2 Model of a Tetragonal Crystal.- A1.3 Stereoscopic Space-Group Drawings.- A.2 Crystallographic Point-Group Study and Recognition Scheme.- A.3 Schoenflies’ Symmetry Notation.- A3.1 Alternating Axis of Symmetry.- A3.2 Notation.- A.4 Generation and Properties of X-Rays.- A4.1 X-Rays and White Radiation.- A4.2 Characteristic X-Rays.- A4.3 Absorption of X-Rays.- A4.4 Filtered Radiation.- A.5 Crystal Perfection and Intensity Measurement.- A5.1 Crystal Perfection.- A5.2 Intensity of Reflected Beam.- A5.3 Intensity Measurements.- A5.3.1 Film Measurements.- A5.3.2 Diffractometer Geometry and Data Collection.- A5.4 Data Processing.- A5.4.1 Introduction.- A5.4.2 Standard Deviation of Intensity.- A5.4.3 Absorption Corrections.- A5.4.4 Scaling.- A5.4.5 Merging Equivalent Reflections.- A5.5 Synchrotron Radiation.- A.6 Transformations.- A.7 Comments on Some Orthorhombic and Monoclinic Space Groups.- A7.1 Orthorhombic Space Groups.- A7.2 Monoclinic Space Groups.- A.8 Vector Algebraic Relationships in Reciprocal Space.- A8.1 Introduction.- A8.2 Reciprocal Lattice.- A8.2.1 Interplanar Spacings.- A8.2.2 Reciprocity of Unit Cell Volumes.- A8.2.3 Angle between Bravais Lattice Planes.- A8.3 X-Ray Diffraction and the Reciprocal Lattice.- A8.3.1 Bragg’s Equation.- A8.4 Laue Photographs.- A8.5 Crystal Setting.- A8.5.1 Setting Technique.- A.9 Intensity Statistics.- A9.1 Weighted Reciprocal Lattice.- A9.1.1 Laue Symmetry.- A9.1.2 Systematic Absences.- A9.1.3 Accidental Absences.- A9.1.4 Enhanced Averages.- A9.1.5 Special Distributions.- A.10 Enantiomorph Selection.- Solutions.