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 ScrewAxes.- 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 LaueMethod.- 3.5.2 Oscillation Method.- 3.5.3 Ewald's Construction.- 3.5.4 Weissenberg Method.- 3.5.5 Precession Method.- Problems.- 4 Intensity of Scattering of X-Rays by Crystals.- 4.1 Introduction.- 4.2 Path Difference.- 4.3 Combination of Two Waves.- 4.4 Argand Diagram.- 4.5 Combination of N Waves.- 4.6 Combined Scattering of X-Rays from the Contents of the Unit Cell.- 4.6.1 Phase Difference.- 4.6.2 Scattering by Atoms.- 4.7 Structure Factor.- 4.8 Intensity Expressions.- 4.9 Phase Problem in Structure Analysis.- 4.10 Applications of the Structure Factor Equation.- 4.10.1 Friedel'sLaw.- 4.10.2 Structure Factor for a Centrosymmetric Crystal.- 4.10.3 Limiting Conditions and Systematic Absences.- 4.10.4 Determination of Unit-Cell Type.- 4.10.5 Structure Factors and Symmetry Elements.- 4.10.6 Limiting Conditions from Screw-Axis Symmetry.- 4.10.7 Centrosymmetric Zones.- 4.10.8 Limiting Conditions from Glide-Plane Symmetry.- 4.11 Preliminary Structure Analysis.- 4.11.1 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 F0 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.- Problems.- 7 Some Further Topics.- 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-Determining Formulae.- 7.2.4 Amplitude Symmetry and Phase Symmetry.- 7.2.5 ?2 Listing.- 7.2.6 Symbolic-Addition Procedure.- 7.2.7 Calculatio