Structure Determination by X-Ray Crystallography, Softcover reprint of the original 1st ed. 1977

Crystallography may be described as the science of the structure of materi­ als, using this word in its widest sense, and its ramifications are apparent over a broad front of current scientific endeavor. It is not surprising, therefore, to find that most universities offer some aspects of crystallography in their undergraduate courses in the physical sciences. It is the principal aim of this book to present an introduction to structure determination by X-ray crystal­ lography that is appropriate mainly to both final-year undergraduate studies in crystallography, chemistry, and chemical physics, and introductory post­ graduate work in this area of crystallography. We believe that the book will be of interest in other disciplines, such as physics, metallurgy, biochemistry, and geology, where crystallography has an important part to play. In the space of one book, it is not possible either to cover all aspects of crystallography or to treat all the subject matter completely rigorously. In particular, certain mathematical results are assumed in order that their applications may be discussed. At the end of each chapter, a short bibliog­ raphy is given, which may be used to extend the scope of the treatment given here. In addition, reference is made in the text to specific sources of information. We have chosen not to discuss experimental methods extensively, as we consider that this aspect of crystallography is best learned through practical experience, but an attempt has been made to simulate the interpretive side of experimental crystallography in both examples and exercises.

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.- 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 ?2Listing.- 7.2.6 Symbolic-Addition Procedure.- 7.2.7 Calculation of E Maps.- 7.3 Least-Squares Refinement.- 7.3.1 Unit-Cell Dimensions.- 7.3.2 Atomic Parameters.- 7.4 Molecular Geometry.- 7.5 Accuracy.- 7.6 Correctness of a Structure Analysis.- 7.7 Anomalous Scattering.- 7.8 Limitations of X-Ray Structure Analysis.- Problems.- 8 Examplesof Crystal Structure Analysis.- 8.1 Introduction.- 8.2 Crystal Structure of 2-Bromobenzo[b]indeno[3,2-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 ?2Listing.- 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.- A.1.1 Stereoviews.- A.1.2 Model of a Tetragonal Crystal.- A.2 Crystallographic Point-Group Study and Recognition Scheme.- A.3 Schoenflies’ Symmetry Notation.- A.3.1 Alternating Axis of Symmetry.- A.3.2 Notation.- A.4 Generation and Properties of X-Rays.- A.4.1 X-Rays and White Radiation.- A.4.2 Characteristic X-Rays.- A.4.3 Absorption of X-Rays.- A.4.4 Filtered Radiation.- A.5 Crystal Perfection and Intensity Measurement.- A.5.1 Crystal Perfection.- A.5.2 Intensity of Reflected Beam.- A.5.3 Intensity Measurements.- A.5.4 Intensity Corrections.- A.6 Transformations.- A.7 Comments on Some Orthorhombic and Monoclinic Space Groups.- A.7.1 Orthorhombic Space Groups.- A.7.2 Monoclinic Space Groups.- Solutions.