Crystals and Crystal Structures

RICHARD J. D. TILLEY D. Sc, Ph. D, is Emeritus Professor in the School of Engineering at the University of Cardiff, Wales, U.K. He has published extensively in the area of solid-state materials science, including nine textbooks (translated into multiple languages), 23 chapters and encyclopedia entries and more than 200 journal papers.

• Preface ixChapter 1: Crystals and Crystal Structures 11.1 Crystal Families and Crystal Systems 11.2 Unit Cells and Miller Indices 41.3 The Determination of Crystal Structures 61.4 The Description of Crystal Structures 61.5 Crystal Structures: Metals 81.5.1 The Cubic Close-packed (A1) Structure of Copper 91.5.2 The Body-Centred Cubic (A2) Structure of Tungsten 91.5.3 The Hexagonal (A3) Structure of Magnesium 101.6 Crystal Structures: Binary Compounds 101.6.1 The Halite (Rock Salt, NaCl) Structure 101.6.2 The Rutile Structure 111.6.3 The Fluorite Structure 111.7 The Cubic Perovskite Structure 121.8 The Structure of Urea 131.9 The Density of a Crystal 14Answers to Introductory Questions 15Problems and Exercises 16Chapter 2: Lattices, Planes and Directions 192.1 Two-dimensional Lattices 192.2 Unit Cells 222.3 The Reciprocal Lattice in Two Dimensions 222.4 Three-dimensional Lattices 262.5 Rhombohedral, Hexagonal and Cubic Lattices 292.6 Alternative Unit Cells 302.7 The Reciprocal Lattice in Three Dimensions 312.8 Lattice Planes and Miller Indices 342.9 Hexagonal Lattices and Miller-Bravais Indices 372.10 Miller Indices and Planes in Crystals 372.11 Directions 392.12 Lattice Geometry 41Answers to Introductory Questions 44Problems and Exercises 44Chapter 3: Two-dimensional Patterns and Tiling 493.1 The Symmetry of an Isolated Shape: Point Symmetry 493.2 Rotation Symmetry of a Plane Lattice 523.3 The Symmetry of the Plane Lattices 533.4 The Ten Plane Crystallographic Point Symmetry Groups 553.5 The Symmetry of Patterns: The 17 Plane Groups 573.6 Two-dimensional Crystal Structures 633.7 General and Special Positions 663.8 Tesselations 68Answers to Introductory Questions 71Problems and Exercises 71Chapter 4: Symmetry in Three Dimensions 754.1 The Mirror Plane and Axes of Rotation 754.2 Axes of Inversion: Rotoinversion 774.3 Axes of Inversion: Rotoreflection 804.4 The Hermann-Mauguin Symbols for Point Groups 814.5 The Symmetry of the Bravais Lattices 834.6 The Crystallographic Point Groups 84Answers to Introductory Questions 87Problems and Exercises 88Chapter 5: Symmetry and Physical Properties 935.1 Properties and Symmetry 935.2 Point Groups and Physical Properties 945.3 Specification of Physical Properties 965.4 Refractive Index 975.5 Optical Activity 1005.5.1 Specific Rotation 1005.5.2 Crystal Symmetry and Optical Activity 1015.5.3 Optical Activity in Homogeneous Crystals 1015.5.4 Optical Activity in Crystals Containing Molecules 1025.5.5 Optical Activity and Chiral Molecules 1035.5.6 Optical Activity, Chemical Reactivity and Symmetry 1035.6 The Pyroelectric Effect 1045.6.1 Pyroelectric and Ferroelectric Crystals 1045.6.2 Crystallographic Aspects of Pyro- and Ferroelectric Behaviour 1065.7 Dielectric Properties 1095.7.1 Dielectrics 1095.7.2 Isotropic Materials 1105.7.3 Non-isotropic Materials 1105.8 Magnetic Point Groups and Colour Symmetry 111Answers to Introductory Questions 113Problems and Exercises 114Chapter 6: Building Crystal Structures from Lattices and Space Groups 1176.1 Symmetry of Three-dimensional Patterns: Space Groups 1176.2 The Crystallographic Space Groups 1196.3 Space Group Symmetry Symbols 1216.4 The Graphical Representation of the Space Groups 1256.5 Building a Structure from a Space Group: Cs3P7 1286.6 The Structure of Diopside, MgCaSi2O6 1316.7 The Structure of Alanine, C3H7NO2 134Answers to Introductory Questions 139Problems and Exercises 139Chapter 7: Diffraction and Crystal Structure Determination 1437.1 The Occurrence of Diffracted Beams: Bragg s Law 1447.2 The Geometry of the Diffraction Pattern 1457.3 Particle Size 1497.4 The Intensities of Diffracted Beams 1507.5 The Atomic Scattering Factor 1517.6 The Structure Factor 1527.7 Structure Factors and Intensities 1567.8 Numerical Evaluation of Structure Factors 1587.9 Symmetry and Reflection Intensities 1597.10 The Temperature Factor 1617.11 Powder X-ray Diffraction 1637.12 Neutron Diffraction 1687.13 Structure Determination Using X-ray Diffraction 1697.14 Solving the Phase Problem 1717.15 Electron Microscopy 1727.15.1 Diffraction Patterns and Structure Images 1727.15.2 Diffraction and Fourier Transforms 1777.16 Protein Crystallography 1787.16.1 The Phase Problem 1787.16.2 The Crystallinity Problem: SFX 1827.16.3 The Crystallinity Problem: Single Particle Cryo-EM 183Answers to Introductory Questions 185Problems and Exercises 186Chapter 8: The Depiction of Crystal Structures 1898.1 The Size of Atoms 1898.2 Sphere Packing 1908.3 Metallic Radii 1938.4 Ionic Radii 1948.5 Covalent Radii 1978.6 Van der Waals Radii 1988.7 Ionic Structures and Structure Building Rules 1988.8 The Bond Valence Model 1998.9 Structures in Terms of Non-metal (Anion) Packing 2028.10 Structures in Terms of Metal (Cation) Packing 2038.11 Cation-Centred Polyhedral Representations of Crystals 2048.12 Polyhedral Representations of Crystals and Diffusion Paths 2078.13 Structures as Nets 2108.14 Organic Structures 2128.15 Protein Structures 2128.15.1 Proteins: Primary Structure 2128.15.2 Proteins: Secondary, Tertiary and Quaternary Structure 214Answers to Introductory Questions 218Problems and Exercises 219Chapter 9: Defects, Modulated Structures and Quasicrystals 2239.1 Defects and Occupancy Factors 2239.2 Defects and Unit Cell Parameters 2259.3 Defects and Density 2269.4 Modular Structures 2279.5 Polytypes 2319.6 Crystallographic Shear (CS) Phases 2339.7 Planar Intergrowths and Polysomes 2379.8 Incommensurately ModulatedStructures 2429.9 Quasicrystals 248Answers to Introductory Questions 252Problems and Exercises 253Appendices 257Appendix A Vector Addition and Subtraction 257Appendix B Crystallographic Data for Some Inorganic Crystal Structures 259Appendix C Schoenflies Symbols 263Appendix D The 230 Space Groups 267Appendix E Complex Numbers 271Appendix F Complex Amplitudes 273Answers to Problems and Exercises 275Bibliography 283Index 289