Crystals and Crystal Structures
Description
Written in a friendly, non-mathematical style, the updated second edition of Crystals and Crystal Structures offers a comprehensive exploration of the key elements of crystals and crystal structures. Starting with the basics, it includes information on multiple areas of crystallography, including modulated structures, quasicrystals and protein crystallography, and interdisciplinary applications as diverse as the relationship between physical properties and symmetry.
To enhance comprehension of the material presented, the book contains a variety of problems and exercises. The revised second edition offers new material and updates in the field including:
* An introduction to the use of high intensity X-ray analysis of protein structures
* Advances in imaging, scanning electron microscopy, and cryo-electron microscopy
* The relationship between symmetry and physical properties highlighting new findings and an introduction to tensor notation in describing these relationships in a concise fashion
* Nanoparticles as well as crystallographic aspects, defects, surface defects and the impact of these crystallographic features on properties
* Perovskite structures and their variations and the inclusion of their wide-ranging properties
Written for students ofcrystallography, chemistry, physics, materials science, biosciences and geology, Crystals and Crystal Structures, Second Edition provides an understanding of the subject and enables students to read scientific papers and articles describing a crystal structure or use crystallographic databases.
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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.
Content
Preface ix
Chapter 1: Crystals and Crystal Structures 1
1.1 Crystal Families and Crystal Systems 1
1.2 Unit Cells and Miller Indices 4
1.3 The Determination of Crystal Structures 6
1.4 The Description of Crystal Structures 6
1.5 Crystal Structures: Metals 8
1.5.1 The Cubic Close-packed (A1) Structure of Copper 9
1.5.2 The Body-Centred Cubic (A2) Structure of Tungsten 9
1.5.3 The Hexagonal (A3) Structure of Magnesium 10
1.6 Crystal Structures: Binary Compounds 10
1.6.1 The Halite (Rock Salt, NaCl) Structure 10
1.6.2 The Rutile Structure 11
1.6.3 The Fluorite Structure 11
1.7 The Cubic Perovskite Structure 12
1.8 The Structure of Urea 13
1.9 The Density of a Crystal 14
Answers to Introductory Questions 15
Problems and Exercises 16
Chapter 2: Lattices, Planes and Directions 19
2.1 Two-dimensional Lattices 19
2.2 Unit Cells 22
2.3 The Reciprocal Lattice in Two Dimensions 22
2.4 Three-dimensional Lattices 26
2.5 Rhombohedral, Hexagonal and Cubic Lattices 29
2.6 Alternative Unit Cells 30
2.7 The Reciprocal Lattice in Three Dimensions 31
2.8 Lattice Planes and Miller Indices 34
2.9 Hexagonal Lattices and Miller-Bravais Indices 37
2.10 Miller Indices and Planes in Crystals 37
2.11 Directions 39
2.12 Lattice Geometry 41
Answers to Introductory Questions 44
Problems and Exercises 44
Chapter 3: Two-dimensional Patterns and Tiling 49
3.1 The Symmetry of an Isolated Shape: Point Symmetry 49
3.2 Rotation Symmetry of a Plane Lattice 52
3.3 The Symmetry of the Plane Lattices 53
3.4 The Ten Plane Crystallographic Point Symmetry Groups 55
3.5 The Symmetry of Patterns: The 17 Plane Groups 57
3.6 Two-dimensional Crystal Structures 63
3.7 General and Special Positions 66
3.8 Tesselations 68
Answers to Introductory Questions 71
Problems and Exercises 71
Chapter 4: Symmetry in Three Dimensions 75
4.1 The Mirror Plane and Axes of Rotation 75
4.2 Axes of Inversion: Rotoinversion 77
4.3 Axes of Inversion: Rotoreflection 80
4.4 The Hermann-Mauguin Symbols for Point Groups 81
4.5 The Symmetry of the Bravais Lattices 83
4.6 The Crystallographic Point Groups 84
Answers to Introductory Questions 87
Problems and Exercises 88
Chapter 5: Symmetry and Physical Properties 93
5.1 Properties and Symmetry 93
5.2 Point Groups and Physical Properties 94
5.3 Specification of Physical Properties 96
5.4 Refractive Index 97
5.5 Optical Activity 100
5.5.1 Specific Rotation 100
5.5.2 Crystal Symmetry and Optical Activity 101
5.5.3 Optical Activity in Homogeneous Crystals 101
5.5.4 Optical Activity in Crystals Containing Molecules 102
5.5.5 Optical Activity and Chiral Molecules 103
5.5.6 Optical Activity, Chemical Reactivity and Symmetry 103
5.6 The Pyroelectric Effect 104
5.6.1 Pyroelectric and Ferroelectric Crystals 104
5.6.2 Crystallographic Aspects of Pyro- and Ferroelectric Behaviour 106
5.7 Dielectric Properties 109
5.7.1 Dielectrics 109
5.7.2 Isotropic Materials 110
5.7.3 Non-isotropic Materials 110
5.8 Magnetic Point Groups and Colour Symmetry 111
Answers to Introductory Questions 113
Problems and Exercises 114
Chapter 6: Building Crystal Structures from Lattices and Space Groups 117
6.1 Symmetry of Three-dimensional Patterns: Space Groups 117
6.2 The Crystallographic Space Groups 119
6.3 Space Group Symmetry Symbols 121
6.4 The Graphical Representation of the Space Groups 125
6.5 Building a Structure from a Space Group: Cs3P7 128
6.6 The Structure of Diopside, MgCaSi2O6 131
6.7 The Structure of Alanine, C3H7NO2 134
Answers to Introductory Questions 139
Problems and Exercises 139
Chapter 7: Diffraction and Crystal Structure Determination 143
7.1 The Occurrence of Diffracted Beams: Bragg s Law 144
7.2 The Geometry of the Diffraction Pattern 145
7.3 Particle Size 149
7.4 The Intensities of Diffracted Beams 150
7.5 The Atomic Scattering Factor 151
7.6 The Structure Factor 152
7.7 Structure Factors and Intensities 156
7.8 Numerical Evaluation of Structure Factors 158
7.9 Symmetry and Reflection Intensities 159
7.10 The Temperature Factor 161
7.11 Powder X-ray Diffraction 163
7.12 Neutron Diffraction 168
7.13 Structure Determination Using X-ray Diffraction 169
7.14 Solving the Phase Problem 171
7.15 Electron Microscopy 172
7.15.1 Diffraction Patterns and Structure Images 172
7.15.2 Diffraction and Fourier Transforms 177
7.16 Protein Crystallography 178
7.16.1 The Phase Problem 178
7.16.2 The Crystallinity Problem: SFX 182
7.16.3 The Crystallinity Problem: Single Particle Cryo-EM 183
Answers to Introductory Questions 185
Problems and Exercises 186
Chapter 8: The Depiction of Crystal Structures 189
8.1 The Size of Atoms 189
8.2 Sphere Packing 190
8.3 Metallic Radii 193
8.4 Ionic Radii 194
8.5 Covalent Radii 197
8.6 Van der Waals Radii 198
8.7 Ionic Structures and Structure Building Rules 198
8.8 The Bond Valence Model 199
8.9 Structures in Terms of Non-metal (Anion) Packing 202
8.10 Structures in Terms of Metal (Cation) Packing 203
8.11 Cation-Centred Polyhedral Representations of Crystals 204
8.12 Polyhedral Representations of Crystals and Diffusion Paths 207
8.13 Structures as Nets 210
8.14 Organic Structures 212
8.15 Protein Structures 212
8.15.1 Proteins: Primary Structure 212
8.15.2 Proteins: Secondary, Tertiary and Quaternary Structure 214
Answers to Introductory Questions 218
Problems and Exercises 219
Chapter 9: Defects, Modulated Structures and Quasicrystals 223
9.1 Defects and Occupancy Factors 223
9.2 Defects and Unit Cell Parameters 225
9.3 Defects and Density 226
9.4 Modular Structures 227
9.5 Polytypes 231
9.6 Crystallographic Shear (CS) Phases 233
9.7 Planar Intergrowths and Polysomes 237
9.8 Incommensurately Modulated
Structures 242
9.9 Quasicrystals 248
Answers to Introductory Questions 252
Problems and Exercises 253
Appendices 257
Appendix A Vector Addition and Subtraction 257
Appendix B Crystallographic Data for Some Inorganic Crystal Structures 259
Appendix C Schoenflies Symbols 263
Appendix D The 230 Space Groups 267
Appendix E Complex Numbers 271
Appendix F Complex Amplitudes 273
Answers to Problems and Exercises 275
Bibliography 283
Index 289
Chapter 1
Crystals and Crystal Structures
What is a crystal system?
What are unit cells?
What information is needed to specify a crystal structure?
Crystals are homogeneous solids that possess a long-range three-dimensional array of ordered atoms. That is, the arrangement of the atoms in one small volume of a crystal is identical (excepting localised mistakes or defects that can arise during crystal growth or that are inserted deliberately) to that in any other similar but remote part of the crystal. Crystallography is the study of crystals and describes the ways in which the component atoms are arranged in crystals and how the long-range order is achieved. Many chemical (including biochemical) and physical properties of solids depend upon crystal structure, and knowledge of crystallography is essential if the properties of materials are to be understood and exploited.
1.1 Crystal Families and Crystal Systems
Crystallography first developed as an observational science: an adjunct to the study of minerals. Minerals were (and still are) described by their morphology or habit, the shape that a mineral specimen may exhibit, which may vary from an amorphous mass to a well-formed gemstone. Indeed, the regular and beautiful shapes of naturally occurring crystals attracted attention from the earliest times, and the relationship between crystal shape and the disposition of crystal faces of a well-formed crystal provides an obvious means of classification. For example, some crystals resemble cubes or octahedra, whilst others are brick-like or form prisms with a hexagonal cross-section (Figure 1.1).
The external shape of a well-formed crystal reflects the internal order of the solid, especially the presence of internal symmetry. Symmetry will be developed later (especially in Chapters 3 and 4), but for the moment we can note that the most important symmetries displayed by crystals, and used in their classification, are the mirror plane, across which two parts of the crystal are related by reflection, and rotation axes. There are four different rotation axes: the diad or two-fold, in which successive rotations by (360/2)° leave the crystal unchanged; the triad or three-fold, in which successive rotations by (360/3)° leave the crystal unchanged; the tetrad or four-fold, in which successive rotations by (360/4)° leave the crystal unchanged; and the hexad or six-fold, in which successive rotations by (360/6)° leave the crystal unchanged. Careful measurement of mineral specimens using these symmetry criteria have allowed crystals to be classified in terms of six crystal families - anorthic, monoclinic, orthorhombic, tetragonal, hexagonal, and isometric or cubic - later expanded slightly by crystallographers into seven crystal systems (Table 1.1).
Figure 1.1 (a) Quartz (SiO2) crystals, showing hexagonal morphology; (b) pyrite (FeS2, fool's gold) crystals showing cubic and octahedral morphology.
The crystal systems are sets of reference axes, which have a direction as well as a magnitude, and hence are vectors.1 The allocation of a crystal to a particular system is made on the basis of the internal symmetries that are inferred from the crystal habit, which includes the apparent external symmetry. For instance, a crystal that resembled a hexagonal prism would be allocated to the hexagonal crystal system. It is common sense to allocate one reference axis to lie parallel to the hexagonal prism length and the other two axes to be normal to two chosen faces of the hexagonal prism. The axis lying along the prism length is really the defining axis for this designation because it is the unique hexad axis about which the crystal could be rotated by successive 60° turns to reproduce the same shape. Similarly, if a crystal has the form of a brick with a square cross-section, it is assigned to the tetragonal system, and the axes are positioned parallel to the three edges of the crystal. The defining axis this time is a tetrad perpendicular to the square cross-section, and this time the crystal can be rotated by successive 90° turns to regenerate the same shape. Thus the allocation of the reference axes and the placement of a crystal into a crystal family depends upon the external symmetry of the crystal.
The three reference axes for each family, allocated with respect to the crystal symmetry, are labelled a, b and c, and the angles between the positive directions of the axes are a, ß, and ?, where a lies between +b and +c, ß lies between +a and +c, and ? lies between +a and +b (Figure 1.2). The angles are chosen to be greater or equal to 90° except for the trigonal crystal system, as described below. In the figures, the a-axis is represented as projecting out of the plane of the page, towards the reader, the b-axis points to the right and the c-axis points towards the top of the page. This arrangement is a right-handed coordinate system. Measurements on mineral specimens could give absolute values for the inter-axial angles, but only relative axial lengths could be derived. These relative lengths are written a, b and c.
Table 1.1 The seven crystal systems
Crystal family Crystal system Required symmetrya Axial relationships Anorthic Triclinic None a ? b ? c, a ? 90°, ß ? 90°, ? ? 90° Monoclinic Monoclinic 1 diad or 1 mirror plane a ? b ? c, a = 90°, ß ? 90°, ? = 90° Orthorhombic Orthorhombic 3 diads or1 diad + 2 mirror planes a ? b ? c, a = 90°, ß = 90°, ? = 90° Tetragonal Tetragonal 1 tetrad a = b ? c, a = 90°, ß = 90°, ? = 90° Hexagonal Trigonal 1 triad a = b = c, a = ß = ? (rhombohedral axes); or
a' = b' ? c, a' = 90°, ß' = 90°, ?' = 120° (hexagonal axes) Hexagonal 1 hexad a = b ? c, a = 90°, ß = 90°, ? = 120° Isometric Cubic 3 tetrads a = b = c, a = 90°, ß = 90°, ? = 90°
a Symmetry nomenclature is expanded in Chapters 3 and 4.
The most symmetric of the crystal systems is the cubic or isometric system, in which the three tetrad axes are arranged at 90° to each other and the axial lengths are identical. These form the familiar Cartesian axes. The tetragonal system is similar, with mutually perpendicular axes, but two of these, usually designated a (= b), are of equal length, whilst the third, the tetrad axis, usually designated c, is longer or shorter than the other two. The orthorhombic system has three mutually perpendicular axes of different lengths parallel to the three diads. The monoclinic system is also defined by three unequal axes. Two of these, conventionally chosen as a and c, are at an oblique angle, ß, whilst the third c, normally parallel to the diad axis or mirror normal, is perpendicular to the plane containing a and b. The least symmetrical crystal system is the triclinic, which has three unequal axes at oblique angles.
Figure 1.2 Reference axes used to characterise the seven crystal systems.
The hexagonal crystal system has two axes of equal length, designated a (= b), at an angle, ?, of 120°. The c-axis lies perpendicular to the plane containing a and b and lies parallel to the hexad axis. The trigonal system has three axes of equal length, each enclosing equal angles a (= ß = ?), forming a rhombohedron. The axes are called rhombohedral axes and the triad axis is allocated to the body diagonal of the rhombohedron. Crystals described in terms of rhombohedral axes are often more conveniently described in terms of a hexagonal set of axes. In this case, the hexagonal c-axis is parallel to the rhombohedral body diagonal, which is parallel to the triad axis (Figure 1.3). The relationship between the two sets of axes is given by the vector equations:
where the subscripts R and H stand for rhombohedral and hexagonal respectively. (Note that in these equations the vectors a, b and c are added vectorially, not arithmetically [see Appendix A].) The arithmetical relationships between...
