The basics of crystallography and diffraction (3rd Edition)

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$The basics of crystallography and diffraction (3rd Edition)$

3 Bravais lattices and crystal systems 3.1 Introduction The definitions of the motif, the repeating ‘unit of pattern’, and the lattice, an array of points in space in which each point has an identical environment, hold in three dimensions exactly as they do in two dimensions. However, in three dimensions there are additional symmetry elements that need to be considered: both point symmetry elements to describe the symmetry of the three-dimensional motif (or indeed any crystal or three-dimensional object) and also translational symmetry elements, which are required (like glide lines in the two-dimensional case) to describe all the possible patterns which arise by combining motifs of different symmetries with their appropriate lattices. Clearly, these considerations suggest that the subject is going to be rather more complicated and ‘difficult’; it is obvious that there are going to be many more threedimensional patterns (or space groups) than the seventeen two-dimensional patterns (or plane groups or the eighty two-sided patterns—Chapter 2), and to work through all of these systematically would take up many pages! However, it is not necessary to do so; all that is required is an understanding of the principles involved (Chapter 2), the operation and significance of the additional symmetry elements, and the main results. These main results may be stated straight away. The additional point symmetry elements required are centres of symmetry, mirror planes (instead of lines) and inversion axes; the additional translational symmetry elements are glide planes (instead of lines) and screw axes. The application and permutation of all symmetry elements to patterns in space give rise to 230 space groups (instead of seventeen plane groups) distributed among fourteen space lattices (instead of five plane lattices) and thirty-two point group symmetries (instead of ten plane point group symmetries). In this chapter the concept of space (or Bravais) lattices and their symmetries is discussed and, deriving from this, the classification of crystals into seven systems. 3.2 The fourteen space (Bravais) lattices The systematic work of describing and enumerating the space lattices was done initially by Frankenheim ∗ who, in 1835, proposed that there were fifteen in all. Unfortunately for Frankenheim, two of his lattices were identical, a fact first pointed out by Bravais ∗ in 1848. It was, to take a two-dimensional analogy, as if Frankenheim had failed to notice ∗ Denotes biographical notes available in Appendix 3.
3.2 The fourteen space (Bravais) lattices 85 (see Fig. 2.4(b)) that the rhombic or diamond and the rectangular centred plane lattices were identical! Hence, to this day, the fourteen space lattices are usually, and perhaps unfairly, called Bravais lattices. The unit cells of the Bravais lattices are shown in Fig. 3.1. The different shapes and sizes of these cells may be described in terms of three cell edge lengths or axial distances, a a a a a a a a a Simple cubic (P) Body-centred cubic (I) Face-centred cubic (F) a c a a c a a c b a c b Simple tetragonal (P) Body-centred tetragonal (I) Simple orthorhombic (P) Body-centred orthorhombic (I) c c a b a b a α a a α α a c 120° a Base-centred orthorhombic (C) Face-centred orthorhombic (F) Rhombohedral (R) Hexagonal (P) β a c b β a c b c β α a γ b Simple monoclinic (P) Base-centred monoclinic (C) Triclinic (P) Fig. 3.1. The fourteen Bravais lattices (from Elements of X-Ray Diffraction, (2nd edn), by B. D. Cullity, Addison-Wesley, 1978).
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INTERNATIONAL UNION OF CRYSTALLOGRA
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The Basics of Crystallography and D
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Preface to the First Edition (1997)
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Preface to Third Edition (2009) vii
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Contents X-ray photograph of zinc b
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Contents xi 6 The reciprocal lattic
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Contents xiii 11.4.1 Determining or
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X-ray photograph of deoxyribonuclei
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1 Crystals and crystal structures 1
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1.1 The nature of the crystalline s
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1.2 Constructing crystals from hexa
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1.3 Unit cells of the hcp and ccp s
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1.4 Constructing crystals from squa
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1.6 Interstitial structures 1.6 Int
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2r A 1.6 Interstitial structures 13
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1.6 Interstitial structures 15 a√
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1.6 Interstitial structures 17 (a)
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1.8 Representing crystals in projec
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1.9 Stacking faults and twins 21 Fi
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1.9 Stacking faults and twins 23 if
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1.9 Stacking faults and twins 25 tw
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1.10 The crystal chemistry of inorg
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1.10 The crystal chemistry of inorg
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1.11 Some more complex crystal stru
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Page 70 and 71: Exercises 53 (c) (b) (a) Fig. 1.42.
Page 72 and 73: 2 Two-dimensional patterns, lattice
Page 74 and 75: 2.2 Two-dimensional patterns and la
Page 76 and 77: R 2.3 Two-dimensional symmetry elem
Page 78 and 79: 2.4 The five plane lattices 61 mirr
Page 80 and 81: R R R R R R R R R R 2.4 The five pl
Page 82 and 83: 2.7 Symmetry in art and design: cou
Page 85: (b) p1 pg p2 cm pm p2mm p2mg p2gg c
Page 88: What is the highest order of rotati
Page 91 and 92: 2.7 Symmetry in art and design: cou
Page 93 and 94: 2.8 Layer symmetry and examples in
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Page 97 and 98: 2.9 Non-periodic patterns and tilin
Page 99 and 100: 2.9 Non-periodic patterns and tilin
Page 101 and 102: Exercises 81 Fig. 2.20. A plane pat
Page 103: Exercises 83 (a) (b) (c) (d) (e) (f
Page 107 and 108: 3.2 The fourteen space (Bravais) la
Page 109 and 110: 3.3 The symmetry of the fourteen Br
Page 111: System Bravais lattices Axial lengt
Page 114 and 115: 3.4 The coordination or environment
Page 116 and 117: Exercises 95 Fig. 3.10. Epidermal c
Page 118 and 119: 4 Crystal symmetry: point groups, s
Page 120 and 121: 4.2 The thirty-two crystal classes
Page 122 and 123: 4.3 Centres and inversion axes of s
Page 124 and 125: 4.3 Centres and inversion axes of s
Page 126 and 127: 4.4 Crystal symmetry and properties
Page 128 and 129: 4.5 Translational symmetry elements
Page 130 and 131: 4.5 Translational symmetry elements
Page 132 and 133: 4.6 Space groups 111 the optical ac
Page 134 and 135: 4.6 Space groups 113 P b a 2 C 8 2y
Page 136 and 137: 4.6 Space groups 115 (b) 5 P 2 1 /c
Page 138 and 139: 4.6 Space groups 117 that the coord
Page 140 and 141: 4.7 Bravais lattices, space groups
Page 142 and 143: 4.8 The crystal structures and spac
Page 148 and 149: 4.9 Quasiperiodic crystals or cryst
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Page 152 and 153: 5 Describing lattice planes and dir
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5.3 Indexing lattice planes—Mille
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5.3 Indexing lattice planes—Mille
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5.5 Lattice plane spacings, Miller
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5.6 Zones, zone axes and the zone l
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5.7 Indexing in the trigonal and he
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5.8 Transforming Miller indices and
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5.8 Transforming Miller indices and
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5.10 A simple method for inverting
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Exercises Exercises 149 5.1 Write d
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6.2 Reciprocal lattice vectors 151
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6.3 Reciprocal lattice unit cells 1
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6.3 Reciprocal lattice unit cells 1
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6.4 Reciprocal lattice cells for cu
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6.5 Proofs of some geometrical rela
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Hence: 6.6 Lattice planes and recip
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6.7 Summary 163 with their normal a
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7 The diffraction of light 7.1 Intr
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7.2 Simple observations of the diff
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7.3 The nature of light: coherence,
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7.4 Geometry of diffraction pattern
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7.5 The resolving power of optical
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Exercises 191 pattern (see Figs. 1.
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8.2 Laue’s analysis of X-ray diff
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8.3 Laue’s analysis of X-ray diff
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8.3 Bragg’s analysis of X-ray dif
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8.4 Ewald’s synthesis: the reflec
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8.4 Ewald’s synthesis: the reflec
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9 The diffraction of X-rays 9.1 Int
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9.1 Introduction 205 (a) u u u u f
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9.2 The intensities of X-ray diffra
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9.3 The broadening of diffracted be
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9.4 Fixed θ, varying λ X-ray tech
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9.5 Fixed λ, varying θ X-ray tech
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[010] 9.5 Fixed λ, varying θ X-ra
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9.5 Fixed λ, varying θ X-ray tech
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9.6 X-ray diffraction from single c
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9.6 X-ray diffraction from single c
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9.7 X-ray (and neutron) diffraction
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9.7 X-ray (and neutron) diffraction
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9.8 Practical considerations: X-ray
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9.8 Practical considerations: X-ray
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Exercises 241 of the reciprocal lat
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10 X-ray diffraction of polycrystal
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10.2 X-ray diffraction 245 (a) S D
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10.2 X-ray diffraction 247 effect,
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10.2 X-ray diffraction 249 Square r
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10.2 X-ray diffraction 251 Focusing
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10.3 Applications of X-ray diffract
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10.3 Applications of X-ray diffract
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10.4 Preferred orientation and its
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10.5 X-ray diffraction of DNA: simu
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10.5 X-ray diffraction of DNA: simu
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10.6 The Rietveld method for struct
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10.6 The Rietveld method for struct
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Exercises 271 Given that the X-ray
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11 Electron diffraction and its app
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11.2 The Ewald reflecting sphere co
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11.3 The analysis of electron diffr
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11.3 The analysis of electron diffr
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11.4 Applications of electron diffr
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11.5 Kikuchi and electron backscatt
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220 11.6 Image formation and resolu
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11.6 Image formation and resolution
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Exercises 293 and hence, using the
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Exercises 295 11.5 Given the lattic
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12.1 Introduction 297 Fig. 12.1. A
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12.2 Construction of the stereograp
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12.2 Construction of the stereograp
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12.3 Manipulation of the stereograp
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12.4 Stereographic projections of n
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12.5 Stereographic projections of n
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12.5 Applications of the stereograp
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13 Fourier analysis in diffraction
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13.1 Introduction—Fourier series
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13.2 Fourier analysis in crystallog
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13.2 Fourier analysis in crystallog
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13.3 Analysis of the Fraunhofer dif
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13.4 Abbe theory of image formation
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13.4 Abbe theory of image formation
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Appendix 1 Computer programs, model
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Appendix 1 335 CRYSTALS is availabl
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Appendix 1 337 Fig. A1.2. Models sh
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Appendix 2 Polyhedra in crystallogr
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Appendix 2 341 Fig. A2.2. The five
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Appendix 2 343 Of these thirteen po
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Appendix 2 345 (a) (b) (c) Fig. A2.
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Appendix 2 347 next Brillouin zone
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Appendix 3 Biographical notes on cr
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Appendix 3 351 it did nevertheless
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Appendix 3 353 Lawrence. Gwen immer
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Appendix 3 355 mathematics and phys
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Appendix 3 357 Martin Julian Buerge
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Appendix 3 359 Fourier’s main ach
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Appendix 3 361 of high humidity) ha
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Appendix 3 363 dome covering the Fo
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Appendix 3 365 Christiaan Huygens 1
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Appendix 3 367 Cambridge was purcha
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Appendix 3 369 Kathleen Lonsdale (a
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Appendix 3 371 Isaac Newton was bor
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Appendix 3 373 optical activity, cr
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Appendix 3 375 Jean Baptiste Louis
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Appendix 3 377 In 1925 Wulff died a
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Appendix 3 379 both from boyhood as
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Appendix 3 381 the Royal Society as
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Appendix 4 383 Hexagonal: cos ρ =
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Appendix 5 A simple introduction to
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Appendix 5 387 3b r b a 5a Fig. A5.
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Appendix 5 389 (all the other terms
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Appendix 5 391 ’imaginary’ axis
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Appendix 6 393 Fig. A6.1. Writing d
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these two values in the structure f
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Table A6.1 Appendix 6 397 Extinctio
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Appendix 6 399 etc. in the row thro
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Answers to exercises Chapter 1 1.1
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Answers to exercises 403 common clo
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Answers to exercises 405 (c) p = 61
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Answers to exercises 407 Diffractio
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Answers to exercises 409 C/C * rota
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Answers to exercises 411 11.5 In Fi
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Answers to exercises 413 -x Small c
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Further Reading 415 Books which cov
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Further Reading 417 Williams, D. B.
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Further Reading 419 impressions is
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Index Note: Illustrations are indic
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Index 423 point group symbols for 1
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Index 425 Hanawalt groups 254 Hanaw
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Index 427 model building 5, 337-8 m
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Index 429 rotation-reflection (mirr
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Index 431 stacking sequence (in clo
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