Crystal Structures. Symmetry Relationships between. Applications of Crystallographic Group Theory in Crystal Chemistry OXFORD.

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1 Symmetry Relationships between Crystal Structures Applications of Crystallographic Group Theory in Crystal Chemistry Ulrich Miiller Fachbereich Chemie, Philipps-Universitat Marburg, Germany with texts adapted from Hans Wondratschek and Hartmut Barnighausen OXFORD UNIVERSITY PRESS

2 Contents List of symbols xvi 1 Introduction The symmetry principle in crystal chemistry Introductory examples 4 1 Crystallographic Foundations 9 2 Basics of crystallography, part Introductory remarks Crystals and lattices Appropriate coordinate systems, crystal coordinates Lattice directions, net planes, and reciprocal lattice Calculation of distances and angles 16 3 Mappings Mappings in crystallography An example Symmetry operations Affine mappings Application of (w +1) x (n + 1) matrices Affine mappings of vectors Isometries Types of isometries Changes of the coordinate system Origin shift Basis change General transformation of the coordinate system The effect of coordinate transformations on mappings Several consecutive transformations of the coordinate system Calculation of origin shifts from coordinate transfor mations Transformation of further crystallographic quantities 39 Exercises 40

3 xii Contents 4 Basics of crystallography, part The description of crystal symmetry in International Tables A: Positions Crystallographic symmetry operations Geometric interpretation of the matrix-column pair (W,w) of a crystallographic symmetry operation Derivation of the matrix-column pair of an isometry 47 Exercises 48 5 Group theory Two examples of groups Basics of group theory Coset decomposition of a group Conjugation Factor groups and homomorphisms Action of a group on a set 59 Exercises 61 6 Basics of crystallography, part Space groups and point groups Molecular symmetry The space group and its point group Classification of the space groups The lattice of a space group Space-group symbols Hermann-Mauguin symbols Schoenflies symbols Description of space-group symmetry in International Tables A Diagrams of the symmetry elements Lists ofthe Wyckoffpositions Symmetry operations of the general position Diagrams of the general positions General and special positions of the space groups The general position of a space group The special positions of a space group 6.6 The difference between space group and space-group type 84 Exercises Subgroups and supergroups of point and space groups Subgroups of the point groups of molecules Subgroups of the space groups Maximal translationengleiche subgroups Maximal non-isomorphic klassengleiche subgroups Maximal isomorphic subgroups Minimal supergroups of the space groups Layer groups and rod groups 96 Exercises 99

4 Contents xiii 8 Conjugate subgroups, normalizers and equivalent descriptions of crystal structures Conjugate subgroups of space groups Normalizers of space groups The number of conjugate subgroups. Subgroups on a par Standardized description of crystal structures Equivalent descriptions of crystal structures Chirality Wrongly assigned space groups Isotypism 117 Exercises How to handle space groups Wyckoff positions of space groups Relations between the Wyckoff positions in group-subgroup relations Non-conventional settings of space groups Orthorhombic space groups Monoclinic space groups Tetragonal space groups Rhombohedral space groups Hexagonal space groups 129 Exercises II Symmetry Relations between Space Groups as a Tool to Disclose Connections between Crystal Structures The group-theoretical presentation of crystal-chemical relationships Symmetry relations between related crystal structures The space group of a structure is a translationengleiche maxi mal subgroup of the space group of another structure The maximal subgroup is klassengleiche The maximal subgroup is isomorphic The subgroup is neither translationengleiche nor klassengleiche The space groups of two structures have a common supergroup Large families of structures 151 Exercises Pitfalls when setting up group-subgroup relations Origin shifts Subgroups on a par Wrong cell transformations Different paths of symmetry reduction Forbidden addition of symmetry operations 165 Exercises 166

5 xiv Contents 13 Derivation of crystal structures from closest packings of spheres Occupation of interstices in closest packings of spheres Occupation of octahedral interstices in the hexagonal-closest packing of spheres Rhombohedral hettotypes Hexagonal and trigonal hettotypes of the hexagonalclosest packing of spheres Occupation of octahedral and tetrahedral interstices in the cubicclosest packing of spheres Hettotypes of the NaCl type with doubled unit cell Hettotypes of the CaF2 type with doubled unit cell 180 Exercises Crystal structures of molecular compounds Symmetry reduction due to reduced point symmetry of building blocks Molecular packings after the pattern of sphere packings The packing in tetraphenylphosphonium salts 191 Exercises Symmetry relations at phase transitions Phase transitions in the solid state First- and second-order phase transitions Structural classification of phase transitions On the theory of phase transitions Lattice vibrations The Landau theory of continuous phase transitions Domains and twinned crystals Can a reconstructive phase transition proceed via a common subgroup? Growth and transformation twins Antiphase domains 211 Exercises Topotactic reactions Symmetry relations among topotactic reactions Topotactic reactions among lanthanoid halides 220 Exercises Group-subgroup relations as an aid for structure determination What space group should be chosen? Solving the phase problem of protein structures Superstructure reflections, suspicious structural features Detection of twinned crystals 230 Exercises 233

6 Contents xv 18 Prediction of possible structure types Derivation of hypothetical structure types with the aid of group-subgroup relations Enumeration of possible structure types The total number of possible structures The number of possible structures depending on sym metry Combinatorial computation of distributions of atoms among given positions Derivation of possible crystal structure types for a given mole cular structure 249 Exercises Historical remarks 255 Appendices 259 A Isomorphic subgroups 261 Exercises 267 B On the theory of phase transitions 269 B.l Thermodynamic aspects concerning phase transitions 269 B.2 About Landau theory 271 B.3 Renormalization-group theory 274 B.4 Discontinuous phase transitions 276 C Symmetry species 279 D Solutions to the exercises 281 References 301 Glossary 323 Index 327

Setting up Trees of Group-Subgroup Relations

International School on Mathematical and Theoretical Crystallography June 5 Université Henri Poincaré Nancy Setting up Trees of Group-Subgroup Relations Ulrich Müller Fachbereich Chemie, Philipps-Universität