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E-raamat: Symmetry Relationships Between Crystal Structures: Applications of Crystallographic Group Theory in Crystal Chemistry [Oxford Scholarship Online e-raamatud]

(, Fachbereich Chemie, Philipps-Universität Marburg, Germany)
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Symmetry Relationships Between Crystal Structures: Applications of Crystallographic Group Theory in Crystal ChemistrySuurem pilt
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In crystal chemistry and crystal physics, the relations between the symmetry groups (space groups) of crystalline solids are of special importance. Part 1 of this book presents the necessary mathematical foundations and tools: the fundamentals of crystallography with special emphasis on symmetry, the theory of the crystallographic groups, and the formalisms of the needed crystallographic computations. Part 2 gives an insight into applications to problems in crystal chemistry. With the aid of numerous examples, it is shown how crystallographic group theory can be used to make evident relationships between crystal structures, to set up a systematic order in the huge amount of known crystal structures, to predict crystal structures, to analyse phase transitions and topotactic reactions in the solid state, to understand the formation of domains and twins in crystals, and to avoid errors in crystal structure determinations.

A broad range of end-of-chapter exercises offers the possibility to apply the learned material. Worked-out solutions to the exercises can be found at the end of the book.
List of symbols
xvi
1 Introduction
1(8)
1.1 The symmetry principle in crystal chemistry
2(2)
1.2 Introductory examples
4(5)
I Crystallographic Foundations
9(122)
2 Basics of crystallography, part 1
11(8)
2.1 Introductory remarks
11(1)
2.2 Crystals and lattices
11(2)
2.3 Appropriate coordinate systems, crystal coordinates
13(2)
2.4 Lattice directions, net planes, and reciprocal lattice
15(1)
2.5 Calculation of distances and angles
16(3)
3 Mappings
19(22)
3.1 Mappings in crystallography
19(1)
3.1.1 An example
19(1)
3.1.2 Symmetry operations
19(1)
3.2 Affine mappings
20(3)
3.3 Application of (n + 1) x (n + 1) matrices
23(1)
3.4 Affine mappings of vectors
24(1)
3.5 Isometries
25(2)
3.6 Types of isometries
27(3)
3.7 Changes of the coordinate system
30(11)
3.7.1 Origin shift
30(1)
3.7.2 Basis change
31(1)
3.7.3 General transformation of the coordinate system
32(1)
3.7.4 The effect of coordinate transformations on mappings
33(3)
3.7.5 Several consecutive transformations of the coordinate system
36(2)
3.7.6 Calculation of origin shifts from coordinate transformations
38(1)
3.7.7 Transformation of further crystallographic quantities
39(1)
Exercises
40(1)
4 Basics of crystallography, part 2
41(8)
4.1 The description of crystal symmetry in International Tables A: Positions
41(1)
4.2 Crystallographic symmetry operations
41(4)
4.3 Geometric interpretation of the matrix-column pair (W, w) of a crystallographic symmetry operation
45(2)
4.4 Derivation of the matrix-column pair of an isometry
47(2)
Exercises
48(1)
5 Group theory
49(14)
5.1 Two examples of groups
49(2)
5.2 Basics of group theory
51(2)
5.3 Coset decomposition of a group
53(3)
5.4 Conjugation
56(1)
5.5 Factor groups and homomorphisms
57(2)
5.6 Action of a group on a set
59(4)
Exercises
61(2)
6 Basics of crystallography, part 3
63(24)
6.1 Space groups and point groups
63(6)
6.1.1 Molecular symmetry
63(3)
6.1.2 The space group and its point group
66(1)
6.1.3 Classification of the space groups
67(2)
6.2 The lattice of a space group
69(1)
6.3 Space-group symbols
70(6)
6.3.1 Hermann-Mauguin symbols
70(4)
6.3.2 Schoenflies symbols
74(2)
6.4 Description of space-group symmetry in International Tables A
76(5)
6.4.1 Diagrams of the symmetry elements
76(3)
6.4.2 Lists of the Wyckoff positions
79(1)
6.4.3 Symmetry operations of the general position
80(1)
6.4.4 Diagrams of the general positions
80(1)
6.5 General and special positions of the space groups
81(3)
6.5.1 The general position of a space group
82(1)
6.5.2 The special positions of a space group
83(1)
6.6 The difference between space group and space-group type
84(3)
Exercises
85(2)
7 Subgroups and supergroups of point and space groups
87(14)
7.1 Subgroups of the point groups of molecules
87(2)
7.2 Subgroups of the space groups
89(5)
7.2.1 Maximal translationengleiche subgroups
91(2)
7.2.2 Maximal non-isomorphic klassengleiche subgroups
93(1)
7.2.3 Maximal isomorphic subgroups
93(1)
7.3 Minimal supergroups of the space groups
94(2)
7.4 Layer groups and rod groups
96(5)
Exercises
99(2)
8 Conjugate subgroups, normalizers and equivalent descriptions of crystal structures
101(20)
8.1 Conjugate subgroups of space groups
101(2)
8.2 Normalizers of space groups
103(3)
8.3 The number of conjugate subgroups. Subgroups on a par
106(4)
8.4 Standardized description of crystal structures
110(1)
8.5 Equivalent descriptions of crystal structures
110(3)
8.6 Chirality
113(2)
8.7 Wrongly assigned space groups
115(2)
8.8 Isotypism
117(4)
Exercises
119(2)
9 How to handle space groups
121(10)
9.1 Wyckoff positions of space groups
121(1)
9.2 Relations between the Wyckoff positions in group-subgroup relations
122(1)
9.3 Non-conventional settings of space groups
123(8)
9.3.1 Orthorhombic space groups
123(2)
9.3.2 Monoclinic space groups
125(2)
9.3.3 Tetragonal space groups
127(2)
9.3.4 Rhombohedral space groups
129(1)
9.3.5 Hexagonal space groups
129(1)
Exercises
130(1)
II Symmetry Relations between Space Groups as a Tool to Disclose Connections between Crystal Structures
131(128)
10 The group-theoretical presentation of crystal-chemical relationships
133(4)
11 Symmetry relations between related crystal structures
137(22)
11.1 The space group of a structure is a translationengleiche maximal subgroup of the space group of another structure
137(4)
11.2 The maximal subgroup is klassengleiche
141(4)
11.3 The maximal subgroup is isomorphic
145(3)
11.4 The subgroup is neither translationengleiche nor klassengleiche
148(1)
11.5 The space groups of two structures have a common supergroup
149(2)
11.6 Large families of structures
151(8)
Exercises
156(3)
12 Pitfalls when setting up group-subgroup relations
159(8)
12.1 Origin shifts
160(2)
12.2 Subgroups on a par
162(1)
12.3 Wrong cell transformations
162(1)
12.4 Different paths of symmetry reduction
163(2)
12.5 Forbidden addition of symmetry operations
165(2)
Exercises
166(1)
13 Derivation of crystal structures from closest packings of spheres
167(18)
13.1 Occupation of interstices in closest packings of spheres
167(1)
13.2 Occupation of octahedral interstices in the hexagonal-closest packing of spheres
168(10)
13.2.1 Rhombohedral hettotypes
168(6)
13.2.2 Hexagonal and trigonal hettotypes of the hexagonal-closest packing of spheres
174(4)
13.3 Occupation of octahedral and tetrahedral interstices in the cubic-closest packing of spheres
178(7)
13.3.1 Hettotypes of the NaC1 type with doubled unit cell
178(2)
13.3.2 Hettotypes of the CaF2 type with doubled unit cell
180(3)
Exercises
183(2)
14 Crystal structures of molecular compounds
185(12)
14.1 Symmetry reduction due to reduced point symmetry of building blocks
186(1)
14.2 Molecular packings after the pattern of sphere packings
187(4)
14.3 The packing in tetraphenylphosphonium salts
191(6)
Exercises
195(2)
15 Symmetry relations at phase transitions
197(20)
15.1 Phase transitions in the solid state
197(3)
15.1.1 First- and second-order phase transitions
198(1)
15.1.2 Structural classification of phase transitions
199(1)
15.2 On the theory of phase transitions
200(5)
15.2.1 Lattice vibrations
200(2)
15.2.2 The Landau theory of continuous phase transitions
202(3)
15.3 Domains and twinned crystals
205(2)
15.4 Can a reconstructive phase transition proceed via a common subgroup?
207(3)
15.5 Growth and transformation twins
210(1)
15.6 Antiphase domains
211(6)
Exercises
214(3)
16 Topotactic reactions
217(10)
16.1 Symmetry relations among topotactic reactions
218(2)
16.2 Topotactic reactions among lanthanoid halides
220(7)
Exercises
224(3)
17 Group-subgroup relations as an aid for structure determination
227(8)
17.1 What space group should be chosen?
228(1)
17.2 Solving the phase problem of protein structures
228(1)
17.3 Superstructure reflections, suspicious structural features
229(1)
17.4 Detection of twinned crystals
230(5)
Exercises
233(2)
18 Prediction of possible structure types
235(20)
18.1 Derivation of hypothetical structure types with the aid of group-subgroup relations
235(4)
18.2 Enumeration of possible structure types
239(6)
18.2.1 The total number of possible structures
239(2)
18.2.2 The number of possible structures depending on symmetry
241(4)
18.3 Combinatorial computation of distributions of atoms among given positions
245(4)
18.4 Derivation of possible crystal structure types for a given molecular structure
249(6)
Exercises
253(2)
19 Historical remarks
255(4)
Appendices
259(42)
A Isomorphic subgroups
261(8)
Exercises
267(2)
B On the theory of phase transitions
269(10)
B.1 Thermodynamic aspects concerning phase transitions
269(2)
B.2 About Landau theory
271(3)
B.3 Renormalization-group theory
274(2)
B.4 Discontinuous phase transitions
276(3)
C Symmetry species
279(2)
D Solutions to the exercises
281(20)
References 301(22)
Glossary 323(4)
Index 327
Ulrich Müller was born in Colombia in 1940. He studied chemistry in Germany. His Ph.D. work (1964 - 1966) was performed in Inorganic Chemistry, partly at the University of Stuttgart, Germany, partly at Purdue University, Indiana, USA. After post-doctoral work at the University of Karlsruhe, Germany, he was appointed as professor of Inorganic Chemistry at the University of Marburg, Gemany, in 1972. From 1992 to 1999 he was professor of solid state chemistry at the University of Kassel, Germany, and then returned to the University of Marburg. He is now retired since 2005. He is the author of several textbooks in chemistry for beginners and advanced students.
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