Crystal Symmetries (eBook)

Front Cover 1
Crystal Symmetries: Shubnikov Centennial Papers 4
Copyright Page 5
Table of Contents 6
Preface 10
CHAPTER 1. A. V. SHUBNIKOV AND HIS IDEAS IN 
12 
REFERENCES 16
CHAPTER 
18 
CHAPTER 
26 
HISTORICAL BACKGROUND 26
OPERATIONS IN THE SYMMETRY OF SIMILARITY 26
SIMILARITY SYMMETRY GROUPS 29
SUMMARY 31
REFERENCES 32
CHAPTER 
34 
REFERENCES 38
CHAPTER 
40 
1. INTRODUCTION 40
2. SYMMETRY OF SCALAR AND VECTORS 40
3. THE KINETIC COEFFICIENT SYMMETRY PRINCIPLE (THE ONSAGER PRINCIPLE) 41
4. GEOMETRICAL IMAGE REFLECTING SYMMETRY OF SPACE AND TIME 42
REFERENCES 45
CHAPTER 
46 
1. SUMMARY 46
2. OVERVIEW 47
3. THE STRUCTURE INVARIANTS 48
4. THE STRUCTURE SEMINVARIANTS 48
5. THE ALGEBRAIC APPROACH 50
6. THE FUNDAMENTAL PRINCIPLE OF DIRECT METHODS 50
7. THE NEIGHBORHOOD PRINCIPLE 50
8. THE PROBABILISTIC APPROACH 51
9. CONCLUDING REMARKS 56
REFERENCES 57
CHAPTER 
58 
1. INTRODUCTION 58
2. CENTROSYMMETRIC SPACE GROUPS 58
3. NON-CENTROSYMMETRIC SPACE GROUPS 61
4. APPLICATION 66
REFERENCES 67
CHAPTER 
68 
1. CONTRIBUTION OF A.V. SHUBNIKOV TO THE THEORY OF SYMMETRY 68
2. ORTHOGONAL-PERMUTATION SYMMETRY GROUPS IN CLASSICAL CRYSTAL PHYSICS 68
3. ANTISYMMETYU, COLOUR SYMMETRY AND THEIR MAGNETIC INTERPRETATIONS 69
4. MAGNETIC, COMPLEX AND PHASE SYMMETRY IN CRYSTAL PHYSICS AND IN STRUCTURE ANALYSIS OF CRYSTALS 72
5. SUPERSPACE SYMMETRY OF MOLECULARLY MODULATED POLYSYSTEM CRYSTALS 80
6. THE PROBLEM OF INCOMMENSURABILITY IN CRYSTAL PHYSICS 81
7. THE PROBLEM OF ICOSAHEDRIC QUASICRYSTALS [51-55] 84
REFERENCES 84
CHAPTER 
86 
1. INTRODUCTION 86
2. THE ASSESSMENT OF THE UNIQUENESS OF THE CRYSTAL STRUCTURE ANALYSIS 86
3. THE REGULARITY CONDITION OF A (r, R)-SYSTEM 87
4. DENSEST PACKING OF REGULAR POLYTOPES 88
5. PACKINGS OF RIGID BALLS 89
6. SUBGROUPS OF THE ORTHOGONAL GROUP O(n) 89
7. DERIVATION OF SPACE GROUPS 90
8. AN UPPER BOUND FOR THE NUMBER OF FACES OF A DIRICHLET DOMAIN 91
9. CHARACTERIZATION OF Z-REDUCED QUADRATIC FORMS 92
10. CHARACTERIZATION OF THE TYPES OF MAXIMAL PARALLELOTOPES 93
11. A PROGRAM LIBRARY FOR MATHEMATICAL CRYSTALLOGRAPHY 96
REFERENCES 96
CHAPTER 
98 
REFERENCES 101
CHAPTER 
104 
INTRODUCTION 104
DESCRIPTION OF THE MODEL 104
ANALYSIS OF CRYSTAL STRUCTURE 107
DISCUSSION 111
REFERENCES 112
CHAPTER 
114 
1. INTRODUCTION 114
2. AGEOMETRICAL METHOD FOR OBTAINING THE LOW-SYMMETRY PHASES ARISING UPON PHASE TRANSITIONS 115
3. THERMODYNAMIC DESCRIPTION OF PHASE TRANSITIONS 116
4. THE PHASE DIAGRAMS WITH INCOMMENSURATE PHASES TAKEN INTO ACCOUNT 118
5. PHYSICAL FIELDS LOWERING THE SYMMETRY 121
6. COMPLETE CONDENSATE OF NORMAL MODES 123
REFERENCES 124
CHAPTER 
126 
REFERENCES 128
CHAPTER 14. ORDER–DISORDER STRUCTURES 130
1. THE NOTION OF OD STRUCTURES 130
2. BASIC IDEAS OF OD THEORY 130
3. OD STRUCTURES AND POLYTYPES 135
4. DATABASE OF OD STRUCTURES 137
5. CONCLUDING REMARKS 137
REFERENCES 137
CHAPTER 
140 
REFERENCES 145
CHAPTER 
148 
1. INTRODUCTION 148
2. NIGGLI'S CRITERION 148
3. DEFINITION OF THE 44 CHARACTERS 149
4. NIGGLI AND THE CELL - TYPE CRITERION 150
REFERENCES 152
CHAPTER 
154 
1. INTRODUCTION 154
2. REMEMBRANCE OF THE THINGS KNOWN 154
3. CORRELATION OF REPRESENTATIONS WITH SUBGROUPS 155
4. LATTICES OF NORMAL SUBGROUPS OF IRREDUCIBLE SPACE GROUPS 156
5. REDUCIBLE SPACE GROUPS 158
6. A BRIEF REMARK ON ORIGIN CHOICES 161
7. LATTICES OF NORMAL SUBGROUPS OF LAYER AND ROD GROUPS AND THEIR RELATION TO LATTICES OF NORMAL SUBGROUPS OF SPACE GROUPS 161
8. VIEWS FOR THE FUTURE 165
REFERENCES 165
CHAPTER 
168 
1. INTRODUCTION 168
2. RESULTS 169
3. DISCUSSION 180
REFERENCES 181
CHAPTER 
182 
1. INTRODUCTION 182
2. THE GROUPS OF COLOR SYMMETRY 182
3. PHASE TRANSITIONS AND COMPLETE CONDENSATE OF THE STATIONARY VECTORS 183
4. EXAMPLES OF CONSTRUCTION OF THE COMPLETE CONDENSATE OF STATIONARY VECTORS 185
5. SUBGROUPS OF THE G-GROUP AND THE IRREDUCIBLE REPRESENTATIONS OF G 187
6. GROUPS OF THE COLOR P-SYMMETRY ASSOCIATED WITH MULTIDIMENSIONAL IR's 191
7. W-SYMMETRY GROUPS ASSOCIATED WITH MULTIDIMENSIONAL IR's 192
8. CONCLUSION 194
REFERENCES 195
CHAPTER 
198 
REFERENCES 203
CHAPTER 
206 
1. INTRODUCTION 206
2. COLOR GROUPS 207
3. FURTHER DEVELOPMENTS 209
4. COLORED PATTERNS 211
5. COLOR SYMMETRY AND SYMMETRY THEORY 213
REFERENCES 214
CHAPTER 
216 
SECTION 1 216
SECTION 2 216
SECTION 3 217
SECTION 4 220
SECTION 5 221
SECTION 6 222
REFERENCES 222
CHAPTER 
224 
REFERENCES 227
CHAPTER 
230 
1 INTRODUCTION 230
2. THE NOTION OF POLYTYPISM AND THE POSITION OF POLYTYPES IN THE STRUCTURAL SYSTEMATICS OF CRYSTALLINE SUBSTANCES 230
3. POLYTYPE STRUCTURES 233
4. POLYTYPE SYMBOLIC NOTATIONS 245
5. THE SCIENTIFIC SIGNIFICANCE OF CRYSTAL STRUCTURE POLYTYPISM 249
REFERENCES 250
CHAPTER 
254 
REFERENCES 257
CHAPTER 
258 
REFERENCES 277
CHAPTER 
278 
INTRODUCTION 278
GENERAL GEOMETRIC AND WEIGHT PARAMETERS 279
STRUCTURE ORGANIZATION OF DNA 281
DIRECT RESTORATION OF THE PHAGE STRUCTURE 284
TEMPERATURE TRANSITIONS IN PHAGE T7 285
INFLUENCE OF THE IONIC STRENGTH 286
CONCLUSION 288
REFERENCES 288
CHAPTER 
290 
INTRODUCTION 290
SPIN HAMILTONIAN FORMALISM 290
MOLECULAR MOTIONS 291
EPR TIME-SCALE 291
MAGNETIC RELAXATION 292
SPIN HAMILTONIAN INVARIANTS IN SINGLE CRYSTALS 293
SPIN HAMILTONIAN INVARIANTS IN ISOTROPIC LIQUIDS 294
INVARIANCE RELATIONS BETWEEN CRYSTALS AND LIQUIDS 295
CONCLUSIONS 296
REFERENCES 296
CHAPTER 
298 
1. INTRODUCTION 298
2. PROPER WAVES IN CRYSTALS 298
3. BOUNDARY PROBLEMS IN CRYSTAL OPTICS 302
4. THE DETERMINATION OF OPTICAL PARAMETERS OF CRYSTALS 307
5. CONCLUSION 314
REFERENCES 314
CHAPTER 
318 
REFERENCES 321
CHAPTER 
324 
REFERENCES 330