Symmetry Relationships between Crystal Structures: Applications of Crystallographic Group Theory in Crystal Chemistry (International Union of Crystallography Texts on Crystallography)

This Text Presents The Basic Information Needed To Understand And To Organise The Huge Amount Of Known Structures Of Crystalline Solids. Its Basis Is Crystallographic Group Theory (space Group Theory), With Special Emphasis On The Relations Between The Symmetry Properties Of Crystals. Machine Generated Contents Note: 1. Introduction -- 1.1. Symmetry Principle In Crystal Chemistry -- 1.2. Introductory Examples -- I. Crystallographic Foundations -- 2. Basics Of Crystallography, Part 1 -- 2.1. Introductory Remarks -- 2.2. Crystals And Lattices -- 2.3. Appropriate Coordinate Systems, Crystal Coordinates -- 2.4. Lattice Directions, Net Planes, And Reciprocal Lattice -- 2.5. Calculation Of Distances And Angles -- 3. Mappings -- 3.1. Mappings In Crystallography -- 3.1.1. Example -- 3.1.2. Symmetry Operations -- 3.2. Affine Mappings -- 3.3. Application Of (n + 1) X (n + 1) Matrices -- 3.4. Affine Mappings Of Vectors -- 3.5. Isometries -- 3.6. Types Of Isometries -- 3.7. Changes Of The Coordinate System -- 3.7.1. Origin Shift -- 3.7.2. Basis Change -- 3.7.3. General Transformation Of The Coordinate System -- 3.7.4. Effect Of Coordinate Transformations On Mappings -- 3.7.5. Several Consecutive Transformations Of The Coordinate System -- 3.7.6. Calculation Of Origin Shifts From Coordinate Transformations -- 3.7.7. Transformation Of Further Crystallographic Quantities -- Exercises -- 4. Basics Of Crystallography, Part 2 -- 4.1. Description Of Crystal Symmetry In International Tables A: Positions -- 4.2. Crystallographic Symmetry Operations -- 4.3. Geometric Interpretation Of The Matrix-column Pair (w, W) Of A Crystallographic Symmetry Operation -- 4.4. Derivation Of The Matrix-column Pair Of An Isometry -- Exercises -- 5. Group Theory -- 5.1. Two Examples Of Groups -- 5.2. Basics Of Group Theory -- 5.3. Coset Decomposition Of A Group -- 5.4. Conjugation -- 5.5. Factor Groups And Homomorphisms -- 5.6. Action Of A Group On A Set -- Exercises -- 6. Basics Of Crystallography, Part 3 -- 6.1. Space Groups And Point Groups -- 6.1.1. Molecular Symmetry -- 6.1.2. Space Group And Its Point Group -- 6.1.3. Classification Of The Space Groups -- 6.2. Lattice Of A Space Group -- 6.3. Space-group Symbols -- 6.3.1. Hermann-mauguin Symbols -- 6.3.2. Schoenflies Symbols -- 6.4. Description Of Space-group Symmetry In International Tables A -- 6.4.1. Diagrams Of The Symmetry Elements -- 6.4.2. Lists Of The Wyckoff Positions -- 6.4.3. Symmetry Operations Of The General Position -- 6.4.4. Diagrams Of The General Positions -- 6.5. General And Special Positions Of The Space Groups -- 6.5.1. General Position Of A Space Group -- 6.5.2. Special Positions Of A Space Group -- 6.6. Difference Between Space Group And Space-group Type -- Exercises -- 7. Subgroups And Supergroups Of Point And Space Groups -- 7.1. Subgroups Of The Point Groups Of Molecules -- 7.2. Subgroups Of The Space Groups -- 7.2.1. Maximal Translationengleiche Subgroups -- 7.2.2. Maximal Non-isomorphic Klassengleiche Subgroups -- 7.2.3. Maximal Isomorphic Subgroups -- 7.3. Minimal Supergroups Of The Space Groups -- 7.4. Layer Groups And Rod Groups -- Exercises -- 8. Conjugate Subgroups, Normalizers And Equivalent Descriptions Of Crystal Structures -- 8.1. Conjugate Subgroups Of Space Groups -- 8.2. Normalizers Of Space Groups -- 8.3. Number Of Conjugate Subgroups. Subgroups On A Par -- 8.4. Standardized Description Of Crystal Structures -- 8.5. Equivalent Descriptions Of Crystal Structures -- 8.6. Chirality -- 8.7. Wrongly Assigned Space Groups -- 8.8. Isotypism -- Exercises -- 9. How To Handle Space Groups -- 9.1. Wyckoff Positions Of Space Groups -- 9.2. Relations Between The Wyckoff Positions In Group-subgroup Relations -- 9.3. Non-conventional Settings Of Space Groups -- 9.3.1. Orthorhombic Space Groups -- 9.3.2. Monoclinic Space Groups -- 9.3.3. Tetragonal Space Groups -- 9.3.4. Rhombohedral Space Groups -- 9.3.5. Hexagonal Space Groups -- Exercises -- Ii. Symmetry Relations Between Space Groups As A Tool To Disclose Connections Between Crystal Structures -- 10. Group-theoretical Presentation Of Crystal-chemical Relationships -- 11. Symmetry Relations Between Related Crystal Structures -- 11.1. Space Group Of A Structure Is A Translationengleiche Maximal Subgroup Of The Space Group Of Another Structure -- 11.2. Maximal Subgroup Is Klassengleiche -- 11.3. Maximal Subgroup Is Isomorphic -- 11.4. Subgroup Is Neither Translationengleiche Nor Klassengleiche -- 11.5. Space Groups Of Two Structures Have A Common Supergroup -- 11.6. Large Families Of Structures -- Exercises -- 12. Pitfalls When Setting Up Group-subgroup Relations -- 12.1. Origin Shifts -- 12.2. Subgroups On A Par -- 12.3. Wrong Cell Transformations -- 12.4. Different Paths Of Symmetry Reduction -- 12.5. Forbidden Addition Of Symmetry Operations -- Exercises -- 13. Derivation Of Crystal Structures From Closest Packings Of Spheres -- 13.1. Occupation Of Interstices In Closest Packings Of Spheres -- 13.2. Occupation Of Octahedral Interstices In The Hexagonal-closest Packing Of Spheres -- 13.2.1. Rhombohedral Hettotypes -- 13.2.2. Hexagonal And Trigonal Hettotypes Of The Hexagonal-closest Packing Of Spheres -- 13.3. Occupation Of Octahedral And Tetrahedral Interstices In The Cubic-closest Packing Of Spheres -- 13.3.1. Hettotypes Of The Nac1 Type With Doubled Unit Cell -- 13.3.2. Hettotypes Of The Caf2 Type With Doubled Unit Cell -- Exercises -- 14. Crystal Structures Of Molecular Compounds -- 14.1. Symmetry Reduction Due To Reduced Point Symmetry Of Building Blocks -- 14.2. Molecular Packings After The Pattern Of Sphere Packings -- 14.3. Packing In Tetraphenylphosphonium Salts -- Exercises -- 15. Symmetry Relations At Phase Transitions -- 15.1. Phase Transitions In The Solid State -- 15.1.1. First- And Second-order Phase Transitions -- 15.1.2. Structural Classification Of Phase Transitions -- 15.2. On The Theory Of Phase Transitions -- 15.2.1. Lattice Vibrations -- 15.2.2. Landau Theory Of Continuous Phase Transitions -- 15.3. Domains And Twinned Crystals -- 15.4. Can A Reconstructive Phase Transition Proceed Via A Common Subgroup? -- 15.5. Growth And Transformation Twins -- 15.6. Antiphase Domains -- Exercises -- 16. Topotactic Reactions -- 16.1. Symmetry Relations Among Topotactic Reactions -- 16.2. Topotactic Reactions Among Lanthanoid Halides -- Exercises -- 17. Group-subgroup Relations As An Aid For Structure Determination -- 17.1. What Space Group Should Be Chosen? -- 17.2. Solving The Phase Problem Of Protein Structures -- 17.3. Superstructure Reflections, Suspicious Structural Features -- 17.4. Detection Of Twinned Crystals -- Exercises -- 18. Prediction Of Possible Structure Types -- 18.1. Derivation Of Hypothetical Structure Types With The Aid Of Group-subgroup Relations -- 18.2. Enumeration Of Possible Structure Types -- 18.2.1. Total Number Of Possible Structures -- 18.2.2. Number Of Possible Structures Depending On Symmetry -- 18.3. Combinatorial Computation Of Distributions Of Atoms Among Given Positions -- 18.4. Derivation Of Possible Crystal Structure Types For A Given Molecular Structure -- Exercises -- 19. Historical Remarks -- Appendices -- A. Isomorphic Subgroups -- Exercises -- B. On The Theory Of Phase Transitions -- B.1. Thermodynamic Aspects Concerning Phase Transitions -- B.2. About Landau Theory -- B.3. Renormalization-group Theory -- B.4. Discontinuous Phase Transitions -- C. Symmetry Species -- D. Solutions To The Exercises. Ulrich Müller ; With Texts Adapted From Hans Wondratschek And Hartmut Bärnighausen. Includes Bibliographical References (p. [301]-321) And Index.