Mathematical Crystallography

Monte B. Boisen, Gerald V. Gibbs

• Publisher: De Gruyter
• Year: 2018
• Print ISBN: 9780939950195
• eBook ISBN: 9781501508912
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Geology & Earth Sciences

Volume 15 of Reviews in Mineralogy is written with two goals in mind. The first is to derive the 32 crystallographic point groups, the 14 Bravais lattice types and the 230 crystallographic space group types. The second is to develop the mathematical tools necessary for these derivations in such a manner as to lay the mathematical foundation needed to solve numerous basic problems in crystallography and to avoid extraneous discourses. To demonstrate how these tools can be employed, a large number of examples are solved and problems are given. The book is, by and large, self-contained. In particular, topics usually omitted from the traditional courses in mathematics that are essential to the study of crystallography are discussed. For example, the techniques needed to work in vector spaces with noncartesian bases are developed. Unlike the traditional group-theoretical approach, isomorphism is not the essential ingredient in crystallographic classification schemes. Because alternative classification schemes must be used, the notions of equivalence relations and classes which are fundamental to such schemes are defined, discussed and illustrated. For example, we will find that the classification of the crystallographic space groups into the traditional 230 types is defined in terms of their matrix representations. Therefore, the derivation of these groups from the point groups will be conducted using the 37 distinct matrix groups rather than the 32 point groups they represent.

Table of contents

• COPYRIGHT: FIRST EDITION 1985; REVISED EDITION 1990
• REVIEWS IN MINERALOGY
• Preface to the Revised Edition of Mathematical Crystallography
• PREFACE
• ACKNOWLEDGEMENTS
• EXPLANATION OF SYMBOLS
• CONTENTS
• CHAPTER 1. MODELING SYMMETRICAL PATTERNS AND GEOMETRIES OF MOLECULES AND CRYSTALS
• CHAPTER 2. SOME GEOMETRICAL ASPECTS OF CRYSTALS
• CHAPTER 3. POINT ISOMETRIES - VEHICLES FOR DESCRIBING SYMMETRY
• CHAPTER 4. THE MONAXIAL CRYSTALLOGRAPHIC POINT GROUPS
• CHAPTER 5. THE POLYAXIAL CRYSTALLOGRAPHIC POINT GROUPS
• CHAPTER 6. THE BRAVAIS LATTICE TYPES
• CHAPTER 7. THE CRYSTALLOGRAPHIC SPACE GROUPS
• APPENDIX 1. MAPPINGS
• APPENDIX 2. MATRIX METHODS
• APPENDIX 3. CONSTRUCTION AND INTERPRETATION OF MATRICES REPRESENTING POINT ISOMETRIES
• APPENDIX 4. POTPOURRI
• APPENDIX 5. SOME PROPERTIES OF LATTICE PLANES
• APPENDIX 6. INTERSECTION ANGLES BETWEEN ROTATION AXES
• APPENDIX 7. EQUIVALENCE RELATIONS, COSETS AND FACTOR GROUPS
• APPENDIX 8. ISOMORPHISMS
• REFERENCES
• Solutions to Problems
• INDEX