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Symmetry Relationships between Crystal StructuresApplications of Crystallographic Group Theory in Crystal Chemistry$
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Ulrich Müller

Print publication date: 2013

Print ISBN-13: 9780199669950

Published to Oxford Scholarship Online: December 2013

DOI: 10.1093/acprof:oso/9780199669950.001.0001

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Prediction of possible structure types

Prediction of possible structure types

(p.235) 18 Prediction of possible structure types
Symmetry Relationships between Crystal Structures

Ulrich Müller

Oxford University Press

After selection of an arsitotype (e.g., a hexagonal closest packing of spheres) and a structural principle (e.g., the occupation of octahedral interstices in this packing), it is possible to derive all possible crystal structures that meet these initial conditions. A Bärnighausen tree reveals which space groups may appear. For each space group of the tree it can be calculated how many different structure types can adopt this space group. This is done with mathematical enumeration theorems and combinatorial computations. Given a molecular structure and a packing principle, one can derive which space groups are possible. Such considerations can even help to solve the crystal structures of disordered crystals exhibiting diffuse X-ray diffraction streaks.

Keywords:   space groups, enumeration theorems, molecular structure

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