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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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Basics of crystallography, part 3

Basics of crystallography, part 3

(p.62) (p.63) 6 Basics of crystallography, part 3
Symmetry Relationships between Crystal Structures

Ulrich Müller

Oxford University Press

The molecular symmetry is called the point group of the molecule. A point group can be represented by a set of matrices W. The point group of a crystal is the symmetry group of the bundle of normals on the crystal faces. The set of all translations of a crystal is its translation group. The point group types of crystals are the 32 crystal classes. The point group types of the crystal lattices are the 7 crystal systems. Two space groups belong to one out of 219 affine space group types if their sets W,w coincide, and they belong to one out of 230 space group types if their sets W,w coincide referred to a right-handed coordinate system. Space groups are designated by short Hermann-Mauguin symbols like Cmcm, P63/mcm, or F4132 or by full Hermann-Mauguin symbols like C2/m2/c21 /m or P63/m2/m2/c. The letter in the first position represents the lattice type (centring). The following symmetry symbols specify the kinds of symmetry operations in three symmetry directions. The direction follows from the position in the symbol. Detailed information for every space-group type is listed in International Tables for Crystallography.A general position in a space group is a position with the site symmetry 1, a special position has a higher site symmetry. A space group is the symmetry of a specific crystal structure, including its specific lattice. A space group type has arbitrary lattice parameters; it is one out of 230 possible ways to combine symmetry operations in space.

Keywords:   point group, crystal class, crystal system, affine space group type, molecular symmetry, Hermann-Mauguin symbol, general position, special position

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