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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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(p.19) 3 Mappings
Symmetry Relationships between Crystal Structures

Ulrich Müller

Oxford University Press

A mapping is an instruction by which for each point in space there is a uniquely determined image point. An affine mapping is a mapping which maps parallel straight lines onto parallel straight lines. It can be represented by a set of three equations or, more concisely, by a 3 × 3 matrix W and a column w, a matrix-column pair W,w. Matrix and column can be combined to a 4 × 4 matrix, the augmented matrix. An isometry is an affine mapping that leaves all distances unchanged. A symmetry operation is an isometry that maps an object onto itself. The determinant of W specifies any volume change. An isometry has det(W) = 1 and leaves the metric tensor unchanged. Different kinds of isometries are the identity, translations, rotations, screw rotations, the inversion, rotoinversions, reflections, and glide reflections. The set of all symmetry operations of a crystal structure is its space group. A change of the coordinate system may involve an origin shift and/or a basis change and requires corresponding computations; formulae and examples are given.

Keywords:   mapping, affine mapping, matrix-column pair, augmented matrix, isometry, symmetry operation, space group, rotoinversions

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