*Ulrich Müller*

- Published in print:
- 2013
- Published Online:
- December 2013
- ISBN:
- 9780199669950
- eISBN:
- 9780191775086
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199669950.003.0004
- Subject:
- Physics, Crystallography, Condensed Matter Physics / Materials

A space group consists of an infinity of crystallographic symmetry operations which are represented by matrix-column pairs W,w. However, the number of different matrices W always is finite. Rotations ...
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A space group consists of an infinity of crystallographic symmetry operations which are represented by matrix-column pairs W,w. However, the number of different matrices W always is finite. Rotations are restricted to rotation angles of 360°/N with the orders of N = 1, 2, 3, 4, and 6. Symmetry operations are designated by Hermann-Mauguin symbols. These include: 1 for the identity; the number N for rotations; Np for screw rotations; N̄ for rotoinversions; m for reflections; a, b, c, d, e, and n for glide reflections. A plane perpendicular to an axis is specified by a fraction sign like 4/m. The geometric meaning of a matrix-column pair W,w can be inferred from the determinant and the trace of W.Less

A space group consists of an infinity of crystallographic symmetry operations which are represented by matrix-column pairs *W*,*w*. However, the number of different matrices *W* always is finite. Rotations are restricted to rotation angles of 360°/*N* with the orders of *N* = 1, 2, 3, 4, and 6. Symmetry operations are designated by Hermann-Mauguin symbols. These include: 1 for the identity; the number *N* for rotations; *N*_{p} for screw rotations; *N̄* for rotoinversions; *m* for reflections; *a*, *b*, *c*, *d*, *e*, and *n* for glide reflections. A plane perpendicular to an axis is specified by a fraction sign like 4/*m*. The geometric meaning of a matrix-column pair *W*,*w* can be inferred from the determinant and the trace of *W*.

*Ulrich Müller*

- Published in print:
- 2013
- Published Online:
- December 2013
- ISBN:
- 9780199669950
- eISBN:
- 9780191775086
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199669950.003.0003
- Subject:
- Physics, Crystallography, Condensed Matter Physics / Materials

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 ...
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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.Less

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.