Crystallographic point group
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In [[crystallography]], a '''crystallographic point group''' is a set of [[symmetry]] operations, like rotations or reflections, that leave a point fixed while moving each atom of the [[crystal]] to the position of an atom of the same kind. That is, an infinite crystal would look exactly the same before and after any of the operations in its point group. In the classification of crystals, each point group corresponds to a '''crystal class'''.
There are infinitely many 3D point groups; in crystallography, however, they are restricted to be compatible with the discrete translation symmetries of a crystal lattice. This [[crystallographic restriction theorem|crystallographic restriction]] of the infinite families of general [[point group]]s results in 32 crystallographic point groups.
The point group of a crystal, among other things, determines some of the crystal's [[crystal optics|optical properties]], such as whether it is [[birefringence|birefringent]], or whether it shows the [[Pockels effect]].
==Notation==
The point groups are denoted by their component symmetries. There are a few standard notations used by crystallographers, mineralogists, and physicists.
For the correspondence of the two systems below, see '''[[crystal system]]'''.
===Schönflies notation===
{{main|Schoenflies notation}}
{{details|Point groups in three dimensions}}
In [[Arthur Moritz Schönflies|Schönflies]] notation, point groups are denoted by a letter symbol with a subscript. The symbols used in crystallography mean the following:
*The letter ''O'' (for [[octahedron]]) indicates that the group has the symmetry of an octahedron (or [[cube]]), with (''O<sub>h</sub>'') or without (''O'') improper operations (those that change handedness).
*The letter ''T'' (for [[tetrahedron]]) indicates that the group has the symmetry of a tetrahedron. ''T<sub>d</sub>'' includes improper operations, ''T'' excludes improper operations, and ''T<sub>h</sub>'' is ''T'' with the addition of an inversion.
*''C<sub>n</sub>'' (for [[cyclic group|cyclic]]) indicates that the group has an ''n''-fold rotation axis. ''C<sub>nh</sub>'' is ''C<sub>n</sub>'' with the addition of a mirror (reflection) plane perpendicular to the axis of rotation. ''C<sub>nv</sub>'' is ''C<sub>n</sub>'' with the addition of a mirror plane parallel to the axis of rotation.
*''S<sub>n</sub>'' (for ''Spiegel'', German for [[mirror]]) denotes a group that contains only an ''n''-fold [[rotation-reflection axis]].
*''D<sub>n</sub>'' (for [[dihedral]], or two-sided) indicates that the group has an ''n''-fold rotation axis plus a twofold axis perpendicular to that axis. ''D<sub>nh</sub>'' has, in addition, a mirror plane perpendicular to the ''n''-fold axis. ''D<sub>nv</sub>'' has, in addition to the elements of ''D<sub>n</sub>'', mirror planes parallel to the ''n''-fold axis.
Due to the [[crystallographic restriction theorem]], ''n'' = 1, 2, 3, 4, or 6.
=== Hermann-Mauguin notation===
An abbreviated form of the [[Hermann-Mauguin notation]] commonly used for [[space group]]s also serves to describe crystallographic point groups. Group names are
*1, <u style="text-decoration:overline">1</u>
*2, m, <sup>2</sup>⁄<sub>m</sub>
*222, mm2, mmm
*4,<u style="text-decoration:overline">4</u>, <sup>4</sup>⁄<sub>m</sub>, 422, 4mm, <u style="text-decoration:overline">4</u>2m, <sup>4</sup>⁄<sub>m</sub>mm
*3, <u style="text-decoration:overline">3</u>, 32, 3m, <u style="text-decoration:overline">3</u>m
*6, <u style="text-decoration:overline">6</u>, <sup>6</sup>⁄<sub>m</sub>, 622, 6mm, <u style="text-decoration:overline">6</u>2m, <sup>6</sup>⁄<sub>m</sub>mm
*23, m<u style="text-decoration:overline">3</u>, 432, <u style="text-decoration:overline">4</u>3m, m<u style="text-decoration:overline">3</u>m
==See also==
* [[Molecular symmetry]]
* [[Point group]]
* [[Space group]]
* [[Point groups in three dimensions]]
* [[Crystal system]]
==External links==
*[http://newton.ex.ac.uk/research/qsystems/people/goss/symmetry/Solids.html Pictorial overview of the 32 groups]
[[Category:Symmetry]]
[[Category:Crystallography]]
[[de:Punktgruppe]]
[[fr:Groupe ponctuel de symétrie]]
[[he:חבורת סימטריות נקודתית]]
[[ko:결정학적 점군]]
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