 Crystallographic point group

In crystallography, a crystallographic point group is a set of symmetry operations, like rotations or reflections, that leave a central point fixed while moving other directions and faces of the crystal to the positions of features of the same kind. For a true crystal (as opposed to a quasicrystal), the group must also be consistent with maintenance of the threedimensional translational symmetry that defines crystallinity. The macroscopic properties of a 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 is also known as a crystal class.
There are infinitely many threedimensional point groups; However, the crystallographic restriction of the infinite families of general point groups results in there being only 32 crystallographic point groups. These 32 point groups are oneandthe same as the 32 types of morphological (external) crystalline symmetries derived in 1830 by Johann Friedrich Christian Hessel from a consideration of observed crystal forms.
The point group of a crystal, among other things, determines directional variation of the physical properties that arise from its structure, including optical properties such as whether it is birefringent, or whether it shows the Pockels effect.
Contents
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 article: Schoenflies notationFor more details on this topic, see Point groups in three dimensions.In 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_{h}) 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_{d} includes improper operations, T excludes improper operations, and T_{h} is T with the addition of an inversion.
 The letter I (for icosahedron) indicates that the group has the symmetry of an icosahedron (or dodecahedron), either with (I_{h}) or without (I) improper operations.
 C_{n} (for cyclic) indicates that the group has an nfold rotation axis. C_{nh} is C_{n} with the addition of a mirror (reflection) plane perpendicular to the axis of rotation. C_{nv} is C_{n} with the addition of a mirror plane parallel to the axis of rotation.
 S_{2n} (for Spiegel, German for mirror) denotes a group that contains only a 2nfold rotationreflection axis.
 D_{n} (for dihedral, or twosided) indicates that the group has an nfold rotation axis plus a twofold axis perpendicular to that axis. D_{nh} has, in addition, a mirror plane perpendicular to the nfold axis. D_{nv} has, in addition to the elements of D_{n}, mirror planes parallel to the nfold axis.
Due to the crystallographic restriction theorem, n = 1, 2, 3, 4, or 6 in 2 or 3dimensional space.
n 1 2 3 4 6 C_{n} C_{1} C_{2} C_{3} C_{4} C_{6} C_{nv} C_{1v}=C_{1h} C_{2v} C_{3v} C_{4v} C_{6v} C_{nh} C_{1h} C_{2h} C_{3h} C_{4h} C_{6h} D_{n} D_{1}=C_{2} D_{2} D_{3} D_{4} D_{6} D_{nh} D_{1h}=C_{2v} D_{2h} D_{3h} D_{4h} D_{6h} D_{nd} D_{1d}=C_{2h} D_{2d} D_{3d} D_{4d} D_{6d} S_{2n} S_{2} S_{4} S_{6} S_{8} S_{12} D_{4d} and D_{6d} are actually forbidden because they contain improper rotations with n=8 and 12 respectively. The 27 point groups in the table plus T, T_{d}, T_{h}, O and O_{h} constitute 32 crystallographic point groups
Hermann–Mauguin notation
An abbreviated form of the Hermann–Mauguin notation commonly used for space groups also serves to describe crystallographic point groups. Group names are
1 1 2 ^{2}⁄_{m} 222 m mm2 mmm 3 3 32 3m 3m 4 4 ^{4}⁄_{m} 422 4mm 42m ^{4}⁄_{m}mm 6 6 ^{6}⁄_{m} 622 6mm 62m ^{6}⁄_{m}mm 23 m3 432 43m m3m The correspondence between the four notations is:
HermannMauguin Schoenflies Orbifold Coxeter Order 1 C_{1} 11 [ ]^{+} 1 2 C_{2} 22 [2]^{+} 2 3 C_{3} 33 [3]^{+} 3 4 C_{4} 44 [4]^{+} 4 6 C_{6} 66 [6]^{+} 6 mm2 C_{2v} *22 [2] 4 3m C_{3v} *33 [3] 6 4mm C_{4v} *44 [4] 8 6mm C_{6v} *66 [6] 12 m C_{s} = C_{1h} *11 [ ] 2 2/m C_{2h} 2* [2,2^{+}] 4 6 C_{3h} 3* [2,3^{+}] 6 4/m C_{4h} 4* [2,4^{+}] 8 6/m C_{6h} 6* [2,6^{+}] 12 1 C_{i} = S_{2} 1x [1^{+},2^{+}] 2 4 S_{4} 2x [2^{+},4^{+}] 4 3 S_{6} 3x [2^{+},6^{+}] 6 222 D_{2} 222 [2,2]^{+} 4 32 D_{3} 322 [3,2]^{+} 6 422 D_{4} 422 [4,2]^{+} 8 622 D_{6} 622 [6,2]^{+} 12 mmm D_{2h} *222 [2,2] 8 62m D_{3h} *322 [3,2] 12 4/m mm D_{4h} *422 [4,2] 16 6/m mm D_{6h} *622 [6,2] 24 42m D_{2d} 2*2 [2^{+},4] 8 3m D_{3d} 2*3 [2^{+},6] 12 23 T 332 [3,3]^{+} 12 43m T_{d} *332 [3,3] 24 m3 T_{h} 3*2 [3^{+},4] 24 432 O 432 [4,3]^{+} 24 m3m O_{h} *432 [4,3] 48 See also
External links
Categories: Symmetry
 Crystallography
 Discrete groups
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