Crystallographic Point Group

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## Notation

### Schoenflies notation

### Hermann-Mauguin notation

### The correspondence between different notations

## Deriving the crystallographic point group (crystal class) from the space group

## See also

## References

## External links

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Crystallographic Point Group

In crystallography, a **crystallographic point group** is a set of symmetry operations, corresponding to one of the point groups in three dimensions, such that each operation would leave the structure of a crystal unchanged i.e. the same kinds of atoms would be placed in similar positions as before the transformation. For example, in a primitive cubic crystal system, a rotation of the unit cell by 90 degree around an axis that is perpendicular to two parallel faces of the cube, intersecting at its center, is a symmetry operation that projects each atom to the location of one of its neighbor leaving the overall structure of the crystal unaffected.

In the classification of crystals, each point group defines a so-called **(geometric) crystal class**. There are infinitely many three-dimensional point groups. However, the crystallographic restriction on the general point groups results in there being only 32 crystallographic point groups. These 32 point groups are one-and-the-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 determines, among other things, the directional variation of physical properties that arise from its structure, including optical properties such as birefringency, or electro-optical features such as the Pockels effect. For a periodic crystal (as opposed to a quasicrystal), the group must maintain the three-dimensional translational symmetry that defines crystallinity.

The point groups are named according to their component symmetries. There are several standard notations used by crystallographers, mineralogists, and physicists.

For the correspondence of the two systems below, see **crystal system**.

In Schoenflies notation, point groups are denoted by a letter symbol with a subscript. The symbols used in crystallography mean the following:

*C*(for cyclic) indicates that the group has an_{n}*n*-fold rotation axis.*C*is_{nh}*C*with the addition of a mirror (reflection) plane perpendicular to the axis of rotation._{n}*C*is_{nv}*C*with the addition of n mirror planes parallel to the axis of rotation._{n}*S*(for_{2n}*Spiegel*, German for mirror) denotes a group with only a*2n*-fold rotation-reflection axis.*D*(for dihedral, or two-sided) indicates that the group has an_{n}*n*-fold rotation axis plus*n*twofold axes perpendicular to that axis.*D*has, in addition, a mirror plane perpendicular to the_{nh}*n*-fold axis.*D*has, in addition to the elements of_{nd}*D*, mirror planes parallel to the_{n}*n*-fold axis.- The letter
*T*(for tetrahedron) indicates that the group has the symmetry of a tetrahedron.*T*includes improper rotation operations,_{d}*T*excludes improper rotation operations, and*T*is_{h}*T*with the addition of an inversion. - The letter
*O*(for octahedron) indicates that the group has the symmetry of an octahedron (or cube), with (*O*) or without (_{h}*O*) improper operations (those that change handedness).

Due to the crystallographic restriction theorem, *n* = 1, 2, 3, 4, or 6 in 2- or 3-dimensional 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

An abbreviated form of the Hermann-Mauguin notation commonly used for space groups also serves to describe crystallographic point groups. Group names are

Class | Group names | |||||||
---|---|---|---|---|---|---|---|---|

Cubic | 23 | m3 | 432 | 43m | m3m | |||

Hexagonal | 6 | 6 | ^{6}/_{m} |
622 | 6mm | 6m2 | ^{6}/_{m}mm
| |

Trigonal | 3 | 3 | 32 | 3m | 3m | |||

Tetragonal | 4 | 4 | ^{4}/_{m} |
422 | 4mm | 42m | ^{4}/_{m}mm
| |

Orthorhombic | 222 | mm2 | mmm | |||||

Monoclinic | 2 | ^{2}/_{m} |
m | |||||

Triclinic | 1 | 1 | Subgroup relations of the 32 crystallographic point groups (rows represent group orders from bottom to top as: 1,2,3,4,6,8,12,16,24, and 48.) |

Crystal system | Hermann-Mauguin | Shubnikov^{[1]} |
Schoenflies | Orbifold | Coxeter | Order | |
---|---|---|---|---|---|---|---|

(full) | (short) | ||||||

Triclinic | 1 | 1 | C_{1} |
11 | [ ]^{+} |
1 | |

1 | 1 | C_{i} = S_{2} |
× | [2^{+},2^{+}] |
2 | ||

Monoclinic | 2 | 2 | C_{2} |
22 | [2]^{+} |
2 | |

m | m | C_{s} = C_{1h} |
* | [ ] | 2 | ||

2/m | C_{2h} |
2* | [2,2^{+}] |
4 | |||

Orthorhombic | 222 | 222 | D_{2} = V |
222 | [2,2]^{+} |
4 | |

mm2 | mm2 | C_{2v} |
*22 | [2] | 4 | ||

mmm | D = _{2h}V_{h} |
*222 | [2,2] | 8 | |||

Tetragonal | 4 | 4 | C_{4} |
44 | [4]^{+} |
4 | |

4 | 4 | S_{4} |
2× | [2^{+},4^{+}] |
4 | ||

4/m | C_{4h} |
4* | [2,4^{+}] |
8 | |||

422 | 422 | D_{4} |
422 | [4,2]^{+} |
8 | ||

4mm | 4mm | C_{4v} |
*44 | [4] | 8 | ||

42m | 42m | D = _{2d}V_{d} |
2*2 | [2^{+},4] |
8 | ||

4/mmm | D_{4h} |
*422 | [4,2] | 16 | |||

Trigonal | 3 | 3 | C_{3} |
33 | [3]^{+} |
3 | |

3 | 3 | C_{3i} = S_{6} |
3× | [2^{+},6^{+}] |
6 | ||

32 | 32 | D_{3} |
322 | [3,2]^{+} |
6 | ||

3m | 3m | C_{3v} |
*33 | [3] | 6 | ||

3 | 3m | D_{3d} |
2*3 | [2^{+},6] |
12 | ||

Hexagonal | 6 | 6 | C_{6} |
66 | [6]^{+} |
6 | |

6 | 6 | C_{3h} |
3* | [2,3^{+}] |
6 | ||

6/m | C_{6h} |
6* | [2,6^{+}] |
12 | |||

622 | 622 | D_{6} |
622 | [6,2]^{+} |
12 | ||

6mm | 6mm | C_{6v} |
*66 | [6] | 12 | ||

6m2 | 6m2 | D_{3h} |
*322 | [3,2] | 12 | ||

6/mmm | D_{6h} |
*622 | [6,2] | 24 | |||

Cubic | 23 | 23 | T |
332 | [3,3]^{+} |
12 | |

3 | m3 | T_{h} |
3*2 | [3^{+},4] |
24 | ||

432 | 432 | O |
432 | [4,3]^{+} |
24 | ||

43m | 43m | T_{d} |
*332 | [3,3] | 24 | ||

3 | m3m | O_{h} |
*432 | [4,3] | 48 |

- Leave out the Bravais type
- Convert all symmetry elements with translational components into their respective symmetry elements without translation symmetry (Glide planes are converted into simple mirror planes; Screw axes are converted into simple axes of rotation)
- Axes of rotation, rotoinversion axes and mirror planes remain unchanged.

**^**"Archived copy". Archived from the original on 2013-07-04. Retrieved .CS1 maint: archived copy as title (link)

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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