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 periodic crystal (as opposed to a quasicrystal), the group must also be consistent with maintenance of the three-dimensional 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 three-dimensional 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 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, 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.

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.

Schoenflies notation

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

• Cn (for cyclic) indicates that the group has an n-fold rotation axis. Cnh is Cn with the addition of a mirror (reflection) plane perpendicular to the axis of rotation. Cnv is Cn with the addition of n mirror planes parallel to the axis of rotation.
• S2n (for Spiegel, German for mirror) denotes a group that contains only a 2n-fold rotation-reflection axis.
• Dn (for dihedral, or two-sided) indicates that the group has an n-fold rotation axis plus n twofold axes perpendicular to that axis. Dnh has, in addition, a mirror plane perpendicular to the n-fold axis. Dnd has, in addition to the elements of Dn, mirror planes parallel to the n-fold axis.
• The letter T (for tetrahedron) indicates that the group has the symmetry of a tetrahedron. Td includes improper rotation operations, T excludes improper rotation operations, and Th is 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 (Oh) or without (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
Cn C1 C2 C3 C4 C6
Cnv C1v=C1h C2v C3v C4v C6v
Cnh C1h C2h C3h C4h C6h
Dn D1=C2 D2 D3 D4 D6
Dnh D1h=C2v D2h D3h D4h D6h
Dnd D1d=C2h D2d D3d D4d D6d
S2n S2 S4 S6 S8 S12

D4d and D6d are actually forbidden because they contain improper rotations with n=8 and 12 respectively. The 27 point groups in the table plus T, Td, Th, O and Oh 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

Class Group names
Cubic 23 m3 432 43m m3m
Hexagonal 6 6 6m 622 6mm 6m2 6mmm
Trigonal 3 3 32 3m 3m
Tetragonal 4 4 4m 422 4mm 42m 4mmm
Orthorhombic 222 mm2 mmm
Monoclinic 2 2m 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.)

The correspondence between different notations

Crystal system Hermann-Mauguin Shubnikov[1] Schoenflies Orbifold Coxeter Order
(full) (short)
Triclinic 1 1 ${\displaystyle 1\ }$ C1 11 [ ]+ 1
1 1 ${\displaystyle {\tilde {2}}}$ Ci = S2 × [2+,2+] 2
Monoclinic 2 2 ${\displaystyle 2\ }$ C2 22 [2]+ 2
m m ${\displaystyle m\ }$ Cs = C1h * [ ] 2
${\displaystyle {\tfrac {2}{m}}}$ 2/m ${\displaystyle 2:m\ }$ C2h 2* [2,2+] 4
Orthorhombic 222 222 ${\displaystyle 2:2\ }$ D2 = V 222 [2,2]+ 4
mm2 mm2 ${\displaystyle 2\cdot m\ }$ C2v *22 [2] 4
${\displaystyle {\tfrac {2}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}}$ mmm ${\displaystyle m\cdot 2:m\ }$ D2h = Vh *222 [2,2] 8
Tetragonal 4 4 ${\displaystyle 4\ }$ C4 44 [4]+ 4
4 4 ${\displaystyle {\tilde {4}}}$ S4 [2+,4+] 4
${\displaystyle {\tfrac {4}{m}}}$ 4/m ${\displaystyle 4:m\ }$ C4h 4* [2,4+] 8
422 422 ${\displaystyle 4:2\ }$ D4 422 [4,2]+ 8
4mm 4mm ${\displaystyle 4\cdot m\ }$ C4v *44 [4] 8
42m 42m ${\displaystyle {\tilde {4}}\cdot m}$ D2d = Vd 2*2 [2+,4] 8
${\displaystyle {\tfrac {4}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}}$ 4/mmm ${\displaystyle m\cdot 4:m\ }$ D4h *422 [4,2] 16
Trigonal 3 3 ${\displaystyle 3\ }$ C3 33 [3]+ 3
3 3 ${\displaystyle {\tilde {6}}}$ C3i = S6 [2+,6+] 6
32 32 ${\displaystyle 3:2\ }$ D3 322 [3,2]+ 6
3m 3m ${\displaystyle 3\cdot m\ }$ C3v *33 [3] 6
3${\displaystyle {\tfrac {2}{m}}}$ 3m ${\displaystyle {\tilde {6}}\cdot m}$ D3d 2*3 [2+,6] 12
Hexagonal 6 6 ${\displaystyle 6\ }$ C6 66 [6]+ 6
6 6 ${\displaystyle 3:m\ }$ C3h 3* [2,3+] 6
${\displaystyle {\tfrac {6}{m}}}$ 6/m ${\displaystyle 6:m\ }$ C6h 6* [2,6+] 12
622 622 ${\displaystyle 6:2\ }$ D6 622 [6,2]+ 12
6mm 6mm ${\displaystyle 6\cdot m\ }$ C6v *66 [6] 12
6m2 6m2 ${\displaystyle m\cdot 3:m\ }$ D3h *322 [3,2] 12
${\displaystyle {\tfrac {6}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}}$ 6/mmm ${\displaystyle m\cdot 6:m\ }$ D6h *622 [6,2] 24
Cubic 23 23 ${\displaystyle 3/2\ }$ T 332 [3,3]+ 12
${\displaystyle {\tfrac {2}{m}}}$3 m3 ${\displaystyle {\tilde {6}}/2}$ Th 3*2 [3+,4] 24
432 432 ${\displaystyle 3/4\ }$ O 432 [4,3]+ 24
43m 43m ${\displaystyle 3/{\tilde {4}}}$ Td *332 [3,3] 24
${\displaystyle {\tfrac {4}{m}}}$3${\displaystyle {\tfrac {2}{m}}}$ m3m ${\displaystyle {\tilde {6}}/4}$ Oh *432 [4,3] 48