Crystallographic point group

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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 (perhaps followed by a translation) 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 many crystals in the 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 moves each atom to the location of another atom of the same kind, 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.

Notation

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.

Schoenflies notation

Main page: Physics: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 with 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

Main page: 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

System Group names Cubic 23 m3 432 43m m3m
Hexagonal 6 6 6m 622 6mm 6m2 6/mmm
Trigonal 3 3 32 3m 3m
Tetragonal 4 4 4m 422 4mm 42m 4/mmm
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 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
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  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 $\displaystyle{ \tilde{4} }$ 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  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 $\displaystyle{ \tilde{6} }$ C3i = S6 [2+,6+] 6
32 32 $\displaystyle{ 3:2\ }$ D3 322 [3,2]+ 6
3m 3m $\displaystyle{ 3 \cdot m\ }$ C3v *33  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 $\displaystyle{ 3:m\ }$ 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  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

Isomorphisms

Many of the crystallographic point groups share the same internal structure. For example, the point groups 1, 2, and m contain different geometric symmetry operations, (inversion, rotation, and reflection, respectively) but all share the structure of the cyclic group C2. All isomorphic groups are of the same order, but not all groups of the same order are isomorphic. The point groups which are isomorphic are shown in the following table:

Hermann-Mauguin Schoenflies Order Abstract group
1 C1 1 C1 $\displaystyle{ G_1^1 }$
1 Ci = S2 2 C2 $\displaystyle{ G_2^1 }$
2 C2 2
m Cs = C1h 2
3 C3 3 C3 $\displaystyle{ G_3^1 }$
4 C4 4 C4 $\displaystyle{ G_4^1 }$
4 S4 4
2/m  C2h 4 D2 = C2 × C2 $\displaystyle{ G_4^2 }$
222 D2 = V 4
mm2 C2v  4
3 C3i = S6 6 C6 $\displaystyle{ G_6^1 }$
6 C6 6
6 C3h 6
32 D3 6 D3 $\displaystyle{ G_6^2 }$
3m C3v 6
mmm D2h = Vh 8 D2 × C2 $\displaystyle{ G_8^3 }$
4/m C4h 8 C4 × C2 $\displaystyle{ G_8^2 }$
422 D4 8 D4 $\displaystyle{ G_8^4 }$
4mm C4v 8
42m D2d = Vd 8
6/m C6h 12 C6 × C2 $\displaystyle{ G_{12}^2 }$
23 T 12 A4 $\displaystyle{ G_{12}^5 }$
3m D3d 12 D6 $\displaystyle{ G_{12}^3 }$
622 D6 12
6mm C6v 12
6m2 D3h 12
4/mmm D4h 16 D4 × C2 $\displaystyle{ G_{16}^9 }$
6/mmm D6h 24 D6 × C2 $\displaystyle{ G_{24}^5 }$
m3 Th 24 A4 × C2 $\displaystyle{ G_{24}^{10} }$
432 O   24 S4 $\displaystyle{ G_{24}^{7} }$
43m Td 24
m3m Oh 48 S4 × C2 $\displaystyle{ G_{48}^7 }$

This table makes use of cyclic groups (C1, C2, C3, C4, C6), dihedral groups (D2, D3, D4, D6), one of the alternating groups (A4), and one of the symmetric groups (S4). Here the symbol " × " indicates a direct product.

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

1. Leave out the Bravais type
2. 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)
3. Axes of rotation, rotoinversion axes and mirror planes remain unchanged.