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, theoretically, each of the operations could be applied to a particular crystal and the result would have the same kinds of atoms in the same positions as before, perhaps after a translation. (It doesn't matter whether any actual crystal exists having the necessary structure.) Because translations may be needed, there is not necessarily any point in the crystal where applying the point group would preserve the structure. For a periodic crystal (as opposed to a quasicrystal), the group must maintain the three-dimensional translational symmetry that defines crystallinity. The geometric properties of a crystal must look exactly the same before and after applying any of the operations in its point group. 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.

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

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

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 23m343243mm3m
Hexagonal 666m6226mm6m26mmm
Trigonal 33323m3m
Tetragonal 444m4224mm42m4mmm
Orthorhombic 222mm2mmm
Monoclinic 22mm
Triclinic 11 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 11$1\$ C111[ ]+1
11${\tilde {2}}$ Ci = S2×[2+,2+]2
Monoclinic 22$2\$ C222+2
mm$m\$ Cs = C1h*[ ]2
${\tfrac {2}{m}}$ 2/m$2:m\$ C2h2*[2,2+]4
Orthorhombic 222222$2:2\$ D2 = V222[2,2]+4
mm2mm2$2\cdot m\$ C2v*224
${\tfrac {2}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$ mmm$m\cdot 2:m\$ D2h = Vh*222[2,2]8
Tetragonal 44$4\$ C444+4
44${\tilde {4}}$ S4[2+,4+]4
${\tfrac {4}{m}}$ 4/m$4:m\$ C4h4*[2,4+]8
422422$4:2\$ D4422[4,2]+8
4mm4mm$4\cdot m\$ C4v*448
42m42m${\tilde {4}}\cdot m$ D2d = Vd2*2[2+,4]8
${\tfrac {4}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$ 4/mmm$m\cdot 4:m\$ D4h*422[4,2]16
Trigonal 33$3\$ C333+3
33${\tilde {6}}$ C3i = S6[2+,6+]6
3232$3:2\$ D3322[3,2]+6
3m3m$3\cdot m\$ C3v*336
3${\tfrac {2}{m}}$ 3m${\tilde {6}}\cdot m$ D3d2*3[2+,6]12
Hexagonal 66$6\$ C666+6
66$3:m\$ C3h3*[2,3+]6
${\tfrac {6}{m}}$ 6/m$6:m\$ C6h6*[2,6+]12
622622$6:2\$ D6622[6,2]+12
6mm6mm$6\cdot m\$ C6v*6612
6m26m2$m\cdot 3:m\$ D3h*322[3,2]12
${\tfrac {6}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$ 6/mmm$m\cdot 6:m\$ D6h*622[6,2]24
Cubic 2323$3/2\$ T332[3,3]+12
${\tfrac {2}{m}}$ 3m3${\tilde {6}}/2$ Th3*2[3+,4]24
432432$3/4\$ O432[4,3]+24
43m43m$3/{\tilde {4}}$ Td*332[3,3]24
${\tfrac {4}{m}}$ 3${\tfrac {2}{m}}$ m3m${\tilde {6}}/4$ Oh*432[4,3]48

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.