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.

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 article: Schoenflies notation

Further information: Point groups in three dimensions

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

• C_n (for cyclic) indicates that the group has an n-fold 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 n mirror planes parallel to the axis of rotation.
• S_2n (for Spiegel, German for mirror) denotes a group with only a 2n-fold rotation-reflection axis.
• D_n (for dihedral, or two-sided) indicates that the group has an n-fold rotation axis plus n twofold axes perpendicular to that axis. D_nh has, in addition, a mirror plane perpendicular to the n-fold axis. D_nd has, in addition to the elements of D_n, mirror planes parallel to the n-fold axis.
• The letter T (for tetrahedron) indicates that the group has the symmetry of a tetrahedron. T_d includes improper rotation operations, T excludes improper rotation operations, and T_h 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 (O_h) 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

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

Main article: 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	6⁄m	622	6mm	6m2	6/mmm
Trigonal	3	3		32	3m	3m
Tetragonal	4	4	4⁄m	422	4mm	42m	4/mmm
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.)

The correspondence between different notations

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

Isomorphisms

See also: Crystal structure § Crystal systems

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 Z₂. 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	Z1	$G_{1}^{1}$
1	Ci = S2	2	Z2	$G_{2}^{1}$
2	C2	2
m	Cs = C1h	2
3	C3	3	Z3	$G_{3}^{1}$
4	C4	4	Z4	$G_{4}^{1}$
4	S4	4
2/m	C2h	4	D2 = Z2 × Z2	$G_{4}^{2}$
222	D2 = V	4
mm2	C2v	4
3	C3i = S6	6	Z6	$G_{6}^{1}$
6	C6	6
6	C3h	6
32	D3	6	D3	$G_{6}^{2}$
3m	C3v	6
mmm	D2h = Vh	8	D2 × Z2	$G_{8}^{3}$
4/m	C4h	8	Z4 × Z2	$G_{8}^{2}$
422	D4	8	D4	$G_{8}^{4}$
4mm	C4v	8
42m	D2d = Vd	8
6/m	C6h	12	Z6 × Z2	$G_{12}^{2}$
23	T	12	A4	$G_{12}^{5}$
3m	D3d	12	D6	$G_{12}^{3}$
622	D6	12
6mm	C6v	12
6m2	D3h	12
4/mmm	D4h	16	D4 × Z2	$G_{16}^{9}$
6/mmm	D6h	24	D6 × Z2	$G_{24}^{5}$
m3	Th	24	A4 × Z2	$G_{24}^{10}$
432	O	24	S4	$G_{24}^{7}$
43m	Td	24
m3m	Oh	48	S4 × Z2	$G_{48}^{7}$

This table makes use of cyclic groups (Z₁, Z₂, Z₃, Z₄, Z₆), 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.

See also

• Molecular symmetry
• Point group
• Space group
• Point groups in three dimensions
• Crystal system

External links

• Point-group symbols in International Tables for Crystallography (2006). Vol. A, ch. 12.1, pp. 818-820
• Names and symbols of the 32 crystal classes in International Tables for Crystallography (2006). Vol. A, ch. 10.1, p. 794
• Pictorial overview of the 32 groups