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 true 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.

Schönflies notation

Main article: Schoenflies notation

For more details on this topic, see Point groups in three dimensions.

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

• 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).
• The letter T (for tetrahedron) indicates that the group has the symmetry of a tetrahedron. T_d includes improper operations, T excludes improper operations, and T_h is T with the addition of an inversion.
• The letter I (for icosahedron) indicates that the group has the symmetry of an icosahedron (or dodecahedron), either with (I_h) or without (I) improper operations.
• 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 a mirror plane parallel to the axis of rotation.
• S_2n (for Spiegel, German for mirror) denotes a group that contains 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 a twofold axis perpendicular to that axis. D_nh has, in addition, a mirror plane perpendicular to the n-fold axis. D_nv has, in addition to the elements of D_n, mirror planes parallel to the n-fold axis.

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

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

1	1
2		2⁄m	222	m	mm2	mmm
3	3		32	3m	3m
4	4	4⁄m	422	4mm	42m	4⁄mmm
6	6	6⁄m	622	6mm	62m	6⁄mmm
23	m3	432	43m	m3m

The correspondence between the four notations is:

Hermann-Mauguin	Schoenflies	Orbifold	Coxeter	Order
1	C1	11	[ ]+	1
2	C2	22	[2]+	2
3	C3	33	[3]+	3
4	C4	44	[4]+	4
6	C6	66	[6]+	6
mm2	C2v	*22	[2]	4
3m	C3v	*33	[3]	6
4mm	C4v	*44	[4]	8
6mm	C6v	*66	[6]	12
m	Cs = C1h	*11	[ ]	2
2/m	C2h	2*	[2,2+]	4
6	C3h	3*	[2,3+]	6
4/m	C4h	4*	[2,4+]	8
6/m	C6h	6*	[2,6+]	12
1	Ci = S2	1x	[1+,2+]	2
4	S4	2x	[2+,4+]	4
3	S6	3x	[2+,6+]	6
222	D2	222	[2,2]+	4
32	D3	322	[3,2]+	6
422	D4	422	[4,2]+	8
622	D6	622	[6,2]+	12
mmm	D2h	*222	[2,2]	8
62m	D3h	*322	[3,2]	12
4/m mm	D4h	*422	[4,2]	16
6/m mm	D6h	*622	[6,2]	24
42m	D2d	2*2	[2+,4]	8
3m	D3d	2*3	[2+,6]	12
23	T	332	[3,3]+	12
43m	Td	*332	[3,3]	24
m3	Th	3*2	[3+,4]	24
432	O	432	[4,3]+	24
m3m	Oh	*432	[4,3]	48

See also

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

External links

• Pictorial overview of the 32 groups
• Point Groups - Flow Chart