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 (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 degrees around an axis that is perpendicular to one of the faces of the cube 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

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
C_n	C₁	C₂	C₃	C₄	C₆
C_nv	C_1v=C_1h	C_2v	C_3v	C_4v	C_6v
C_nh	C_1h	C_2h	C_3h	C_4h	C_6h
D_n	D₁=C₂	D₂	D₃	D₄	D₆
D_nh	D_1h=C_2v	D_2h	D_3h	D_4h	D_6h
D_nd	D_1d=C_2h	D_2d	D_3d	D_4d	D_6d
S_2n	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

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.)

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

Crystal family	Crystal system	Group names
Cubic		23	m3		432	43m	m3m
Hexagonal	Hexagonal	6	6	⁶⁄_m	622	6mm	6m2	6/mmm
Hexagonal	Trigonal (Rhombohedral)	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

The correspondence between different notations

Crystal family	Crystal system	Hermann-Mauguin		Shubnikov[1]	Schoenflies	Orbifold	Coxeter	Order
Crystal family	Crystal system	(full)	(short)	Shubnikov[1]	Schoenflies	Orbifold	Coxeter	Order
Triclinic		1	1	$1\$	C₁	11	[ ]⁺	1
Triclinic		1	1	${\tilde {2}}$	C_i = S₂	×	[2⁺,2⁺]	2
Monoclinic		2	2	$2\$	C₂	22	[2]⁺	2
		m	m	$m\$	C_s = C_1h	*	[ ]	2
		${\tfrac {2}{m}}$	2/m	$2:m\$	C_2h	2*	[2,2⁺]	4
Orthorhombic		222	222	$2:2\$	D₂ = V	222	[2,2]⁺	4
		mm2	mm2	$2\cdot m\$	C_2v	*22	[2]	4
		${\tfrac {2}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$	mmm	$m\cdot 2:m\$	D_2h = V_h	*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\$	C_4h	4*	[2,4⁺]	8
		422	422	$4:2\$	D₄	422	[4,2]⁺	8
		4mm	4mm	$4\cdot m\$	C_4v	*44	[4]	8
		42m	42m	${\tilde {4}}\cdot m$	D_2d = V_d	2*2	[2⁺,4]	8
		${\tfrac {4}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$	4/mmm	$m\cdot 4:m\$	D_4h	*422	[4,2]	16
Hexagonal	Trigonal	3	3	$3\$	C₃	33	[3]⁺	3
		3	3	${\tilde {6}}$	C_3i = S₆	3×	[2⁺,6⁺]	6
		32	32	$3:2\$	D₃	322	[3,2]⁺	6
		3m	3m	$3\cdot m\$	C_3v	*33	[3]	6
		3 ${\tfrac {2}{m}}$	3m	${\tilde {6}}\cdot m$	D_3d	2*3	[2⁺,6]	12
	Hexagonal	6	6	$6\$	C₆	66	[6]⁺	6
		6	6	$3:m\$	C_3h	3*	[2,3⁺]	6
		${\tfrac {6}{m}}$	6/m	$6:m\$	C_6h	6*	[2,6⁺]	12
		622	622	$6:2\$	D₆	622	[6,2]⁺	12
		6mm	6mm	$6\cdot m\$	C_6v	*66	[6]	12
		6m2	6m2	$m\cdot 3:m\$	D_3h	*322	[3,2]	12
		${\tfrac {6}{m}}{\tfrac {2}{m}}{\tfrac {2}{m}}$	6/mmm	$m\cdot 6:m\$	D_6h	*622	[6,2]	24
Cubic		23	23	$3/2\$	T	332	[3,3]⁺	12
		${\tfrac {2}{m}}$ 3	m3	${\tilde {6}}/2$	T_h	3*2	[3⁺,4]	24
		432	432	$3/4\$	O	432	[4,3]⁺	24
		43m	43m	$3/{\tilde {4}}$	T_d	*332	[3,3]	24
		${\tfrac {4}{m}}$ 3 ${\tfrac {2}{m}}$	m3m	${\tilde {6}}/4$	O_h	*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:[2]

Hermann-Mauguin	Schoenflies	Order	Abstract group
1	C₁	1	C₁	$G_{1}^{1}$
1	C_i = S₂	2	C₂	$G_{2}^{1}$
2	C₂	2
m	C_s = C_1h	2
3	C₃	3	C₃	$G_{3}^{1}$
4	C₄	4	C₄	$G_{4}^{1}$
4	S₄	4	C₄	$G_{4}^{1}$
2/m	C_2h	4	D₂ = C₂ × C₂	$G_{4}^{2}$
222	D₂ = V	4
mm2	C_2v	4
3	C_3i = S₆	6	C₆	$G_{6}^{1}$
6	C₆	6
6	C_3h	6
32	D₃	6	D₃	$G_{6}^{2}$
3m	C_3v	6	D₃	$G_{6}^{2}$
mmm	D_2h = V_h	8	D₂ × C₂	$G_{8}^{3}$
4/m	C_4h	8	C₄ × C₂	$G_{8}^{2}$
422	D₄	8	D₄	$G_{8}^{4}$
4mm	C_4v	8
42m	D_2d = V_d	8
6/m	C_6h	12	C₆ × C₂	$G_{12}^{2}$
23	T	12	A₄	$G_{12}^{5}$
3m	D_3d	12	D₆	$G_{12}^{3}$
622	D₆	12
6mm	C_6v	12
6m2	D_3h	12
4/mmm	D_4h	16	D₄ × C₂	$G_{16}^{9}$
6/mmm	D_6h	24	D₆ × C₂	$G_{24}^{5}$
m3	T_h	24	A₄ × C₂	$G_{24}^{10}$
432	O	24	S₄	$G_{24}^{7}$
43m	T_d	24	S₄	$G_{24}^{7}$
m3m	O_h	48	S₄ × C₂	$G_{48}^{7}$

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

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

References

"(International Tables) Abstract". Archived from the original on 2013-07-04. Retrieved 2011-11-25.
Novak, I (1995-07-18). "Molecular isomorphism". European Journal of Physics. IOP Publishing. 16 (4): 151–153. Bibcode:1995EJPh...16..151N. doi:10.1088/0143-0807/16/4/001. ISSN 0143-0807. S2CID 250887121.

External links

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[1] "(International Tables) Abstract". Archived from the original on 2013-07-04. Retrieved 2011-11-25.

[2] Novak, I (1995-07-18). "Molecular isomorphism". European Journal of Physics. IOP Publishing. 16 (4): 151–153. Bibcode:1995EJPh...16..151N. doi:10.1088/0143-0807/16/4/001. ISSN 0143-0807. S2CID 250887121.

Crystal systems
Bravais lattice Crystallographic point group
Seven 3D systems	triclinic (anorthic) monoclinic orthorhombic tetragonal trigonal & hexagonal cubic (isometric)
Four 2D systems	oblique rectangular square hexagonal