2.4: Crystallographic Point Groups

The proper and improper rotations that are compatible with translational periodicity generate 32 crystallographic point groups among the 7 crystal classes:

Crystal System	Notation		Lattice Directions	Order	Characteristics
Triclinic	\mathcal{C}_{i}	\overline{1}		2	Centrosymmetric
	\mathcal{C}_{1}	1		1	Polar, Chiral
Monoclinic	\mathcal{C}_{2h}	2/m	[010] or [001]	4	Centrosymmetric
	\mathcal{C}_{2}	2		2	Polar, Chiral
	\mathcal{C}_{s}	m		2	Polar
Orthorhombic	\mathcal{D}_{2h}	mmm	[100][010][001]	8	Centrosymmetric
	\mathcal{D}_{2}	222		4	Chiral
	\mathcal{C}_{2v}	mm2, m2m, 2mm		4	Polar
Tetragonal	\mathcal{D}_{4h}	4/mmm	[001]{100}{110}	16	Centrosymmetric
	\mathcal{D}_{4}	422		8	Chiral
	\mathcal{C}_{4v}	4mm		8	Polar
	\mathcal{D}_{2d}	\overline{4}2m, \overline{4}m2		8
	\mathcal{C}_{4h}	4/m	[001]	8	Centrosymmetric
	\mathcal{S}_{4}	\overline{4}		4
	\mathcal{C}_{4}	4		4	Polar, Chiral
Trigonal	\mathcal{D}_{3d}	\overline{3}m, \overline{3}m1, \overline{3}1m	[001]{100}{\overline{1}10}	12	Centrosymmetric
	\mathcal{D}_{3}	32, 321, 312		6	Chiral
	\mathcal{C}_{3v}	3m, 3m1, 31m		6	Polar
	\mathcal{S}_{6}	\overline{3}	[001]	6	Centrosymmetric
	\mathcal{C}_{3}	3		3	Polar, Chiral
Hexagonal	\mathcal{D}_{6h}	6/mmm	[001]{100}{\overline{1}10}	24	Centrosymmetric
	\mathcal{D}_{6}	622		12	Chiral
	\mathcal{C}_{6v}	6mm		12	Polar
	\mathcal{D}_{3h}	\overline{6}2m, \overline{6}m2		12
	\mathcal{C}_{6h}	6/m	[001]	12	Centrosymmetric
	\mathcal{C}_{3h}	\overline{6}		6
	\mathcal{C}_{6}	6		6	Polar, Chiral
Cubic	\mathcal{O}_{h}	m\overline{3}m	{100}{111}{110}	48	Centrosymmetric
	\mathcal{O}	432		24	Chiral
	\mathcal{T}_{d}	\overline{4}3m		24
	\mathcal{T}_{h}	\overline{m}3	{100}{111}	24	Centrosymmetric
	\mathcal{T}	23		12	Chiral

When a crystal grows, its shape will adopt one of these 32 point symmetries and its physical properties will conform to the same point symmetry. For each crystallographic point group, the table includes (i) the Schönflies and International symbols; (ii) lattice directions corresponding to the order of symmetry operations in the International symbol; (iii) the order, which equals the number of symmetry operations in the set; and (iv) important characteristics of the group.

Lattice Directions

As noted above, the International notation is significant for crystals in which symmetry operations have definite orientations with respect to the unit cell parameters. The list below elaborates on the order of the different symbols in the International notation relative to unit cell directions:

Triclinic	One symbol; no directions are necessary.
Monoclinic	One symbol for 2-fold axis along b [010].
Orthorhombic	1^st symbol for 2-fold axis along a [100]. 2^nd symbol for 2-fold axis along b [010]. 3^rd symbol for 2-fold axis along c [001].
Tetragonal	1^st symbol for 4-fold axis along c [001]. 2^nd symbol for 2-fold axes along a and b {100} = [100] and [010]. 3^rd symbol for 2-fold axes along a+b and a–b {110} = [110] and [1\overline{1}0].
Trigonal	1^st symbol for 3-fold axis along c [001]. 2^nd symbol for 2-fold axes along a and b {100} = [100] and [010]. 3^rd symbol for 2-fold axes along lines 30° from a and b, i.e., −a+b, 2a+b, a+2b {\overline{1}10} = [\overline{1}10], [210], [120]. (NOTE: \overline{1}, read “bar one”, means –1.)
Hexagonal	1^st symbol for 6-fold axis along c [001]. 2^nd symbol for 2-fold axes along a and b {100} = [100] and [010]. 3^rd symbol for 2-fold axes along lines 30° from a and b, i.e., −a+b, 2a+b, a+2b {\overline{1}10} = [\overline{1}10], [210], [120].
Cubic	1^st symbol for 4- or 2-fold axes along a, b, and c {100} = [100], [010], [001]. 2^nd symbol for 3-fold axes along a+b+c, −a−b+c, −a+b−c, a−b−c (body-diagonals) {111} = [111], [\overline{1}\overline{1}1], [\overline{1}1\overline{1}],[1\overline{1}\overline{1}\rbrack. 3^rd symbol for symmetry axes along a+b, a−b, b+c, b−c, c+a, c−a (face-diagonals) {110} = [110], [1\overline{1}0], [011], [01\overline{1}], [101], [\overline{1}01].

For some point groups, there are multiple different International symbols arising from how the different rotation axes are oriented in space, such as \(\mathcal{D}_{2d} = \overline{4}2m\) or \(\overline{4}m2\). This point group belongs to the tetragonal crystal system because it contains a 4-fold roto-inversion or roto-reflection axis that is parallel with the c-axis. The second symbol refers to the equivalent a- and b-axes and the third part corresponds to the equivalent a+b- and a–b-directions. As a second example, in the trigonal system, there are three different International symbols for \(\mathcal{C}_{3v}\): \(3m1\) and \(31m\) differ in the orientations of the vertical mirror planes with respect to the unit cell directions a and b; and \(3m\) would be applied to a randomly oriented molecule because it is not necessary to specify any orientation of mirror planes.

Characteristics

The first row for each crystal class is the point group with the highest order, called the holohedral group of that class. Each of these seven groups is the point group of the corresponding lattice and is centrosymmetric, which means they include inversion symmetry. There are four additional centrosymmetric point groups within the tetragonal, trigonal, hexagonal, and cubic systems. The 11 centrosymmetric groups, called Laue groups or Laue classes, are important because coherent diffraction patterns of crystals always appear centrosymmetric about an origin point, which coincides with the undiffracted incident beam of X-rays, neutrons, or electrons. Therefore, the point symmetry of every diffraction pattern will belong to one of these 11 point groups. The remaining 21 noncentrosymmetric crystallographic point groups have their own significant features, such as being:

• Chiral (Enantiomorphic): These 11 groups have no roto-inversions (\(\overline{n}\)) or roto-reflections (S_n). The remaining 21 groups are achiral.
• Polar: These 10 groups do not have a unique origin point, but they do have a central axis. The five cyclic crystallographic point groups 1, 2, 3, 4, and 6 are both chiral and polar.

There are five noncentrosymmetric point groups that are neither chiral nor polar: \(\overline{4}\), \(\overline{4}2m\), \(\overline{6}\), \(\overline{6}m2\), and \(\overline{4}3m\).