Practical Electron Microscopy and Database

Space groups represent the ways that the macroscopic and microscopic symmetry elements (operations) can be self-consistently arranged in space. There are totally 230 space groups. The space groups add the centering information and microscopic elements to the point groups. Figure 4549a schematically shows the relationship between the 7 crystal systems, 14 Bravais Lattices, 32 point groups, and 230 space groups. The seven holohedric point groups F⁰ and all their subgroups (merohedral groups) form the 32 crystallographic point groups.

Figure 4549a. The relationship between the 7 crystal systems,
14 Bravais Lattices, 32 point groups, and 230 space groups.

Nicolaus Steno first showed in 1669 that the angles between the faces of crystals are constant, independently of the regularity of a given crystal morphology. The analysis of crystal morphologies led to the formulation of a complete set of 32 symmetry classes, called “point groups” as shown in Table 4549a. The morphologies of all crystals obey the 32 point groups. Possible symmetry elements are 1-, 2-, 3-, 4-, and 6-fold rotations, mirror plane m, inversion center and a combination of rotation axis with inversion center (inversion axis). Table 4549b shows the relation between three-dimensional crystal families, crystal systems, and lattice systems. 32 point groups indicate that there are only 32 possible combinations of symmetry elements. Furthermore, Table 4549c lists the symmetry groups for the crystal systems together with the point groups with Hermann–Mauguin notation.

Table 4549a. 32 point groups (Notation: Schönflies).

Table 4549b. The relation between three-dimensional crystal families, crystal systems, and lattice systems.

Table 4549c. Symmetry groups for the crystal systems.

System	Essential symmetry	Lattice symmetry	Diffraction (Laue) symmetry	Point Groups (Hermann–Mauguin notation)
				X		X⊥m (X/m)	X||m (Xm)	()	X⊥2	X⊥m||m
Triclinic	None			1	-1	--	--	--	--	--
Monoclinic		2/m	2/m	2	m	2/m	--	--	--	--
Orthorhombic	222 or 2mm	mmm	mmm	--	--	--	mm2	--	222	mmm
Tetragonal		4/mmm	4/m	4	-4	4/m	--	--	--	--
			4/mmm	--	--	--	4mm	42m	422	4/mmm
Trigonal			3	3	-3	--	--	--	--	--
			-3m1	--	--	--	3m	-3m1	321	--
			-31m	--	--	--		-31m	312	--
Hexagonal		6/mmm	6/m	6	-6	6/m	--	--	--	--
			6/mmm	--	--	--	6mm	-62m	622	6/mmm
Cubic	23	m3m	m-3	23	m-3	--	--	--	--	--
			m-3m	--	--	--	--	4-3m	432	m-3m
*  X: Major X-fold rotation axis alone.    : Inversion axis alone, namely, the major X-fold inversion axes.     X/m: Rotation axis with a symmetry plane normal to it.     Xm: Rotation axis with a symmetry plane that is not normal to it (usually a vertical symmetry plane).    : Inversion axis with a symmetry plane not normal to it.     X⊥2: Rotation axis with a diad normal to it (The two-fold rotation axis is perpendicular to the major axis).     X/mm: Rotation axis with a symmetry plane normal to it and another not so.

Of the 32 point groups, 11 crystal classes are centrosymmetric and thus possess no polar properties. Of the remaining 21 non-centrosymmetric classes, 20 classes show electrical polarity when a stress is applied.

Figure 4549b shows the schematic illustration of the different classes of crystal systems and their properties. In ferroelectric substances, the number of distinct spontaneous polarization directions depends on its point group symmetry.

Seven crystal systems
32 point groups
No center of symmetry (21 point groups)	Center of symmetry (11 point groups)
Piezoelectric (20 point groups)	Non-piezoelectric (1 point groups)
Polar or pyroelectric (10 point groups)	Non-pyroelectric
Ferroelectric (Only if polarization Is reversible)	Non-ferroelectric

Figure 4549b. Schematic illustration of the different classes of crystal systems and their properties.