Crystal system

In crystallography, a crystal system is a set of point groups (a group of geometric symmetries with at least one fixed point). A lattice system is a set of Bravais lattices (an infinite array of discrete points). Space groups (symmetry groups of a configuration in space) are classified into crystal systems according to their point groups, and into lattice systems according to their Bravais lattices. Crystal systems that have space groups assigned to a common lattice system are combined into a crystal family.

The seven crystal systems are triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. Informally, two crystals are in the same crystal system if they have similar symmetries (though there are many exceptions).

Overview

Crystals can be classified in three ways: lattice systems, crystal systems and crystal families. The various classifications are often confused: in particular the trigonal crystal system is often confused with the rhombohedral lattice system, and the term "crystal system" is sometimes used to mean "lattice system" or "crystal family".

Crystal families (6 in three dimensions)
A crystal family is determined by lattices and point groups. It is formed by combining crystal systems that have space groups assigned to a common lattice system. In three dimensions, the hexagonal and trigonal crystal systems are combined into one hexagonal crystal family.
Crystal systems (7 in three dimensions)	Lattice systems (7 in three dimensions)
A crystal system is a set of point groups in which the point groups themselves and their corresponding space groups are assigned to a lattice system. Of the 32 crystallographic point groups that exist in three dimensions, most are assigned to only one lattice system, in which case both the crystal and lattice systems have the same name. However, 5 point groups have 7 corresponding space groups assigned to the rhombohedral lattice system and 18 corresponding space groups assigned to the hexagonal lattice system. These point groups are assigned to the trigonal crystal system.	A lattice system is a group of lattices with the same set of lattice point groups. In three dimensions, the 14 Bravais lattices are grouped into seven lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic.

In three dimensions

Summary

Five of the crystal systems are essentially the same as five of the lattice systems. The hexagonal and trigonal crystal systems differ from the hexagonal and rhombohedral lattice systems. These are combined into the hexagonal crystal family.

The relation between three-dimensional crystal families, crystal systems and lattice systems is shown in the following table:

Crystal families	Point group classification		Space groups	Lattice classification
Crystal families	Crystal systems	Point groups	Space groups	Bravais lattices	Lattice systems
Triclinic	Triclinic	2	2	1	Triclinic
Monoclinic	Monoclinic	3	13	2	Monoclinic
Orthorhombic	Orthorhombic	3	59	4	Orthorhombic
Tetragonal	Tetragonal	7	68	2	Tetragonal
Hexagonal	Trigonal	5	7	1	Rhombohedral
	Trigonal	5	18	1	Hexagonal
	Hexagonal	7	27	1	Hexagonal
Cubic	Cubic	5	36	3	Cubic
Total		32	230	14	Total

Note: there is no "trigonal" lattice system. To avoid confusion of terminology, the term "trigonal lattice" is not used.

Crystal classes

The 7 crystal systems consist of 32 crystal classes (corresponding to the 32 crystallographic point groups) as shown in the following table below:

Crystal family	Crystal system	Required symmetries	32 crystal classes (Hermann-Mauguin)
Triclinic		None	1	1
Monoclinic		1 twofold axis of rotation or 1 mirror plane	2		²⁄_m		m
Orthorhombic		3 twofold axes of rotation or 1 twofold axis of rotation and 2 mirror planes				222		mm2	mmm
Tetragonal		1 fourfold axis of rotation	4	4	⁴⁄_m	422	4mm	42m	4/mmm
Hexagonal	Trigonal	1 threefold axis of rotation	3	3		32	3m	3m
Hexagonal	Hexagonal	1 sixfold axis of rotation	6	6	⁶⁄_m	622	6mm	6m2	6/mmm
Cubic		4 threefold axes of rotation	23	m3		432	43m	m3m

Bravais lattices

There are seven different kinds of lattice systems, and each kind of lattice system has four different kinds of centerings (primitive, base-centered, body-centered, face-centered). However, not all of the combinations are unique; some of the combinations are equivalent while other combinations are not possible due to symmetry reasons. This reduces the number of unique lattices to the 14 Bravais lattices.

The distribution of the 14 Bravais lattices into 7 lattice systems is given in the following table.

Crystal family	Lattice system	Point group (Hermann-Mauguin)	14 Bravais lattices
Crystal family	Lattice system	Point group (Hermann-Mauguin)	Primitive (P)	Base-centered (S)	Body-centered (I)	Face-centered (F)
Triclinic (a)		${\bar {1}}$	aP
Monoclinic (m)		$2/m$	mP	mS
Orthorhombic (o)		$mmm$	oP	oS	oI	oF
Tetragonal (t)		$4/mmm$	tP		tI
Hexagonal (h)	Rhombohedral	${\bar {3}}m$	hR
Hexagonal (h)	Hexagonal	$6/mmm$	hP
Cubic (c)		$m{\bar {3}}m$	cP		cI	cF

These lattices are classified by the space group of the lattice itself, viewed as a collection of points; there are 14 Bravais lattices in three dimensions; each belongs to one lattice system only. They^{[clarification needed]} represent the maximum symmetry a structure with the given translational symmetry can have.

All crystalline materials (not including quasicrystals) must, by definition, fit into one of these arrangements.

For convenience a Bravais lattice is depicted by a unit cell which is a factor 1, 2, 3, or 4 larger than the primitive cell. Depending on the symmetry of a crystal or other pattern, the fundamental domain is again smaller, up to a factor 48.

In other dimensions

Two-dimensional space

In two-dimensional space, the crystal families, crystal systems, and lattice systems are the same. There are four crystal families (oblique, rectangular, square, and hexagonal).^[1]^[2]

Crystal family	Crystallographic point groups	Plane groups	Bravais lattices
Oblique (monoclinic)	1, 2	2	mp
Rectangular (orthorhombic)	m, 2mm	7	op, oc
Square (tetragonal)	4, 4mm	3	tp
Hexagonal	3, 6, 3m, 6mm	5	hp
Total	10	17	5

Four-dimensional space

In four-dimensional space there are 23 crystal families, 33 crystal systems and 33 lattice systems. The relation between four-dimensional crystal families, crystal systems, and lattice systems is shown in the following table.^[3]^[4]

Enantiomorphic systems are marked with an asterisk. The number of enantiomorphic pairs is given in parentheses. Here the term "enantiomorphic" has a different meaning than in the table for three-dimensional crystal classes. The latter means, that enantiomorphic point groups describe chiral (enantiomorphic) structures. In the current table, "enantiomorphic" means that a group itself (considered as a geometric object) is enantiomorphic, like enantiomorphic pairs of three-dimensional space groups P3₁ and P3₂, P4₁22 and P4₃22. Starting from four-dimensional space, point groups also can be enantiomorphic in this sense.

Crystal families in 4D space
No. of crystal family	Crystal family	Point group classification		Space groups	Lattice classification
No. of crystal family	Crystal family	Crystal system	Point groups	Space groups	Bravais lattices	Lattice system
I	Hexaclinic	Hexaclinic	2	2	1	Hexaclinic P
II	Triclinic	Triclinic	3	13	2	Triclinic P, S
III	Diclinic	Diclinic	2	12	3	Diclinic P, S, D
IV	Monoclinic	Monoclinic	4	207	6	Monoclinic P, S, S, I, D, F
V	Orthogonal	Non-axial orthogonal	2	2	1	Orthogonal KU
		Non-axial orthogonal	2	112	8	Orthogonal P, S, I, Z, D, F, G, U
		Axial orthogonal	3	887	8	Orthogonal P, S, I, Z, D, F, G, U
VI	Tetragonal monoclinic	Tetragonal monoclinic	7	88	2	Tetragonal monoclinic P, I
VII	Hexagonal monoclinic	Trigonal monoclinic	5	9	1	Hexagonal monoclinic R
		Trigonal monoclinic	5	15	1	Hexagonal monoclinic P
		Hexagonal monoclinic	7	25	1	Hexagonal monoclinic P
VIII	Ditetragonal diclinic*	Ditetragonal diclinic*	1 (+1)	1 (+1)	1 (+1)	Ditetragonal diclinic P*
IX	Ditrigonal (dihexagonal) diclinic*	Ditrigonal diclinic*	2 (+2)	2 (+2)	1 (+1)	Ditrigonal diclinic P*
X	Tetragonal orthogonal	Inverse tetragonal orthogonal	5	7	1	Tetragonal orthogonal KG
		Inverse tetragonal orthogonal	5	351	5	Tetragonal orthogonal P, S, I, Z, G
		Proper tetragonal orthogonal	10	1312	5	Tetragonal orthogonal P, S, I, Z, G
XI	Hexagonal orthogonal	Trigonal orthogonal	10	81	2	Hexagonal orthogonal R, RS
		Trigonal orthogonal	10	150	2	Hexagonal orthogonal P, S
		Hexagonal orthogonal	12	240	2	Hexagonal orthogonal P, S
XII	Ditetragonal monoclinic*	Ditetragonal monoclinic*	1 (+1)	6 (+6)	3 (+3)	Ditetragonal monoclinic P, S, D*
XIII	Ditrigonal (dihexagonal) monoclinic*	Ditrigonal monoclinic*	2 (+2)	5 (+5)	2 (+2)	Ditrigonal monoclinic P, RR
XIV	Ditetragonal orthogonal	Crypto-ditetragonal orthogonal	5	10	1	Ditetragonal orthogonal D
		Crypto-ditetragonal orthogonal	5	165 (+2)	2	Ditetragonal orthogonal P, Z
		Ditetragonal orthogonal	6	127	2	Ditetragonal orthogonal P, Z
XV	Hexagonal tetragonal	Hexagonal tetragonal	22	108	1	Hexagonal tetragonal P
XVI	Dihexagonal orthogonal	Crypto-ditrigonal orthogonal*	4 (+4)	5 (+5)	1 (+1)	Dihexagonal orthogonal G*
		Crypto-ditrigonal orthogonal*	4 (+4)	5 (+5)	1	Dihexagonal orthogonal P
		Dihexagonal orthogonal	11	20
		Ditrigonal orthogonal	11	41
		Ditrigonal orthogonal	11	16	1	Dihexagonal orthogonal RR
XVII	Cubic orthogonal	Simple cubic orthogonal	5	9	1	Cubic orthogonal KU
		Simple cubic orthogonal	5	96	5	Cubic orthogonal P, I, Z, F, U
		Complex cubic orthogonal	11	366	5	Cubic orthogonal P, I, Z, F, U
XVIII	Octagonal*	Octagonal*	2 (+2)	3 (+3)	1 (+1)	Octagonal P*
XIX	Decagonal	Decagonal	4	5	1	Decagonal P
XX	Dodecagonal*	Dodecagonal*	2 (+2)	2 (+2)	1 (+1)	Dodecagonal P*
XXI	Diisohexagonal orthogonal	Simple diisohexagonal orthogonal	9 (+2)	19 (+5)	1	Diisohexagonal orthogonal RR
		Simple diisohexagonal orthogonal	9 (+2)	19 (+3)	1	Diisohexagonal orthogonal P
		Complex diisohexagonal orthogonal	13 (+8)	15 (+9)	1	Diisohexagonal orthogonal P
XXII	Icosagonal (icosahedral)	Icosagonal	7	20	2	Icosagonal P, SN
XXIII	Hypercubic	Octagonal hypercubic	21 (+8)	73 (+15)	1	Hypercubic P
		Octagonal hypercubic	21 (+8)	107 (+28)	1	Hypercubic Z
		Dodecagonal hypercubic	16 (+12)	25 (+20)	1	Hypercubic Z
Total	23 (+6)	33 (+7)	227 (+44)	4783 (+111)	64 (+10)	33 (+7)

The names here are given according to Whittaker.^[3] They are almost the same as in Brown et al.,^[4] with exception for names of the crystal families 9, 13, and 22. The names for these three families according to Brown et al. are given in parentheses.

References

↑ Giacovazzo, Carmelo (10 February 2011). Fundamentals of Crystallography (3rd ed.). Oxford University Press. ISBN 978-0-19-957366-0.
↑ Hahn, Theo (2005). International Tables for Crystallography Volume A: Space-Group Symmetry (5th ed.). Table 2.1.2.1: Springer.{{cite book}}: CS1 maint: location (link)
1 2 Whittaker, E. J. W. (1985). An Atlas of Hyperstereograms of the Four-Dimensional Crystal Classes. Oxford: Clarendon Press. ISBN 978-0-19-854432-6. OCLC 638900498.
1 2 Brown, H.; Bülow, R.; Neubüser, J.; Wondratschek, H.; Zassenhaus, H. (1978). Crystallographic Groups of Four-Dimensional Space. New York: Wiley. ISBN 978-0-471-03095-9. OCLC 939898594.

Works cited

Hahn, Theo, ed. (2002). International Tables for Crystallography, Volume A: Space Group Symmetry. Vol. A (5th ed.). Berlin, New York: Springer-Verlag. doi:10.1107/97809553602060000100. ISBN 978-0-7923-6590-7.

External links

[Giacovazzo-1] Giacovazzo, Carmelo (10 February 2011). Fundamentals of Crystallography (3rd ed.). Oxford University Press. ISBN 978-0-19-957366-0.

[ITA-2] Hahn, Theo (2005). International Tables for Crystallography Volume A: Space-Group Symmetry (5th ed.). Table 2.1.2.1: Springer.{{cite book}}: CS1 maint: location (link)

[Whittaker-3] 1 2 Whittaker, E. J. W. (1985). An Atlas of Hyperstereograms of the Four-Dimensional Crystal Classes. Oxford: Clarendon Press. ISBN 978-0-19-854432-6. OCLC 638900498.

[Brown-4] 1 2 Brown, H.; Bülow, R.; Neubüser, J.; Wondratschek, H.; Zassenhaus, H. (1978). Crystallographic Groups of Four-Dimensional Space. New York: Wiley. ISBN 978-0-471-03095-9. OCLC 939898594.

[1]

[2]

[3]

[4]

v t e 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