Crystal system

In crystallography, the terms crystal system, crystal family, and lattice system each refer to one of several classes of space groups, lattices, point groups, or crystals. Informally, two crystals are in the same crystal system if they have similar symmetries, although there are many exceptions to this.

The diamond crystal structure belongs to the face-centered cubic lattice, with a repeated two-atom pattern.

Crystal systems, crystal families and lattice systems are similar but slightly different, and there is widespread confusion between them: 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".

Space groups and crystals are divided into seven crystal systems according to their point groups, and into seven lattice systems according to their Bravais lattices. Five of the crystal systems are essentially the same as five of the lattice systems, but the hexagonal and trigonal crystal systems differ from the hexagonal and rhombohedral lattice systems. The six crystal families are formed by combining the hexagonal and trigonal crystal systems into one hexagonal family, in order to eliminate this confusion.

Overview

Hexagonal hanksite crystal, with threefold c-axis symmetry

A lattice system is a class of lattices with the same set of lattice point groups, which are subgroups of the arithmetic crystal classes. The 14 Bravais lattices are grouped into seven lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic.

In a crystal system, a set of point groups and their corresponding space groups are assigned to a lattice system. Of the 32 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, five point groups are assigned to two lattice systems, rhombohedral and hexagonal, because both exhibit threefold rotational symmetry. These point groups are assigned to the trigonal crystal system. In total there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic.

A crystal family is determined by lattices and point groups. It is formed by combining crystal systems which have space groups assigned to a common lattice system. In three dimensions, the crystal families and systems are identical, except the hexagonal and trigonal crystal systems, which are combined into one hexagonal crystal family. In total there are six crystal families: triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, and cubic.

Spaces with less than three dimensions have the same number of crystal systems, crystal families and lattice systems. In one-dimensional space, there is one crystal system. In 2D space, there are four crystal systems: oblique, rectangular, square, and hexagonal.

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

Crystal family (6)	Crystal system (7)	Required symmetries of point group	Point groups	Space groups	Bravais lattices	Lattice system
Triclinic		None	2	2	1	Triclinic
Monoclinic		1 twofold axis of rotation or 1 mirror plane	3	13	2	monoclinic
Orthorhombic		3 twofold axes of rotation or 1 twofold axis of rotation and 2 mirror planes	3	59	4	Orthorhombic
Tetragonal		1 fourfold axis of rotation	7	68	2	Tetragonal
Hexagonal	Trigonal	1 threefold axis of rotation	5	7	1	Rhombohedral
	Trigonal	1 threefold axis of rotation	5	18	1	Hexagonal
	Hexagonal	1 sixfold axis of rotation	7	27	1	Hexagonal
Cubic		3 fourfold axes of rotation	5	36	3	Cubic
6	7	Total	32	230	14	7

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	Point group / Crystal class	Schönflies	Hermann–Mauguin	Orbifold	Coxeter	Point symmetry	Order	Abstract group
triclinic		pedial	C₁	1	11	[ ]⁺	enantiomorphic polar	1	trivial $\mathbb {Z} _{1}$
triclinic		pinacoidal	C_i (S₂)	1	1x	[2,1⁺]	centrosymmetric	2	cyclic $\mathbb {Z} _{2}$
monoclinic		sphenoidal	C₂	2	22	[2,2]⁺	enantiomorphic polar	2	cyclic $\mathbb {Z} _{2}$
		domatic	C_s (C_1h)	m	*11	[ ]	polar	2	cyclic $\mathbb {Z} _{2}$
		prismatic	C_2h	2/m	2*	[2,2⁺]	centrosymmetric	4	Klein four $\mathbb {V} =\mathbb {Z} _{2}\times \mathbb {Z} _{2}$
orthorhombic		rhombic-disphenoidal	D₂ (V)	222	222	[2,2]⁺	enantiomorphic	4	Klein four $\mathbb {V} =\mathbb {Z} _{2}\times \mathbb {Z} _{2}$
		rhombic-pyramidal	C_2v	mm2	*22	[2]	polar	4	Klein four $\mathbb {V} =\mathbb {Z} _{2}\times \mathbb {Z} _{2}$
		rhombic-dipyramidal	D_2h (V_h)	mmm	*222	[2,2]	centrosymmetric	8	$\mathbb {V} \times \mathbb {Z} _{2}$
tetragonal		tetragonal-pyramidal	C₄	4	44	[4]⁺	enantiomorphic polar	4	cyclic $\mathbb {Z} _{4}$
		tetragonal-disphenoidal	S₄	4	2x	[2⁺,2]	non-centrosymmetric	4	cyclic $\mathbb {Z} _{4}$
		tetragonal-dipyramidal	C_4h	4/m	4*	[2,4⁺]	centrosymmetric	8	$\mathbb {Z} _{4}\times \mathbb {Z} _{2}$
		tetragonal-trapezohedral	D₄	422	422	[2,4]⁺	enantiomorphic	8	dihedral $\mathbb {D} _{8}=\mathbb {Z} _{4}\rtimes \mathbb {Z} _{2}$
		ditetragonal-pyramidal	C_4v	4mm	*44	[4]	polar	8	dihedral $\mathbb {D} _{8}=\mathbb {Z} _{4}\rtimes \mathbb {Z} _{2}$
		tetragonal-scalenohedral	D_2d (V_d)	42m or 4m2	2*2	[2⁺,4]	non-centrosymmetric	8	dihedral $\mathbb {D} _{8}=\mathbb {Z} _{4}\rtimes \mathbb {Z} _{2}$
		ditetragonal-dipyramidal	D_4h	4/mmm	*422	[2,4]	centrosymmetric	16	$\mathbb {D} _{8}\times \mathbb {Z} _{2}$
hexagonal	trigonal	trigonal-pyramidal	C₃	3	33	[3]⁺	enantiomorphic polar	3	cyclic $\mathbb {Z} _{3}$
		rhombohedral	C_3i (S₆)	3	3x	[2⁺,3⁺]	centrosymmetric	6	cyclic $\mathbb {Z} _{6}=\mathbb {Z} _{3}\times \mathbb {Z} _{2}$
		trigonal-trapezohedral	D₃	32 or 321 or 312	322	[3,2]⁺	enantiomorphic	6	dihedral $\mathbb {D} _{6}=\mathbb {Z} _{3}\rtimes \mathbb {Z} _{2}$
		ditrigonal-pyramidal	C_3v	3m or 3m1 or 31m	*33	[3]	polar	6	dihedral $\mathbb {D} _{6}=\mathbb {Z} _{3}\rtimes \mathbb {Z} _{2}$
		ditrigonal-scalenohedral	D_3d	3m or 3m1 or 31m	2*3	[2⁺,6]	centrosymmetric	12	dihedral $\mathbb {D} _{12}=\mathbb {Z} _{6}\rtimes \mathbb {Z} _{2}$
	hexagonal	hexagonal-pyramidal	C₆	6	66	[6]⁺	enantiomorphic polar	6	cyclic $\mathbb {Z} _{6}=\mathbb {Z} _{3}\times \mathbb {Z} _{2}$
		trigonal-dipyramidal	C_3h	6	3*	[2,3⁺]	non-centrosymmetric	6	cyclic $\mathbb {Z} _{6}=\mathbb {Z} _{3}\times \mathbb {Z} _{2}$
		hexagonal-dipyramidal	C_6h	6/m	6*	[2,6⁺]	centrosymmetric	12	$\mathbb {Z} _{6}\times \mathbb {Z} _{2}$
		hexagonal-trapezohedral	D₆	622	622	[2,6]⁺	enantiomorphic	12	dihedral $\mathbb {D} _{12}=\mathbb {Z} _{6}\rtimes \mathbb {Z} _{2}$
		dihexagonal-pyramidal	C_6v	6mm	*66	[6]	polar	12	dihedral $\mathbb {D} _{12}=\mathbb {Z} _{6}\rtimes \mathbb {Z} _{2}$
		ditrigonal-dipyramidal	D_3h	6m2 or 62m	*322	[2,3]	non-centrosymmetric	12	dihedral $\mathbb {D} _{12}=\mathbb {Z} _{6}\rtimes \mathbb {Z} _{2}$
		dihexagonal-dipyramidal	D_6h	6/mmm	*622	[2,6]	centrosymmetric	24	$\mathbb {D} _{12}\times \mathbb {Z} _{2}$
cubic		tetartoidal	T	23	332	[3,3]⁺	enantiomorphic	12	alternating $\mathbb {A} _{4}$
		diploidal	T_h	m3	3*2	[3⁺,4]	centrosymmetric	24	$\mathbb {A} _{4}\times \mathbb {Z} _{2}$
		gyroidal	O	432	432	[4,3]⁺	enantiomorphic	24	symmetric $\mathbb {S} _{4}$
		hextetrahedral	T_d	43m	*332	[3,3]	non-centrosymmetric	24	symmetric $\mathbb {S} _{4}$
		hexoctahedral	O_h	m3m	*432	[4,3]	centrosymmetric	48	$\mathbb {S} _{4}\times \mathbb {Z} _{2}$

The point symmetry of a structure can be further described as follows. Consider the points that make up the structure, and reflect them all through a single point, so that (x,y,z) becomes (−x,−y,−z). This is the 'inverted structure'. If the original structure and inverted structure are identical, then the structure is centrosymmetric. Otherwise it is non-centrosymmetric. Still, even in the non-centrosymmetric case, the inverted structure can in some cases be rotated to align with the original structure. This is a non-centrosymmetric achiral structure. If the inverted structure cannot be rotated to align with the original structure, then the structure is chiral or enantiomorphic and its symmetry group is enantiomorphic.[1]

A direction (meaning a line without an arrow) is called polar if its two directional senses are geometrically or physically different. A symmetry direction of a crystal that is polar is called a polar axis.[2] Groups containing a polar axis are called polar. A polar crystal possesses a unique polar axis (more precisely, all polar axes are parallel). Some geometrical or physical property is different at the two ends of this axis: for example, there might develop a dielectric polarization as in pyroelectric crystals. A polar axis can occur only in non-centrosymmetric structures. There cannot be a mirror plane or twofold axis perpendicular to the polar axis, because they would make the two directions of the axis equivalent.

The crystal structures of chiral biological molecules (such as protein structures) can only occur in the 65 enantiomorphic space groups (biological molecules are usually chiral).

Bravais lattices

There are seven different kinds of crystal systems, and each kind of crystal 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 lattice systems and crystal families is given in the following table.

Crystal family	Lattice system	Schönflies	14 Bravais lattices
Crystal family	Lattice system	Schönflies	Primitive	Base-centered	Body-centered	Face-centered
triclinic		C_i
monoclinic		C_2h
orthorhombic		D_2h
tetragonal		D_4h
hexagonal	rhombohedral	D_3d
hexagonal	hexagonal	D_6h
cubic		O_h

In geometry and crystallography, a Bravais lattice is a category of translative symmetry groups (also known as lattices) in three directions.

Such symmetry groups consist of translations by vectors of the form

R = n₁a₁ + n₂a₂ + n₃a₃,

where n₁, n₂, and n₃ are integers and a₁, a₂, and a₃ are three non-coplanar vectors, called primitive vectors.

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 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.

The Bravais lattices were studied by Moritz Ludwig Frankenheim in 1842, who found that there were 15 Bravais lattices. This was corrected to 14 by A. Bravais in 1848.

In four-dimensional space

‌The four-dimensional unit cell is defined by four edge lengths (a, b, c, d) and six interaxial angles (α, β, γ, δ, ε, ζ). The following conditions for the lattice parameters define 23 crystal families

Crystal families in 4D space
No.	Family	Edge lengths	Interaxial angles
1	Hexaclinic	a ≠ b ≠ c ≠ d	α ≠ β ≠ γ ≠ δ ≠ ε ≠ ζ ≠ 90°
2	Triclinic	a ≠ b ≠ c ≠ d	α ≠ β ≠ γ ≠ 90° δ = ε = ζ = 90°
3	Diclinic	a ≠ b ≠ c ≠ d	α ≠ 90° β = γ = δ = ε = 90° ζ ≠ 90°
4	Monoclinic	a ≠ b ≠ c ≠ d	α ≠ 90° β = γ = δ = ε = ζ = 90°
5	Orthogonal	a ≠ b ≠ c ≠ d	α = β = γ = δ = ε = ζ = 90°
6	Tetragonal monoclinic	a ≠ b = c ≠ d	α ≠ 90° β = γ = δ = ε = ζ = 90°
7	Hexagonal monoclinic	a ≠ b = c ≠ d	α ≠ 90° β = γ = δ = ε = 90° ζ = 120°
8	Ditetragonal diclinic	a = d ≠ b = c	α = ζ = 90° β = ε ≠ 90° γ ≠ 90° δ = 180° − γ
9	Ditrigonal (dihexagonal) diclinic	a = d ≠ b = c	α = ζ = 120° β = ε ≠ 90° γ ≠ δ ≠ 90° cos δ = cos β − cos γ
10	Tetragonal orthogonal	a ≠ b = c ≠ d	α = β = γ = δ = ε = ζ = 90°
11	Hexagonal orthogonal	a ≠ b = c ≠ d	α = β = γ = δ = ε = 90°, ζ = 120°
12	Ditetragonal monoclinic	a = d ≠ b = c	α = γ = δ = ζ = 90° β = ε ≠ 90°
13	Ditrigonal (dihexagonal) monoclinic	a = d ≠ b = c	α = ζ = 120° β = ε ≠ 90° γ = δ ≠ 90° cos γ = −1/2cos β
14	Ditetragonal orthogonal	a = d ≠ b = c	α = β = γ = δ = ε = ζ = 90°
15	Hexagonal tetragonal	a = d ≠ b = c	α = β = γ = δ = ε = 90° ζ = 120°
16	Dihexagonal orthogonal	a = d ≠ b = c	α = ζ = 120° β = γ = δ = ε = 90°
17	Cubic orthogonal	a = b = c ≠ d	α = β = γ = δ = ε = ζ = 90°
18	Octagonal	a = b = c = d	α = γ = ζ ≠ 90° β = ε = 90° δ = 180° − α
19	Decagonal	a = b = c = d	α = γ = ζ ≠ β = δ = ε cos β = −1/2 − cos α
20	Dodecagonal	a = b = c = d	α = ζ = 90° β = ε = 120° γ = δ ≠ 90°
21	Diisohexagonal orthogonal	a = b = c = d	α = ζ = 120° β = γ = δ = ε = 90°
22	Icosagonal (icosahedral)	a = b = c = d	α = β = γ = δ = ε = ζ cos α = −1/4
23	Hypercubic	a = b = c = d	α = β = γ = δ = ε = ζ = 90°

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 parenthesis.

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 are 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 systems in 4D space
No. of crystal family	Crystal family	Crystal system	No. of crystal system	Point groups	Space groups	Bravais lattices	Lattice system
I	Hexaclinic		1	2	2	1	Hexaclinic P
II	Triclinic		2	3	13	2	Triclinic P, S
III	Diclinic		3	2	12	3	Diclinic P, S, D
IV	Monoclinic		4	4	207	6	Monoclinic P, S, S, I, D, F
V	Orthogonal	Non-axial orthogonal	5	2	2	1	Orthogonal KU
		Non-axial orthogonal	5	2	112	8	Orthogonal P, S, I, Z, D, F, G, U
		Axial orthogonal	6	3	887	8	Orthogonal P, S, I, Z, D, F, G, U
VI	Tetragonal monoclinic		7	7	88	2	Tetragonal monoclinic P, I
VII	Hexagonal monoclinic	Trigonal monoclinic	8	5	9	1	Hexagonal monoclinic R
		Trigonal monoclinic	8	5	15	1	Hexagonal monoclinic P
		Hexagonal monoclinic	9	7	25	1	Hexagonal monoclinic P
VIII	Ditetragonal diclinic*		10	1 (+1)	1 (+1)	1 (+1)	Ditetragonal diclinic P*
IX	Ditrigonal diclinic*		11	2 (+2)	2 (+2)	1 (+1)	Ditrigonal diclinic P*
X	Tetragonal orthogonal	Inverse tetragonal orthogonal	12	5	7	1	Tetragonal orthogonal KG
		Inverse tetragonal orthogonal	12	5	351	5	Tetragonal orthogonal P, S, I, Z, G
		Proper tetragonal orthogonal	13	10	1312	5	Tetragonal orthogonal P, S, I, Z, G
XI	Hexagonal orthogonal	Trigonal orthogonal	14	10	81	2	Hexagonal orthogonal R, RS
		Trigonal orthogonal	14	10	150	2	Hexagonal orthogonal P, S
		Hexagonal orthogonal	15	12	240	2	Hexagonal orthogonal P, S
XII	Ditetragonal monoclinic*		16	1 (+1)	6 (+6)	3 (+3)	Ditetragonal monoclinic P, S, D*
XIII	Ditrigonal monoclinic*		17	2 (+2)	5 (+5)	2 (+2)	Ditrigonal monoclinic P, RR
XIV	Ditetragonal orthogonal	Crypto-ditetragonal orthogonal	18	5	10	1	Ditetragonal orthogonal D
		Crypto-ditetragonal orthogonal	18	5	165 (+2)	2	Ditetragonal orthogonal P, Z
		Ditetragonal orthogonal	19	6	127	2	Ditetragonal orthogonal P, Z
XV	Hexagonal tetragonal		20	22	108	1	Hexagonal tetragonal P
XVI	Dihexagonal orthogonal	Crypto-ditrigonal orthogonal*	21	4 (+4)	5 (+5)	1 (+1)	Dihexagonal orthogonal G*
		Crypto-ditrigonal orthogonal*	21	4 (+4)	5 (+5)	1	Dihexagonal orthogonal P
		Dihexagonal orthogonal	23	11	20
		Ditrigonal orthogonal	22	11	41
		Ditrigonal orthogonal	22	11	16	1	Dihexagonal orthogonal RR
XVII	Cubic orthogonal	Simple cubic orthogonal	24	5	9	1	Cubic orthogonal KU
		Simple cubic orthogonal	24	5	96	5	Cubic orthogonal P, I, Z, F, U
		Complex cubic orthogonal	25	11	366	5	Cubic orthogonal P, I, Z, F, U
XVIII	Octagonal*		26	2 (+2)	3 (+3)	1 (+1)	Octagonal P*
XIX	Decagonal		27	4	5	1	Decagonal P
XX	Dodecagonal*		28	2 (+2)	2 (+2)	1 (+1)	Dodecagonal P*
XXI	Diisohexagonal orthogonal	Simple diisohexagonal orthogonal	29	9 (+2)	19 (+5)	1	Diisohexagonal orthogonal RR
		Simple diisohexagonal orthogonal	29	9 (+2)	19 (+3)	1	Diisohexagonal orthogonal P
		Complex diisohexagonal orthogonal	30	13 (+8)	15 (+9)	1	Diisohexagonal orthogonal P
XXII	Icosagonal		31	7	20	2	Icosagonal P, SN
XXIII	Hypercubic	Octagonal hypercubic	32	21 (+8)	73 (+15)	1	Hypercubic P
		Octagonal hypercubic	32	21 (+8)	107 (+28)	1	Hypercubic Z
		Dodecagonal hypercubic	33	16 (+12)	25 (+20)	1	Hypercubic Z
Total	23 (+6)	33 (+7)		227 (+44)	4783 (+111)	64 (+10)	33 (+7)

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References

Flack, Howard D. (2003). "Chiral and Achiral Crystal Structures". Helvetica Chimica Acta. 86 (4): 905–921. CiteSeerX 10.1.1.537.266. doi:10.1002/hlca.200390109.
Hahn (2002), p. 804
Whittaker, E. J. W. (1985). An Atlas of Hyperstereograms of the Four-Dimensional Crystal Classes. Oxford & New York: Clarendon Press.
Brown, H.; Bülow, R.; Neubüser, J.; Wondratschek, H.; Zassenhaus, H. (1978). Crystallographic Groups of Four-Dimensional Space. New York: Wiley.

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

External links

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[1] Flack, Howard D. (2003). "Chiral and Achiral Crystal Structures". Helvetica Chimica Acta. 86 (4): 905–921. CiteSeerX 10.1.1.537.266. doi:10.1002/hlca.200390109.

[2] Hahn (2002), p. 804

[Whittaker-3] Whittaker, E. J. W. (1985). An Atlas of Hyperstereograms of the Four-Dimensional Crystal Classes. Oxford & New York: Clarendon Press.

[Brown-4] Brown, H.; Bülow, R.; Neubüser, J.; Wondratschek, H.; Zassenhaus, H. (1978). Crystallographic Groups of Four-Dimensional Space. New York: Wiley.