Crystal system

The diamond crystal structure belongs to the face-centered cubic lattice, with a repeated 2-atom pattern.

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 tend to be in the same crystal system if they have similar symmetries, though there are many exceptions to this.

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 7 crystal systems according to their point groups, and into 7 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 three-fold c-axis symmetry

A lattice system is generally identified as a set of lattices with the same shape according to the relative lengths of the cell edges (a, b, c) and the angles between them (α, β, γ). Each lattice is assigned to one of the following classifications (lattice types) based on the positions of the lattice points within the cell: primitive (P), body-centered (I), face-centered (F), base-centered (A, B, or C), and rhombohedral (R). The 14 unique combinations of lattice systems and lattice types are collectively known as the Bravais lattices. Associated with each lattice system is a set of point groups, sometimes called lattice point groups, which are subgroups of the arithmetic crystal classes. In total there are seven lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic.

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 point groups that exist in three dimensions, most are assigned to only one lattice system, in which case the crystal system and lattice system both have the same name. However, five point groups are assigned to two lattice systems, rhombohedral and hexagonal, because both lattice systems 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 are identical to the crystal systems 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 zero- and one-dimensional space, there is one crystal system. In two-dimensional 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	Crystal system	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 two 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		4 threefold axes of rotation	5	36	3	Cubic
Total: 6	7		32	230	14	7

Caution: There is no "trigonal" lattice system. To avoid confusion of terminology, don't use the term "trigonal lattice".

Crystal classes

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

crystal family	crystal system	point group / crystal class	Schönflies	Hermann-Mauguin	Orbifold	Coxeter	Point symmetry	Order	Abstract group
triclinic		triclinic-pedial	C₁	1	11	[ ]⁺	enantiomorphic polar	1	trivial $\mathbb{Z}_1$
triclinic		triclinic-pinacoidal	C_i	1	1x	[2,1⁺]	centrosymmetric	2	cyclic $\mathbb{Z}_2$
monoclinic		monoclinic-sphenoidal	C₂	2	22	[2,2]⁺	enantiomorphic polar	2	cyclic $\mathbb{Z}_2$
		monoclinic-domatic	C_s	m	*11	[ ]	polar	2	cyclic $\mathbb{Z}_2$
		monoclinic-prismatic	C_2h	2/m	2*	[2,2⁺]	centrosymmetric	4	Klein four $\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2$
orthorhombic		orthorhombic-sphenoidal	D₂	222	222	[2,2]⁺	enantiomorphic	4	Klein four $\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2$
		orthorhombic-pyramidal	C_2v	mm2	*22	[2]	polar	4	Klein four $\mathbb{V} = \mathbb{Z}_2\times\mathbb{Z}_2$
		orthorhombic-bipyramidal	D_2h	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-trapezoidal	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-scalenoidal	D_2d	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	S₆ (C_3i)	3	3x	[2⁺,3⁺]	centrosymmetric	6	cyclic $\mathbb{Z}_6 = \mathbb{Z}_3\times\mathbb{Z}_2$
		trigonal-trapezoidal	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-scalahedral	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-trapezoidal	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		tetrahedral	T	23	332	[3,3]⁺	enantiomorphic	12	alternating $\mathbb{A}_4$
		hextetrahedral	T_d	43m	*332	[3,3]	non-centrosymmetric	24	symmetric $\mathbb{S}_4$
		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 $\mathbb{S}_4$
		hexoctahedral	O_h	m3m	*432	[4,3]	centrosymmetric	48	$\mathbb{S}_4\times\mathbb{Z}_2$

Point symmetry can be thought of in the following fashion: consider the coordinates which make up the structure, and project 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 for non-centrosymmetric case, inverted structure in some cases can be rotated to align with the original structure. This is the case of non-centrosymmetric achiral structure. If the inverted structure cannot be rotated to align with the original structure, then the structure is chiral (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 polar symmetry direction of a crystal is called a polar axis.^[2] Groups containing a polar axis are called polar. A polar crystal possess a "unique" axis (found in no other directions) such that some geometrical or physical property is different at the two ends of this axis. It may develop a dielectric polarization, e.g. in pyroelectric crystals. A polar axis can occur only in non-centrosymmetric structures. There should also not be a mirror plane or 2-fold axis perpendicular to the polar axis, because they will make both directions of the axis equivalent.

The crystal structures of chiral biological molecules (such as protein structures) can only occur in the 11 enantiomorphic point groups (biological molecules are usually chiral).

Lattice systems

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

The 7 lattice systems	The 14 Bravais Lattices
triclinic (parallelepiped)
monoclinic (right prism with parallelogram base; here seen from above)	simple	base-centered

orthorhombic (cuboid)	simple	base-centered	body-centered	face-centered
orthorhombic (cuboid)
tetragonal (square cuboid)	simple	body-centered
tetragonal (square cuboid)
rhombohedral (trigonal trapezohedron)
hexagonal (centered regular hexagon)
cubic (isometric; cube)	simple	body-centered	face-centered
cubic (isometric; cube)

In geometry and crystallography, a Bravais lattice is a category of symmetry groups for translational symmetry in three directions, or correspondingly, a category of translation lattices.

Such symmetry groups consist of translations by vectors of the form

\mathbf{R} = n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2 + n_3 \mathbf{a}_3,

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 space group of the translation lattice itself; there are 14 Bravais lattices in three dimensions; each can apply in one lattice system only. They represent the maximum symmetry a structure with the translational symmetry concerned can have.

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

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 (1801–1869), in 1842, who found that there were 15 Bravais lattices. This was corrected to 14 by A. Bravais in 1848.

Crystal systems in four-dimensional space

The four-dimensional unit cell is defined by four edge lengths ( $a, b, c, d$ ) and six interaxial angles ( $\alpha, \beta, \gamma, \delta, \epsilon, \zeta$ ). The following conditions for the lattice parameters define 23 crystal families:

1 Hexaclinic: $a\ne b \ne c \ne d, \alpha \ne \beta \ne \gamma \ne \delta \ne \epsilon \ne \zeta \ne 90 ^\circ$

2 Triclinic: $a\ne b \ne c \ne d, \alpha \ne \beta \ne \gamma \ne 90 ^\circ, \delta = \epsilon = \zeta = 90 ^\circ$

3 Diclinic: $a\ne b \ne c \ne d, \alpha \ne 90 ^\circ, \beta = \gamma = \delta = \epsilon = 90 ^\circ, \zeta \ne 90 ^\circ$

4 Monoclinic: $a\ne b \ne c \ne d, \alpha \ne 90 ^\circ, \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ$

5 Orthogonal: $a\ne b \ne c \ne d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ$

6 Tetragonal Monoclinic: $a\ne b = c \ne d, \alpha \ne 90 ^\circ, \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ$

7 Hexagonal Monoclinic: $a\ne b = c \ne d, \alpha \ne 90 ^\circ, \beta = \gamma = \delta = \epsilon = 90 ^\circ, \zeta = 120 ^\circ$

8 Ditetragonal Diclinic: $a = d \ne b = c, \alpha = \zeta = 90 ^\circ, \beta = \epsilon \ne 90 ^\circ, \gamma \ne 90 ^\circ, \delta = 180 ^\circ - \gamma$

9 Ditrigonal (Dihexagonal) Diclinic: $a = d \ne b = c, \alpha = \zeta = 120 ^\circ, \beta = \epsilon \ne 90 ^\circ, \gamma \ne \delta \ne 90 ^\circ, cos \delta = cos \beta - cos \gamma$

10 Tetragonal Orthogonal: $a\ne b = c \ne d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ$

11 Hexagonal Orthogonal: $a\ne b = c \ne d, \alpha = \beta = \gamma = \delta = \epsilon = 90 ^\circ, \zeta = 120 ^\circ$

12 Ditetragonal Monoclinic: $a = d \ne b = c, \alpha = \gamma = \delta = \zeta = 90 ^\circ, \beta = \epsilon \ne 90 ^\circ$

13 Ditrigonal (Dihexagonal) Monoclinic: $a = d \ne b = c, \alpha = \zeta = 120 ^\circ, \beta = \epsilon \ne 90 ^\circ, \gamma = \delta \ne 90 ^\circ, cos \gamma = -\tfrac{1}{2} cos \beta$

14 Ditetragonal Orthogonal: $a = d \ne b = c, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ$

15 Hexagonal Tetragonal: $a = d \ne b = c, \alpha = \beta = \gamma = \delta = \epsilon = 90 ^\circ, \zeta = 120 ^\circ$

16 Dihexagonal Orthogonal: $a = d \ne b = c, \alpha = \zeta = 120 ^\circ, \beta = \gamma = \delta = \epsilon = 90 ^\circ,$

17 Cubic Orthogonal: $a = b = c \ne d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ$

18 Octagonal: $a = b = c = d, \alpha = \gamma = \zeta \ne 90 ^\circ, \beta = \epsilon = 90 ^\circ, \delta = 180 ^\circ - \alpha$

19 Decagonal: $a = b = c = d, \alpha = \gamma = \zeta \ne \beta = \delta = \epsilon, cos \beta = -0.5 - cos \alpha$

20 Dodecagonal: $a = b = c = d, \alpha = \zeta = 90 ^\circ, \beta = \epsilon = 120 ^\circ, \gamma = \delta \ne 90 ^\circ$

21 Di-isohexagonal Orthogonal: $a = b = c = d, \alpha = \zeta = 120 ^\circ, \beta = \gamma = \delta = \epsilon = 90 ^\circ$

22 Icosagonal (Icosahedral): $a = b = c = d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta, cos \alpha = -\tfrac{1}{4}$

23 Hypercubic: $a = b = c = d, \alpha = \beta = \gamma = \delta = \epsilon = \zeta = 90 ^\circ$

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 asterisk. The number of enantiomorphic pairs are given in parentheses. Here the term "enantiomorphic" has different meaning than in table for three-dimensional crystal classes. The latter means, that enantiomorphic point groups describe chiral (enantiomorphic) structures. In the current table, "enantiomorphic" means, that group itself (considered as 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.

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	Di-isohexagonal Orthogonal	Simple Di-isohexagonal Orthogonal	29	9 (+2)	19 (+5)	1	Di-isohexagonal Orthogonal RR
		Simple Di-isohexagonal Orthogonal	29	9 (+2)	19 (+3)	1	Di-isohexagonal Orthogonal P
		Complex Di-isohexagonal Orthogonal	30	13 (+8)	15 (+9)	1	Di-isohexagonal 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)

Notes

↑ Howard D. Flack (2003). "Chiral and Achiral Crystal Structures". Helvetica Chimica Acta 86: 905–921. doi:10.1002/hlca.200390109.
↑ E. Koch , W. Fischer , U. Müller , in ‘International Tables for Crystallography, Vol. A, Space-Group Symmetry’, 5th edn., Ed. T. Hahn, Kluwer Academic Publishers, Dordrecht, 2002, Chapt. 10, p. 804.
1 2 E. J. W. Whittaker, An atlas of hyperstereograms of the four-dimensional crystal classes. Clarendon Press (Oxford Oxfordshire and New York) 1985.
1 2 H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus, Crystallographic Groups of Four-Dimensional Space. Wiley, NY, 1978.

References

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

External links

The 7 crystal systems

triclinic monoclinic orthorhombic tetragonal trigonal hexagonal cubic (isometric)

Mineral identification

Cleavage Crystal habit Crystal system Fracture Lustre Mohs scale Specific gravity Streak

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