Crystal system

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

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).

Classifications

Further information: Space group § Classification systems

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".

Lattice system

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

Crystal system

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, 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.

Crystal family

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.

Hexagonal hanksite crystal, with threefold c-axis symmetry

Comparison

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

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

Crystal classes

Main article: Crystallographic point group

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 ${\displaystyle \mathbb {Z} _{1}}$
		pinacoidal	Ci (S2)	1	1x	[2,1+]	centrosymmetric	2	cyclic ${\displaystyle \mathbb {Z} _{2}}$
monoclinic		sphenoidal	C2	2	22	[2,2]+	enantiomorphic polar	2	cyclic ${\displaystyle \mathbb {Z} _{2}}$
		domatic	Cs (C1h)	m	*11	[ ]	polar	2	cyclic ${\displaystyle \mathbb {Z} _{2}}$
		prismatic	C2h	2/m	2*	[2,2+]	centrosymmetric	4	Klein four ${\displaystyle \mathbb {V} =\mathbb {Z} _{2}\times \mathbb {Z} _{2}}$
orthorhombic		rhombic-disphenoidal	D2 (V)	222	222	[2,2]+	enantiomorphic	4	Klein four ${\displaystyle \mathbb {V} =\mathbb {Z} _{2}\times \mathbb {Z} _{2}}$
		rhombic-pyramidal	C2v	mm2	*22	[2]	polar	4	Klein four ${\displaystyle \mathbb {V} =\mathbb {Z} _{2}\times \mathbb {Z} _{2}}$
		rhombic-dipyramidal	D2h (Vh)	mmm	*222	[2,2]	centrosymmetric	8	${\displaystyle \mathbb {V} \times \mathbb {Z} _{2}}$
tetragonal		tetragonal-pyramidal	C4	4	44	[4]+	enantiomorphic polar	4	cyclic ${\displaystyle \mathbb {Z} _{4}}$
		tetragonal-disphenoidal	S4	4	2x	[2+,2]	non-centrosymmetric	4	cyclic ${\displaystyle \mathbb {Z} _{4}}$
		tetragonal-dipyramidal	C4h	4/m	4*	[2,4+]	centrosymmetric	8	${\displaystyle \mathbb {Z} _{4}\times \mathbb {Z} _{2}}$
		tetragonal-trapezohedral	D4	422	422	[2,4]+	enantiomorphic	8	dihedral ${\displaystyle \mathbb {D} _{8}=\mathbb {Z} _{4}\rtimes \mathbb {Z} _{2}}$
		ditetragonal-pyramidal	C4v	4mm	*44	[4]	polar	8	dihedral ${\displaystyle \mathbb {D} _{8}=\mathbb {Z} _{4}\rtimes \mathbb {Z} _{2}}$
		tetragonal-scalenohedral	D2d (Vd)	42m or 4m2	2*2	[2+,4]	non-centrosymmetric	8	dihedral ${\displaystyle \mathbb {D} _{8}=\mathbb {Z} _{4}\rtimes \mathbb {Z} _{2}}$
		ditetragonal-dipyramidal	D4h	4/mmm	*422	[2,4]	centrosymmetric	16	${\displaystyle \mathbb {D} _{8}\times \mathbb {Z} _{2}}$
hexagonal	trigonal	trigonal-pyramidal	C3	3	33	[3]+	enantiomorphic polar	3	cyclic ${\displaystyle \mathbb {Z} _{3}}$
		rhombohedral	C3i (S6)	3	3x	[2+,3+]	centrosymmetric	6	cyclic ${\displaystyle \mathbb {Z} _{6}=\mathbb {Z} _{3}\times \mathbb {Z} _{2}}$
		trigonal-trapezohedral	D3	32 or 321 or 312	322	[3,2]+	enantiomorphic	6	dihedral ${\displaystyle \mathbb {D} _{6}=\mathbb {Z} _{3}\rtimes \mathbb {Z} _{2}}$
		ditrigonal-pyramidal	C3v	3m or 3m1 or 31m	*33	[3]	polar	6	dihedral ${\displaystyle \mathbb {D} _{6}=\mathbb {Z} _{3}\rtimes \mathbb {Z} _{2}}$
		ditrigonal-scalenohedral	D3d	3m or 3m1 or 31m	2*3	[2+,6]	centrosymmetric	12	dihedral ${\displaystyle \mathbb {D} _{12}=\mathbb {Z} _{6}\rtimes \mathbb {Z} _{2}}$
	hexagonal	hexagonal-pyramidal	C6	6	66	[6]+	enantiomorphic polar	6	cyclic ${\displaystyle \mathbb {Z} _{6}=\mathbb {Z} _{3}\times \mathbb {Z} _{2}}$
		trigonal-dipyramidal	C3h	6	3*	[2,3+]	non-centrosymmetric	6	cyclic ${\displaystyle \mathbb {Z} _{6}=\mathbb {Z} _{3}\times \mathbb {Z} _{2}}$
		hexagonal-dipyramidal	C6h	6/m	6*	[2,6+]	centrosymmetric	12	${\displaystyle \mathbb {Z} _{6}\times \mathbb {Z} _{2}}$
		hexagonal-trapezohedral	D6	622	622	[2,6]+	enantiomorphic	12	dihedral ${\displaystyle \mathbb {D} _{12}=\mathbb {Z} _{6}\rtimes \mathbb {Z} _{2}}$
		dihexagonal-pyramidal	C6v	6mm	*66	[6]	polar	12	dihedral ${\displaystyle \mathbb {D} _{12}=\mathbb {Z} _{6}\rtimes \mathbb {Z} _{2}}$
		ditrigonal-dipyramidal	D3h	6m2 or 62m	*322	[2,3]	non-centrosymmetric	12	dihedral ${\displaystyle \mathbb {D} _{12}=\mathbb {Z} _{6}\rtimes \mathbb {Z} _{2}}$
		dihexagonal-dipyramidal	D6h	6/mmm	*622	[2,6]	centrosymmetric	24	${\displaystyle \mathbb {D} _{12}\times \mathbb {Z} _{2}}$
cubic		tetartoidal	T	23	332	[3,3]+	enantiomorphic	12	alternating ${\displaystyle \mathbb {A} _{4}}$
		diploidal	Th	m3	3*2	[3+,4]	centrosymmetric	24	${\displaystyle \mathbb {A} _{4}\times \mathbb {Z} _{2}}$
		gyroidal	O	432	432	[4,3]+	enantiomorphic	24	symmetric ${\displaystyle \mathbb {S} _{4}}$
		hextetrahedral	Td	43m	*332	[3,3]	non-centrosymmetric	24	symmetric ${\displaystyle \mathbb {S} _{4}}$
		hexoctahedral	Oh	m3m	*432	[4,3]	centrosymmetric	48	${\displaystyle \mathbb {S} _{4}\times \mathbb {Z} _{2}}$

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

Main article: Bravais lattice

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 (Schönflies notation)	14 Bravais lattices
			Primitive (P)	Base-centered (S)	Body-centered (I)	Face-centered (F)
Triclinic (a)		Ci	aP
Monoclinic (m)		C2h	mP	mS
Orthorhombic (o)		D2h	oP	oS	oI	oF
Tetragonal (t)		D4h	tP		tI
Hexagonal (h)	Rhombohedral	D3d	hR
	Hexagonal	D6h	hP
Cubic (c)		Oh	cP		cI	cF

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^{[ 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.

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 other dimensions

Two-dimensional space

In two-dimensional space, there are four crystal systems (oblique, rectangular, square, hexagonal), four crystal families (oblique, rectanguar, square, hexagonal), and four lattice systems (oblique, rectangular, square, and hexagonal). ^{[3] [4]}

Crystal family	Crystal system	Crystallographic point groups	No. of plane groups	Bravais lattices
Oblique (monoclinic)	Oblique	1, 2	2	mp
Rectangular (orthorhombic)	Rectangular	m, 2mm	7	op, oc
Square (tetragonal)	Square	4, 4mm	3	tp
Hexagonal	Hexagonal	3, 6, 3m, 6mm	5	hp
Total	4	10	17	5

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. ^[5] They are almost the same as in Brown et al., ^[6] 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.

The relation between four-dimensional crystal families, crystal systems, and lattice systems is shown in the following table. ^{[5] [6]} 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 systems in 4D space

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

See also

• Crystal cluster  – Group of crystals formed in an open space with form determined by their internal crystal structure
• Crystal structure  – Ordered arrangement of atoms, ions, or molecules in a crystalline material
• List of space groups
• Polar point group  – symmetry in geometry and crystallographyPages displaying wikidata descriptions as a fallback

References

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.
3. Giacovazzo, Carmelo (10 February 2011). Fundamentals of Crystallography (3rd ed.). Oxford University Press. ISBN   978-0-19-957366-0.
4. Hahn, Theo (2005). International Tables for Crystallography Volume A: Space-Group Symmetry (5th ed.). Table 2.1.2.1: Springer.
5. 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.
6. 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

• Overview of the 32 groups
• Mineral galleries – Symmetry
• all cubic crystal classes, forms, and stereographic projections (interactive java applet)
• Crystal system at the Online Dictionary of Crystallography
• Crystal family at the Online Dictionary of Crystallography
• Lattice system at the Online Dictionary of Crystallography
• Conversion Primitive to Standard Conventional for VASP input files Archived 2021-11-26 at the Wayback Machine
• Learning Crystallography