Crystal system explained

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. Space groups 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

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.

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

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

See 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 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

Two dimensional space has the same number of crystal systems, crystal families, and lattice systems. In 2D space, there are four crystal systems: oblique, rectangular, square, and hexagonal.

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 = −cos 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 = − − 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 = −
23	Hypercubic	a = b = c = d	α = β = γ = δ = ε = ζ = 90°

The names here are given according to Whittaker.^[2] They are almost the same as in Brown et al.,^[3] 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.^{[2] [3]} 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.

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
					112	8	Orthogonal P, S, I, Z, D, F, G, U
		Axial orthogonal	6	3	887
VI	Tetragonal monoclinic		7	7	88	2	Tetragonal monoclinic P, I
VII	Hexagonal monoclinic	Trigonal monoclinic	8	5	9	1	Hexagonal monoclinic R
					15	1	Hexagonal monoclinic P
		Hexagonal monoclinic	9	7	25
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
					351	5	Tetragonal orthogonal P, S, I, Z, G
		Proper tetragonal orthogonal	13	10	1312
XI	Hexagonal orthogonal	Trigonal orthogonal	14	10	81	2	Hexagonal orthogonal R, RS
					150	2	Hexagonal orthogonal P, S
		Hexagonal orthogonal	15	12	240
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
					165 (+2)	2	Ditetragonal orthogonal P, Z
		Ditetragonal orthogonal	19	6	127
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*
					5 (+5)	1	Dihexagonal orthogonal P
		Dihexagonal orthogonal	23	11	20
		Ditrigonal orthogonal	22	11	41
					16	1	Dihexagonal orthogonal RR
XVII	Cubic orthogonal	Simple cubic orthogonal	24	5	9	1	Cubic orthogonal KU
					96	5	Cubic orthogonal P, I, Z, F, U
		Complex cubic orthogonal	25	11	366
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
					19 (+3)	1	Diisohexagonal orthogonal P
		Complex diisohexagonal orthogonal	30	13 (+8)	15 (+9)
XXII	Icosagonal		31	7	20	2	Icosagonal P, SN
XXIII	Hypercubic	Octagonal hypercubic	32	21 (+8)	73 (+15)	1	Hypercubic P
					107 (+28)	1	Hypercubic Z
		Dodecagonal hypercubic	33	16 (+12)	25 (+20)
Total	23 (+6)	33 (+7)		227 (+44)	4783 (+111)	64 (+10)	33 (+7)

Works cited

• Book: Hahn . Theo . International Tables for Crystallography, Volume A: Space Group Symmetry . . Berlin, New York . 5th . 978-0-7923-6590-7 . 10.1107/97809553602060000100 . 2002 . A. International Tables for Crystallography .

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
• Learning Crystallography

Notes and References

1. Flack. Howard D.. 2003. Chiral and Achiral Crystal Structures. Helvetica Chimica Acta. 86. 4. 905–921. 10.1.1.537.266. 10.1002/hlca.200390109.
2. Book: Whittaker, E. J. W.. An Atlas of Hyperstereograms of the Four-Dimensional Crystal Classes. Clarendon Press. 1985. 978-0-19-854432-6. Oxford. 638900498.
3. Book: Brown. H.. Crystallographic Groups of Four-Dimensional Space. Bülow. R.. Neubüser. J.. Wondratschek. H.. Zassenhaus. H.. Wiley. 1978. 978-0-471-03095-9. New York. 939898594.