1. Overview

2. Crystal classes

3. Bravais lattices

4. In four-dimensional space

5. See also

6. References

7. External links

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.

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

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:

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:

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

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

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

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{{what|date=January 2019}} 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

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.

See also

• {{annotated link|Crystal cluster}}
• {{annotated link|Crystal structure}}
• {{annotated link|List of space groups}}
• {{annotated link|Polar point group}}

References

• {{cite book |editor1-last=Hahn |editor1-first=Theo |title=International Tables for Crystallography, Volume A: Space Group Symmetry |url=http://it.iucr.org/A/ |publisher=Springer-Verlag |location=Berlin, New York |edition=5th |isbn=978-0-7923-6590-7 |doi=10.1107/97809553602060000100 |year=2002 |volume=A|series=International Tables for Crystallography }}

External links

• Overview of the 32 groups
• [https://web.archive.org/web/20050624024940/http://mineral.galleries.com/minerals/symmetry/symmetry.htm 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

• Maple Avenue Historic District (disambiguation)
• Maple Avenue Maple Lane Historic District
• Maple Avenue/Maple Lane Historic District
• Maple Avenue (TV series)
• Maple Baronets
• Maple Bay (disambiguation)
• Maple Beach
• Maple Beach, Pennsylvania
• Maple Beach, Wisconsin
• Maple bladder-gall mite
• Maple bladder gall mite
• Maple bladdergall mite
• Maple Blues Award
• Maple, Burlington
• Maple Bush No. 224, Saskatchewan
• Maple Canyon
• Maple Canyon (Sanpete County, Utah)
• Maple City
• Maple City, KS
• Maple class tug
• Maple Club
• MapleCore Ltd.
• Maple Creek No. 111, Saskatchewan
• Maple Creek, SK
• Maplecrest sempar school