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.
A lattice system is a class of lattices with the same point group. In three dimensions there are seven lattice systems: triclinic, monoclinic, orthorhombic, tetragonal, rhombohedral, hexagonal, and cubic. The lattice system of a crystal or space group is determined by its lattice but not always by its point group.
A crystal system is a class of point groups. Two point groups are placed in the same crystal system if the sets of possible lattice systems of their space groups are the same. For many point groups there is only one possible lattice system, and in these cases the crystal system corresponds to a lattice system and is given the same name. However, for the five point groups in the trigonal crystal class there are two possible lattice systems for their point groups: rhombohedral or hexagonal. In three dimensions there are seven crystal systems: triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. The crystal system of a crystal or space group is determined by its point group but not always by its lattice.
A crystal family also consists of point groups and is formed by combining crystal systems whenever two crystal systems have space groups with the same lattice. In three dimensions a crystal family is almost the same as a crystal system (or lattice system), except that the hexagonal and trigonal crystal systems are combined into one hexagonal family. In three dimensions there are six crystal families: triclinic, monoclinic, orthorhombic, tetragonal, hexagonal, and cubic. The crystal family of a crystal or space group is determined by either its point group or its lattice, and crystal families are the smallest collections of point groups with this property.
In dimensions less than three there is no essential difference between crystal systems, crystal families, and lattice systems. There are 1 in dimension 0, 1 in dimension 1, and 4 in dimension 2, called 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|
|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|
|Hexagonal||1 sixfold axis of rotation||7||27|
|Cubic||4 threefold axes of rotation||5||36||3||Cubic|
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||Group structure|
|triclinic||triclinic-pedial||C1||1||11||[ ]+||enantiomorphic polar||1||trivial|
|tetragonal-scalenoidal||D2d||Template:Overline2m or Template:Overlinem2||2*2||[2+,4]||non-centrosymmetric||8||dihedral|
|trigonal-trapezoidal||D3||32 or 321 or 312||322||[3,2]+||enantiomorphic||6||dihedral|
|ditrigonal-pyramidal||C3v||3m or 3m1 or 31m||*33||||polar||6||dihedral|
|ditrigonal-scalahedral||D3d||Template:Overlinem or Template:Overlinem1 or Template:Overline1m||2*3||[2+,6]||centrosymmetric||12||dihedral|
|ditrigonal-dipyramidal||D3h||Template:Overlinem2 or Template:Overline2m||*322||[2,3]||non-centrosymmetric||12||dihedral|
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.
A direction 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. 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.
Lattice systems Edit
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|
|monoclinic (right prism with parallelogram base; here seen from above)||simple||base-centered|
|Monoclinic, simple||Monoclinic, centered|
|Orthohombic, simple||Orthohombic, base-centered||Orthohombic, body-centered||Orthohombic, face-centered|
|tetragonal (square cuboid)||simple||body-centered|
|Tetragonal, simple||Tetragonal, body-centered|
| rhombohedral |
|hexagonal (centered regular hexagon)||Hexagonal|
|Cubic, simple||Cubic, body-centered||Cubic, face-centered|
Such symmetry groups consist of translations by vectors of the form
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.
Crystal systems in four-dimensional spaceEdit
The four-dimensional unit cell is defined by four edge lengths () and six interaxial angles (). The following conditions for the lattice parameters define 23 crystal families:
6 Tetragonal Monoclinic:
7 Hexagonal Monoclinic:
8 Ditetragonal Diclinic:
9 Ditrigonal (Dihexagonal) Diclinic:
10 Tetragonal Orthogonal:
11 Hexagonal Orthogonal:
12 Ditetragonal Monoclinic:
13 Ditrigonal (Dihexagonal) Monoclinic:
14 Ditetragonal Orthogonal:
15 Hexagonal Tetragonal:
16 Dihexagonal Orthogonal:
17 Cubic Orthogonal:
21 Di-isohexagonal Orthogonal:
22 Icosagonal (Icosahedral):
The names here are given according to Whittaker. They are almost the same as in Brown et al, 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. 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 </br >Crystal family||Crystal family||Crystal system||No. of </br>Crystal system||Point groups||Space groups||Bravais lattices||Lattice system|
|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|
|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|
|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|
|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|
|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|
|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*|
|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|
|19 (+3)||1||Di-isohexagonal Orthogonal P|
|Complex Di-isohexagonal 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)|
- ↑ Howard D. Flack (2003). "Chiral and Achiral Crystal Structures". Helvetica Chimica Acta 86: 905–921. doi:10.1002/hlca.200390109 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.
- ↑ 3.0 3.1 E. J. W. Whittaker, An atlas of hyperstereograms of the four-dimensional crystal classes. Clarendon Press (Oxford Oxfordshire and New York) 1985.
- ↑ 4.0 4.1 H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus, Crystallographic Groups of Four-Dimensional Space. Wiley, NY, 1978.
- 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
doi:10.1107/97809553602060000100. ISBN 978-0-7923-6590-7 ISBN 978-0-7923-6590-7. http://it.iucr.org/A/.
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