Crystal system
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, 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 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.
Contents
Overview[edit]
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:
Crystal family (6) | Crystal system (7) | 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 2 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 |
18 | 1 | Hexagonal | ||||
Hexagonal | 1 sixfold axis of rotation | 7 | 27 | |||
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[edit]
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 | C_{1} | 1 | 11 | [ ]^{+} | enantiomorphic polar | 1 | trivial | |
pinacoidal | C_{i} (S_{2}) | 1 | 1x | [2,1^{+}] | centrosymmetric | 2 | cyclic | ||
monoclinic | sphenoidal | C_{2} | 2 | 22 | [2,2]^{+} | enantiomorphic polar | 2 | cyclic | |
domatic | C_{s} (C_{1h}) | m | *11 | [ ] | polar | 2 | cyclic | ||
prismatic | C_{2h} | 2/m | 2* | [2,2^{+}] | centrosymmetric | 4 | Klein four | ||
orthorhombic | rhombic-disphenoidal | D_{2} (V) | 222 | 222 | [2,2]^{+} | enantiomorphic | 4 | Klein four | |
rhombic-pyramidal | C_{2v} | mm2 | *22 | [2] | polar | 4 | Klein four | ||
rhombic-dipyramidal | D_{2h} (V_{h}) | mmm | *222 | [2,2] | centrosymmetric | 8 | |||
tetragonal | tetragonal-pyramidal | C_{4} | 4 | 44 | [4]^{+} | enantiomorphic polar | 4 | cyclic | |
tetragonal-disphenoidal | S_{4} | 4 | 2x | [2^{+},2] | non-centrosymmetric | 4 | cyclic | ||
tetragonal-dipyramidal | C_{4h} | 4/m | 4* | [2,4^{+}] | centrosymmetric | 8 | |||
tetragonal-trapezohedral | D_{4} | 422 | 422 | [2,4]^{+} | enantiomorphic | 8 | dihedral | ||
ditetragonal-pyramidal | C_{4v} | 4mm | *44 | [4] | polar | 8 | dihedral | ||
tetragonal-scalenohedral | D_{2d} (V_{d}) | 42m or 4m2 | 2*2 | [2^{+},4] | non-centrosymmetric | 8 | dihedral | ||
ditetragonal-dipyramidal | D_{4h} | 4/mmm | *422 | [2,4] | centrosymmetric | 16 | |||
hexagonal | trigonal | trigonal-pyramidal | C_{3} | 3 | 33 | [3]^{+} | enantiomorphic polar | 3 | cyclic |
rhombohedral | C_{3i} (S_{6}) | 3 | 3x | [2^{+},3^{+}] | centrosymmetric | 6 | cyclic | ||
trigonal-trapezohedral | D_{3} | 32 or 321 or 312 | 322 | [3,2]^{+} | enantiomorphic | 6 | dihedral | ||
ditrigonal-pyramidal | C_{3v} | 3m or 3m1 or 31m | *33 | [3] | polar | 6 | dihedral | ||
ditrigonal-scalenohedral | D_{3d} | 3m or 3m1 or 31m | 2*3 | [2^{+},6] | centrosymmetric | 12 | dihedral | ||
hexagonal | hexagonal-pyramidal | C_{6} | 6 | 66 | [6]^{+} | enantiomorphic polar | 6 | cyclic | |
trigonal-dipyramidal | C_{3h} | 6 | 3* | [2,3^{+}] | non-centrosymmetric | 6 | cyclic | ||
hexagonal-dipyramidal | C_{6h} | 6/m | 6* | [2,6^{+}] | centrosymmetric | 12 | |||
hexagonal-trapezohedral | D_{6} | 622 | 622 | [2,6]^{+} | enantiomorphic | 12 | dihedral | ||
dihexagonal-pyramidal | C_{6v} | 6mm | *66 | [6] | polar | 12 | dihedral | ||
ditrigonal-dipyramidal | D_{3h} | 6m2 or 62m | *322 | [2,3] | non-centrosymmetric | 12 | dihedral | ||
dihexagonal-dipyramidal | D_{6h} | 6/mmm | *622 | [2,6] | centrosymmetric | 24 | |||
cubic | tetartoidal | T | 23 | 332 | [3,3]^{+} | enantiomorphic | 12 | alternating | |
diploidal | T_{h} | m3 | 3*2 | [3^{+},4] | centrosymmetric | 24 | |||
gyroidal | O | 432 | 432 | [4,3]^{+} | enantiomorphic | 24 | symmetric | ||
hextetrahedral | T_{d} | 43m | *332 | [3,3] | non-centrosymmetric | 24 | symmetric | ||
hexoctahedral | O_{h} | m3m | *432 | [4,3] | centrosymmetric | 48 |
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^{[clarification needed]} 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 twofold 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 65 enantiomorphic space groups (biological molecules are usually chiral).
Bravais lattices[edit]
The distribution of the 14 Bravais lattices into lattice systems and crystal families is given in the following table.
Crystal family | Lattice system | Schönflies | 14 Bravais Lattices | |||
---|---|---|---|---|---|---|
Primitive | Base-centered | Body-centered | Face-centered | |||
triclinic | C_{i} | |||||
monoclinic | C_{2h} | |||||
orthorhombic | D_{2h} | |||||
tetragonal | D_{4h} | |||||
hexagonal | rhombohedral | D_{3d} | ||||
hexagonal | D_{6h} | |||||
cubic | O_{h} |
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
- R = n_{1}a_{1} + n_{2}a_{2} + n_{3}a_{3},
where n_{1}, n_{2}, and n_{3} are integers and a_{1}, a_{2}, and a_{3} 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 in 1842, who found that there were 15 Bravais lattices. This was corrected to 14 by A. Bravais in 1848.
In four-dimensional space[edit]
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
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.^{[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 P3_{1} and P3_{2}, P4_{1}22 and P4_{3}22. 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 |
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) |
See also[edit]
References[edit]
This article lacks ISBNs for the books listed in it. (August 2017) |
- ^ 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.
- ^ Hahn (2002), p. 804
- ^ ^{a} ^{b} Whittaker, E. J. W. (1985). An Atlas of Hyperstereograms of the Four-Dimensional Crystal Classes. Oxford & New York: Clarendon Press.
- ^ ^{a} ^{b} Brown, H.; Bülow, R.; Neubüser, J.; Wondratschek, H.; Zassenhaus, H. (1978). Crystallographic Groups of Four-Dimensional Space. New York: Wiley.
- Hahn, Theo, ed. (2002). International Tables for Crystallography, Volume A: Space Group Symmetry. International Tables for Crystallography. A (5th ed.). Berlin, New York: Springer-Verlag. doi:10.1107/97809553602060000100. ISBN 978-0-7923-6590-7.
External links[edit]
- 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