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, 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.
Contents
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 onedimensional space, there is one crystal system. In 2D space, there are four crystal systems: oblique, rectangular, square, and hexagonal.
The relation between threedimensional 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
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  rhombicdisphenoidal  D_{2} (V)  222  222  [2,2]^{+}  enantiomorphic  4  Klein four  
rhombicpyramidal  C_{2v}  mm2  *22  [2]  polar  4  Klein four  
rhombicdipyramidal  D_{2h} (V_{h})  mmm  *222  [2,2]  centrosymmetric  8  
tetragonal  tetragonalpyramidal  C_{4}  4  44  [4]^{+}  enantiomorphic polar  4  cyclic  
tetragonaldisphenoidal  S_{4}  4  2x  [2^{+},2]  noncentrosymmetric  4  cyclic  
tetragonaldipyramidal  C_{4h}  4/m  4*  [2,4^{+}]  centrosymmetric  8  
tetragonaltrapezohedral  D_{4}  422  422  [2,4]^{+}  enantiomorphic  8  dihedral  
ditetragonalpyramidal  C_{4v}  4mm  *44  [4]  polar  8  dihedral  
tetragonalscalenohedral  D_{2d} (V_{d})  42m or 4m2  2*2  [2^{+},4]  noncentrosymmetric  8  dihedral  
ditetragonaldipyramidal  D_{4h}  4/mmm  *422  [2,4]  centrosymmetric  16  
hexagonal  trigonal  trigonalpyramidal  C_{3}  3  33  [3]^{+}  enantiomorphic polar  3  cyclic 
rhombohedral  C_{3i} (S_{6})  3  3x  [2^{+},3^{+}]  centrosymmetric  6  cyclic  
trigonaltrapezohedral  D_{3}  32 or 321 or 312  322  [3,2]^{+}  enantiomorphic  6  dihedral  
ditrigonalpyramidal  C_{3v}  3m or 3m1 or 31m  *33  [3]  polar  6  dihedral  
ditrigonalscalenohedral  D_{3d}  3m or 3m1 or 31m  2*3  [2^{+},6]  centrosymmetric  12  dihedral  
hexagonal  hexagonalpyramidal  C_{6}  6  66  [6]^{+}  enantiomorphic polar  6  cyclic  
trigonaldipyramidal  C_{3h}  6  3*  [2,3^{+}]  noncentrosymmetric  6  cyclic  
hexagonaldipyramidal  C_{6h}  6/m  6*  [2,6^{+}]  centrosymmetric  12  
hexagonaltrapezohedral  D_{6}  622  622  [2,6]^{+}  enantiomorphic  12  dihedral  
dihexagonalpyramidal  C_{6v}  6mm  *66  [6]  polar  12  dihedral  
ditrigonaldipyramidal  D_{3h}  6m2 or 62m  *322  [2,3]  noncentrosymmetric  12  dihedral  
dihexagonaldipyramidal  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]  noncentrosymmetric  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 noncentrosymmetric. Still, even for noncentrosymmetric case, inverted structure in some cases can be rotated to align with the original structure. This is the case of noncentrosymmetric 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 noncentrosymmetric 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
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  Basecentered  Bodycentered  Facecentered  
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 noncoplanar 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 fourdimensional space
The fourdimensional 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 fourdimensional 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 threedimensional 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 threedimensional space groups P3_{1} and P3_{2}, P4_{1}22 and P4_{3}22. Starting from fourdimensional 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  Nonaxial 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  Cryptoditetragonal 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  Cryptoditrigonal 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
References
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 FourDimensional Crystal Classes. Oxford & New York: Clarendon Press.
 ^ ^{a} ^{b} Brown, H.; Bülow, R.; Neubüser, J.; Wondratschek, H.; Zassenhaus, H. (1978). Crystallographic Groups of FourDimensional 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: SpringerVerlag. doi:10.1107/97809553602060000100. ISBN 9780792365907.
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