Crystal system

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The diamond crystal structure belongs to the face-centered cubic lattice, with a repeated two-atom pattern.

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.

Overview[edit]

Hexagonal hanksite crystal, with threefold c-axis symmetry

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.

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 point groups that exist in three dimensions, most are assigned to only one lattice system, in which case the crystal system and lattice system both have the same name. However, five point groups are assigned to two lattice systems, rhombohedral and hexagonal, because both lattice systems 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 are identical to the crystal systems 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 zero- and one-dimensional space, there is one crystal system. In two-dimensional 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 Crystal system 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 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
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 triclinic-pedial C1 1 11 [ ]+ enantiomorphic polar 1 trivial
triclinic-pinacoidal Ci 1 1x [2,1+] centrosymmetric 2 cyclic
monoclinic monoclinic-sphenoidal C2 2 22 [2,2]+ enantiomorphic polar 2 cyclic
monoclinic-domatic Cs m *11 [ ] polar 2 cyclic
monoclinic-prismatic C2h 2/m 2* [2,2+] centrosymmetric 4 Klein four
orthorhombic orthorhombic-sphenoidal D2 222 222 [2,2]+ enantiomorphic 4 Klein four
orthorhombic-pyramidal C2v mm2 *22 [2] polar 4 Klein four
orthorhombic-bipyramidal D2h 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-trapezoidal D4 422 422 [2,4]+ enantiomorphic 8 dihedral
ditetragonal-pyramidal C4v 4mm *44 [4] polar 8 dihedral
tetragonal-scalenoidal D2d 42m or 4m2 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 S6 (C3i) 3 3x [2+,3+] centrosymmetric 6 cyclic
trigonal-trapezoidal 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-scalahedral D3d 3m or 3m1 or 31m 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-trapezoidal D6 622 622 [2,6]+ enantiomorphic 12 dihedral
dihexagonal-pyramidal C6v 6mm *66 [6] polar 12 dihedral
ditrigonal-dipyramidal D3h 6m2 or 62m *322 [2,3] non-centrosymmetric 12 dihedral
dihexagonal-dipyramidal D6h 6/mmm *622 [2,6] centrosymmetric 24
cubic tetrahedral T 23 332 [3,3]+ enantiomorphic 12 alternating
hextetrahedral Td 43m *332 [3,3] non-centrosymmetric 24 symmetric
diploidal Th m3 3*2 [3+,4] centrosymmetric 24
gyroidal O 432 432 [4,3]+ enantiomorphic 24 symmetric
hexoctahedral Oh 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 11 enantiomorphic point groups (biological molecules are usually chiral).

Bravais lattices[edit]

Main article: Bravais lattice

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 Ci Triclinic
monoclinic C2h Monoclinic, simple Monoclinic, centered
orthorhombic D2h Orthorhombic, simple Orthorhombic, base-centered Orthorhombic, body-centered Orthorhombic, face-centered
tetragonal D4h Tetragonal, simple Tetragonal, body-centered
hexagonal rhombohedral D3d Rhombohedral
hexagonal D6h Hexagonal
cubic Oh Cubic, simple Cubic, body-centered Cubic, face-centered

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 = 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 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.

Crystal systems 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:

Crystal families in 4D space
No. Family Edge lengths Interaxial angles
1 Hexaclinic abcd αβγδεζ ≠ 90°
2 Triclinic abcd αβγ ≠ 90°
δ = ε = ζ = 90°
3 Diclinic abcd α ≠ 90°
β = γ = δ = ε = 90°
ζ ≠ 90°
4 Monoclinic abcd α ≠ 90°
β = γ = δ = ε = ζ = 90°
5 Orthogonal abcd α = β = γ = δ = ε = ζ = 90°
6 Tetragonal monoclinic ab = cd α ≠ 90°
β = γ = δ = ε = ζ = 90°
7 Hexagonal monoclinic ab = cd α ≠ 90°
β = γ = δ = ε = 90°
ζ = 120°
8 Ditetragonal diclinic a = db = c α = ζ = 90°
β = ε ≠ 90°
γ ≠ 90°
δ = 180° − γ
9 Ditrigonal (dihexagonal) diclinic a = db = c α = ζ = 120°
β = ε ≠ 90°
γδ ≠ 90°
cos δ = cos β − cos γ
10 Tetragonal orthogonal ab = cd α = β = γ = δ = ε = ζ = 90°
11 Hexagonal orthogonal ab = cd α = β = γ = δ = ε = 90°, ζ = 120°
12 Ditetragonal monoclinic a = db = c α = γ = δ = ζ = 90°
β = ε ≠ 90°
13 Ditrigonal (dihexagonal) monoclinic a = db = c α = ζ = 120°
β = ε ≠ 90°
γ = δ ≠ 90°
cos γ = −1/2cos β
14 Ditetragonal orthogonal a = db = c α = β = γ = δ = ε = ζ = 90°
15 Hexagonal tetragonal a = db = c α = β = γ = δ = ε = 90°
ζ = 120°
16 Dihexagonal orthogonal a = db = c α = ζ = 120°
β = γ = δ = ε = 90°
17 Cubic orthogonal a = b = cd 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 = ζβ = 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 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.

Crystal systems in 4D space
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]

  1. ^ Flack, Howard D. (2003). "Chiral and Achiral Crystal Structures". Helvetica Chimica Acta. 86 (4): 905–921. doi:10.1002/hlca.200390109. 
  2. ^ Hahn (2002), p. 804
  3. ^ a b Whittaker, E. J. W. (1985). An Atlas of Hyperstereograms of the Four-Dimensional Crystal Classes. Oxford & New York: Clarendon Press. 
  4. ^ a b Brown, H.; Bülow, R.; Neubüser, J.; Wondratschek, H.; Zassenhaus, H. (1978). Crystallographic Groups of Four-Dimensional Space. New York: Wiley. 

External links[edit]