Definitions

gyroidal

Crystal system

A crystal system is a category of space groups, which characterize symmetry of structures in three dimensions with translational symmetry in three directions, having a discrete class of point groups. A major application is in crystallography, to categorize crystals, but by itself the topic is one of 3D Euclidean geometry.

Overview

There are 7 crystal systems:

  • Triclinic, all cases not satisfying the requirements of any other system. There is no necessary symmetry other than translational symmetry, although inversion is possible.
  • Monoclinic, requires either 1 twofold axis of rotation or 1 mirror plane.
  • Orthorhombic, requires either 3 twofold axes of rotation or 1 twofold axis of rotation and two mirror planes.
  • Tetragonal, requires 1 fourfold axis of rotation.
  • Rhombohedral, also called trigonal, requires 1 threefold axis of rotation.
  • Hexagonal, requires 1 sixfold axis of rotation.
  • Isometric or cubic, requires 4 threefold axes of rotation.

There are 2, 13, 59, 68, 25, 27, and 36 space groups per crystal system, respectively, for a total of 230. The following table gives a brief characterization of the various crystal systems:

Crystal system '''No. of point groups No. of bravais lattices No. of space groups
Triclinic 2 1 2
Monoclinic 3 2 13
Orthorhombic 3 4 59
Tetragonal 7 2 68
Rhombohedral 5 1 25
Hexagonal 7 1 27
Cubic 5 3 36
Total 32 14 230

Within a crystal system there are two ways of categorizing space groups:

  • by the linear parts of symmetries, i.e. by crystal class, also called crystallographic point group; each of the 32 crystal classes applies for one of the 7 crystal systems
  • by the symmetries in the translation lattice, i.e. by Bravais lattice; each of the 14 Bravais lattices applies for one of the 7 crystal systems.

The 73 symmorphic space groups (see space group) are largely combinations, within each crystal system, of each applicable point group with each applicable Bravais lattice: there are 2, 6, 12, 14, 5, 7, and 15 combinations, respectively, together 61.

Crystallographic point groups

A symmetry group consists of isometric affine transformations; each is given by an orthogonal matrix and a translation vector (which may be the zero vector). Space groups can be grouped by the matrices involved, i.e. ignoring the translation vectors (see also Euclidean group). This corresponds to discrete symmetry groups with a fixed point. There are infinitely many of these point groups in three dimensions. However, only part of these are compatible with translational symmetry: the crystallographic point groups. This is expressed in the crystallographic restriction theorem. (In spite of these names, this is a geometric limitation, not just a physical one.)

The point group of a crystal, among other things, determines the symmetry of the crystal's optical properties. For instance, one knows whether it is birefringent, or whether it shows the Pockels effect, by simply knowing its point group.

Overview of point groups by crystal system

crystal system point group / crystal class Schönflies Hermann-Mauguin orbifold Type
triclinic triclinic-pedial C1 1 11 enantiomorphic polar
triclinic-pinacoidal Ci bar{1} 1x centrosymmetric
monoclinic monoclinic-sphenoidal C2 2 22 enantiomorphic polar
monoclinic-domatic Cs m 1* polar
monoclinic-prismatic C2h 2/m 2* centrosymmetric
orthorhombic orthorhombic-sphenoidal D2 222 222 enantiomorphic
orthorhombic-pyramidal C2v mm2 *22 polar
orthorhombic-bipyramidal D2h mmm *222 centrosymmetric
tetragonal tetragonal-pyramidal C4 4 44 enantiomorphic polar
tetragonal-disphenoidal S4 bar{4} 2x
tetragonal-dipyramidal C4h 4/m 4* centrosymmetric
tetragonal-trapezoidal D4 422 422 enantiomorphic
ditetragonal-pyramidal C4v 4mm *44 polar
tetragonal-scalenoidal D2d bar{4}2m or bar{4}m2 2*2
ditetragonal-dipyramidal D4h 4/mmm *422 centrosymmetric
rhombohedral (trigonal) trigonal-pyramidal C3 3 ! 33 enantiomorphic polar
rhombohedral S6 (C3i) bar{3} 3x centrosymmetric
trigonal-trapezoidal D3 32 or 321 or 312 322 enantiomorphic
ditrigonal-pyramidal C3v 3m or 3m1 or 31m *33 polar
ditrigonal-scalahedral D3d bar{3} m or bar{3} m 1 or bar{3} 1 m 2*3 centrosymmetric
hexagonal hexagonal-pyramidal C6 6 66 enantiomorphic polar
trigonal-dipyramidal C3h bar{6} 3*
hexagonal-dipyramidal C6h 6/m 6* centrosymmetric
hexagonal-trapezoidal D6 622 622 enantiomorphic
dihexagonal-pyramidal C6v 6mm *66 polar
ditrigonal-dipyramidal D3h bar{6}m2 or bar{6}2m *322
dihexagonal-dipyramidal D6h 6/mmm *622 centrosymmetric
cubic tetartohedral T 23 332 enantiomorphic
diploidal Th mbar{3} 3*2 centrosymmetric
gyroidal O 432 432 enantiomorphic
tetrahedral Td bar{4}3m *332
hexoctahedral Oh mbar{3}m *432 centrosymmetric

The crystal structures of biological molecules (such as protein structures) can only occur in the 11 enantiomorphic point groups, as biological molecules are invariably chiral. The protein assemblies themselves may have symmetries other than those given above, because they are not intrinsically restricted by the Crystallographic restriction theorem. For example the Rad52 DNA binding protein has an 11-fold rotational symmetry (in human), however, it must form crystals in one of the 11 enantiomorphic point groups given above.

Classification of lattices

The 7 Crystal systems The 14 Bravais Lattices
triclinic (parallelepiped)

monoclinic (right prism with parallelogram base; here seen from above) simple centered

orthorhombic (cuboid) simple base-centered body-centered face-centered

tetragonal (square cuboid) simple body-centered

rhombohedral
(trigonal) (3-sided trapezohedron)
hexagonal (centered regular hexagon)
cubic
(isometric; cube)
simple body-centered 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

mathbf{R} = n_1 mathbf{a}_1 + n_2 mathbf{a}_2 + n_3 mathbf{a}_3,

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 crystal 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 (1801-1869), in 1842, who found that there were 15 Bravais lattices. This was corrected to 14 by A. Bravais in 1848.

See also

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