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 | Group structure |
---|---|---|---|---|---|---|---|---|---|
triclinic | triclinic-pedial | C1 | 1 | 11 | + | enantiomorphic polar | 1 | trivial | |
triclinic-pinacoidal | Ci | 1 | 1x | centrosymmetric | 2 | cyclic | |||
monoclinic | monoclinic-sphenoidal | C2 | 2 | 22 | + | enantiomorphic polar | 2 | cyclic | |
monoclinic-domatic | Cs | m | *11 | polar | 2 | cyclic | |||
monoclinic-prismatic | C2h | 2/m | 2* | centrosymmetric | 4 | 2×cyclic | |||
orthorhombic | orthorhombic-sphenoidal | D2 | 222 | 222 | + | enantiomorphic | 4 | dihedral | |
orthorhombic-pyramidal | C2v | mm2 | *22 | polar | 4 | dihedral | |||
orthorhombic-bipyramidal | D2h | mmm | *222 | centrosymmetric | 8 | 2×dihedral | |||
tetragonal | tetragonal-pyramidal | C4 | 4 | 44 | + | enantiomorphic polar | 4 | cyclic | |
tetragonal-disphenoidal | S4 | 4 | 2x | non-centrosymmetric | 4 | cyclic | |||
tetragonal-dipyramidal | C4h | 4/m | 4* | centrosymmetric | 8 | 2×cyclic | |||
tetragonal-trapezoidal | D4 | 422 | 422 | + | enantiomorphic | 8 | dihedral | ||
ditetragonal-pyramidal | C4v | 4mm | *44 | polar | 8 | dihedral | |||
tetragonal-scalenoidal | D2d | 42m or 4m2 | 2*2 | non-centrosymmetric | 8 | dihedral | |||
ditetragonal-dipyramidal | D4h | 4/mmm | *422 | centrosymmetric | 16 | 2×dihedral | |||
hexagonal | trigonal | trigonal-pyramidal | C3 | 3 | 33 | + | enantiomorphic polar | 3 | cyclic |
rhombohedral | S6 (C3i) | 3 | 3x | centrosymmetric | 6 | cyclic | |||
trigonal-trapezoidal | D3 | 32 or 321 or 312 | 322 | + | enantiomorphic | 6 | dihedral | ||
ditrigonal-pyramidal | C3v | 3m or 3m1 or 31m | *33 | polar | 6 | dihedral | |||
ditrigonal-scalahedral | D3d | 3m or 3m1 or 31m | 2*3 | centrosymmetric | 12 | dihedral | |||
hexagonal | hexagonal-pyramidal | C6 | 6 | 66 | + | enantiomorphic polar | 6 | cyclic | |
trigonal-dipyramidal | C3h | 6 | 3* | non-centrosymmetric | 6 | cyclic | |||
hexagonal-dipyramidal | C6h | 6/m | 6* | centrosymmetric | 12 | 2×cyclic | |||
hexagonal-trapezoidal | D6 | 622 | 622 | + | enantiomorphic | 12 | dihedral | ||
dihexagonal-pyramidal | C6v | 6mm | *66 | polar | 12 | dihedral | |||
ditrigonal-dipyramidal | D3h | 6m2 or 62m | *322 | non-centrosymmetric | 12 | dihedral | |||
dihexagonal-dipyramidal | D6h | 6/mmm | *622 | centrosymmetric | 24 | 2×dihedral | |||
cubic | tetrahedral | T | 23 | 332 | + | enantiomorphic | 12 | alternating | |
hextetrahedral | Td | 43m | *332 | non-centrosymmetric | 24 | symmetric | |||
diploidal | Th | m3 | 3*2 | centrosymmetric | 24 | 2×alternating | |||
gyroidal | O | 432 | 432 | + | enantiomorphic | 24 | symmetric | ||
hexoctahedral | Oh | m3m | *432 | centrosymmetric | 48 | 2×symmetric |
Point symmetry can be thought of in the following fashion: consider the coordinates which make up the lattice, and project them all through a single point, so that (x,y,z) becomes (-x,-y,-z). This is the 'reciprocal lattice.' If the lattice and reciprocal lattice are identical, then the crystal is centrosymmetric. If the reciprocal lattice can be rotated to align with the lattice, then the crystal is non-centrosymmetric. If the reciprocal lattice can't be rotated to align with the lattice, that is, with some elements which are a mirror image of the lattice, then the crystal is enantiomorphic. If rotation of the original lattice reveals an axis where the two ends are different, then the crystal is polar. H2O is a common example of a polar molecule.
The crystal structures of chiral biological molecules (such as protein structures) can only occur in the 11 enantiomorphic point groups (biological molecules are frequently 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.
Read more about this topic: Crystal System
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