Crystal System - Crystal Classes

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.

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