In crystallography, a crystallographic point group is a set of symmetry operations, corresponding to one of the point groups in three dimensions, such that each operation would leave the structure of a crystal unchanged i.e. the same kinds of atoms would be placed in similar positions as before the transformation. For example, in a primitive cubic crystal system, a rotation of the unit cell by 90 degree around an axis that is perpendicular to two parallel faces of the cube, intersecting at its center, is a symmetry operation that projects each atom to the location of one of its neighbor leaving the overall structure of the crystal unaffected.
In the classification of crystals, each point group defines a so-called (geometric) crystal class. There are infinitely many three-dimensional point groups. However, the crystallographic restriction on the general point groups results in there being only 32 crystallographic point groups. These 32 point groups are one-and-the-same as the 32 types of morphological (external) crystalline symmetries derived in 1830 by Johann Friedrich Christian Hessel from a consideration of observed crystal forms.
The point group of a crystal determines, among other things, the directional variation of physical properties that arise from its structure, including optical properties such as birefringency, or electro-optical features such as the Pockels effect. For a periodic crystal (as opposed to a quasicrystal), the group must maintain the three-dimensional translational symmetry that defines crystallinity.
The point groups are named according to their component symmetries. There are several standard notations used by crystallographers, mineralogists, and physicists.
For the correspondence of the two systems below, see crystal system.
In Schoenflies notation, point groups are denoted by a letter symbol with a subscript. The symbols used in crystallography mean the following:
Due to the crystallographic restriction theorem, n = 1, 2, 3, 4, or 6 in 2- or 3-dimensional space.
n | 1 | 2 | 3 | 4 | 6 |
---|---|---|---|---|---|
C_{n} | C_{1} | C_{2} | C_{3} | C_{4} | C_{6} |
C_{nv} | C_{1v}=C_{1h} | C_{2v} | C_{3v} | C_{4v} | C_{6v} |
C_{nh} | C_{1h} | C_{2h} | C_{3h} | C_{4h} | C_{6h} |
D_{n} | D_{1}=C_{2} | D_{2} | D_{3} | D_{4} | D_{6} |
D_{nh} | D_{1h}=C_{2v} | D_{2h} | D_{3h} | D_{4h} | D_{6h} |
D_{nd} | D_{1d}=C_{2h} | D_{2d} | D_{3d} | D_{4d} | D_{6d} |
S_{2n} | S_{2} | S_{4} | S_{6} | S_{8} | S_{12} |
D_{4d} and D_{6d} are actually forbidden because they contain improper rotations with n=8 and 12 respectively. The 27 point groups in the table plus T, T_{d}, T_{h}, O and O_{h} constitute 32 crystallographic point groups.
An abbreviated form of the Hermann-Mauguin notation commonly used for space groups also serves to describe crystallographic point groups. Group names are
Class | Group names | |||||||
---|---|---|---|---|---|---|---|---|
Cubic | 23 | m3 | 432 | 43m | m3m | |||
Hexagonal | 6 | 6 | ^{6}/_{m} | 622 | 6mm | 6m2 | 6/mmm | |
Trigonal | 3 | 3 | 32 | 3m | 3m | |||
Tetragonal | 4 | 4 | ^{4}/_{m} | 422 | 4mm | 42m | 4/mmm | |
Orthorhombic | 222 | mm2 | mmm | |||||
Monoclinic | 2 | ^{2}/_{m} | m | |||||
Triclinic | 1 | 1 | Subgroup relations of the 32 crystallographic point groups (rows represent group orders from bottom to top as: 1,2,3,4,6,8,12,16,24, and 48.) |
Crystal system | Hermann-Mauguin | Shubnikov^{[1]} | Schoenflies | Orbifold | Coxeter | Order | |
---|---|---|---|---|---|---|---|
(full) | (short) | ||||||
Triclinic | 1 | 1 | C_{1} | 11 | [ ]^{+} | 1 | |
1 | 1 | C_{i} = S_{2} | × | [2^{+},2^{+}] | 2 | ||
Monoclinic | 2 | 2 | C_{2} | 22 | [2]^{+} | 2 | |
m | m | C_{s} = C_{1h} | * | [ ] | 2 | ||
2/m | C_{2h} | 2* | [2,2^{+}] | 4 | |||
Orthorhombic | 222 | 222 | D_{2} = V | 222 | [2,2]^{+} | 4 | |
mm2 | mm2 | C_{2v} | *22 | [2] | 4 | ||
mmm | D_{2h} = V_{h} | *222 | [2,2] | 8 | |||
Tetragonal | 4 | 4 | C_{4} | 44 | [4]^{+} | 4 | |
4 | 4 | S_{4} | 2× | [2^{+},4^{+}] | 4 | ||
4/m | C_{4h} | 4* | [2,4^{+}] | 8 | |||
422 | 422 | D_{4} | 422 | [4,2]^{+} | 8 | ||
4mm | 4mm | C_{4v} | *44 | [4] | 8 | ||
42m | 42m | D_{2d} = V_{d} | 2*2 | [2^{+},4] | 8 | ||
4/mmm | D_{4h} | *422 | [4,2] | 16 | |||
Trigonal | 3 | 3 | C_{3} | 33 | [3]^{+} | 3 | |
3 | 3 | C_{3i} = S_{6} | 3× | [2^{+},6^{+}] | 6 | ||
32 | 32 | D_{3} | 322 | [3,2]^{+} | 6 | ||
3m | 3m | C_{3v} | *33 | [3] | 6 | ||
3 | 3m | D_{3d} | 2*3 | [2^{+},6] | 12 | ||
Hexagonal | 6 | 6 | C_{6} | 66 | [6]^{+} | 6 | |
6 | 6 | C_{3h} | 3* | [2,3^{+}] | 6 | ||
6/m | C_{6h} | 6* | [2,6^{+}] | 12 | |||
622 | 622 | D_{6} | 622 | [6,2]^{+} | 12 | ||
6mm | 6mm | C_{6v} | *66 | [6] | 12 | ||
6m2 | 6m2 | D_{3h} | *322 | [3,2] | 12 | ||
6/mmm | D_{6h} | *622 | [6,2] | 24 | |||
Cubic | 23 | 23 | T | 332 | [3,3]^{+} | 12 | |
3 | m3 | T_{h} | 3*2 | [3^{+},4] | 24 | ||
432 | 432 | O | 432 | [4,3]^{+} | 24 | ||
43m | 43m | T_{d} | *332 | [3,3] | 24 | ||
3 | m3m | O_{h} | *432 | [4,3] | 48 |
Many of the crystallographic point groups share the same internal structure. For example, the point groups 1, 2, and m contain different geometric symmetry operations, (inversion, rotation, and reflection, respectively) but all share the structure of the cyclic group Z_{2}. All isomorphic groups are of the same order, but not all groups of the same order are isomorphic. The point groups which are isomorphic are shown in the following table:^{[2]}
Hermann-Mauguin | Schoenflies | Order | Abstract group | |
---|---|---|---|---|
1 | C_{1} | 1 | Z_{1} | |
1 | C_{i} = S_{2} | 2 | Z_{2} | |
2 | C_{2} | 2 | ||
m | C_{s} = C_{1h} | 2 | ||
3 | C_{3} | 3 | Z_{3} | |
4 | C_{4} | 4 | Z_{4} | |
4 | S_{4} | 4 | ||
2/m | C_{2h} | 4 | D_{2} = Z_{2} × Z_{2} | |
222 | D_{2} = V | 4 | ||
mm2 | C_{2v} | 4 | ||
3 | C_{3i} = S_{6} | 6 | Z_{6} | |
6 | C_{6} | 6 | ||
6 | C_{3h} | 6 | ||
32 | D_{3} | 6 | D_{3} | |
3m | C_{3v} | 6 | ||
mmm | D_{2h} = V_{h} | 8 | D_{2} × Z_{2} | |
4/m | C_{4h} | 8 | Z_{4} × Z_{2} | |
422 | D_{4} | 8 | D_{4} | |
4mm | C_{4v} | 8 | ||
42m | D_{2d} = V_{d} | 8 | ||
6/m | C_{6h} | 12 | Z_{6} × Z_{2} | |
23 | T | 12 | A_{4} | |
3m | D_{3d} | 12 | D_{6} | |
622 | D_{6} | 12 | ||
6mm | C_{6v} | 12 | ||
6m2 | D_{3h} | 12 | ||
4/mmm | D_{4h} | 16 | D_{4} × Z_{2} | |
6/mmm | D_{6h} | 24 | D_{6} × Z_{2} | |
m3 | T_{h} | 24 | A_{4} × Z_{2} | |
432 | O | 24 | S_{4} | |
43m | T_{d} | 24 | ||
m3m | O_{h} | 48 | S_{4} × Z_{2} |
This table makes use of cyclic groups (Z_{1}, Z_{2}, Z_{3}, Z_{4}, Z_{6}), dihedral groups (D_{2}, D_{3}, D_{4}, D_{6}), one of the alternating groups (A_{4}), and one of the symmetric groups (S_{4}). Here the symbol " × " indicates a direct product.