In crystallography, a crystallographic point group is a three dimensional point group whose symmetry operations are compatible with a three dimensional crystallographic lattice. According to the crystallographic restriction it may only contain one-, two-, three-, four- and sixfold rotations or rotoinversions. This reduces the number of crystallographic point groups to 32 (from an infinity of general point groups). These 32 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.
In the classification of crystals, to each space group is associated a crystallographic point group by "forgetting" the translational components of the symmetry operations. That is, by turning screw rotations into rotations, glide reflections into reflections and moving all symmetry elements into the origin. Each crystallographic point group defines the (geometric) crystal class of the crystal.
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.
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.
Main article: Schoenflies notation |
Further information: Point groups in three dimensions |
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.
Main article: Hermann–Mauguin notation |
An abbreviated form of the Hermann–Mauguin notation commonly used for space groups also serves to describe crystallographic point groups. Group names are
Crystal family | Crystal system | Group names | ||||||
---|---|---|---|---|---|---|---|---|
Cubic | 23 | m3 | 432 | 43m | m3m | |||
Hexagonal | 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 |
Crystal family | 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 | ||||
Hexagonal | 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 |
See also: Crystal structure § Crystal systems |
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 C_{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 | C_{1} | |
1 | C_{i} = S_{2} | 2 | C_{2} | |
2 | C_{2} | 2 | ||
m | C_{s} = C_{1h} | 2 | ||
3 | C_{3} | 3 | C_{3} | |
4 | C_{4} | 4 | C_{4} | |
4 | S_{4} | 4 | ||
2/m | C_{2h} | 4 | D_{2} = C_{2} × C_{2} | |
222 | D_{2} = V | 4 | ||
mm2 | C_{2v} | 4 | ||
3 | C_{3i} = S_{6} | 6 | C_{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} × C_{2} | |
4/m | C_{4h} | 8 | C_{4} × C_{2} | |
422 | D_{4} | 8 | D_{4} | |
4mm | C_{4v} | 8 | ||
42m | D_{2d} = V_{d} | 8 | ||
6/m | C_{6h} | 12 | C_{6} × C_{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} × C_{2} | |
6/mmm | D_{6h} | 24 | D_{6} × C_{2} | |
m3 | T_{h} | 24 | A_{4} × C_{2} | |
432 | O | 24 | S_{4} | |
43m | T_{d} | 24 | ||
m3m | O_{h} | 48 | S_{4} × C_{2} |
This table makes use of cyclic groups (C_{1}, C_{2}, C_{3}, C_{4}, C_{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.