Crystallographic point group

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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.

Contents

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.

Notation

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 .

Schoenflies notation

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.

n12346
CnC1C2C3C4C6
CnvC1v=C1hC2vC3vC4vC6v
CnhC1hC2hC3hC4hC6h
DnD1=C2D2D3D4D6
DnhD1h=C2vD2hD3hD4hD6h
DndD1d=C2hD2dD3dD4dD6d
S2nS2S4S6S8S12

D4d and D6d are actually forbidden because they contain improper rotations with n=8 and 12 respectively. The 27 point groups in the table plus T, Td, Th, O and Oh constitute 32 crystallographic point groups.

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 familyCrystal systemGroup names
Cubic 23m343243mm3m
Hexagonal Hexagonal666m6226mm6m26/mmm
Trigonal33323m3m
Tetragonal 444m4224mm42m4/mmm
Orthorhombic 222mm2mmm
Monoclinic 22mm
Triclinic 11

The correspondence between different notations

Crystal family Crystal system Hermann-Mauguin Shubnikov [1] Schoenflies Orbifold Coxeter Order
(full)(short)
Triclinic 11C111[ ]+1
11Ci = S2×[2+,2+]2
Monoclinic 22C222[2]+2
mmCs = C1h*[ ]2
2/mC2h2*[2,2+]4
Orthorhombic 222222D2 = V222[2,2]+4
mm2mm2C2v*22[2]4
mmmD2h = Vh*222[2,2]8
Tetragonal 44C444[4]+4
44S4[2+,4+]4
4/mC4h4*[2,4+]8
422422D4422[4,2]+8
4mm4mmC4v*44[4]8
42m42mD2d = Vd2*2[2+,4]8
4/mmmD4h*422[4,2]16
Hexagonal Trigonal33C333[3]+3
33C3i = S6[2+,6+]6
3232D3322[3,2]+6
3m3mC3v*33[3]6
33mD3d2*3[2+,6]12
Hexagonal66C666[6]+6
66C3h3*[2,3+]6
6/mC6h6*[2,6+]12
622622D6622[6,2]+12
6mm6mmC6v*66[6]12
6m26m2D3h*322[3,2]12
6/mmmD6h*622[6,2]24
Cubic 2323T332[3,3]+12
3m3Th3*2[3+,4]24
432432O432[4,3]+24
43m43mTd*332[3,3]24
3m3mOh*432[4,3]48

Isomorphisms

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 C2. 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
1C11 C1
1Ci = S22 C2
2C22
mCs = C1h2
3C33 C3
4C44 C4
4S44
2/m C2h4 D2 = C2 × C2
 222D2 = V4
mm2C2v 4
3C3i = S66 C6
6C66
6C3h6
32D36 D3
3mC3v6
mmmD2h = Vh8D2 × C2
 4/mC4h8C4 × C2
422D48 D4
4mmC4v8
42mD2d = Vd8
6/mC6h12C6 × C2
23T12 A4
3mD3d12 D6
622D612
6mmC6v12
6m2D3h12
4/mmmD4h16D4 × C2
6/mmmD6h24D6 × C2
m3Th24A4 × C2
432O  24 S4
43mTd24
m3mOh48S4 × C2

This table makes use of cyclic groups (C1, C2, C3, C4, C6), dihedral groups (D2, D3, D4, D6), one of the alternating groups (A4), and one of the symmetric groups (S4). Here the symbol " × " indicates a direct product.

Deriving the crystallographic point group (crystal class) from the space group

  1. Leave out the Bravais lattice type.
  2. Convert all symmetry elements with translational components into their respective symmetry elements without translation symmetry. (Glide planes are converted into simple mirror planes; screw axes are converted into simple axes of rotation.)
  3. Axes of rotation, rotoinversion axes, and mirror planes remain unchanged.

See also

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References

  1. "(International Tables) Abstract". Archived from the original on 2013-07-04. Retrieved 2011-11-25.
  2. Novak, I (1995-07-18). "Molecular isomorphism". European Journal of Physics. 16 (4). IOP Publishing: 151–153. Bibcode:1995EJPh...16..151N. doi:10.1088/0143-0807/16/4/001. ISSN   0143-0807. S2CID   250887121.