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Crystallographic point group

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Template:Short description 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 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 1867 Axel Gadolin, who was unaware of the previous work of Hessel, found the crystallographic point groups independently using stereographic projection to represent the symmetry elements of the 32 groups.<ref name="Authier">Template:Cite book</ref>Template:Rp

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.

NotationEdit

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 notationEdit

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In Schoenflies notation, point groups are denoted by a letter symbol with a subscript. The symbols used in crystallography mean the following:

  • Cn (for cyclic) indicates that the group has an n-fold rotation axis. Cnh is Cn with the addition of a mirror (reflection) plane perpendicular to the axis of rotation. Cnv is Cn with the addition of n mirror planes parallel to the axis of rotation.
  • S2n (for Spiegel, German for mirror) denotes a group with only a 2n-fold rotation-reflection axis.
  • Dn (for dihedral, or two-sided) indicates that the group has an n-fold rotation axis plus n twofold axes perpendicular to that axis. Dnh has, in addition, a mirror plane perpendicular to the n-fold axis. Dnd has, in addition to the elements of Dn, mirror planes parallel to the n-fold axis.
  • The letter T (for tetrahedron) indicates that the group has the symmetry of a tetrahedron. Td includes improper rotation operations, T excludes improper rotation operations, and Th is T with the addition of an inversion.
  • The letter O (for octahedron) indicates that the group has the symmetry of an octahedron, with (Oh) or without (O) improper operations (those that change handedness).

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
Cn C1 C2 C3 C4 C6
Cnv C1v=C1h C2v C3v C4v C6v
Cnh C1h C2h C3h C4h C6h
Dn D1=C2 D2 D3 D4 D6
Dnh D1h=C2v D2h D3h D4h D6h
Dnd D1d=C2h D2d D3d D4d D6d
S2n S2 S4 S6 S8 S12

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 notationEdit

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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 mTemplate:Overline 432 Template:Overline3m mTemplate:Overlinem
Hexagonal Hexagonal 6 Template:Overline 6⁄m 622 6mm Template:Overlinem2 6/mmm
Trigonal 3 Template:Overline 32 3m Template:Overlinem
Tetragonal 4 Template:Overline 4⁄m 422 4mm Template:Overline2m 4/mmm
Orthorhombic 222 mm2 mmm
Monoclinic 2 2⁄m m
Triclinic 1 Template:Overline

The correspondence between different notationsEdit

Crystal family Crystal system Hermann-Mauguin Shubnikov<ref>{{#invoke:citation/CS1|citation CitationClass=web

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Schoenflies Orbifold Coxeter Order
(full) (short)
Triclinic 1 1 <math>1\ </math> C1 11 [ ]+ 1
Template:Overline Template:Overline <math>\tilde{2}</math> Ci = S2 × [2+,2+] 2
Monoclinic 2 2 <math>2\ </math> C2 22 [2]+ 2
m m <math>m\ </math> Cs = C1h * [ ] 2
<math>\tfrac{2}{m}</math> 2/m <math>2:m\ </math> C2h 2* [2,2+] 4
Orthorhombic 222 222 <math>2:2\ </math> D2 = V 222 [2,2]+ 4
mm2 mm2 <math>2 \cdot m\ </math> C2v *22 [2] 4
<math>\tfrac{2}{m}\tfrac{2}{m}\tfrac{2}{m}</math> mmm <math>m \cdot 2:m\ </math> D2h = Vh *222 [2,2] 8
Tetragonal 4 4 <math>4\ </math> C4 44 [4]+ 4
Template:Overline Template:Overline <math>\tilde{4}</math> S4 2× [2+,4+] 4
<math>\tfrac{4}{m}</math> 4/m <math>4:m\ </math> C4h 4* [2,4+] 8
422 422 <math>4:2\ </math> D4 422 [4,2]+ 8
4mm 4mm <math>4 \cdot m\ </math> C4v *44 [4] 8
Template:Overline2m Template:Overline2m <math>\tilde{4}\cdot m</math> D2d = Vd 2*2 [2+,4] 8
<math>\tfrac{4}{m}\tfrac{2}{m}\tfrac{2}{m}</math> 4/mmm <math>m \cdot 4:m\ </math> D4h *422 [4,2] 16
Hexagonal Trigonal 3 3 <math>3\ </math> C3 33 [3]+ 3
Template:Overline Template:Overline <math>\tilde{6}</math> C3i = S6 3× [2+,6+] 6
32 32 <math>3:2\ </math> D3 322 [3,2]+ 6
3m 3m <math>3 \cdot m\ </math> C3v *33 [3] 6
Template:Overline<math>\tfrac{2}{m}</math> Template:Overlinem <math>\tilde{6}\cdot m</math> D3d 2*3 [2+,6] 12
Hexagonal 6 6 <math>6\ </math> C6 66 [6]+ 6
Template:Overline Template:Overline <math>3:m\ </math> C3h 3* [2,3+] 6
<math>\tfrac{6}{m}</math> 6/m <math>6:m\ </math> C6h 6* [2,6+] 12
622 622 <math>6:2\ </math> D6 622 [6,2]+ 12
6mm 6mm <math>6 \cdot m\ </math> C6v *66 [6] 12
Template:Overlinem2 Template:Overlinem2 <math>m \cdot 3:m\ </math> D3h *322 [3,2] 12
<math>\tfrac{6}{m}\tfrac{2}{m}\tfrac{2}{m}</math> 6/mmm <math>m \cdot 6:m\ </math> D6h *622 [6,2] 24
Cubic 23 23 <math>3/2\ </math> T 332 [3,3]+ 12
<math>\tfrac{2}{m}</math>Template:Overline mTemplate:Overline <math>\tilde{6}/2</math> Th 3*2 [3+,4] 24
432 432 <math>3/4\ </math> O 432 [4,3]+ 24
Template:Overline3m Template:Overline3m <math>3/\tilde{4}</math> Td *332 [3,3] 24
<math>\tfrac{4}{m}</math>Template:Overline<math>\tfrac{2}{m}</math> mTemplate:Overlinem <math>\tilde{6}/4</math> Oh *432 [4,3] 48

IsomorphismsEdit

Template:See also

Many of the crystallographic point groups share the same internal structure. For example, the point groups Template:Overline, 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:<ref>Template:Cite journal</ref>

Hermann–Mauguin Schoenflies Order Abstract group
1 C1 1 C1 <math>G_1^1</math>
Template:Overline Ci = S2 2 C2 <math>G_2^1</math>
2 C2 2
m Cs = C1h 2
3 C3 3 C3 <math>G_3^1</math>
4 C4 4 C4 <math>G_4^1</math>
Template:Overline S4 4
2/m  C2h 4 D2 = C2 × C2 <math>G_4^2</math>
 222 D2 = V 4
mm2 C2v  4
Template:Overline C3i = S6 6 C6 <math>G_6^1</math>
6 C6 6
Template:Overline C3h 6
32 D3 6 D3 <math>G_6^2</math>
3m C3v 6
mmm D2h = Vh 8 D2 × C2 <math>G_8^3</math>
 4/m C4h 8 C4 × C2 <math>G_8^2</math>
422 D4 8 D4 <math>G_8^4</math>
4mm C4v 8
Template:Overline2m D2d = Vd 8
6/m C6h 12 C6 × C2 <math>G_{12}^2</math>
23 T 12 A4 <math>G_{12}^5</math>
Template:Overlinem D3d 12 D6 <math>G_{12}^3</math>
622 D6 12
6mm C6v 12
Template:Overlinem2 D3h 12
4/mmm D4h 16 D4 × C2 <math>G_{16}^9</math>
6/mmm D6h 24 D6 × C2 <math>G_{24}^5</math>
mTemplate:Overline Th 24 A4 × C2 <math>G_{24}^{10}</math>
432 O   24 S4 <math>G_{24}^{7}</math>
Template:Overline3m Td 24
mTemplate:Overlinem Oh 48 S4 × C2 <math>G_{48}^7</math>

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 groupEdit

  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 alsoEdit

ReferencesEdit

<references />

External linksEdit

Template:Crystal systems

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