# Point group

A point group is a special type of symmetry group in Euclidean geometry that describes the symmetry of a finite body. All point groups are characterized by the fact that there is a point which is mapped back onto itself by all symmetry operations of the point group. Based on Neumann's principle , the point group determines the macroscopic properties of the body. Further statements can be obtained with the help of representation theory .

The point groups are used in molecular physics and crystallography , where the 32 crystallographic point groups are also called crystal classes. The point groups are designated in the Schoenflies notation . In crystallography, the Hermann Mauguin symbolism is now mainly used.

## Mathematical basics

The symmetry of a body is mathematically described as the set of all possible symmetry operations ( symmetry group ). With symmetry operations are meant Euclidean movements that represent the body on itself. A distinction must be made here between even movements that maintain the orientation and odd movements that reverse the orientation, e.g. B. Reflections on planes.

Possible symmetry operations in point groups in three-dimensional Euclidean vector space, the symmetry operations, which have a fixed point at least: identity mapping , point reflection on a center of inversion , reflection on a mirror plane , rotation about an axis of rotation , as well as a combination thereof rotary reflection or the equivalent of rotary inversion . The translation , the screwing and the glide mirror cannot be elements of a point group because they do not have a fixed point.

If one understands the successive execution of symmetry operations as additive operation, one recognizes that a set of symmetry operations is a (usually non- commutative ) group .

There are both discrete and continuous point groups. The discrete point groups can be divided into two different types:

• Point groups with a maximum of one axis of rotation with a count greater than two,
• Point groups with at least two axes of rotation with a count greater than two.

The discrete point groups with a maximum of one marked -numerous axis of rotation can also be combined with mirror planes and twofold axes of rotation. There are a total of the following options: ${\ displaystyle n}$

group Group symbol (Schönflies) Explanation
Turning group C n An n-fold axis of rotation
C nv 1 C n axis + n mirror planes that contain this axis (v: vertical mirror plane)
C nh 1 C n -axis + 1 mirror plane perpendicular to this axis (h: horizontal mirror plane)
Dihedral group D n 1 C n axis + n C 2 axes perpendicular to it
D nd 1 D n axis + n mirror planes that contain the D n axis and an angle bisector of the C 2 axes (d: diagonal mirror plane)
D nh 1 D n axis + 1 mirror plane perpendicular to it
Rotating mirror group S n 1 n-fold rotating mirror axis

There are special names for individual groups:

• ${\ displaystyle C_ {S} \ equiv C_ {1v} \ equiv C_ {1h} \ equiv S_ {1}}$( Reflection)${\ displaystyle S =}$
• ${\ displaystyle C_ {i} \ equiv S_ {2}}$( Inversion, i.e. point reflection)${\ displaystyle i =}$

The point groups which have at least two axes of rotation with a number greater than two correspond to the symmetry groups of the Platonic solids .

• The tetrahedral groups : . This corresponds to the full symmetry of a tetrahedron .${\ displaystyle T, T_ {d}, T_ {h}}$${\ displaystyle T_ {d}}$
• The Oktaedergruppen : . This corresponds to the full symmetry of an octahedron or hexahedron .${\ displaystyle O, O_ {h}}$${\ displaystyle O_ {h}}$
• The Ikosaedergruppen : . This corresponds to the full symmetry of an icosahedron or dodecahedron .${\ displaystyle I, I_ {h}}$${\ displaystyle I_ {h}}$

The continuous point groups are also called Curie groups . They consist of the cylinder groups (with an infinite axis of rotation) and the spherical groups (with two infinite axes of rotation).

## Point groups in crystallography

The complete possible symmetry of a crystal structure is described with the 230 crystallographic space groups . In addition to the symmetry operations of the point groups, there are also translations in the form of screws and sliding reflections as symmetry operations. To describe the symmetry of a macroscopic single crystal, on the other hand, the point groups are sufficient, since crystals are always convex polyhedra and possible internal translations in the structure cannot be macroscopically recognized.

If you delete all translations in a space group and additionally replace the screw axes and glide mirror planes with corresponding axes of rotation and mirror planes, you get the so-called geometric crystal class or point group of the crystal. Only point groups whose symmetry is compatible with an infinitely extended lattice can therefore be considered as crystal classes. In a crystal, only 6-, 4-, 3-, 2-way axes of rotation are possible (rotations by 60, 90, 120 or 180 and multiples thereof). The three-dimensional point groups in which no or only 2, 3, 4 and / or 6-fold axes of rotation occur are therefore called crystallographic point groups. There are a total of 32 crystallographic point groups, which are also referred to as crystal classes.

### The 32 crystallographic point groups (crystal classes)

Point group (crystal class) Physical Properties Examples
No. Crystal system Surname Schoenflies icon International symbol
( Hermann-Mauguin )
Tepid class Associated room
groups ( no.)
Enantiomorphism Optical activity Pyroelectricity Piezoelectricity ; SHG effect
Full Short
1 triclinic triklin-pedial C 1 1 1 1 1 + + + [ uvw ] + Abelsonite
axinite
2 triclinic-pinacoidal C i ( S 2 ) 1 1 2 - - - - Albite
anorthite
3 monoclinic monoclinic-sphenoid C 2 121 (or 112) 2 2 / m 3-5 + + + [010] (or [001]) + Uranophane
halotrichite
4th monoclinic C s ( C 1 h ) 1 m 1 (or 11 m ) m 6-9 - + + [ u 0 w ] (or [ uv 0]) + Soda
Skolezite
5 monoclinic prismatic C 2 h 12 / m 1 (or 112 / m ) 2 / m 10-15 - - - - Gypsum
cryolite
6th orthorhombic orthorhombic-disphenoidal D 2 ( V ) 222 222 mmm 16-24 + + - + Austinite
epsomite
7th orthorhombic-pyramidal C 2 v mm 2 mm 2 25-46 - + + [001] + Hemimorphite
struvite
8th orthorhombic-dipyramidal D 2 h ( V h ) 2 / m 2 / m 2 / m mmm 47-74 - - - - Topaz
anhydrite
9 tetragonal tetragonal-pyramidal C 4 4th 4th 4 / m 75-80 + + + [001] + Pinnoit
Percleveit- (Ce)
10 tetragonal-disphenoidal S 4 4th 4th 81-82 - + - + Clerk's
seat Cahnit
11 tetragonal-dipyramidal C 4 h 4 / m 4 / m 83-88 - - - - Scheelite
baotite
12 tetragonal-trapezoidal D 4 422 422 4 / mmm 89-98 + + - + Cristobalite
maucherite
13 ditetragonal-pyramidal C 4 v 4 mm 4 mm 99-110 - - + [001] + Lenait
Diaboleit
14th tetragonal-scalenohedral D 2 d ( V d ) 4 2 m or  4 m 2 4 2 m 111-122 - + - + Chalcopyrite
stannite
15th ditetragonal-dipyramidal D 4 h 4 / m 2 / m 2 / m 4 / mmm 123-142 - - - - Rutile
zircon
16 trigonal trigonal-pyramidal C 3 3 3 3 143-146 + + + [001] + Carlinite
gratonite
17th rhombohedral C 3 i ( S 6 ) 3 3 147-148 - - - - Dolomite
Dioptas
18th trigonal-trapezoidal D 3 321 or 312 32 3 m 149-155 + + - + Quartz
tellurium
19th ditrigonal-pyramidal C 3 v 3 m 1 or 31 m 3 m 156-161 - - + [001] + Tourmaline
pyrargyrite
20th ditrigonal-scalenohedral D 3 d 3 2 / m 1 or  3 12 / m 3 m 162-167 - - - - Calcite
corundum
21st hexagonal hexagonal-pyramidal C 6 6th 6th 6 / m 168-173 + + + [001] + Nepheline
zinkenite
22nd trigonal-dipyramidal C 3 h 6th 6th 174 - - - + Penfieldite
Laurelite
23 hexagonal-dipyramidal C 6 h 6 / m 6 / m 175-176 - - - - Apatite
zemannite
24 hexagonal-trapezoidal D 6 622 622 6 / mmm 177-182 + + - + High quartz
pseudorutile
25th dihexagonal-pyramidal C 6 v 6 mm 6 mm 183-186 - - + [001] + Wurtzit
Zinkit
26th ditrigonal-dipyramidal D 3 h 6 m 2 or  6 2 m 6 m 2 187-190 - - - + Bastnasite
benitoite
27 dihexagonal-dipyramidal D 6 h 6 / m 2 / m 2 / m 6 / mmm 191-194 - - - - Graphite
magnesium
28 cubic tetrahedral-pentagon-dodecahedral T 23 23 m 3 195-199 + + - + Ullmannit
sodium bromate
29 disdodecahedral T h 2 / m 3 m 3 200-206 - - - - Pyrite
Potash Alum
30th pentagon-icositetrahedral O 432 432 m 3 m 207-214 + + - - Maghemite
Petzit
31 hexakistrahedral T d 4 3 m 4 3 m 215-220 - - - + Sphalerite
sodalite
32 hexakisoctahedral O h 4 / m 3 2 / m m 3 m 221-230 - - - - Diamond
copper
1. In the information on the physical properties, " - " means forbidden due to the symmetry and " + " means allowed. No statement can be made about the magnitude of the optical activity, pyro- and piezoelectricity and the SHG effect purely due to the symmetry. However, one can assume that there is always at least a weak expression of the property. For the pyroelectricity, the direction of the pyroelectric vector is given, if available.

### Remarks

The relationship between the space group and the point group of a crystal results as follows: The set of all translations of a space group form a normal divisor of . The point group of the crystal is the point group that is isomorphic to the factor group . The point group describes the symmetry of a crystal at the gamma point, i.e. its macroscopic properties. At other points in the Brillouin zone , the symmetry of the crystal is described by the star group of the corresponding wave vector . These are usually different for room groups that belong to the same point group. ${\ displaystyle T}$${\ displaystyle R}$${\ displaystyle R}$ ${\ displaystyle R / T}$

The "prohibition" of 5-, 7- and higher-number axes of rotation only applies to three-dimensional periodic crystals. Such axes of rotation occur both in molecules and in solids in the quasicrystals . Until the discovery of quasicrystals and the subsequent redefinition of the term crystal , the prohibition on crystals was accepted as universally valid.

According to Friedel's law, the diffraction pattern of crystals in structural analyzes using X-ray diffraction always contains an inversion center in the absence of anomalous scattering . Therefore, crystals from the diffraction data cannot be assigned directly to one of the 32 crystal classes, but only to one of the 11 centrosymmetric crystallographic point groups, which are also referred to as Lau groups . Through the identification of the Lau group, the affiliation of the crystal to one of the seven crystal systems is clarified.

## Point groups in molecular physics

Point groups and molecular symmetry
Schoenflies H. / M. Symmetry elements Molecular examples
Point groups of low symmetry
C 1 ${\ displaystyle 1 \}$ I / E = C 1 CHFClBr
C s ≡ S 1 ${\ displaystyle m \}$ σ ≡ S 1 BFClBr, SOBrCl
C i ≡ S 2 ${\ displaystyle {\ bar {1}}}$ i ≡ S 2 1,2-dibromo-1,2-dichloroethane, meso- tartaric acid
flat turning groups SO (2)
C 2 ${\ displaystyle 2 \}$ C 2 H 2 O 2 , S 2 Cl 2
C 3 ${\ displaystyle 3 \}$ C 3 Triphenylmethane , N (GeH 3 ) 3
C 4 ${\ displaystyle 4 \}$ C 4
C 5 ${\ displaystyle 5 \}$ C 5 15-crown-5
C 6 ${\ displaystyle 6 \}$ C 6 α-cyclodextrin
Turning groups with vertical mirror planes
C 2v ≡ D 1h ${\ displaystyle 2mm \}$ C 2 , 2σ v H 2 O , SO 2 Cl 2 , o- / m- dichlorobenzene
C 3v ${\ displaystyle 3m \}$ C 3 , 3σ v NH 3 , CHCl 3 , CH 3 Cl , POCl 3
C 4v ${\ displaystyle 4mm \}$ C 4 , 4σ v SF 5 Cl, XeOF 4
C 5v - C 5 , 5σ v Corannulene , C 5 H 5 In
C 6v ${\ displaystyle 6mm \}$ C 6 , 6σ v Benzene-hexamethylbenzene-chromium (0)
C ∞v - C , ∞σ v linear molecules like HCN , COS
Turning groups with horizontal mirror planes
C 2h ≡ D 1d ≡ S 2v ${\ displaystyle 2 / m \}$ C 2 , σ h , i Oxalic acid , trans -butene
C 3h ≡ S 3 ${\ displaystyle 3 / m \}$ C 3 , σ h Boric acid
C 4h ${\ displaystyle 4 / m \}$ C 4 , σ h , i Polycycloalkane C 12 H 20
C 6h ${\ displaystyle 6 / m \}$ C 6 , σ h , i Hexa-2-propenyl-benzene
Rotating mirror groups
S 4 ${\ displaystyle {\ bar {4}}}$ S 4 12-crown-4, tetraphenylmethane , Si (OCH 3 ) 4
S 6 ≡ C 3i ${\ displaystyle {\ bar {3}}}$ S 6 18-crown-6, hexacyclopropylethane
Dihedral groups
D 2 ≡ S 1v ${\ displaystyle 222 \}$ 3C 2 Twistan
D 3 ${\ displaystyle 32 \}$ C 3 , 3C 2 Tris-chelate complexes
D 4 ${\ displaystyle 422 \}$ C 4 , 4C 2 -
D 6 ${\ displaystyle 622 \}$ C 6 , 6C 2 Hexaphenylbenzene
Dieder groups with horizontal mirror planes
D 2h ${\ displaystyle mmm \}$ S 2 , 3C 2 , 2σ v , σ h , i Ethene , p- dichlorobenzene
D 3h ${\ displaystyle {\ bar {6}} 2m}$ S 3 , C 3 , 3C 2 , 3σ v , σ h BF 3 , PCl 5
D 4h ${\ displaystyle 4 / mmm \}$ S 4 , C 4 , 4C 2 , 4σ v , σ h , i XeF 4
D 5h - S 5 , C 5 , 5C 2 , 5σ v , σ h IF 7
D 6h ${\ displaystyle 6 / mmm \}$ S 6 , C 6 , 6C 2 , 6σ v , σ h , i benzene
D h - S 2 , C , ∞C 2, ∞σ v, σh, i linear molecules like carbon dioxide , ethyne
Dieder groups with diagonal mirror planes
D 2d ≡ S 4v ${\ displaystyle {\ bar {4}} 2m \}$ S 4 , 2C 2 , 2σ d Propadiene , cyclooctatetraene , B 2 Cl 4
D 3d ≡ S 6v ${\ displaystyle {\ bar {3}} m \}$ S 6 , C 3 , 3C 2 , 3σ d , i Cyclohexane
D 4d ≡ S 8v - S 8 , C 4 , 4C 2 , 4σ d Cyclo- sulfur (S 8 )
D 5d ≡ S 10v - S 10 , C 5 , 5C 2 , 5σ d Ferrocene
Tetrahedral groups
T ${\ displaystyle 23 \}$ 4C 3 , 3C 2 Pt (PF 3 ) 4
T h ${\ displaystyle m3}$ 4S 6 , 4C 3 , 3C 2 , 3σ h , i Fe (C 6 H 5 ) 6
T d ${\ displaystyle {\ bar {4}} 3m}$ 3S 4 , 4C 3 , 3C 2 , 6σ d CH 4 , P 4 , adamantane
Octahedral groups
O ${\ displaystyle 432 \}$ 3C 4 , 4C 3 , 6C 2 -
O h ${\ displaystyle m3m \}$ 4S 6 , 3S 4 , 3C 4 , 4C 3 , 6C 2 , 3σ h , 6σ d , i SF 6 , cubane
Icosahedral groups
I. - 12S 10 , 10S 6 , 6C 5 , 10C 3 , 15C 2 -
I h - 12S 10 , 10S 6 , 6C 5 , 10C 3 , 15C 2 , 15σ v , i Fullerene-C60 , fullerene-C20 ( pentagon dodecahedron )
spatial rotation groups SO (3)
K h - ∞C , ∞σ, i monatomic particles such as helium , elementary particles

## Applications

The properties of a crystal generally depend on the direction. Therefore, all material properties are described by a corresponding tensor . There is a fixed relationship between the point group of a crystal and the shape of the respective property tensor or the number of its independent components. Here are two examples:

In point groups with a center of inversion, all components of an odd tensor are identically zero. Therefore there is no pyro effect, no piezo effect and also no optical activity in these point groups.

The elastic constants are a 4th order tensor. This generally has 3 4 = 81 components. In the cubic crystal system there are only three independent components that differ from zero: C 1111 (= C 2222 = C 3333 ), C 1122 (= C 2233 = C 1133 ) and C 1212 (= C 1313 = C 2323 ). All other components are zero.

In molecular and solid-state physics, the symmetry of the molecule or crystal can be used to determine the number of infrared and Raman-active modes and their deflection patterns. An assignment of the measured frequencies to the respective modes is not possible with group theoretical methods. If this assignment can be carried out, the binding energies between the atoms can be calculated from the frequencies.