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 threedimensional 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:
group  Group symbol (Schönflies)  Explanation 

Turning group  C _{n}  An nfold 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 nfold rotating mirror axis 
There are special names for individual groups:
 ( Reflection)
 ( Inversion, i.e. point reflection)
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 .
 The Oktaedergruppen : . This corresponds to the full symmetry of an octahedron or hexahedron .
 The Ikosaedergruppen : . This corresponds to the full symmetry of an icosahedron or dodecahedron .
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 socalled 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, 2way axes of rotation are possible (rotations by 60, 90, 120 or 180 and multiples thereof). The threedimensional point groups in which no or only 2, 3, 4 and / or 6fold 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 ( HermannMauguin ) 
Tepid class  Associated room groups ( no.) 
Enantiomorphism  Optical activity  Pyroelectricity  Piezoelectricity ; SHG effect  
Full  Short  
1  triclinic  triklinpedial  C _{1}  1  1  1  1  +  +  + [ uvw ]  + 
Abelsonite axinite 
2  triclinicpinacoidal  C _{i} ( S _{2} )  1  1  2         
Albite anorthite 

3  monoclinic  monoclinicsphenoid  C _{2}  121 (or 112)  2  2 / m  35  +  +  + [010] (or [001])  + 
Uranophane halotrichite 
4th  monoclinic  C _{s} ( C _{1 h} )  1 m 1 (or 11 m )  m  69    +  + [ u 0 w ] (or [ uv 0])  + 
Soda Skolezite 

5  monoclinic prismatic  C _{2 h}  12 / m 1 (or 112 / m )  2 / m  1015         
Gypsum cryolite 

6th  orthorhombic  orthorhombicdisphenoidal  D _{2} ( V )  222  222  mmm  1624  +  +    + 
Austinite epsomite 
7th  orthorhombicpyramidal  C _{2 v}  mm 2  mm 2  2546    +  + [001]  + 
Hemimorphite struvite 

8th  orthorhombicdipyramidal  D _{2 h} ( V _{h} )  2 / m 2 / m 2 / m  mmm  4774         
Topaz anhydrite 

9  tetragonal  tetragonalpyramidal  C _{4}  4th  4th  4 / m  7580  +  +  + [001]  + 
Pinnoit Percleveit (Ce) 
10  tetragonaldisphenoidal  S _{4}  4th  4th  8182    +    + 
Clerk's seat Cahnit 

11  tetragonaldipyramidal  C _{4 h}  4 / m  4 / m  8388         
Scheelite baotite 

12  tetragonaltrapezoidal  D _{4}  422  422  4 / mmm  8998  +  +    + 
Cristobalite maucherite 

13  ditetragonalpyramidal  C _{4 v}  4 mm  4 mm  99110      + [001]  + 
Lenait Diaboleit 

14th  tetragonalscalenohedral  D _{2 d} ( V _{d} )  4 2 m or 4 m 2  4 2 m  111122    +    + 
Chalcopyrite stannite 

15th  ditetragonaldipyramidal  D _{4 h}  4 / m 2 / m 2 / m  4 / mmm  123142         
Rutile zircon 

16  trigonal  trigonalpyramidal  C _{3}  3  3  3  143146  +  +  + [001]  + 
Carlinite gratonite 
17th  rhombohedral  C _{3 i} ( S _{6} )  3  3  147148         
Dolomite Dioptas 

18th  trigonaltrapezoidal  D _{3}  321 or 312  32  3 m  149155  +  +    + 
Quartz tellurium 

19th  ditrigonalpyramidal  C _{3 v}  3 m 1 or 31 m  3 m  156161      + [001]  + 
Tourmaline pyrargyrite 

20th  ditrigonalscalenohedral  D _{3 d}  3 2 / m 1 or 3 12 / m  3 m  162167         
Calcite corundum 

21st  hexagonal  hexagonalpyramidal  C _{6}  6th  6th  6 / m  168173  +  +  + [001]  + 
Nepheline zinkenite 
22nd  trigonaldipyramidal  C _{3 h}  6th  6th  174        + 
Penfieldite Laurelite 

23  hexagonaldipyramidal  C _{6 h}  6 / m  6 / m  175176         
Apatite zemannite 

24  hexagonaltrapezoidal  D _{6}  622  622  6 / mmm  177182  +  +    + 
High quartz pseudorutile 

25th  dihexagonalpyramidal  C _{6 v}  6 mm  6 mm  183186      + [001]  + 
Wurtzit Zinkit 

26th  ditrigonaldipyramidal  D _{3 h}  6 m 2 or 6 2 m  6 m 2  187190        + 
Bastnasite benitoite 

27  dihexagonaldipyramidal  D _{6 h}  6 / m 2 / m 2 / m  6 / mmm  191194         
Graphite magnesium 

28  cubic  tetrahedralpentagondodecahedral  T  23  23  m 3  195199  +  +    + 
Ullmannit sodium bromate 
29  disdodecahedral  T _{h}  2 / m 3  m 3  200206         
Pyrite Potash Alum 

30th  pentagonicositetrahedral  O  432  432  m 3 m  207214  +  +     
Maghemite Petzit 

31  hexakistrahedral  T _{d}  4 3 m  4 3 m  215220        + 
Sphalerite sodalite 

32  hexakisoctahedral  O _{h}  4 / m 3 2 / m  m 3 m  221230         
Diamond copper 


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.
The "prohibition" of 5, 7 and highernumber axes of rotation only applies to threedimensional 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 Xray 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
Schoenflies  H. / M.  Symmetry elements  Molecular examples 

Point groups of low symmetry  
C _{1}  I / E = C _{1}  CHFClBr  
C _{s} ≡ S _{1}  σ ≡ S _{1}  BFClBr, SOBrCl  
C _{i} ≡ S _{2}  i ≡ S _{2}  1,2dibromo1,2dichloroethane, meso tartaric acid  
flat turning groups SO (2)  
C _{2}  C _{2}  H _{2} O _{2} , S _{2} Cl _{2}  
C _{3}  C _{3}  Triphenylmethane , N (GeH _{3} ) _{3}  
C _{4}  C _{4}  
C _{5}  C _{5}  15crown5  
C _{6}  C _{6}  αcyclodextrin  
Turning groups with vertical mirror planes  
C _{2v} ≡ D _{1h}  C _{2} , 2σ _{v}  H _{2} O , SO _{2} Cl _{2} , o / m dichlorobenzene  
C _{3v}  C _{3} , 3σ _{v}  NH _{3} , CHCl _{3} , CH _{3} Cl , POCl _{3}  
C _{4v}  C _{4} , 4σ _{v}  SF _{5} Cl, XeOF _{4}  
C _{5v}    C _{5} , 5σ _{v}  Corannulene , C _{5} H _{5} In 
C _{6v}  C _{6} , 6σ _{v}  Benzenehexamethylbenzenechromium (0)  
C _{∞v}    C _{∞} , ∞σ _{v}  linear molecules like HCN , COS 
Turning groups with horizontal mirror planes  
C _{2h} ≡ D _{1d} ≡ S _{2v}  C _{2} , σ _{h} , i  Oxalic acid , trans butene  
C _{3h} ≡ S _{3}  C _{3} , σ _{h}  Boric acid  
C _{4h}  C _{4} , σ _{h} , i  Polycycloalkane C _{12} H _{20}  
C _{6h}  C _{6} , σ _{h} , i  Hexa2propenylbenzene  
Rotating mirror groups  
S _{4}  S _{4}  12crown4, tetraphenylmethane , Si (OCH _{3} ) _{4}  
S _{6} ≡ C _{3i}  S _{6}  18crown6, hexacyclopropylethane  
Dihedral groups  
D _{2} ≡ S _{1v}  3C _{2}  Twistan  
D _{3}  C _{3} , 3C _{2}  Trischelate complexes  
D _{4}  C _{4} , 4C _{2}    
D _{6}  C _{6} , 6C _{2}  Hexaphenylbenzene  
Dieder groups with horizontal mirror planes  
D _{2h}  S _{2} , 3C _{2} , 2σ _{v} , σ _{h} , i  Ethene , p dichlorobenzene  
D _{3h}  S _{3} , C _{3} , 3C _{2} , 3σ _{v} , σ _{h}  BF _{3} , PCl _{5}  
D _{4h}  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}  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}  S _{4} , 2C _{2} , 2σ _{d}  Propadiene , cyclooctatetraene , B _{2} Cl _{4}  
D _{3d} ≡ S _{6v}  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  4C _{3} , 3C _{2}  Pt (PF _{3} ) _{4}  
T _{h}  4S _{6} , 4C _{3} , 3C _{2} , 3σ _{h} , i  Fe (C _{6} H _{5} ) _{6}  
T _{d}  3S _{4} , 4C _{3} , 3C _{2} , 6σ _{d}  CH _{4} , P _{4} , adamantane  
Octahedral groups  
O  3C _{4} , 4C _{3} , 6C _{2}    
O _{h}  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  FullereneC60 , fullereneC20 ( 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 solidstate physics, the symmetry of the molecule or crystal can be used to determine the number of infrared and Ramanactive 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.
literature
 Wolfgang Demtröder: Molecular Physics . Oldenbourg, Munich 2003, ISBN 3486249746 .
 Will Kleber , HansJoachim Bautsch , Joachim Bohm , Detlef Klimm: Introduction to crystallography . 19th edition. Oldenbourg Wissenschaftsverlag, Munich 2010, ISBN 9783486590753 .
 Hahn, Theo (Ed.): International Tables for Crystallography Vol. A D. Reidel publishing Company, Dordrecht 1983, ISBN 9027714452
 Hollas, J. Michael: The symmetry of molecules , Walter de Gruyter, Berlin 1975, ISBN 3110046377
Web links
 Definition of the point group (IUCr)
 Geometric crystal class (IUCr)
Individual evidence
 ^ The Nobel Prize in Chemistry 2011. In: Nobelprize.org. Retrieved October 21, 2011 .