Point group

A point group is a special type of a symmetry group of Euclidean geometry, which describes the symmetry of a finite field. All point groups are characterized by the fact that there is a point that is mapped back onto itself by all symmetry operations of the point group. Due to the Neumann principle, the point group determines the macroscopic properties of the body. Other statements can be obtained by using the representation theory.

This uses the point groups in molecular physics and crystallography, where the 32 crystallographic point groups are also called crystal classes. Be referred to the point groups in the Schoenflies notation. In crystallography, the Hermann- Mauguin symbols are now mainly used.

Mathematical Foundations

The symmetry group of a field is mathematically described as a set of all possible symmetry operations. With symmetry operations Euclidean motions are meant to depict the body up. A distinction must be straight movements that preserve orientation and odd, which reverse the orientation, such as reflections in planes.

Possible symmetry operations in point groups in three-dimensional Euclidean vector space are the symmetry operations that have at least one fixed point: identity mapping, point reflection at a center of inversion, a reflection in a mirror plane, rotation about an axis of rotation, as well as a combination of both a rotation-reflection and a rotation inversion. The translation, screw connection, and the glide can be elements of a point group, as they have no fixed point.

When one conceives the cascading performing symmetry operations as the additive combination, you can see that a lot of symmetry operations is a (generally non-commutative ) group.

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

• Point groups with a maximum of a rotary axis with a multiplicity greater than two,
• Point groups is greater with at least two axes of rotation with a multiplicity two.

The discrete point groups with a maximum of an excellent n -fold rotation axis can be additionally combined with mirror planes and twofold axes of rotation. Overall, there are the following options:

For some groups there are special names:

• CS ≡ ≡ C1v C1h ≡ S1 ( S = Reflection )
• Ci ≡ S2 ( i = inversion, that is, point reflection )

The point groups that have at least two axes of rotation with a multiplicity greater than two, corresponding to the symmetry groups of the Platonic solids.

• The tetrahedral groups T, Td, Th Td It corresponds to the full symmetry of a tetrahedron.

• The Oktaedergruppen: O, Oh. Oh It corresponds to the full symmetry of an octahedron or hexahedron.

• The Ikosaedergruppen: I, Ih. This corresponds to the full Ih symmetry of an icosahedron or dodecahedron.

The continuous point groups are also called Curie groups. They consist of the cylinder groups ( with a unendlichzähligen axis of rotation) and the ball groups (with two unendlichzähligen rotational axes ).

Point groups in crystallography

The complete potential symmetry of a crystal structure is described with the 230 crystallographic space groups. Come addition to the symmetry operations of the point groups also translations in the form of Screw holes and glide reflections as symmetry operations before. To describe the symmetry of a macroscopic single crystal, however, satisfy the point groups, as it always is for crystals to convex polyhedra and possible internal translations are macroscopically not visible in the structure.

Can be spread in a space group of all translations and also replaced the screw axes and glide planes by appropriate rotation axes and mirror planes, one obtains the so-called geometric crystal class or point group of the crystal. As crystal classes come along only those point groups in question, whose symmetry is compatible with an infinite lattice. In a crystal are only 6 -, 4 -, 3 -, 2- fold rotation axes are possible (turns 60, 90, 120 or 180 and are each multiples thereof). The three-dimensional point group in which none or only 2 -, 3 -, 4 - and / or 6 - occur fold rotation axes is referred to, therefore, as a crystallographic point groups. In total there are 32 crystallographic point groups are also referred to as crystal classes.

The 32 crystallographic point groups (crystal classes)

Comments

The relationship between the space and the point group of a crystal is derived as follows: The set of all translations T of a space group R form a normal subgroup of R. The point group of the crystal is that of the point group, which is isomorphic to the factor group R / T. The point group describes the symmetry of a crystal at the gamma point, ie its macroscopic properties. At Andren points of the Brillouin zone, the symmetry of the crystal is described by the Stern group of the corresponding wave vector. These are great for space groups that belong to the same point group, usually different.

The "ban" of 5 -, 7 - and höherzähligen axes of rotation applies only to three-dimensionally -periodic crystals. Such rotation axes occur both in molecules and in solids in the quasicrystals. Until the discovery of quasicrystals and the subsequent redefinition of the ban on crystal crystals was assumed to be universally valid.

The diffraction pattern of crystals with structural analysis using X-ray diffraction always contains according to the Friedel 's law in the absence of anomalous scattering an inversion center. Therefore, crystals can not be directly attributed to one of the 32 crystal classes, but only one of the 11 centrosymmetric crystallographic point groups, which are also known as Laue groups from the diffraction data. Through the identification of the Laue group of the crystal is also the affiliation clarified one of the seven crystal systems.

Point groups in molecular physics

Applications

The properties of the crystal will depend in general 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 form of the respective Eigenschaftstensors or the number of its related components. Here are two examples:

In point groups with an inversion center all components of an odd tensor are identically zero. Therefore, there is no pyroelectric effects in these groups of points, no piezoelectric effect, and no optical activity.

The elastic constants are a tensor of fourth stage. This generally has 34 = 81 components. In the cubic crystal system, there are only three independent nonzero components: C1111 ( = C2222 = C3333 ), C1122 ( = C2233 = C1133 ) and C1212 ( C2323 = C1313 = ). All other components are zero.

In the molecular and solid state physics one can determine the number of infrared and Raman- active modes and their displacement pattern of the symmetry of the molecule or crystal. An assignment of the observed frequencies to the respective modes is not possible with group theoretical methods. Can this assignment to perform, it can be calculated from the frequencies of the binding energies between the atoms.