Wyckoff positions

A gravity or Wyckoff positions classifies all points of a unit cell with respect to those elements of their space group symmetry, which have a fixed point. In each space group the point positions in one general location and specific locations are classified.

Description

Apart from pure displacements of each symmetry element of the space group forms a point P of the unit cell to a symmetrical equivalent point P ' from. If the point P not on a fixed point of the symmetry operation of the space group, it has a maximum of many symmetrically equivalent points in the unit cell. This point lies in a general position. A point in a general position has no special site symmetry. The number of all these mutually symmetrical equivalent points are called the multiplicity of the point position. In the centered unit cell but also the shift of the Zentrierungsvektoren is taken into account additionally.

Is the point P, however, is a fixed point of one or more elements of the space group symmetry, the symmetry operations with respect to this symmetrical equivalent points P 'to the point P, are even identical. Such point locations are called special situations. The multiplicity of a special situation is reduced accordingly. But it is always a divisor of the multiplicity of the general position. The symmetry of a particular location is higher. It is the point group symmetry of all the operations that can fix this issue. This point group is a subgroup of the point group of the crystal. The site symmetry may thus correspond to a maximum of the point group of the crystal.

All possible point positions of a unit cell were first described by Wyckoff in his book The Analytical Expression of the Results of the Theory of Space Groups. Therefore they are also called Wyckoff positions. The number of Wyckoff positions is finite. There are a total of 1731 in the three -dimensional space group, wherein the space group 27, the most Wyckoff positions.

Wyckoff has these layers with small Latin letters referred to, beginning with a for the highly symmetric special location. ( Only in the general situation has the letter α. ) The order, however, is partly arbitrary, and must therefore be looked up. Generally, the multiplicity of gravity is indicated with (eg 4a), the position of symmetry less frequently. In International Tables all point locations are indicated with their multiplicities and symmetries location for each space group.

There is space groups in which there are no specific documents, such as P1, P31, etc. The assumption that if only screw axes and glide planes are indicated in the space group symbol, then there would be no special layers, is wrong. For example, there are quite or special situations. ( Incidentally, it is also wrong to assume if no screw axes or glide plane are given in the space group symbol that it also does not exist in the space group. )

Example

The monoclinic space group ( Hermann- Mauguin symbols ) is the only symmetry operation a mirror plane perpendicular to the b axis. The axes of the crystal lattice are placed so that the mirror plane lies exactly in the xz-plane of the coordinate system. Each point (x, y, z) of the unit cell is imaged by the mirror on the point ( x, y, z). If the point is not in the unit cell, taking the point in the unit cell, which is moved to this point a grating vector ( x, 1 -y, z).

Thus, in this space group of the unit cell, each point (x, y, z) to (x, 1-y, z) is mapped. This location is the general situation and has the multiplicity 2 has no special site symmetry.

Is a point with coordinates of the form (x, 0, y) so it is mapped to (x, -0, y ), ie to itself. These points are thus at a particular location. She has the multiplicity 1, since these points lie exactly in the mirror plane, they have the site symmetry m.

Because of not only translational (x, 0, z) is a mirror plane, it (x, 1, Z); (x, 2, z ), etc. are too. But now produce two parallel planes, a third mirror, which is located in the middle between them. This is evident in both the general situation: a point (x, 0.5, z) is represented by the origin mirror plane at (x, -0.5, z). This pixel is but translation is equivalent to its archetype. Therefore, the points are of the type ( x, 0.5, z) is also a special situation of multiplicity 1 and the site symmetry m.

To summarize, an overview of all the point positions of the space group P1M1 according to the International Tables for Crystallography:

Applications

Application, see the point positions in the complete description of a crystal structure. This indicates the point on which the individual layers of atoms sit. Example SrTiO3:

Space group ( Hermann- Mauguin symbols or OH1 in the Schoenflies symbolism ) No 221 The atoms are located at the following special positions.

Knowing the lattice constants, the density and the stoichiometry of the crystal, it may be the number of the individual atoms in the unit cell to calculate for each element. With the information about the point positions can draw also conclusions as to the point on which documents the individual atoms are: A comparison of the number of atoms of an element in the unit cell with the multiplicities often restricts the possible atomic positions for this element already very strong (see example above). Further restrictions can be found if one takes into account the required minimum distances between the atoms to each other.

The symmetry of a particular location also determines the symmetry of the crystal field on this point. Measurement methods that do not see the crystal as a whole, but only the close vicinity of a single atom, do not register the symmetries of the space group, but the site symmetry of each atom with their measurement methods. These measurement methods include nuclear magnetic resonance (NMR) spectroscopy, Mössbauer spectroscopy and EXAFS. These measurement methods can be used to study pseudo-balanced structures and phase transitions.