Point location

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A point position or Wyckoff position classifies all points of a unit cell with respect to those symmetry elements of their space group that have a fixed point . In each room group, the point locations are divided into a general location and special locations.

description

Apart from pure displacements, each symmetry element of the space group maps a point P of the unit cell onto a symmetrically equivalent point P '. If the point P does not lie on a fixed point of one of the symmetry operations of the space group, then it has a maximum of many symmetrically equivalent points in the unit cell. This point lies on a general situation . A point on a general plane has no special positional symmetry. The number of all these symmetrically equivalent points is called the multiplicity of the point position. In the centered unit cells, however, the shift around the centering vectors is also taken into account.

If, however, the point P is a fixed point of one or more symmetry elements of the space group, the points P 'which are symmetrically equivalent with respect to these symmetry operations are identical to the point P itself. Such point positions are called special positions . The multiplicity of a specific location is reduced accordingly. But it is always a factor in the multiplicity of the general situation. The symmetry of a particular layer is higher. It is the point group of all of the symmetry operations that leave this point fixed. This point group is a subgroup of the point group of the crystal. The positional symmetry can therefore correspond at most to the point group of the crystal.

There are room groups in which there are no special locations. These include the tricline space group P 1 and space groups that only contain screw axes or glide planes , such as B. P 3 1 or Cc . If, on the other hand, the space group contains screw axes perpendicular to glide mirror planes, an inversion center with positional symmetry 1 and thus a special position is created. Examples are the space groups P 2 1 / c or I 4 1 / acd .

Wyckoff positions and Wyckoff symbols

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 . Hence they are also called Wyckoff positions. The number of possible Wyckoff positions is finite. There are a total of 1731 in the three-dimensional space groups, whereby the space group Pmmm has the most Wyckoff positions with 27.

Wyckoff has designated these positions with small Latin letters (Wyckoff symbols), starting with a for the highly symmetrical special position. (Only in Pmmm does the general position have the letter α ). However, the order is partially arbitrary and must therefore be looked up. In general, the multiplicity of the point position is also specified (e.g. 4 a ), more rarely also the position symmetry. In the International Tables , all point positions with their multiplicities and position symmetries are given for each space group.

example

The monoclinic space group P 1 m 1 ( Hermann-Mauguin symbolism ) has as the only symmetry operation a mirror plane perpendicular to the b axis. The axes of the crystal lattice are placed in such a way that the mirror plane lies exactly in the xz plane of the axis cross . Each point (x, y, z) of the unit cell is mapped to the point (x, −y, z) by the reflection. If this point is no longer in the unit cell, one takes the point in the unit cell that is shifted to this point by a grid vector: (x, 1 − y, z).

In this space group, every point of the unit cell (x, y, z) is mapped to (x, 1 − y, z). This position is the general position and has the multiplicity 2. It has no special position symmetry.

If a point has coordinates of the form (x, 0, z) then it is mapped onto (x, −0, z), i.e. onto itself. These points are therefore on a special position. It has the multiplicity 1. Since these points lie exactly in the mirror plane, they have the positional symmetry m .

Due to the translation invariance, not only is (x, 0, z) a mirror plane, but (x, 1, z); (x, 2, z) etc. are too. But now two parallel mirror planes create a third one, which lies in the middle between them. This can also be seen from the general position: a point (x, 0.5, z) is mapped onto (x, −0.5, z) through the original mirror plane. However, this image point is equivalent in translation to its original image. Therefore, the points of the type (x, 0.5, z) also form a special position of multiplicity 1 and position symmetry m .

To summarize, an overview of all point positions of the space group P 1 m 1 according to the International Tables for Crystallography  :

Multiplicity Wyckoff letter Positional symmetry Coordinates annotation
2 c 1 (x, y, z), (x, 1 − y, z) general situation
1 b m (x, 0.5, z) special location
1 a m (x, 0, z) special location

Applications

The point positions are used in the complete description of a crystal structure. Here it is indicated on which point positions the individual types of atoms sit. Example strontium titanate (SrTiO 3 ):

Space group Pm 3 m (Hermann-Mauguin symbolism or O h 1 in Schoenflies symbolism ) No. 221. The atoms are in the following special positions.

atom Multiplicity WP Positional symmetry Coordinates
Sr 1 a m 3 m (0, 0, 0)
Ti 1 b m 3 m (0.5, 0.5, 0.5)
O 3 c 4 / mmm (0, 0.5, 0.5) (0.5, 0, 0.5) (0.5, 0.5, 0)

If one knows the lattice constants, the density and the stoichiometry of the crystal, one can calculate the number of individual atoms in the unit cell for each element. With the information about the point positions, one can also draw conclusions about the point positions on which the individual atoms are located: A comparison of the number of atoms of an element in the unit cell with the multiplicities often severely restricts the possible atomic positions for this element (see example above). Further restrictions can be found if one takes into account the necessary minimum distances between the atoms.

The symmetry of a particular position also determines the symmetry of the crystal field at 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 with their measurement methods, but rather the positional symmetry of the respective atom. These measurement methods include nuclear magnetic resonance spectroscopy (NMR), Mössbauer spectroscopy and EXAFS . These measurement methods can be used to investigate pseudo-symmetrical structures and phase transitions.

literature

Web links

Individual evidence

  1. Ralph W. Gr. Wyckoff: The Analytical Expression of the Results of the Theory of Space Groups . Washington 1922.