Primitive cell

An elementary cell or unit cell is the parallelepiped formed by the basis vectors, a crystal lattice. Their volume is the scalar triple product of the basis vectors. The entire crystal can be built up to think of moving the unit cell in all three directions of the crystal lattice. The covering of the space by the unit cell is gapless, overlap- free. The two-dimensional counterpart in Oberflächenkristallographie is the elementary mesh.

• 4.1 The asymmetric unit

Description

The crystal structure is a three - dimensional periodic repetition of a motif. The set of all translation vectors that will bring a crystal with them to cover, on a point lattice, the crystal lattice of G. The points of this grid do not represent atoms, they merely describe the periodicity of the structure.

Any three translation vectors, from T, which do not lie in a plane forming a crystallographic base. The set of all linear combinations of these integer -based vectors are the grating B, which is a subset of G of the crystal lattice generally.

These basis vectors are the coordinate system will be described with the aid of the crystal. They also define a volume element V, the fundamental mesh of the grid B:

This volume element is the unit cell of the grid, as described by the vectors. It has the shape of a parallelepiped.

The vectors in G are uniquely determined by a symmetry property of the crystal. The vectors of the crystal lattice B used to describe the crystal. Therefore, they may choose from the set G suitable. However, for this selection, there are standards. (See below)

Application

All points of the space can be uniquely assigned to a unit cell. It is shifted by a vector from the origin of the crystal lattice. Two points in space with respect to the crystal lattice equivalent if they occupy relative to the origin of its unit cell the same position. Thus, the crystal lattice divides the space into equivalence classes. Each equivalence class consists of all points that are different from a given point, only by a translation vector of the crystal lattice.

The atoms which are in a unit cell, are the basis of the crystal. The description of the crystal, it is sufficient to specify the position of the atoms in the unit cell of the base. These atoms can be considered as representative of an equivalence class. In the discussion of crystal structures, the term atom of the base is often implicitly used in this sense.

The primitive unit cell

If you have chosen the basis vectors so that the lattice formed by them B is identical with the crystal lattice G, then we call this basic primitive. These vectors then describe a primitive unit cell. The coordinates of the points in the crystal lattice are integers.

Each primitive unit cell contains only one point of the crystal lattice. It is the unit cell with the smallest possible volume.

In the picture all points of the crystal lattice are shown. Only one vertex (0,0,0 ) belongs to the unit cell.

The centered unit cell

Especially if one wants to use one -axis system, which is adapted to the symmetry elements of the space group of the crystal, one can not help with most crystal systems, to use non- primitive unit cells. The crystal lattice then contains points with non-integer coordinates. A unit cell thus includes a plurality of points in the crystal lattice.

These unit cells are called centered. Their volume is a multiple of the volume of the primitive unit cell. For a description of all possible structures of three-dimensional crystals with a conventional cell one needs 14 different lattice. These are the Bravais lattice.

In the picture all points of the crystal lattice are shown. Only a corner (0,0,0) and the internal point are unit cell. In this case, the vector ( ½, ½, ½ ) is a vector of the crystal lattice, which does not have integer coordinates.

Other cells

Can achieve a complete and overlap-free separation of the space even with cells that do not have the shape of a parallelepiped, and are therefore no elementary cells in the actual sense.

The most famous of these cells is the Wigner -Seitz cell.

For the description of hexagonal close packing of spheres, a 6- square prism is used as a cell in the literature often. This prism is not a unit cell. It is not intended to, most likely, to the crystallographic description of the structure, but only to their illustration.

The asymmetric unit

So far was considered the only symmetry operation translation. In a crystal, but may also exist other symmetry operations: rotation, the point of reflection, the rotation inversion, the screwing, and the glide. The set of all symmetry operations of a crystal form its space group. These symmetry operations form the crystal onto itself. In particular, but also a part of unit cell can be imaged by such an operation on another part of the unit cell. In this case the two parts of the unit cell are symmetrical equivalent to each other. A volume element of the crystal, from which the crystal can be formed using all the symmetry operations of the space group is called asymmetric unit ( engl. asymmetric unit). It is usually smaller than the primitive unit cell. For each space group an asymmetric unit is given in the International Tables.

On the problem of the different terms

The language used is not always clear and internationally not uniform. In German crystallographers unit cell is the common term that is used interchangeably with English unit cell. Synonymous also French maille élémentaire and Italian cella are elementary. These terms are usually used in the sense of " conventional cell" (see below), but may also denote a primitive cell. It is noteworthy that the French term not found in older writers: Bravais used parallélogramme générateur or maille parallélogramme in two dimensions and parallélopipède générateur or noyau ( "core" ) in three dimensions. Mallard simply wrote maille, maille Friedel simple. Clearly, only the terms " primitive cell" and " conventional cell." The Commission for Crystallographic Nomenclature of the International Union of Crystallography gives the following definitions: