Unit cell

The components of the crystal structure: lattice, unit cell and base

A unit cell or unit cell is the three basic vectors , , formed of a crystal lattice parallelepiped . Their volume is the late product of the basis vectors. The entire crystal can be thought of as being made up of the displacement of the unit cell in all three directions of the crystal lattice by integer multiples of the basis vectors. The unit cells cover the space without gaps or overlap. The two-dimensional equivalent in surface crystallography is the elementary mesh . ${\ displaystyle {\ vec {a}}}$${\ displaystyle {\ vec {b}}}$${\ displaystyle {\ vec {c}}}$ ${\ displaystyle {\ vec {a}} \ cdot ({\ vec {b}} \ times {\ vec {c}})}$

description

Point grating with a crystallographic base
Cubic, primitive crystal lattice with the unit cell and the three basis vectors in blue

The crystal structure is a three-dimensional periodic repetition of a motif. The translation vectors that bring a crystal into alignment are called lattice vectors . They form a point lattice , precisely the crystal lattice G. The points of this lattice do not represent atoms, they merely describe the periodicity of the structure.

Three arbitrary lattice vectors , , which do not lie in a plane, form a crystallographic base . The set of all integer linear combinations of these basis vectors form a lattice B, which is generally a subset of the crystal lattice G. ${\ displaystyle {\ vec {a}}}$${\ displaystyle {\ vec {b}}}$${\ displaystyle {\ vec {c}}}$

${\ displaystyle B: = \ left \ {\ left.u {\ vec {a}} + v {\ vec {b}} + w {\ vec {c}} \; \ right | \, u, v, w \ in \ mathbb {Z} \ right \} \ subseteqq G}$

The three basis vectors also define a volume element V, the fundamental mesh of the lattice B:

${\ displaystyle V: = \ left \ {\ left.x {\ vec {a}} + y {\ vec {b}} + z {\ vec {c}} \; \ right | \, 0 \ leq x , y, z <1 \ right \}}$

This volume element is the unit cell of by the vectors , , grating described. It has the shape of a parallelepiped . If the unit cell contains exactly one grid point of G, then it is called a primitive unit cell . In this case the grid B is equal to the grid G, otherwise it is a true subset. ${\ displaystyle {\ vec {a}}}$${\ displaystyle {\ vec {b}}}$${\ displaystyle {\ vec {c}}}$

The basis vectors form the coordinate system with which the crystal is described. The coordinates can be expressed both as fractional coordinates and in Cartesian coordinates .

The vectors in G are uniquely determined by a symmetry property of the crystal. The vectors of the crystal lattice B are used to describe the crystal. They can therefore be selected appropriately from the set G. However, there are standards for this selection. (see below)

application

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

The atoms that lie in a unit cell form the basis of the crystal. To describe the crystal it is sufficient to indicate the position of the base atoms in the unit cell. These atoms can also be viewed as representatives of an equivalence class. When discussing crystal structures, the term atom of the base is often used tacitly in this sense.

The primitive unit cell

Cubic primitive unit cell.

If the basis vectors have been chosen so that the lattice B formed by them is identical to the crystal lattice G, this basis is called primitive . These vectors then describe a primitive unit cell. The coordinates of the points of 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.

All points of the crystal lattice are shown in the picture. Only one corner point (0,0,0) belongs to the unit cell.

The centered unit cell

Cubic body centered unit cell.

In particular, if you want to use an axis system that is adapted to the symmetry elements of the space group of the crystal, you cannot avoid using non-primitive unit cells in most crystal systems. The crystal lattice then also contains points with non-integer coordinates. A unit cell thus contains several points of the crystal lattice.

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

All points of the crystal lattice are shown in the picture. Only one corner point (0,0,0) and the inner point belong to the unit cell. In this case the vector ( ½, ½, ½ ) is a vector of the crystal lattice that has no integer coordinates.

Other cells

Representation of a hexagonal unit cell (dark lines).

A gap-free and overlap-free division of space can also be achieved with cells that do not have the shape of a parallelepiped and are therefore not unit cells in the true sense.

The best known of these cells is the Wigner-Seitz cell .

In the literature, a hexagonal prism is often used as a cell to describe hexagonal close packing of spheres. This prism is not a unit cell. As a rule, it is not used for the crystallographic description of the structure, but only to illustrate it.

The asymmetrical unit

So far, translation has been considered the only symmetry operation. The: in a crystal but also other symmetry operations can exist rotation , the point reflection , the rotation inversion , the screwing and glide . The set of all symmetry operations of a crystal form its space group . These symmetry operations also map the crystal onto itself. In particular, however, part of the unit cell can also be mapped onto another part of the unit cell by such an operation. In this case, the two parts of the unit cell are symmetrically equivalent to each other. A volume element of the crystal from which the crystal can be formed using all symmetry operations of the space group is called an asymmetric unit. It is usually smaller than the primitive unit cell. An asymmetric unit is specified in the International Tables for each room group.

On the problem of the different terms

The use of language is not always clear and also not uniform internationally. In German crystallographer is elementary cell of the equivalent to the usual English term unit cell is used. French maille élémentaire and Italian cella elementare are also synonymous . These terms are mostly used in the sense of "conventional cell" (see below), but can also designate a primitive cell. It is noteworthy that the French term does not appear in older authors: Bravais used parallélogramme générateur or maille parallélogramme in two dimensions and parallélopipède générateur or noyau ("core") in three dimensions. Mallard just wrote maille , Friedel wrote maille simple . Only the terms “primitive cell” and “conventional cell” are clear. The Commission for Crystallographic Nomenclature of the International Union of Crystallography gives the following definitions:

Primitive cell
A primitive cell (French: maille primitive ) is a unit cell that is spanned by the basis vectors of a primitive basis of the direct grid. This means that each grid vector can be represented as an integer linear combination of the three basis vectors.
Unit cell
The unit cell (dt. Unit cell , French. Maille ) is, b of the three vectors a, c crystallographic basis of the direct lattice spanned parallelepiped. If the base primitive, so (i.e., the unit cell "primitive cell" primitive cell ). If the base is not primitive, the unit cell is a multiple cell . The multiplicity results from the ratio of its volume to the volume of the primitive cell.
Conventional cell
The conventional cell (French: maille conventionnelle ) is the cell for each grid that fulfills the following conditions:
• Their base vectors define a right-handed axis system.
• Their edges run along axes of symmetry of the grid.
• It is the smallest cell that meets the above conditions.
Crystals with the same type of conventional cell belong to the same crystal family .

Individual evidence

1. ^ IUCr Online Dictionary of Crystallography: Primitive cell
2. ^ IUCr Online Dictionary of Crystallography: Unit cell
3. ^ IUCr Online Dictionary of Crystallography: Conventional cell