# Lattice parameters

A lattice parameter or a lattice constant , sometimes also called a cell parameter , is either a length specification or an angle that is required to describe a lattice , in particular the smallest unit of the lattice, the unit cell . The lattice parameter is either a side length of the unit cell or an angle between the edges of the cell. Lattice parameters are important in crystallography and optics .

## definition

The grid is generated by periodically shifting a unit cell by the same distance in a certain spatial direction (grid vector ):

• A single grating parameter is sufficient for a one-dimensional optical grating , namely the specification of the distance between adjacent ( parallel ) grating elements.
• In two dimensions, there are two different grid vectors and three necessary grid parameters - two lengths and the angle between the grid vectors.
The lattice constants of a three-dimensional lattice
• A maximum of six parameters are required to describe a three-dimensional grid, three lengths and three angles. These six parameters that define the unit cell are often referred to as a , b , c and α, β, γ. Three of them, a , b and c , are the lengths of the lattice vectors that span the unit cell. The other three, α, β and γ, are the angles between these vectors, namely
• α is the angle between  b and  c,
• β is the angle between  a and  c,
• γ is the angle between  a and  b .

The description of a grid by grid parameters is ambiguous, different sets of grid parameters can describe the same grid. Therefore, the conventional cell is generally used as the unit cell . With this choice of the unit cell, individual lattice parameters can already be fixed in the individual crystal systems , so that the number of independent lattice parameters is reduced. Therefore you need

## determination

The transmission electron microscope or the scanning tunneling microscope can be used to directly measure the parameters of crystalline substances . In most cases, however, the lattice parameters are determined using diffraction methods , for example with X-ray diffraction . In X-ray structure analysis , determining the lattice parameters is the first step in determining the complete crystal structure .

The cell parameters of surface structures can be determined with the help of the diffraction of slow electrons ( Low Energy Electron Diffraction , LEED).

## Examples

The lattice parameter of silicon , which forms a diamond structure , was measured with very high accuracy and is 543.102 0511 (89)  pm . The exact measurement was carried out in advance of the redefinition of the kilogram and the mole .

The mass density of a crystalline substance can be determined from the lattice parameters. In the simple case of cubic lattices, the density is:

${\ displaystyle \ rho = n \ cdot {\ frac {A_ {r} \ cdot u} {a ^ {3}}}}$

With

The bond length in the diamond structure is , in the face-centered cubic lattice , in the body-centric cubic lattice and in the primitive cubic lattice . ${\ displaystyle l = a \ cdot {\ tfrac {\ sqrt {3}} {4}}}$${\ displaystyle l = a \ cdot {\ tfrac {\ sqrt {2}} {2}}}$${\ displaystyle l = a \ cdot {\ tfrac {\ sqrt {3}} {2}}}$${\ displaystyle l = a}$

The lattice parameter of iron with a body-centric cubic lattice is 286.65 pm. For the lattice parameter of face-centered cubic structures, the examples nickel 352.4 pm, copper 361.48 pm, silver 408.53 pm, and gold 407.82 pm may be mentioned.