Lattice parameters

This article deals with the lattice constant in crystals. For information on the grating constant for optical gratings, see optical grating

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

• to describe a cubic lattice only one lattice parameter,
• to describe a tetragonal , hexagonal and trigonal grid two grid parameters each,
• for an orthogonal grid three grid parameters,
• for a monoclinic lattice four lattice parameters,
• for a triclinic grid, six grid parameters.

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 number  n of atoms per unit cell:
  • 8 for the diamond structure
  • 4 for the face-centered cubic lattice
  • 2 for the body-centric cubic lattice
  • 1 for the cubic primitive lattice
• the relative atomic mass  A _r
• the atomic mass unit  u
• the lattice parameter  a .

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.

See also

• Miller indices
• Vegard's rule

Individual evidence

1. ↑ CODATA Recommended Values. National Institute of Standards and Technology, accessed July 8, 2019 .  Lattice parameters of silicon. The numbers in brackets denote the uncertainty in the last digits of the value; this uncertainty is given as the estimated standard deviation of the specified numerical value from the actual value.
2. ↑ The measurement of the lattice constant was carried out with silicon with a naturally occurring isotopic composition at a temperature of 22.5 ° C in a vacuum, cf. P. 33 ( Memento of the original from October 15, 2011 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. (PDF file; 855 kB) and p. 676 (PDF file; 2.02 MB). 
3. ↑ Kilograms and moles: Counting atoms Communication from the Physikalisch-Technische Bundesanstalt accessed on November 25, 2018