﻿ Lattice constant

# Lattice constant

A lattice parameter and a lattice constant, also sometimes called the cell parameter is either a specified length or an angle which is required for the description of a grid, in particular the smallest unit of the lattice, the unit cell. The lattice parameters is either a side length of the unit cell, or an angle between the edge of the cell. Lattice parameters are important in crystallography and optics.

## Definition

The grating is produced by periodic displacement of a unit cell in each case at the same distance in a given direction in space (lattice vector).

For a one-dimensional (optical ) lattice is a lattice parameter is sufficient to specify the distance between adjacent (parallel ) grid elements. In two dimensions, there are two different grating vectors and grating parameters necessary three - two lengths and the angle between the lattice vectors. A maximum of six parameters to describe a three-dimensional grid, three lengths and three angles needed. These six parameters that define the unit cell, often as a, b, c, and α, β, γ indicated. Three of them, the lengths a, b, and c, which describe the distance between two lattice planes that coincide with the side surfaces of the unit cells. The other three, α, β, γ designate the angle between the grating vectors which are based by means of translation of the unit cell, the structure and that α is the angle between b and c, β the angle between the a and c, and γ is the angle between a and b. The description of a grid by grid parameter is not unique. Different sets of lattice parameters can describe the same lattice. Therefore, it is generally used as a unit cell of the conventional cell. With this choice of the unit cell can in the individual crystal systems have already been defined single lattice parameter, so that the number of independent lattice parameter is reduced. Therefore, one needs to describe the cubic lattice only one lattice parameter, describing the tetragonal, hexagonal and trigonal two, for the three orthogonal, monoclinic and triclinic four six.

## Determination

Lattice parameters can be measured in the simplest case directly or with the microscope. For example, a grating having 250 lines per centimeter, has a lattice constant of.

For direct measurement of parameters of crystalline materials the transmission electron microscope or scanning tunneling microscope may be used. Usually the calculation of the lattice parameter is carried but by diffraction methods, for example with X-ray diffraction. In the X-ray diffraction to determine the lattice parameters of the first step is to determine the complete crystal structure.

The cell parameters of surface structures can be determined by means of low energy electron diffraction ( LEED, Low Energy Electron Diffraction ).

## Examples

The lattice parameter of silicon, which forms a diamond crystal structure was measured with very high accuracy and is 0504 543.102 (89 ) pm The exact measurement was carried out in view of a possible new definition of the kilogram and the mole is.

The mass density of a crystalline material can be determined from the lattice parameters. In the simple case of a cubic lattice, the density: the number n of atoms per unit cell, the relative atomic mass Ar, the atomic mass unit u and calculate the lattice parameter a. The number n is 8 for diamond mesh, 4 for cubic face- centered lattice, 2 for cubic body- centered lattice and one for simple cubic lattice.

The bond length in the diamond lattice is the cubic face- centered lattice, the cubic body- centered lattice and the simple cubic lattice.

The lattice parameter of iron with a cubic body- centered 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 407.82 Gold be mentioned pm.

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