Reciprocal lattice

The reciprocal lattice has a structure of describing the diffraction crystallography on crystals, for example, in the Laue conditions. It is used in the X-ray, electron and neutron diffraction. This concept was - extended in solid state physics to the reciprocal space - with a slightly different definition. There it is used in the quantum mechanical description of physical processes in a crystal by quasi-particles.

Definition

A 3-dimensional grid point is defined by three basic vectors, and. This grid is also called real or direct lattice. The basis vectors of the reciprocal lattice for this grid, and are derived from the equations:

Here, the volume of the unit cell. This generally applies to the crystallographic definition:

Plotting the basis vectors of the real Gittes ( in Cartesian coordinates) in the columns of a matrix A, it can be by transposition and inversion calculate a matrix containing as columns the basis vectors of the reciprocal lattice. In crystallographic definition ( without factor):

The difference between the two definitions is due to the different representation of the scattering process. In crystallography, the incident and scattered wave is usually described by unit vectors respectively. In some cases, the definition is used, λ is the wavelength of the radiation used. In solid-state physics, the wave vectors for the description of waves generally used. Hereinafter the crystallographic definition is used.

Properties of the basis vectors

• A basis vector bi of the reciprocal lattice is perpendicular to the other two vectors of the real lattice. Its length depends on the angles between the basis vectors a. Are all basis vectors perpendicular to each other ( cubic, tetragonal and orthorhombic lattice ), its length is 1/ai.
• The coordinates of a point of the reciprocal lattice are typically referred to with (h, k, l).
• The reciprocal to a reciprocal lattice grid is the corresponding real lattice again.
• The reciprocal lattice of a Bravais lattice belonging to the same crystal system as the real grating, but may have a different centering:

Use in crystallography

Conjunction with the Miller indices

A vector of the reciprocal space (h, k, l) is perpendicular to the set of lattice planes with Miller indices (h, k, l). The length of the vector is equal to the reciprocal of the lattice plane spacing. In the reciprocal lattice but also the points whose coordinates have a common multiple, one meaning: The lattice planes labeled ( 100) and ( 200) are parallel to each other, the lattice planes of the family ( 200) but have only half as large distance as the the band (100).

Bragg equation

The Bragg equation gives a relationship between the interplanar spacing d hkl and the diffraction angle θ. The Bragg equation is valid only for the case run (h, k, l) that the incident and the scattered beam symmetrically to the " reflective " Netzebenenenschaar and reads:

But in this form it provides to each other any statements about the directions of the lattice planes and the incident and scattered wave. Describing the incident wave and the scattered wave with with, this is the Bragg equation in vector form:

It is the vector (h, k, l) and of the reciprocal lattice of the diffraction vector.

General does this equation: an X-ray beam is scattered if and only if the diffraction vector is equal to a reciprocal lattice vector. This relationship is shown graphically with the Ewald sphere.

Historical

The polar grid ( " réseau polaire " ) as a precursor of the reciprocal lattice has already been treated by Auguste Bravais in the context of his work on point grid.

Josiah Willard Gibbs introduced in 1881 the concept of the reciprocal system (" reciprocal system" ) as a purely mathematical construction in his book Vector Analysis a. Its definition is the same as the crystallographic above. Paul Peter Ewald was the first to use this grid for the description of x-ray reflections. Then he built the theory further. But it was only because of a work by John Desmond Bernal, this construction for the description of Bragg reflections was generally known and established.

Notes to the reciprocal lattice in solid state physics

In solid-state physics quasiparticles used to describe the crystal. These have a wave vector and a quasi-momentum. Therefore, they are described in reciprocal space. In that the reciprocal lattice space plays an important role.

Thus, the substantial portion of the reciprocal lattice, which hosts the representations of the quasiparticles, the first Brillouin zone. This is the Wigner- Seitz cell of the reciprocal lattice.

In this specification, the Bragg equation a momentum conservation law for quasiparticles is:

In the interaction between the quasiparticle difference of the pulses before and after the interaction must correspond to the times of a reciprocal lattice vector.