# Reciprocal grid

The reciprocal lattice (Latin reciprocus , “in relation to one another”, “alternately”) is a construction of crystallography and solid state physics .

In crystallography, the reciprocal lattice describes the X-ray, electron and neutron diffraction on crystals , e.g. B. in the Laue condition . In contrast to the microscopic image, the X-ray diffraction image of a crystal is not the direct image of the crystal lattice itself, but the image of the reciprocal lattice that is assigned to the crystal lattice.

In solid-state physics, the reciprocal lattice is used with a slightly different definition (factor ) and is called reciprocal space . As the associated Fourier space of the crystal lattice, it is of outstanding importance. In contrast to the vectors of the crystal lattice, the vectors of the reciprocal lattice have the dimension of an inverse length. ${\ displaystyle 2 \ pi}$

## Definitions

A 3-dimensional grid of points is described by three basis vectors , and . This grid is also called a real or direct grid. The basis vectors , and the reciprocal of this grid mesh emerge from the equations: ${\ displaystyle {\ vec {a}} _ {1}}$${\ displaystyle {\ vec {a}} _ {2}}$${\ displaystyle {\ vec {a}} _ {3}}$${\ displaystyle {\ vec {b}} _ {1}}$${\ displaystyle {\ vec {b}} _ {2}}$${\ displaystyle {\ vec {b}} _ {3}}$

Crystallography Solid state physics
${\ displaystyle {\ vec {b}} _ {1} = {\ frac {{\ vec {a}} _ {2} \ times {\ vec {a}} _ {3}} {{\ vec {a }} _ {1} \ cdot ({\ vec {a}} _ {2} \ times {\ vec {a}} _ {3})}}}$ ${\ displaystyle {\ vec {b}} _ {1} = 2 \ pi \, {\ frac {{\ vec {a}} _ {2} \ times {\ vec {a}} _ {3}} { {\ vec {a}} _ {1} \ cdot ({\ vec {a}} _ {2} \ times {\ vec {a}} _ {3})}}}$
${\ displaystyle {\ vec {b}} _ {2} = {\ frac {{\ vec {a}} _ {3} \ times {\ vec {a}} _ {1}} {{\ vec {a }} _ {1} \ cdot ({\ vec {a}} _ {2} \ times {\ vec {a}} _ {3})}}}$ ${\ displaystyle {\ vec {b}} _ {2} = 2 \ pi \, {\ frac {{\ vec {a}} _ {3} \ times {\ vec {a}} _ {1}} { {\ vec {a}} _ {1} \ cdot ({\ vec {a}} _ {2} \ times {\ vec {a}} _ {3})}}}$
${\ displaystyle {\ vec {b}} _ {3} = {\ frac {{\ vec {a}} _ {1} \ times {\ vec {a}} _ {2}} {{\ vec {a }} _ {1} \ cdot ({\ vec {a}} _ {2} \ times {\ vec {a}} _ {3})}}}$ ${\ displaystyle {\ vec {b}} _ {3} = 2 \ pi \, {\ frac {{\ vec {a}} _ {1} \ times {\ vec {a}} _ {2}} { {\ vec {a}} _ {1} \ cdot ({\ vec {a}} _ {2} \ times {\ vec {a}} _ {3})}}}$

Here is the volume of the unit cell . ${\ displaystyle {\ vec {a}} _ {1} \ cdot ({\ vec {a}} _ {2} \ times {\ vec {a}} _ {3})}$

This generally applies:

• for the crystallographic definition ${\ displaystyle {\ vec {a}} _ {i} \ cdot {\ vec {b}} _ {j} = \ delta _ {ij}}$
• for the definition used in solid state physics: ${\ displaystyle {\ vec {a}} _ {i} \ cdot {\ vec {b}} _ {j} = 2 \ pi \ delta _ {ij}}$

The difference between the two definitions is due to the different representation of the spreading process:

• In crystallography, the incident or scattered wave is usually described by unit vectors or . In some cases the definition is also used, where λ is the wavelength of the radiation used.${\ displaystyle {\ vec {e}} _ {0}}$${\ displaystyle {\ vec {e}} _ {s}}$${\ displaystyle {\ vec {k}} = {\ frac {\ vec {e}} {\ lambda}}}$
• In solid-state physics, wave vectors are generally used to describe waves .${\ displaystyle {\ vec {k}} = 2 \ pi {\ frac {\ vec {e}} {\ lambda}}}$

In the following, the crystallographic definition is used unless otherwise noted.

If you enter the basic vectors of the real grid (in Cartesian coordinates ) into the columns of a matrix , a matrix can be calculated by transposition and inversion , which contains the basic vectors of the reciprocal grid as columns. In crystallographic definition (without the factor ): ${\ displaystyle A}$${\ displaystyle B}$${\ displaystyle 2 \ pi}$

${\ displaystyle B = (A ^ {T}) ^ {- 1}}$

### Properties of the basis vectors

• A basis vector b i of the reciprocal grid is perpendicular to the other two vectors of the real grid. Its length depends on the angles between the real basis vectors  a . If these are all perpendicular to each other (cubic, tetragonal and orthorhombic grids), then its length is 1 / a i .
• The coordinates of a point of the reciprocal lattice are usually denoted by h, k, l.
• The lattice reciprocal to a reciprocal lattice is again the corresponding real lattice. (This only applies to the crystallographic definition without the factor 2π.)
• The reciprocal lattice of a Bravais lattice belongs to the same crystal system as the real lattice, but can have a different centering:
Real Bravais grids and their reciprocal grids
Real grid Reciprocal grid
designation Abbr. designation Abbr.
Primitive P Primitive P
Base
-centered (face-centered on one side)
A , B , C Base
-centered (face-centered on one side)
A , B , C
Centered surface
(face centered on all sides)
F. Inside-centered
(space-centered)
I.
Inside-centered
(space-centered)
I. Centered surface
(face centered on all sides)
F.

## Use in crystallography

### Relation to the Miller indices

A vector (h, k, l) of reciprocal space is perpendicular to the family of lattice planes with Miller indices (hkl). The length of the vector is equal to the reciprocal of the distance between the lattice planes.

From this it follows that in the reciprocal lattice the points whose coordinates have a common multiple also have a meaning: the families of lattice planes labeled (100) and (200) lie parallel to one another, but the lattice planes of the family have (200) only half as great a distance as that of the flock (100).

### Bragg equation and Laue condition

The Bragg equation provides a relationship between the lattice plane spacing and the diffraction angle . It only applies if the incident and the scattered ray are symmetrical to the "reflecting" set of lattice planes  and reads: ${\ displaystyle d_ {hkl}}$${\ displaystyle \ vartheta}$${\ displaystyle (h, k, l)}$

${\ displaystyle n \ lambda = 2d_ {hkl} \, \ sin (\ vartheta)}$

In this form, it does not provide any information about the directions of the network planes and the incident and scattered waves to each other.

If one describes the incident wave with and the scattered wave with , one obtains the Laue condition equivalent to the Bragg equation : ${\ displaystyle {\ vec {k}} _ {0} = {\ frac {{\ vec {e}} _ {0}} {\ lambda}}}$${\ displaystyle {\ vec {k}} _ {s} = {\ frac {{\ vec {e}} _ {s}} {\ lambda}}}$

${\ displaystyle {\ vec {k}} = {\ vec {k}} _ {s} - {\ vec {k}} _ {0} = {\ vec {G}} _ {h, k, l} }$

It is

• ${\ displaystyle {\ vec {k}}}$ the diffraction vector and
• ${\ displaystyle {\ vec {G}} _ {h, k, l}}$ the vector (h, k, l) of the reciprocal lattice.

In general, this equation means: an X-ray beam is scattered exactly when the diffraction vector is equal to a reciprocal grating vector. This connection is clearly shown with the Ewaldkugel . ${\ displaystyle {\ vec {k}}}$

### Historical

The polar grid ("réseau polaire") as a precursor to the reciprocal grid was already discussed by Auguste Bravais in his work on point grids .

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

## Use in solid state physics

In this section the wave vectors are again fundamentally defined with a factor and the same applies to the conventions of the reciprocal lattice. ${\ displaystyle {\ vec {k}} = {\ frac {2 \ pi} {\ lambda}} {\ vec {e}}}$${\ displaystyle 2 \ pi}$

In general, a lattice-periodic function:

${\ displaystyle n ({\ vec {r}} + {\ vec {a}} _ {i}) = n ({\ vec {r}})}$

has a Fourier decomposition with the lattice vectors with wave vectors as Fourier components, which consist of the vectors of the reciprocal lattice: ${\ displaystyle {\ vec {a}} _ {i}}$${\ displaystyle {\ vec {G}} _ {j}}$

${\ displaystyle n ({\ vec {r}}) = \ sum _ {{\ vec {G}} _ {j}} n _ {{\ vec {G}} _ {j}} \ exp {(\ mathrm {i} {\ vec {G}} _ {j} {\ vec {r}})}}$

because the following applies: ${\ displaystyle {\ vec {G}} _ {i} {\ vec {a}} _ {j} = 2 \ pi \ delta _ {ij} \,}$

${\ displaystyle \ exp {(\ mathrm {i} {\ vec {G}} _ {i} {\ vec {a}} _ {j})} = 1}$

This space of wave vectors spanned by the vectors of the reciprocal lattice is also called reciprocal space, but the term reciprocal lattice is often used synonymously.

As the Fourier space of the lattice, reciprocal space is of fundamental importance in solid state physics. The dimension of the vectors of reciprocal space is that of an inverse length. The above-described diffraction of X-rays with the Laue condition (with the wave vectors k, k 'of the photon before and after the scattering and the reciprocal grating vector G) provides a direct image of the reciprocal grating. ${\ displaystyle {\ vec {k}} = {\ vec {k}} '+ {\ vec {G}}}$

### Bloch function

Another example of the importance of reciprocal space or lattice is the Bloch function and Bloch's theorem that the solutions of the Schrödinger equation in the periodic potential of the lattice can be written as the product of a plane wave and a lattice periodic function : ${\ displaystyle u _ {\ vec {k}} ({\ vec {r}})}$

${\ displaystyle \ psi ({\ vec {r}}) = \ mathrm {e} ^ {\ mathrm {i} {\ vec {k}} \ cdot {\ vec {r}}} \ cdot u _ {\ vec {k}} ({\ vec {r}})}$

Since the function is grid-periodic, it can be written as a Fourier sum over vectors of the reciprocal grid. ${\ displaystyle u _ {\ vec {k}} ({\ vec {r}})}$

### Interaction of quasiparticles

Another application is the interaction of quasiparticles such as quantized lattice vibrations ( phonons ). These have a wave vector and an impulse . For example, if an electron with a wave vector scatters with a phonon with a wave vector , the following selection rule applies : ${\ displaystyle {\ vec {k}}}$ ${\ displaystyle \ hbar {\ vec {k}}}$${\ displaystyle {\ vec {k}}}$${\ displaystyle {\ vec {q}}}$

${\ displaystyle {\ vec {k}} + {\ vec {q}} = {\ vec {k}} '+ {\ vec {q}}' + {\ vec {G}}}$

where is a reciprocal lattice vector. ${\ displaystyle {\ vec {G}}}$

The conservation of momentum applies here except for the addition of a vector of the reciprocal lattice, and the momentum considered is also called quasi- impulse or crystal impulse .

Since the wave vector of a quasiparticle like a phonon is only fixed up to vectors of the reciprocal lattice in the lattice, it is sufficient to consider the wave vectors in the first Brillouin zone . It is the Wigner-Seitz cell of the reciprocal lattice. In one dimension, the first Brillouin zone corresponds to wave vectors . If a phonon has a larger wave vector, a vector of the reciprocal lattice can be subtracted from it until the wave vector is in the range without changing the physics. ${\ displaystyle k <{\ frac {\ pi} {a}}}$${\ displaystyle \ left ({\ frac {2 \ pi} {a}} \ right)}$${\ displaystyle \ pm {\ frac {\ pi} {a}}}$