Reciprocal lattice


In physics, the reciprocal lattice represents the Fourier transform of another lattice. In normal usage, the initial lattice is usually a periodic spatial function in real-space and is also known as the direct lattice. While the direct lattice exists in real-space and is what one would commonly understand as a physical lattice, the reciprocal lattice exists in reciprocal space. The reciprocal of a reciprocal lattice is the original direct lattice, since the two are Fourier transforms of each other. Mathematically, direct and reciprocal lattice vectors represent covariant and contravariant vectors, respectively.
The reciprocal lattice plays a very fundamental role in most analytic studies of periodic structures, particularly in the theory of diffraction. In neutron and X-ray diffraction, due to the Laue conditions, the momentum difference between incoming and diffracted X-rays of a crystal is a reciprocal lattice vector. The diffraction pattern of a crystal can be used to determine the reciprocal vectors of the lattice. Using this process, one can infer the atomic arrangement of a crystal.
The Brillouin zone is a Wigner-Seitz cell of the reciprocal lattice.

Mathematical description

Assuming a two-dimensional Bravais lattice
Taking a function where is a vector from the origin to any position, if follows the periodicity of the lattice, e.g. the electronic density in an atomic crystal, it is useful to write as a Fourier series
As follows the periodicity of the lattice, translating by any lattice vector we get the same value, hence
Expressing the above instead in terms of their Fourier series we have
For this to be true,, which only holds when
This criteria restricts the values of to vectors that satisfy this relation. Mathematically, the reciprocal lattice is the set of all vectors that satisfy the above identity for all lattice point position vectors. As such, any function which exhibits the same periodicity of the lattice can be expressed as a Fourier series with angular frequencies taken from the reciprocal lattice.
This reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice, which reveals the Pontryagin duality of their respective vector spaces.
For an infinite two-dimensional lattice, defined by its primitive vectors, its reciprocal lattice can be determined by generating its two reciprocal primitive vectors, through the following formulae,
Where,
Here represents a 90 degree rotation matrix.
For an infinite three-dimensional lattice, defined by its primitive vectors, its reciprocal lattice can be determined by generating its three reciprocal primitive vectors, through the formulae
where
Note that the denominator is the scalar triple product. Using column vector representation of primitive vectors, the formulae above can be rewritten using matrix inversion:
This method appeals to the definition, and allows generalization to arbitrary dimensions. The cross product formula dominates introductory materials on crystallography.
The above definition is called the "physics" definition, as the factor of comes naturally from the study of periodic structures. An equivalent definition, the "crystallographer's" definition, comes from defining the reciprocal lattice to be which changes the definitions of the reciprocal lattice vectors to be
and so on for the other vectors. The crystallographer's definition has the advantage that the definition of
is just the reciprocal magnitude of in the direction of, dropping the factor of. This can simplify certain mathematical manipulations, and expresses reciprocal lattice dimensions in units of spatial frequency. It is a matter of taste which definition of the lattice is used, as long as the two are not mixed.
Each point in the reciprocal lattice corresponds to a set of lattice planes in the real space lattice. The direction of the reciprocal lattice vector corresponds to the normal to the real space planes. The magnitude of the reciprocal lattice vector is given in reciprocal length and is equal to the reciprocal of the interplanar spacing of the real space planes.

Reciprocal lattices of various crystals

Reciprocal lattices for the cubic crystal system are as follows.

Simple cubic lattice

The simple cubic Bravais lattice, with cubic primitive cell of side, has for its reciprocal a simple cubic lattice with a cubic primitive cell of side . The cubic lattice is therefore said to be self-dual, having the same symmetry in reciprocal space as in real space.

Face-centered cubic (FCC) lattice

The reciprocal lattice to an FCC lattice is the body-centered cubic lattice.
Consider an FCC compound unit cell. Locate a primitive unit cell of the FCC; i.e., a unit cell with one lattice point. Now take one of the vertices of the primitive unit cell as the origin. Give the basis vectors of the real lattice. Then from the known formulae, you can calculate the basis vectors of the reciprocal lattice. These reciprocal lattice vectors of the FCC represent the basis vectors of a BCC real lattice. Note that the basis vectors of a real BCC lattice and the reciprocal lattice of an FCC resemble each other in direction but not in magnitude.

Body-centered cubic (BCC) lattice

The reciprocal lattice to a BCC lattice is the FCC lattice.
It can be easily proven that only the Bravais lattices which have 90 degrees between have parallel to their real-space vectors.

Simple hexagonal lattice

The reciprocal to a simple hexagonal Bravais lattice with lattice constants c and a is another simple hexagonal lattice with lattice constants and rotated through 30° about the c axis with respect to the direct lattice. The simple hexagonal lattice is therefore said to be self-dual, having the same symmetry in reciprocal space as in real space.
vectors a 1 = i + j ; a 2 = - i + j ve a 3 = a k

Arbitrary collection of atoms

One path to the reciprocal lattice of an arbitrary collection of atoms comes from the idea of scattered waves in the Fraunhofer limit as a Huygens-style sum of amplitudes from all points of scattering. This sum is denoted by the complex amplitude F in the equation below, because it is also the Fourier transform of an effective scattering potential in direct space:
Here g = q/ is the scattering vector q in crystallographer units, N is the number of atoms, fj is the atomic scattering factor for atom j and scattering vector g, while rj is the vector position of atom j. Note that the Fourier phase depends on one's choice of coordinate origin.
For the special case of an infinite periodic crystal, the scattered amplitude F = M Fhkl from M unit cells turns out to be non-zero only for integer values of, where
when there are j=1,m atoms inside the unit cell whose fractional lattice indices are respectively. To consider effects due to finite crystal size, of course, a shape convolution for each point or the equation above for a finite lattice must be used instead.
Whether the array of atoms is finite or infinite, one can also imagine an "intensity reciprocal lattice" I, which relates to the amplitude lattice F via the usual relation I = F*F where F* is the complex conjugate of F. Since Fourier transformation is reversible, of course, this act of conversion to intensity tosses out "all except 2nd moment" information. For the case of an arbitrary collection of atoms, the intensity reciprocal lattice is therefore:
Here rjk is the vector separation between atom j and atom k. One can also use this to predict the effect of nano-crystallite shape, and subtle changes in beam orientation, on detected diffraction peaks even if in some directions the cluster is only one atom thick. On the down side, scattering calculations using the reciprocal lattice basically consider an incident plane wave. Thus after a first look at reciprocal lattice effects, beam broadening and multiple scattering effects may be important to consider as well.

Generalization of a dual lattice

There are actually two versions in mathematics of the abstract dual lattice concept, for a given lattice L in a real vector space V, of finite dimension.
The first, which generalises directly the reciprocal lattice construction, uses Fourier analysis. It may be stated simply in terms of Pontryagin duality. The dual group V^ to V is again a real vector space, and its closed subgroup L^ dual to L turns out to be a lattice in V^. Therefore, L^ is the natural candidate for dual lattice, in a different vector space.
The other aspect is seen in the presence of a quadratic form Q on V; if it is non-degenerate it allows an identification of the dual space V* of V with V. The relation of V* to V is not intrinsic; it depends on a choice of Haar measure on V. But given an identification of the two, which is in any case well-defined up to a scalar, the presence of Q allows one to speak to the dual lattice to L while staying within V.
In mathematics, the dual lattice of a given lattice L in an abelian locally compact topological group G is the subgroup L∗ of the dual group of G consisting of all continuous characters that are equal to one at each point of L.
In discrete mathematics, a lattice is a locally discrete set of points described by all integral linear combinations of dim = n linearly independent vectors in Rn. The dual lattice is then defined by all points in the linear span of the original lattice with the property that an integer results from the inner product with all elements of the original lattice. It follows that the dual of the dual lattice is the original lattice.
Furthermore, if we allow the matrix B to have columns as the linearly independent vectors that describe the lattice, then the matrix
has columns of vectors that describe the dual lattice.

Reciprocal space

Reciprocal space is the space in which the Fourier transform of a spatial function is represented. A Fourier transform takes us from "real space" to reciprocal space or vice versa. Reciprocal space comes into play regarding wave-mechanics: As a plane wave can be written by an oscillatory term with wave vector and angular frequency, it can be regarded as both a function of and . In space, the periodicity oscillates with - therefore for a given phase, and are reciprocal to each other: and.
A reciprocal lattice is a periodic set of points in this space, and contains the points that compose the Fourier transform of a periodic spatial lattice. The Brillouin zone is a volume within this space that contains all the unique k-vectors that represent the periodicity of classical or quantum waves allowed in a periodic structure.