Reciprocal lattice

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In crystallography, the reciprocal lattice of a Bravais lattice is the set of all vectors K such that

$e^{i\mathbf{K}\cdot\mathbf{R}}=1$

for all lattice point position vectors R. The reciprocal lattice is itself a Bravais lattice, and the reciprocal of the reciprocal lattice is the original lattice.

For a three dimensional lattice, defined by its primitive vectors $(\mathbf{a_{1}}, \mathbf{a_{2}}, \mathbf{a_{3}})$ , its reciprocal lattice can be determined by generating its three reciprocal primitive vectors, through the formula,

$\mathbf{b_{1}}=2 \pi \frac{\mathbf{a_{2}} \times \mathbf{a_{3}}}{\mathbf{a_{1}} \cdot (\mathbf{a_{2}} \times \mathbf{a_{3}})}$

$\mathbf{b_{2}}=2 \pi \frac{\mathbf{a_{3}} \times \mathbf{a_{1}}}{\mathbf{a_{1}} \cdot (\mathbf{a_{2}} \times \mathbf{a_{3}})}$

$\mathbf{b_{3}}=2 \pi \frac{\mathbf{a_{1}} \times \mathbf{a_{2}}}{\mathbf{a_{1}} \cdot (\mathbf{a_{2}} \times \mathbf{a_{3}})}$

Using column vector representation of (reciprocal) primitive vectors, the formula above can be rewritten using matrix inversion:

$\left[\mathbf{b_{1}}\mathbf{b_{2}}\mathbf{b_{3}}\right]^T = 2\pi\left[\mathbf{a_{1}}\mathbf{a_{2}}\mathbf{a_{3}}\right]^{-1}$

This method appeals to the definition, and allows generalization to arbitrary dimensions. Curiously, the cross product formula dominates introductory materials on crystallography.

The above definition is called the "physics" definition, as the factor of $2π$ comes naturally from the study of periodic structures. An equivalent definition, is the "crystallographer's" definition, comes from defining the reciprocal lattice to be $e^{2 \pi i\mathbf{K}\cdot\mathbf{R}}=1$ which changes the definitions of the reciprocal lattice vectors to be

$\mathbf{b_{1}}=\frac{\mathbf{a_{2}} \times \mathbf{a_{3}}}{\mathbf{a_{1}} \cdot (\mathbf{a_{2}} \times \mathbf{a_{3}})}$

and so on for the other vectors. The crystallographer's definition has the advantage that the definition of $\mathbf{b_{1}}$ is just 1 over the length of the normal to the plane defined by $\mathbf{a_{2}} \times \mathbf{a_{3}}$ , dropping the factor of $2π$ , and this can simplify some mathematical manipulations. It is a matter of taste which definition of the lattice is used, as long as the two are not mixed.

Each point (hkl) in the reciprocal lattice corresponds to a set of lattice planes (hkl) in the real space lattice. The direction of the reciprocal lattice vector corresponds to the normal to the real space planes, and the magnitude of the reciprocal lattice vector is equal to the reciprocal of the interplanar spacing of the real space planes.

The reciprocal lattice plays a fundamental role in most analytic studies of periodic structures, particularly in the theory of diffraction. In X-ray diffraction, the momentum difference between incoming and diffracted X-rays of a crystal is a reciprocal lattice vector. The X-ray 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 primitive unit cell of the reciprocal lattice.

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Reciprocal lattices of various crystals

Reciprocal lattices for the cubic crystal system are as follows.

Simple cubic lattice

We find that the reciprocal simple cubic Bravais lattice, with cubic primitive cell of side $a$ , has for its reciprocal a simple cubic lattice with a cubic primitive cell of side $\begin{matrix}\frac{2\pi}{a}\end{matrix}$ ( $\begin{matrix}\frac{1}{a}\end{matrix}$ in the crystallographer's definition). The cubic lattice is therefore said to be dual, having its reciprocal lattice being identical (up to a numerical factor).

Face-centered cubic lattice

The reciprocal lattice to an FCC lattice is the BCC lattice.

Body-centered cubic lattice

The reciprocal lattice to an BCC lattice is the FCC lattice.

Generally, only for the Bravais lattices which have 90 degrees between $(\mathbf{a_{1}}, \mathbf{a_{2}}, \mathbf{a_{3}})$ (cubic, tetragonal, orthorhombic) have $(\mathbf{b_{1}}, \mathbf{b_{2}}, \mathbf{b_{3}})$ parallel to their real-space vectors.

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Mathematics of the 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 (of the same dimension).

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 (volume element) 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.