Reciprocal space

In physics, the reciprocal lattice represents the Fourier transform of another lattice (usually a Bravais lattice). In normal usage, this first lattice (whose transform is represented by the reciprocal 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 (also known as momentum space or less commonly K-space, due to the relationship between the Pontryagin duals momentum and position.) The reciprocal lattice of a reciprocal lattice, then, is the original direct lattice again, since the two lattices are Fourier Transforms of each other.

The reciprocal lattice plays a 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.

Assuming a Bravais lattice

${\displaystyle \mathbf {R} _{\mathbf {n} }={n_{1}}\cdot \mathbf {a} _{1}+{n_{2}}\cdot \mathbf {a} _{2}}$ where ${\displaystyle n_{1},n_{2}\in \mathbb {Z} }$.

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