1 Answers
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 lattice of a reciprocal lattice is equivalent to the original direct lattice, because the defining equations are symmetrical with respect to the vectors in real and reciprocal space. Mathematically, direct and reciprocal lattice vectors represent covariant and contravariant vectors, respectively.
The reciprocal lattice is the set of all vectors G m {\displaystyle \mathbf {G} _{m}} , that are wavevectors of plane waves in the Fourier series of a spatial function whose periodicity is the same as that of a direct lattice R n {\displaystyle \mathbf {R} _{n}}. Each plane wave in this Fourier series has the same phase or phases that are differed by multiples of 2 π {\displaystyle 2\pi } at each direct lattice point.
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.