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In crystallography, the reciprocal lattice of a
Bravais lattice is the set of all vectors K such that
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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 dimension lattice, defined by its primitive vectors , its reciprocal lattice can be determined by generating its three
reciprocal primitive vectors, through the formula,
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In particular, 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 2πa. The cubic lattice is therefore said to be dual, having its reciprocal lattice being identical
(up to a numerical factor).
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
arrangment of a crystal.
The Brillouin zone is a primitive unit cell of the reciprocal
lattice.
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