Duality of spaces and the origin of integral reflection conditions
Abstract
The dualism between direct and reciprocal space is at the origin of well-known relations between basis vectors in the two spaces. We show that when a coordinate system corresponding to a non-primitive unit cell is adopted, this dualism has to be handled with care. In particular, the reciprocal of a non-primitive unit cell is not a unit cell but a region in reciprocal space that does not represent a unit of repetition by translation. The basis vectors do not correspond to reciprocal-space cell lengths, contrary to what is stated even in the CIF core dictionary. The corresponding unit cell is a multiple of this region. The broken correspondence between basis vectors and unit cell is at the origin of the integral reflection conditions.
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