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The fundamental property of a crystal is its triple periodicity and a crystal
may be generated by repeating a certain unit of pattern through the translations
of a certain lattice called the *direct* lattice. The macroscopic
geometric properties of a crystal are a direct consequence of the existence of
this lattice on a microscopic scale. Let us for instance consider the natural
faces of a crystal. These faces are parallel to sets of lattice planes. The
lateral extension of these faces depends on the local physico-chemical
conditions during growth but not on the geometric properties of the lattice. To
describe the morphology of a crystal, the simplest way is to associate, with
each set of lattice planes parallel to a natural face, a vector drawn from a
given origin and normal to the corresponding lattice planes. To complete the
description it suffices to give to each vector a length directly related to the
spacing of the lattice planes. As we shall see in the next section this polar
diagram is the geometric basis for the *reciprocal lattice* .

On the other hand, the basic tool to study a crystal is the diffraction of a
wave with a wavelength of the same order of magnitude as that of the lattice
spacings. The nature of the diffraction pattern is governed by the triple
periodicity and the positions of the diffraction spots depend directly on the
properties of the lattice. This operation transforms the *direct space*
into an associated space, the *reciprocal space* , and we shall see that the
diffraction spots of a crystal are associated with the nodes of its reciprocal
lattice.

The reciprocal lattice is therefore an essential concept for the study of crystal lattices and their diffraction properties. This concept and the relation of the direct and reciprocal lattices through the Fourier transform was first introduced in crystallography by P. P. Ewald (1921).

**Copyright © 1981, 1998 International Union of
Crystallography**