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1. Introduction

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).

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