Periodicity, Crystalline Lattices, Symbols, and Notations

Abstract

AbstractThis chapter builds on the fact (explained in the previous chapter) that the material elements (ions, atoms, molecules or groups of them) that constitute crystalline matter, the motif, are repeated in an orderly manner in the three-dimensional space. When a motif is systematically repeated, the result is a periodic pattern. The concepts of fundamental translation and crystalline lattice are introduced, a concept that allows the study of the crystal from a mathematical point of view, facilitating its study from the point of view of symmetry, and indicating the types of plane and three-dimensional lattices (Bravais lattices). The geometrical elements of the lattice (nodes, rows, and planes) and their identification using vectors, symbols, and Miller indices are explained, from vectors and coordinates in relation to the crystallographic coordinates a, b, and c (or cell parameters, lattice parameters, and fundamental translations). The idea that the lattice has an infinite extension and the cell is the smallest space of the lattice limited by the fundamental translations is emphasized, which allows us to describe the symmetry or the atomic content of a crystal. Bearing in mind that a direction can be common to a beam of planes, the concepts of zone, zone axis, and tautozonality, defined by Weiss at the beginning of the nineteenth century, are introduced to facilitate the morphological study of crystals. Given the existence of periodicity, it is the same to consider the zone axis as an edge common to a series of planes, or as an external line parallel to them, since all lines equivalent to a given line are parallel to it. The concepts of interplanar spacing of a family of planes and reticular density are introduced, along with their relationship with the cell parameters and Miller indices of those planes. Interplanar spacing is very important in relation to the crystal structure and is obtained experimentally (for example, from X-ray diffraction). To finish the chapter, the reciprocal lattice is discussed, which is of special importance in the interpretation of the diffraction of X-rays, electrons, and neutrons by crystals.