Chapter 3 Plastic Deformation of Single Crystals 3.1 Elements of Crystallography A crystalline material is one in which the atoms are situated in a pattern that repeats itself periodically in three dimensions. The actual arrangement of atoms that defines the crystal structure is described with respect to a three-dimensional lattice formed by three straight lines (see Fig. 3.1). In describing the crystal structure, we must distinguish between the pattern of repetition (lattice type) and what is repeated (i.e., the unit cell). The first direct experimental proof of the lattice structure was made possible by von Laue, who in 1912 diffracted X-ray from copper sulfate crystals. Following the first determination of the internal structure for ionic crystals made by H. Bragg and W. L. Bragg in 1913 and their discovery that the reflection of X-rays differs from ordinary optical reflection, significant progress has been made in the following decade, leading to detailed documentation of the crystal structures together with numerical values of the lattice dimensions of various metals and ionic crystals (for a detailed account of the early history and the distribution of the various structures among the elements of the periodic table, the reader is referred to the English translation of the monograph by Schmid and Boas [95] due to F. A. Hughes & Co. Limited, London, 1950). It is also worth noting that long before the discovery of X-rays, the structural symmetries have been studied and classified by examining the shapes of the crystals formed naturally in the process of crystallization of various materials. Most importantly, all the possible symmetries were described mathematically. In fact, the classification of the symmetries and the arrangement of crystals in thirty-two classes and crystal systems that is in use today dates back to Hessel [46] and Bravais [15] (see also [70]). Specifically, it was recognized that for a crystal there exist, in the undeformed state, preferred directions of deformation. Associated with each crystal class is a group of symmetry transformations, G. When subjected to one of these transfor- mations, the crystal is carried into a configuration which is indistinguishable from © Springer International Publishing AG, part of Springer Nature 2019 O. Cazacu et al., Plasticity–Damage Couplings: From Single Crystal to Polycrystalline Materials, Solid Mechanics and Its Applications 253, https://doi.org/10.1007/978-3-319-92922-4_3 61