Conductors, Semiconductors, Superconductors: An Introduction to Solid State Physics

Book Preface

Abstract The lattice structure of crystals is characterized by specific symmetry properties. Translation symmetry yields the 14 Bravais lattices. Rotation, reflection at a mirror plane, and inversion at a point result in the 32 crystallographic point groups. The diffraction of X-rays by a crystal, initiated in 1912 by Max von Laue, represented the first experimental proof of the regular lattice structure of a crystal. The elements of diffraction theory, including the reciprocal lattice and Brillouin zones, are explained. The chapter ends with a discussion of quasi-crystals and the different types of bonding.

Crystals have always generated a particular fascination, because of the rich variety of their colors and shapes. While the systematic exploration of nature became increasingly important ever since the 17th century, at the same time the science of rocks and minerals developed into an independent branch and a collecting point for the many different individual observations. The amateur rock collectors and the mineralogists hiking with their tools through the mountains and hills in the early days must be looked upon as important forerunners of the modern scientific exploration into the properties of solids. The basic geometric crystallographic concepts for describing the large variety of observations also originated within this field of mineralogy.
In terms of physics, the most important property of crystals is their perfect lattice structure with the regular periodic repetition of exactly the same elementary building blocks in all three spatial dimensions. The elementary building blocks can be atoms or molecules, the latter consisting either of only a few or very many individual atoms. For example, the elementary building blocks of protein crystals contain up to 100 000 atoms. Because of their highly regular periodic lattice structure, crystals always possess a number of prominent symmetry properties. Of particular importance is the “translation symmetry” resulting from the regular periodic lattice configuration of the building blocks in all three spatial dimensions.
In a crystal the location of the building blocks of the lattice is described mathematically by the lattice vectors.

Here, n1,n2, and n3 are integers. a, b,and c are the three fundamental translation vectors. Here and in the following we denote vectors by bold symbols. The integer numbers n1, n2, n3 yield the lattice points of the crystal. (In (2.1) we assume, that the origin is located at a lattice point). The translation vectors a, b, c generate the elementary cell (Fig. 2.1), which in turn builds up the crystal lattice by its spatially periodic repetition. Because of this condition of translation symmetry, the possible configurations of all three-dimensional crystal lattices are highly restricted. As was shown already in 1850 by the Frenchman Auguste Bravais, there are only a total of 14 fundamental types of crystal lattices, which are now referred to as “Bravais lattices” (Fig. 2.2). By selecting the lengths of the three vectors a, b, c of the elementary cell (lattice constants) and the three angles between them, at first one obtains seven fundamental types of crystal lattices. If additional lattice points exist at special locations within the elementary cell (in the center of the elementary cell or in the middle of the external surfaces), one obtains a total of 14 translation lattices.
In general, the crystal structure is more complex than that of one of the 14 Bravais lattices. However, the crystal lattice is exactly replicated by means of a specific symmetry operation. In addition to translation, the following fundamental symmetry operations are important: rotation, reflection at a mirror plane, and inversion at a point. In the case of rotation one distinguishes how often the crystal lattice is exactly reproduced during a complete rotation by 2p. Hence, there exist single-, two-, three-, four-, and sixfold rotational axes, corresponding to a rotation by 2p, 2p/2, 2p/3, 2p/4, and 2p/6, respectively. The combination of rotation, reflection at a mirror plane, and inversion specifies one of the 32 crystallographic point groups. By addition of the translation, one of the 230 space groups is obtained, characterizing the crystal structure. Here, mathematical group theory has provided an important input.