Symmetry

Abstract

All single crystals possess translational symmetry, and most possess other symmetry elements as well. In this chapter we describe the 32 crystallographic point groups used for single crystals. The seven Curie groups used for textured polycrystalline materials are enumerated in the next chapter. We live in a three-dimensional world which means that there are basically four kinds of geometric symmetry operations relating one part of this world to another. The four primary types of symmetry are translation, rotation, reflection, and inversion. As pictured in Fig. 3.1, these symmetry operators operate on a point with coordinates Z1, Z2, Z3 and carry it to a new position. By definition, all crystals have a unit cell that is repeated many times in space, a point Z1, Z2, Z3 is repeated over and over again as one unit cell is translated to the next. A mirror plane perpendicular to one of the principal axes is a two-dimensional symmetry element that reverses the sign of one coordinate. Rotation axes are one-dimensional symmetry elements that change two coordinates, while an inversion center is a zero-dimensional point that changes all three coordinates. In developing an understanding of the macroscopic properties of crystals, we recognize that the scale of physical property measurements is much larger than the unit cell dimensions. It is for this reason that we are not concerned about translational symmetry and work with the 32 point group symmetries rather than the 230 space groups. This greatly simplifies the structure–property relationships in crystal physics. Aside from the identity operator 1, there are only four types of rotational symmetry consistent with the translation symmetry common to all crystals. Fig. 3.2 shows why. Parallelograms, equilateral triangles, squares, and hexagons will pack together to fill space but, pentagons symmetry axes are found in crystals. This is the starting point for generating the 32 crystal classes. When taken in combination with mirror planes and inversion centers, these four types of rotation axes are capable of forming 32 self-consistent three-dimensional symmetry patterns around a point. These are the so-called 32 crystal classes or crystallographic point groups.