Chapter 11 - SYMMETRY OF CRYSTAL LATTICES

Abstract

This chapter discusses the symmetry of crystals, the international notations for symmetry operations, point groups, and space groups. The real affine group is the group that consists of orthogonal transformations plus translations. An operation of this group on a position vector r produces a new vector r' = Rr + t ≡ {R|t} r, where R is an orthogonal transformation by an arbitrary amount, and t is a translation by an arbitrary amount. The notation for the operator {R|t} describes the transformation. A subgroup of the real affine group is the set of pure translations. The space group of a crystal is defined as the group of symmetry operations that leave the crystal invariant. A lattice is defined as the set of all points obtained by starting at arbitrary origin and operating on the origin by a translation tn to obtain an infinite array of points. There are 14 lattices in the three-dimensional space. These are called the 14 space lattices or the 14 Bravais lattices.