We have seen in the previous chapter that crystals are common in nature. In this chapter we will investigate in more detail how to think about such three-dimensional periodic structures. Then we will turn to the interaction of waves with such structures. This will lead us to a discussion of correlation functions in condensed matter.

Crystals

In the previous chapter we defined a crystal as a structure which repeats periodically in space. There is a mathematical framework for dealing with physical quantities in perfect crystals; it is the science of crystallography. We will review some of the elementary concepts from this subject.

Of course, any real material is an imperfect realization of a perfect crystal; real materials always have impurities and defects. Even if a crystalline solid is very close to being strictly periodic in bulk, all materials have a surface where the periodicity fails. However, consider a large chunk of matter, say a cube of edge L where the distance between the atoms is a. The number of atoms in the bulk is of the order of (L/a), but the number on the surface is of order (L/a). If L > > a the fraction on the surface is negligible.

Lattices

The first step to defining a crystal is to define a lattice. This is a set of points in d dimensions which are generated by taking linear combinations of d linearly independent vectors called generators: ak, k = 1, … d with integer coefficients.