**The challenge****
**Consider the solid form of two elements: argon and potassium. We know from §6.2 and §7.3, that argon is described as a molecular solid. However, when we describe potassium, we would naturally discount the theory for argon as irrelevant, and use the theory for a free electron gas described in §6.5. This may seem obvious: after all, one is a metal and the other an insulator. Well the essential question is: ‘Why is it obvious?’ Argon and potassium atoms differ by only a single electron in their outer shells. In describing the internal electronic structure of their atoms, we use the same ‘language’ to describe both argon and potassium. What we would like is a unified picture, which will describe the nature of electronic states in all solids. From this we should be able to deduce that argon is an insulator and potassium is a metal. This is the challenge.

**The problem
**The problem with this approach is that the electronic states in solids are not simple. Some states are like the corresponding states on the isolated atoms, and some appear to be like free-electron states i.e. plane waves. Most states, at least most of the valence states, in solids are in a state somewhere between these two extremes, and the challenge is to find a way to describe such states. The solution to this problem has been ‘known’ for half a century or more, but it is not at all clear to be who it is who has been doing the ‘knowing’. I say this, because while physical descriptions of the nature of electronic states are extremely rare, but dense mathematical tomes are plentiful. In this exposition, It is my intention to keep the mathematical complexities to an absolute minimum. However, the level of mathematical ability required (particularly with regard to the tight-binding theory) is definitely higher than for the topics in the main text. Some familiarity with the apparatus of quantum mechanics is also assumed.

We shall proceed as follows:

- §W1.2 and §W1.3 k-space, the reciprocal lattice and free electron theory of metals. These sections repeat some of the material Chapters 6 and 7 of the main text. The purpose of this repetition is to clearly establish the context in which the following discussion takes place.
- §W1.4 Nearly-free electron theory of metals and insulators. In the free electron theory, all materi-als with mobile valence electrons are metals. In this section we will see that the introduction of a weak periodic potential allows us to understand how such materials can behave as metals, semi-metals, semiconductors, or insulators.
- §W1.5 Tight-binding theory. In this section, we introduce a theory which does not ignore the fact that solids are made of atoms. The result is theory of electronic states in solids which is almost realistic.