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**Example text**

Bismuth is a prototype semimetal; its structure is noncubic and rather complicated. However, the packing density, viewed as the packing of spheres, is comparable to that in metals. When bismuth is melted it becomes disordered, with essentially the structure of a liquid metal, and its electrical properties become that of a metal. Our prototype semiconductors are silicon and germanium. Both silicon and germanium have the diamond structure that is illustrated in Fig. 4. The diamond structure may be generated from the face-centered cubic structure by adding a second interlocking face-centered cubic structure displaced from the first one-quarter of the way along a cube diagonal [111].

We may immediately draw an important conclusion. The space of these vectors is a space of h dimensions; it is of course possible to construct only h mutually orthogonal vectors in h-dimensional space. We conclude that the sum of the squares of the dimensions of the irreducible representations must be less than or equal to the number of elements in the group. It turns out to be an equality and we conclude (1:6) The sum is over all inequivalent irreducible representations. It follows from this that we have constructed all the irreducible representations for the triangle.

We will give only a brief survey of group theory and its applications to symmetry properties. 2 Tinkham’s text also provides proofs for theorems which we will state without proof or with plausibility arguments only. First we will illustrate how a symmetry argument can be used to tell us about electronic eigenstates. This treatment will be very close to the more general treatment that uses group theory. We consider a structure that has a symmetry operation which we denote by R. This could be a rotation or a reflection that takes the crystal into itself.