# 8.7: Summary

- Page ID
- 23909

This teaching and learning package has introduced Brillouin zones and their construction in two and three dimensions.

The relationship between the construction of Brillouin zones and diffraction from Bragg Planes has been shown by a consideration of the Bragg equation.

The examples shown in the package itself demonstrate the principles of the construction in two dimensions. In three dimensions the principles of the construction are the same, but the Brillouin Zones are hard to visualise. Rotating the shapes of Brillouin Zones about a cube axis as in the examples we have given in this package helps to improve the visualisation in comparison with simple line diagrams found in textbooks.

## Going further

### Books

There a number of textbooks which discuss in varying levels of detail Brillouin Zones and their construction and application. The classic text on Brillouin Zones is

- Wave Propagation in Periodic Structures, L. Brillouin (Dover 1953, reprinted 2003)

Other texts are also useful, such as:

- Solid State Physics, J.S. Blakemore (Saunders 1974)
- Electronic Properties of Materials, R.E. Hummel (Springer-Verlag 1992)
- Introduction to Solid State Physics, C. Kittel (Wiley 1996)
- Solid State Physics, N.W. Ashcroft and N. D. Mermin. (Harcourt Brace 1976)

### Websites

- Wikipedia - Brillouin zones

This Wikipedia page has a number of useful external links which should help to reinforce and confirm the concepts introduced in this teaching and learning package.

Further links can be readily found by typing “Brillouin Zones” into Google and searching the web. Examples of such links are: