I have been looking at the IUCr Online Dictionary and I would like to suggest a change to the section on Brillouin Zones. The point is that it is not correct to DEFINE the BZ as a Wigner-Seitz construction, as the whole idea of the BZ is to contain all the allowed quantum wave states. Thus it is really just a unit cell in reciprocal space and can therefore be constructed in an infinite variety of ways. The Wigner-Seitz construction is normally used because it always gives a primitive unit cell (i.e. the smallest volume) that when isolated shows the full symmetry of the reciprocal lattice from which it is constructed. However, this is fine for high symmetry cases but really not useful nor recommended for low symmetry cases. For example the shape of the Wigner-Seitz cell for monoclinic lattices is complicated and depends on the particular mix of lattice parameters and beta angle. In such a circumstance it is much easier and more reliable to use a conventional parallepiped cell to describe the wave states. Triclinic is even more horrible. Even in tetragonal symmetry the shape depends on the c/a ratio. This has been pointed out many years ago for example in the classic book by Bradley and Cracknell. The Bilbao Crystallographic Server also for instance in the section dealing with BZ’s doesn’t bother with Wigner-Seitz constructions for the low-symmetry cases.

If you agree, can we alter this section?

Regards

Mike

Here is André's answer

Dear Mike,

Thank you for your message and your remarks. You are of course right and the definition could do with some improvement. However,

- Kittel does define de B.Z. as the Wigner-Seitz cell

- The solid-state textbooks that I have at hand (Altman, Blakemore) do indeed say that the primitive cell of the reciprocal lattice is sufficient, but that it is not convenient and that the centred primitive cell is usually more convenient.

- After all, the Brillouin Zone (as it is defined) is what one is led to automatically when looking for the solutions of the propagation of a wave in a bounded solid

- It is the definition given by Brillouin in his original paper (see attached)

Personally, I would keep the definition as it is, adding some comments along the lines you have written. If you want to be complete, you could say that this construction is of course much older than solid-state physics, it is due to Dirichlet-Voronoi.

...

With my best wishes,

Andre

You are kindly invited to give your opinion of the definition given by the Online dictionnary