Does the sequence of the $n$th Brillouin zones converges to any particular geometrical form?

I am currently studying solid state physics and we just saw the construction of the Brillouin zones. My question is the following:

Does the sequence of the $$n$$th Brillouin zones converges to any particular geometrical form? From some figures I have seen, it seems to tend to a circle ( for different lattices ) but if this is true, I don't have any idea why and whether it is convergent whatever the form of the lattice at the beginning.

Does somebody have an idea?

• This is an interesting geometry question, but I guess the point is that a high enough order polygon is bound to look like a circle, especially with the symmetries of Brillouin zones enforced. – KF Gauss Feb 15 at 9:15
• @KFGauss - I agree, with the caveat that you start with a high symmetry crystal. Books rarely have nice pictures of the Brillouin zones or Fermi surface of a triclinic crystal... – Jon Custer Feb 15 at 13:54

1 Answer

This was studied by mathematician Gareth A. Jones in the paper Geometric and Asymptotic Properties of Brillouin Zones in Lattices, Bulletin of the London Mathematical Society 16, 241 (1984). Non-paywalled link here.

The relevant conclusion is that, as $$n\rightarrow \infty$$, the $$n$$-th Brillouin zone approaches the shape of a thin spherical shell of radius $$O(n^{1/d})$$, where $$d=1,2,3,\dots$$ is the dimension of the lattice. This fact implies that $$\bigcup\limits_{i=1}^{n}B_n$$ approaches the shape of the $$d-1$$-sphere $$S^{d-1}$$. I.e. in two dimensions you get a circle $$S^1$$ as $$n\rightarrow \infty$$, and in three dimensions you get the regular sphere.

A possibly relevant follow-up paper is Jones. and Lansberg Brillouin Zones and the Fundamental Regions, Physica Status Solidi b 128, 619 (1985). I have not read it, but from its abstract it appears to describe some properties of the $$n$$-th Brillouin zone.