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- Limiting shape for Brillouin zones 1 answer

I am having trouble finding a mathematical definition of the Brouillin zone beyond the first, which are basically the Voronoi cells or Wigner-Seitz cells. We could imagine the set of point closer to a particular integer than any other:

$$ \min_{n \in \mathbb{Z}} \big|\big|x - n \big|\big|$$

While straightforward in one dimension, these shapes can be rather nuanced in higher dimensions.

$$ \min_{\vec{n} \in L.\mathbb{Z}^m} \big|\big|\vec{x} - \vec{n} \big|\big|$$

Even for Hexagons, the Brouillin zones look rather complicated.

It's pretty clear from the image that Broullin zones converge to a circular region as $n \to \infty$ but I can't prove as much without a definition.

Picture is taken from **Wave Propogation in Periodic Structures** by Léon Brillouin.