# Could someone help me understand the connection between these two wikipedia entries? (reciprocal lattice)

1. The Brillouin zone is a primitive cell (more specifically a Wigner-Seitz cell) of the reciprocal lattice, which plays an important role in solid state physics due to Bloch's theorem.

2. A Wigner–Seitz cell, like any primitive cell, is a fundamental domain for the discrete translation symmetry of the lattice. The primitive cell of the reciprocal lattice in momentum space is called the Brillouin zone.

I am trying to understand how Brillouin zone, primitive cell and reciprocal lattice are connected. I have copy pasted two sentences from wikipedia I believe are "key" to understanding this, but can't wrap my head around what they mean. What is meant by primitive cell? - and why does it matter that the Brillouin zone is a primitive cell of the reciprocal lattice?

## 1 Answer

First, let's define all of them:

Primitive cell : A primitive unit cell for a periodic crystal is a unit cell containing exactly one lattice point.

Wigner-Seitz cell : Given a lattice point the set of all points in space that are closer to that given point than to any other lattice point that constitutes the Wigner-Seitz cell of the given lattice point.

Brillouin zone A Brillouin zone is any primitive unit cell of the reciprocal lattice.

Let's see the connection in brief:

First, It's always true that the Wigner-Seitz construction for a lattice gives a primitive unit cell. The primitive unit cell contains exactly one lattice point and this sometimes makes counting much easier.

As Brillouin zone is a primitive unit cell of the reciprocal lattice, its construction is similar to Wigner-Seitz's construction for the direct lattice.

P.S. : It's always better to go through a textbook rather than wondering in Wikipedia.