## Brillouin Zone

A Brillouin Zone is a Particular Choice Of the unit cell of the reciprocal lattice. It is defined as Wigner-Seitz cell of the reciprocal lattice.

A Brillouin Zone is defined as a Wigner-Seitz primitive cell in the reciprocal lattice.

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## Brillouin Zone Construction

The reciprocal lattice basis vectors span a vector space that is commonly referred to as reciprocal space, or often in the context of quantum mechanics, k space. This section covers the construction of Brillouin zones in two dimensions.

The first step is to use the real space lattice vectors to find the reciprocal lattice vectors and construct the reciprocal lattice. One of the points in the reciprocal lattice is then designated to be the origin. When constructing Brillouin zones, they are always centred on a reciprocal lattice point, but it is important to keep in mind that there is nothing special about this point as each reciprocal lattice point is equivalent due to translation symmetry.

Draw a line connecting this origin point to one of its nearest neighbours. This line is a reciprocal lattice vector as it connects two points in the reciprocal lattice.

draw on a perpendicular bisector to the first line. This perpendicular bisector is a Bragg Plane.

Add the Bragg Planes corresponding to the other nearest neighbours.

The locus of points in reciprocal space that have no Bragg Planes between them and the origin defines the first Brillouin Zone. It is equivalent to the Wigner-Seitz unit cell of the reciprocal lattice. In the picture below the first Zone is shaded red.

Now draw on the Bragg Planes corresponding to the next nearest neighbours.

The second Brillouin Zone is the region of reciprocal space in which a point has one Bragg Plane between it and the origin. This area is shaded yellow in the picture below. Note that the areas of the first and second Brillouin Zones are the same.

The construction can quite rapidly become complicated as you move beyond the first few zones, and it is important to be systematic so as to avoid missing out important Bragg Planes.

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