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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.

Diagram showing construction of a Brillouin Zone

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

Diagram showing construction of a Brillouin Zone

Add the Bragg Planes corresponding to the other nearest neighbours.

Diagram showing construction of the first Brillouin Zone

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.

Diagram showing construction of the first Brillouin Zone

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

Diagram showing construction of the second Brillouin Zone

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.

Diagram showing construction of the second Brillouin Zone

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. Click the animation below for an interactive illustration that follows this process to show how to construct the first six Brillouin Zones for the 2-D square reciprocal lattice. Use the arrow buttons to navigate forwards and backwards through the different steps.

2-D square lattice

Note: This animation requires Adobe Flash Player 8 and later, which can be downloaded here.

As a further example, click the on the animation below for an interactive illustration showing how to construct the first six zones for a 2-D hexagonal reciprocal lattice.

2-D hexagonal lattice

Note: This animation requires Adobe Flash Player 8 and later, which can be downloaded here.


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