Reciprocal lattice ↩

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created: 2021-10-31 13:06:01
modified: 2022-01-10 04:13:04

The Wigner-Seitz cell of the reciprocal lattice is called the Brillouin zone.

Motivation

The physical quantities of a lattice are usually lattice periodic: is true for any lattice vector. Let's consider the Fourier transform of this physical quantity:

This can only be fulfilled if for every lattice vector.

has to be true where .

We call the corresponding wavenumber vectors the reciprocal lattice vectors of the lattice. We usually denote them with .

Determining the reciprocal lattice

Let's consider a lattice with primitive vectors. Then the , and vectors of the reciprocal lattice have to fulfill this condition: .

We can easily determine these vectors this way:

1. Let's consider the matrix created from the primitive vectors as column vectors:

2. The matrix has to fulfill this condition:
3. The primitive vectors of the reciprocal lattice are then given by the row vectors of the matrix:

Now that we have the primitive vectors of the reciprocal lattice, we can determine any of the reciprocal lattice vectors:

Where

Note: the primitive vectors of the reciprocal lattice depend on how we choose the primitive vectors of the direct lattice, but the set of reciprocal lattice vectors will be the same regardless.