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Mathematical description

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Demonstration of relation between real and reciprocal lattice. A real space 2D lattice (red dots) with primitive vectors and are shown by blue and green arrows respectively. Atop, plane waves of the form are plotted. From this we see that when is any integer combination of reciprocal lattice vector basis and (i.e. any reciprocal lattice vector), the resulting plane waves have the same periodicity of the lattice - that is that any translation from point (shown orange) to a point ( shown red), the value of the plane wave is the same. These plane waves can be added together and the above relation will still apply.

Assuming a two-dimensional Bravais lattice

where .

Taking a function where is a vector from the origin to any position, if follows the periodicity of the lattice, e.g. the electronic density in an atomic crystal, it is useful to write as a Fourier series

As follows the periodicity of the lattice, translating by any lattice vector we get the same value, hence

Expressing the above instead in terms of their Fourier series we have

For this to be true, for all and all , which only holds when

where .

This criteria restricts the values of to vectors that satisfy this relation. Mathematically, the reciprocal lattice is the set of all vectors that satisfy the above identity for all lattice point position vectors . As such, any function which exhibits the same periodicity of the lattice can be expressed as a Fourier series with angular frequencies taken from the reciprocal lattice.

Just as the real lattice can be generated with integer combinations of its primitive vectors , the reciprocal lattice can be generated by a set of primitive vectors . These satisfy the relation

Where is the Kronecker delta.