User:CrambleSquash/sandbox

Mathematical description


Assuming a two-dimensional Bravais lattice


 * $$\mathbf{R} = n_1 \mathbf{a}_1 + n_2 \mathbf{a}_2$$ where $$n_1, n_2 \in \mathbb{Z}$$.

Taking a function $$f(\mathbf{r})$$ where $$\mathbf{r}$$ is a vector from the origin to any position, if $$f(\mathbf{r})$$ follows the periodicity of the lattice, e.g. the electronic density in an atomic crystal, it is useful to write $$f(\mathbf{r})$$ as a Fourier series


 * $$\sum_m {f_m e^{i \mathbf{G}_m \cdot \mathbf{r}}} = f\left(\mathbf{r}\right) $$

As $$f(\mathbf{r})$$ follows the periodicity of the lattice, translating $$\mathbf{r}$$ by any lattice vector $$\mathbf{R}_n$$ we get the same value, hence


 * $$f(\mathbf{r} + \mathbf{R}_n) = f(\mathbf{r})$$

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


 * $$\sum_m {f_m e^{i \mathbf{G}_m \cdot \mathbf{r}}} =

\sum_m {f_m e^{i \mathbf{G}_m \cdot (\mathbf{r} + \mathbf{R}_n)}} = \sum_m {\left(f_m e^{i \mathbf{G}_m \cdot \mathbf{r}}\right) \left(e^{i \mathbf{G}_m \cdot \mathbf{R}_n}\right)} $$

For this to be true, $$ e^{i \mathbf{G}_m \cdot \mathbf{R}_n} = 1$$ for all $$m$$ and all $$n$$, which only holds when


 * $$\mathbf{G}_m \cdot \mathbf{R}_n = 2\pi N$$ where $$N \in \mathbb{Z}$$.

This criteria restricts the values of $$\mathbf{G}_m$$ to vectors that satisfy this relation. Mathematically, the reciprocal lattice is the set of all vectors $$\mathbf{G}_m$$ that satisfy the above identity for all lattice point position vectors $$\mathbf{R}$$. 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 $$\mathbf{a}_k$$, the reciprocal lattice can be generated by a set of primitive vectors $$\mathbf{b}_l$$. These satisfy the relation


 * $$\mathbf{a}_k \cdot \mathbf{b}_l = \delta_{kl}$$

Where $$\delta_{kl}$$ is the Kronecker delta.