User:CrambleSquash/sandbox

Mathematical description

Demonstration of relation between real and reciprocal lattice. A real space 2D lattice (red dots) with primitive vectors $a_{1}$ and $a_{2}$ are shown by blue and green arrows respectively. Atop, plane waves of the form $e^{iG\cdot r}$ are plotted. From this we see that when $G$ is any integer combination of reciprocal lattice vector basis $b_{1}$ and $b_{2}$ (i.e. any reciprocal lattice vector), the resulting plane waves have the same periodicity of the lattice - that is that any translation from point $r$ (shown orange) to a point $R+r$ ($R$ 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

$\mathbf {R} =n_{1}\mathbf {a} _{1}+n_{2}\mathbf {a} _{2}$ 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.