Reciprocal Lattice Calculator And Visualizer In 2D And 3D

Simple Python program for calculation and visualization of 2d and 3d reciprocal space

-space and real space of a 2d lattice
(unit cell of the famous Si(111) 7×7 reconstruction)

Motivation

While working with spectroscopic methods to determine surface properties in the field of ultrafast electron dynamics, I wanted a tool to calculate and visualize real and reciprocal space unit cells and lattices in two and three dimensions.

Github

Real and reciprocal lattice calculator and visualizer in 2D and 3D. This project aims to just implement the transformation of 2D surface lattice and 3D lattice to their recirpocal counterpart.
https://github.com/lukaswittmann/reciprocal-lattice-calculator-and-visualizer-2d-and-3d
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Theory

Calculation of reciprocal lattices is straight forward and only requires small amount of formulas. These can be found anywhere and are even well explained on Wikipedia.

2D reciprocal space

With the condition for the reciprocal basis

$b_i\cdot a_j = 2\pi\delta_{ij},$

where $a_i$ is the primitive vector matrix for real space and $b_i$ for reciprocal space and $\delta_{ij}$ is the Kronecker delta. With this is easy to multiply with the explicit matrix inverse to get

$\begin{pmatrix} b_{1x} & b_{2x} \\ b_{1y} & b_{2y} \end{pmatrix} = \frac{2\pi}{a_{1x}a_{2y}-a_{1y}a_{2x}} \begin{pmatrix} a_{2y} & -a_{1y} \\ -a_{2x} & a_{1x} \end{pmatrix}.$