Missing prefactor in 3D fourier transform of real lattice
File in which the problem appears
https://gitlab.kwant-project.org/solidstate/lectures/-/blob/master/docs/10_xray.md
Problematic sentence
The above result generalizes directly to three dimensions:
{\mathcal F}_\mathbf{k}\left[\rho(\mathbf{r})\right]=\int \mathrm{d}\mathbf{r}\ \mathrm{e}^{i\mathbf{k}\cdot\mathbf{r}} \rho(\mathbf{r}) = \sum_\mathbf{G}\delta(\mathbf{k}-\mathbf{G}).
Correct version
The above result generalizes directly to three dimensions:
{\mathcal F}_\mathbf{k}\left[\rho(\mathbf{r})\right]=\int \mathrm{d}\mathbf{r}\ \mathrm{e}^{i\mathbf{k}\cdot\mathbf{r}} \rho(\mathbf{r}) = \frac{\left(2\pi\right)^3}{V_{cell}}\sum_\mathbf{G}\delta(\mathbf{k}-\mathbf{G}).