Structure Factor

The structure factor $S_\Gamma$ of a discrete set $\Gamma$ is the Fourier Transform of $\delta$-scatterers of equal strengths on all points of $\Gamma$,

$$S_\Gamma(k)=\int \sum_{x\in\Gamma} \delta(x'-x)e^{-2\pi ikx'}\,dx'=\sum_{x\in\Gamma} e^{-2\pi ikx}.$$

References

Baake, M.; Grimm, U.; and Warrington, D. H. ``Some Remarks on the Visible Points of a Lattice.'' J. Phys. A: Math. General 27, 2669-2674, 1994.