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X-ray Diffraction

The diffraction pattern of X-rays through a crystal reveals the arrangement of atoms, providing information about the crystal's structure and symmetry.
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The statement of the theorem

Let ρ(r)\rho(\mathbf{r}) be the electron density of the crystal lattice. The structure factor F(Q)F(\mathbf{Q}) for a scattering vector Q\mathbf{Q} is defined by the Fourier transform:\nF(Q)=∫unit cellρ(r)e−iQ⋅rd3r\mathbf{F}(\mathbf{Q}) = \int_{\text{unit cell}} \rho(\mathbf{r}) e^{-i \mathbf{Q} \cdot \mathbf{r}} d^3\mathbf{r} \nFor a perfect crystal, the intensity I(Q)I(\mathbf{Q}) observed at scattering vector Q\mathbf{Q} is proportional to the square magnitude of the structure factor, I(Q)∝∣F(Q)∣2I(\mathbf{Q}) \propto |\mathbf{F}(\mathbf{Q})|^2. Constructive interference occurs when Q⋅d=2πn\mathbf{Q} \cdot \mathbf{d} = 2\pi n, where d\mathbf{d} is a lattice vector and n∈Zn \in \mathbb{Z}.
Source: Wikipedia