Scattering experiments are a common method for learning about crystals. Such experiments typically involve a probe (e.g. X-rays or neutrons) and a crystalline solid. A well-characterized probe propagating towards the crystal may interact and scatter away in a particular manner. Mathematical expressions relating the scattering pattern, properties of the probe, properties of the experimental apparatus, and properties of the crystal then allow one to derive desired features of the crystalline sample.
The following derivation is based on chapter 14 of Simons The Oxford Solid State Basics and on the report Atomic Displacement Parameter Nomenclature by Trueblood et al. (available under #External links). It is recommended to consult these sources for a more explicit discussion. Background on the quantum mechanics involved may be found in Sakurai and Napolitanos Modern Quantum Mechanics.
Scattering experiments often consist of a particle with initial crystal momentum  incident on a solid. The particle passes through a potential distributed in space, , and exits with crystal momentum . This situation is described by Fermis golden rule, which gives the probability of transition per unit time, , to the energy eigenstate  from the energy eigenstate   due to the weak perturbation caused by our potential .
By inserting a complete set of position states, then utilizing the plane-wave expression relating position and momentum, we find that the matrix element is simply a Fourier transform of the potential.
Above, the length of the sample is denoted by . We now assume that our solid is a periodic crystal with each unit cell labeled by a lattice position vector  . Position within a unit cell is given by a vector   such that the overall position in the crystal may be expressed as . Because of the translational invariance of our unit cells, the potential distribution of every cell is identical and .