Scattering

Scattering broadly refers to experimental techniques that use the interaction between radiation and matter to elucidate structure. In x-ray scattering, a collimated x-ray beam is directed at a sample of interest. The incident x-rays scatter off of all the atoms/particles in the sample. Because of the wavelike nature of x-rays (which are simply high-energy photons; i.e. electromagnetic rays), the scattered waves interfere with one another, leading to constructive interference at some angles, but destructive interference at other angles. The final end result is a pattern of scattered radiation (as a function of angle with respect to the direct beam) that encodes the microscopic, nanoscopic, and molecular-scale structure of the sample.

Geometry

We define a vector in reciprocal-space as the difference between the incident and scattered x-ray beams. This new vector is the momentum transfer, denoted by q:

${\begin{alignedat}{2}\mathbf {q} &=\mathbf {k} _{o}-\mathbf {k} _{i}\\&=k(\mathbf {s} _{o}-\mathbf {s} _{i})\\&={\frac {2\pi }{\lambda }}(\mathbf {s} _{o}-\mathbf {s} _{i})\end{alignedat}}$

The length of this vector is:

${\begin{alignedat}{2}q=|\mathbf {q} |&=k\sin {\theta }\\&={\frac {2\pi }{\lambda }}\sin {\theta }\\&={\frac {4\pi }{\lambda }}\sin {\theta /2}\end{alignedat}}$

Theory

The mathematical form of scattering is closely related to the Fourier transform. The sample's realspace density distribution is Fourier transformed into an abstract 3D reciprocal-space; scattering probes this inverse space.