Scattering

Example scattering pattern (for AgBH). The rings of scattering at well-defined angles arise from scattering off of the lamellar order in the sample.

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 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.