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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{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{alignat} }

The length of this vector is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} q = |\mathbf{q}| & = k \sin { \theta } \\ & = \frac{2 \pi}{\lambda} \sin{ \theta } \\ & = \frac{4 \pi}{\lambda} \sin{ \theta /2} \end{alignat} }