# Scattering

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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{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: \begin{alignat}{2} q = |\mathbf{q}| & = 2 k \sin { ( 2 \theta_s /2 ) } \\ & = \frac{4 \pi}{\lambda} \sin{ \theta_s } \end{alignat}

Where $2 \theta_s$ is the scattering angle.

## 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. The fundamental equation in scattering is: \begin{alignat}{2} I(\mathbf{q}) & = \left\langle \left| \sum_{n=1}^{N} \rho_{n} e^{i \mathbf{q} \cdot \mathbf{r}_n } \right|^2 \right\rangle \\ \end{alignat}

Where the observed scattering intensity (I) in the 3D reciprocal-space (q) is given by an ensemble average of the intensity for all (N) scattering entities probed by the beam. The wave-matter interaction is given by inner term, which coherently sums (interferes) the complex scattering contributions ( $\rho_n$ denotes the scattering power) of the N entities; the exponential term represents a plane wave (incident radiation). In integral form (for a continuous function of the scattering density $\rho(\mathbf{r})$), we can write an integral over all of real-space: \begin{alignat}{2} I(\mathbf{q}) & = \left| \int\limits_{V} \rho(\mathbf{r}) e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}V \right|^2 \\ \end{alignat}

The inner component can be thought of as the reciprocal-space: \begin{alignat}{2} F(\mathbf{q}) & = \int\limits \rho(\mathbf{r}) e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}V \\ \end{alignat}

This is mathematically identical to the (three-dimensional) Fourier transform.