Fluctuation X-ray scattering (FXS) is an X-ray scattering technique similar to small-angle X-ray scattering (SAXS), but is performed using X-ray exposures below sample rotational diffusion times. This technique, ideally performed with an ultra-bright X-ray light source, such as a free electron laser, results in data containing significantly more information as compared to traditional scattering methods.

FXS can be used for the determination of (large) macromolecular structures, but has also found applications in the characterization of metallic nanostructures, magnetic domains and colloids.

The most general setup of FXS is a situation in which fast diffraction snapshots of models are taken which over a long time period undergo a full 3D rotation. A particularly interesting subclass of FXS is the 2D case where the sample can be viewed as a 2-dimensional system with particles exhibiting random in-plane rotations. In this case, an analytical solution exists relation the FXS data to the structure. In absence of symmetry constraints, no analytical data-to-structure relation for the 3D case is available, although various iterative procedures have been developed.

An FXS experiment consists of collecting a large number of X-ray snapshots of samples in a different random configuration. By computing angular intensity correlations for each image and averaging these over all snapshots, the average 2-point correlation function can be subjected to a finite Legendre transform, resulting in a collection of so-called B_l(q,q') curves, where l is the Legendre polynomial order and q / q' the momentum transfer or inverse resolution of the data.

Given a particle with density distribution ${\displaystyle \rho (\mathbf {r} )}$, the associated three-dimensional complex structure factor ${\displaystyle A(\mathbf {q} )}$ is obtained via a Fourier transform

The intensity function corresponding to the complex structure factor is equal to

where ${\displaystyle ^{*}}$ denotes complex conjugation. Expressing ${\displaystyle I(\mathbf {q} )}$ as a spherical harmonics series, one obtains

The average angular intensity correlation as obtained from many diffraction images ${\displaystyle J_{k}(q,\phi _{q})}$ is then