Fluctuation X-ray scattering

Fluctuation X-ray scattering (FXS)^[1][2] 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.^[3]

FXS can be used for the determination of (large) macromolecular structures,^[4] but has also found applications in the characterization of metallic nanostructures,^[5] magnetic domains^[6] and colloids.^[7]

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.^[8] 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

${\displaystyle A(\mathbf {q} )=\int _{V}\rho (\mathbf {r} )\exp[i\mathbf {qr} ]d\mathbf {r} }$

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

${\displaystyle I(\mathbf {q} )=A(\mathbf {q} )A(\mathbf {q} )^{*}}$

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

${\displaystyle I(\mathbf {q} )=\sum _{l=0}^{\infty }\sum _{m=-l}^{l}I_{lm}(q)Y_{l}^{m}(\theta _{q},\phi _{q})}$

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

${\displaystyle C_{2}(q,q',\Delta \phi _{q})={\frac {1}{2\pi N}}\sum _{Nimages}\int _{0}^{2\pi }J_{k}(q,\phi _{q})J_{k}(q',\phi _{q}+\Delta \phi _{q})d\phi _{q}}$

It can be shown that

${\displaystyle C_{2}(q,q',\Delta \phi _{q})=\sum _{l}B_{l}(q,q')P_{l}(\cos(\theta _{q})\cos(\theta _{q'})+\sin(\theta _{q})\sin(\theta _{q'})\cos[\Delta \phi _{q}])}$

where

${\displaystyle \theta _{q}=\arccos({\frac {q\lambda }{4\pi }})}$

with ${\displaystyle \lambda }$ equal to the X-ray wavelength used, and

${\displaystyle B_{l}(q,q')=\sum _{m=-l}^{l}I_{lm}(q)I_{lm}^{*}(q')}$

${\displaystyle P_{l}(\cdot )}$ is a Legendre Polynome. The set of ${\displaystyle B_{l}(q,q')}$ curves can be obtained via a finite Legendre transform from the observed autocorrelation ${\displaystyle C_{2}(q,q',\Delta \phi _{q})}$ and are thus directly related to the structure ${\displaystyle \rho (\mathbf {r} )}$ via the above expressions.

Additional relations can be obtained by obtaining the real space autocorrelation ${\displaystyle \gamma (\mathbf {r} )}$ of the density:

${\displaystyle \gamma (\mathbf {r} )=\int _{V}\rho (u)\rho (\mathbf {r} -\mathbf {u} )d\mathbf {u} }$

A subsequent expansion of ${\displaystyle \gamma (\mathbf {r} )}$ in a spherical harmonics series, results in radial expansion coefficients that are related to the intensity function via a Hankel transform

${\displaystyle I_{lm}(q)=\int _{0}^{\infty }\gamma _{lm}(r)j_{l}(qr)r^{2}dr}$

A concise overview of these relations has been published elsewhere^[1][3]

A generalized Guinier law describing the low resolution behavior of the data can be derived from the above expressions:

${\displaystyle \log B_{l}(q)-2l\log q\approx \log B_{l}^{*}-{\frac {2q^{2}R_{l}^{2}}{2l+3}}}$

Values of ${\displaystyle B_{l}^{*}}$ and ${\displaystyle R_{l}}$ can be obtained from a least squares analyses of the low resolution data.^[3]

The falloff of the data at higher resolution is governed by Porod laws. It can be shown^[3] that the Porod laws derived for SAXS/WAXS data hold here as well, ultimately resulting in:

${\displaystyle B_{l}(q)\propto q^{-8}}$

for particles with well-defined interfaces.

Currently, there are three routes to determine molecular structure from its corresponding FXS data.

By assuming a specific symmetric configuration of the final model, relations between expansion coefficients describing the scattering pattern of the underlying species can be exploited to determine a diffraction pattern consistent with the measure correlation data. This approach has been shown to be feasible for icosahedral^[9] and helical models.^[10]

By representing the to-be-determined structure as an assembly of independent scattering voxels, structure determination from FXS data is transformed into a global optimisation problem and can be solved using simulated annealing.^[3]

The multi-tiered iterative phasing algorithm (M-TIP) overcomes convergence issues associated with the reverse Monte Carlo procedure and eliminates the need to use or derive specific symmetry constraints as needed by the Algebraic method. The M-TIP algorithm utilizes non-trivial projections that modifies a set of trial structure factors ${\displaystyle A(\mathbf {q} )}$ such that corresponding ${\displaystyle B_{l}(q,q')}$ match observed values. The real-space image ${\displaystyle \rho (\mathbf {r} )}$, as obtained by a Fourier Transform of ${\displaystyle A(\mathbf {q} )}$ is subsequently modified to enforce symmetry, positivity and compactness. The M-TIP procedure can start from a random point and has good convergence properties.^[11]