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 Bl(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
\rho(r)
A(q)
A(q)=\intV\rho(r)\exp[iqr]dr
The intensity function corresponding to the complex structure factor is equal to
I(q)=A(q)A(q)*
where
*
I(q)
I(q)=
| infty | |
| \sum | |
| l=0 |
| l | |
| \sum | |
| m=-l |
Ilm(q)
| m(\theta | |
| Y | |
| q,\phi |
q)
The average angular intensity correlation as obtained from many diffraction images
Jk(q,\phiq)
C2(q,q',\Delta\phiq)=
| 1 | |
| 2\piN |
\sumN
| 2\pi | |
| \int | |
| 0 |
Jk(q,\phiq)Jk(q',\phiq+\Delta\phiq)d\phiq
It can be shown that
C2(q,q',\Delta\phiq)=\sumlBl(q,q')Pl(\cos(\thetaq)\cos(\thetaq')+\sin(\thetaq)\sin(\thetaq')\cos[\Delta\phiq])
where
\thetaq=\arccos(
| qλ | |
| 4\pi |
)
with
λ
Bl(q,q')=
| l | |
| \sum | |
| m=-l |
Ilm(q)
| *(q') | |
| I | |
| lm |
Pl( ⋅ )
Bl(q,q')
C2(q,q',\Delta\phiq)
\rho(r)
Additional relations can be obtained by obtaining the real space autocorrelation
\gamma(r)
\gamma(r)=\intV\rho(u)\rho(r-u)du
A subsequent expansion of
\gamma(r)
Ilm(q)=
| infty | |
| \int | |
| 0 |
\gammalm(r)jl(qr)r2dr
A concise overview of these relations has been published elsewhere
A generalized Guinier law describing the low resolution behavior of the data can be derived from the above expressions:
logBl(q)-2llogq ≈ log
| * | |
| B | |
| l |
-
| |||||||
| 2l+3 |
Values of
| * | |
| B | |
| l |
Rl
The falloff of the data at higher resolution is governed by Porod laws. It can be shown that the Porod laws derived for SAXS/WAXS data hold here as well, ultimately resulting in:
Bl(q)\proptoq-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
A(q)
Bl(q,q')
\rho(r)
A(q)