# 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.

## Overview

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.

## Mathematical background

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]

## Basic relations

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.

## Structure determination from FXS data

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

### Algebraic phasing

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]

### Reverse Monte Carlo

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]

### Multi-tiered iterative phasing

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]

## References

1. Kam, Zvi (1977). "Determination of Macromolecular Structure in Solution by Spatial Correlation of Scattering Fluctuations". Macromolecules. 10 (5): 927–934. Bibcode:1977MaMol..10..927K. doi:10.1021/ma60059a009.
2. Kam, Z.; M. H. Koch, and J. Bordas (1981). "Fluctuation x-ray scattering from biological particles in frozen solution by using synchrotron radiation". Proceedings of the National Academy of Sciences of the United States of America. 78 (6): 3559–3562. Bibcode:1981PNAS...78.3559K. doi:10.1073/pnas.78.6.3559. PMC 319609.
3. Malmerberg, Erik; Cheryl A. Kerfeld and Petrus H. Zwart (2015). "Operational properties of fluctuation X-ray scattering data". IUCrJ. 2 (3): 309–316. doi:10.1107/S2052252515002535. PMC 4420540. PMID 25995839.
4. Liu, Haiguang; Poon, Billy K.; Saldin, Dilano K.; Spence, John C. H.; Zwart, Peter H. (2013). "Three-dimensional single-particle imaging using angular correlations from X-ray laser data". Acta Crystallographica Section A. 69 (4): 365–373. doi:10.1107/S0108767313006016. ISSN 0108-7673.
5. Chen, Gang; Modestino, Miguel A.; Poon, Billy K.; Schirotzek, André; Marchesini, Stefano; Segalman, Rachel A.; Hexemer, Alexander; Zwart, Peter H. (2012). "Structure determination of Pt-coated Au dumbbellsviafluctuation X-ray scattering". Journal of Synchrotron Radiation. 19 (5): 695–700. doi:10.1107/S0909049512023801. ISSN 0909-0495.
6. Su, Run; Seu, Keoki A.; Parks, Daniel; Kan, Jimmy J.; Fullerton, Eric E.; Roy, Sujoy; Kevan, Stephen D. (2011). "Emergent Rotational Symmetries in Disordered Magnetic Domain Patterns". Physical Review Letters. 107 (25): 257204. Bibcode:2011PhRvL.107y7204S. doi:10.1103/PhysRevLett.107.257204. ISSN 0031-9007. PMID 22243108.
7. Wochner, Peter; Gutt, Christian; Autenrieth, Tina; Demmer, Thomas; Bugaev, Volodymyr; Ortiz, Alejandro Díaz; Duri, Agnès; Zontone, Federico; Grübel, Gerhard; Dosch, Helmut (2009). "X-ray cross correlation analysis uncovers hidden local symmetries in disordered matter". Proceedings of the National Academy of Sciences. 106 (28): 11511–11514. Bibcode:2009PNAS..10611511W. doi:10.1073/pnas.0905337106. ISSN 0027-8424. PMC 2703671. PMID 20716512.
8. Kurta, R. P.; Altarelli, M.; Weckert, E.; Vartanyants, I. A. (2012). "X-ray cross-correlation analysis applied to disordered two-dimensional systems". Physical Review B. 85 (18). arXiv:1202.6253. Bibcode:2012PhRvB..85r4204K. doi:10.1103/PhysRevB.85.184204. ISSN 1098-0121.
9. Saldin, D. K.; H.-C. Poon, P. Schwander, M. Uddin, and M. Schmidt (2011). "Reconstructing an icosahedral virus from single-particle diffraction experiments". Optics Express. 19 (18): 17318–17335. arXiv:1107.5212. Bibcode:2011OExpr..1917318S. doi:10.1364/OE.19.017318.CS1 maint: multiple names: authors list (link)
10. Poon, H.-C.; P. Schwander, M. Uddin, & D. K. Saldin (2011). "Fiber Diffraction without Fibers" (PDF). Physical Review Letters. 19 (18): 17318–17335. Bibcode:2013PhRvL.110z5505P. doi:10.1103/PhysRevLett.110.265505.CS1 maint: multiple names: authors list (link)
11. Donatelli, Jeffrey J.; Peter H. Zwart, and James A. Sethian (2015). "Iterative phasing for fluctuation X-ray scattering" (PDF). Proceedings of the National Academy of Sciences of the United States of America. 112 (33): 10286–10291. Bibcode:2015PNAS..11210286D. doi:10.1073/pnas.1513738112. PMC 4547282. PMID 26240348. early edition online ahead of publication