# Correlation methods

Conventional x-ray scattering relies on ensemble averaging to yield a robust, high signal-to-noise image. For instance, scattering data is normally averaged over a certain time duration, to accumulate sufficient statistics. For nominally isotropic samples, the two-dimensional detector image is collapsed (circular average) into a one-dimensional curve. This averaging, however, throws away potentially useful information contained within the *variance* of the x-ray signal.

A variety of emerging techniques focus instead on emphasizing and measuring the variations or fluctuations of an x-ray scattering signal (over time, space, angle, etc.). Such an analysis can, most obviously, return information about heterogeneity. However, careful correlation analysis can also extract subtle information about structure (e.g. local packing motifs) that is normally erased in ensemble averaging.

## Contents

# XCCA

**X-ray cross-correlation analysis** (**XCCA**) is a suite of techniques for analyzing correlations within x-ray scattering datasets. In particular, analysis of angular correlations within the 2D detector image can be used to isolate structural information that would be lost in a conventional circular-averaged 1D curve. Thus, even for nominally isotropic materials (powder-like sample), information about local symmetry (and thus packing motifs or unit cell) can be extracted from the data.

Angular correlation information can also be mined to reconstruct the three-dimensional reciprocal-space from individual 2D detector snapshots. That is, XCCA methods can be exploited to co-align scattering frames, registering them into the 3D scattering volume. This is conceptually similar to reciprocal-space mapping, but instead of directly reconstructing reciprocal-space by merging images, this is done in a statistical sense (because the relative alignment of frames is not known).

Correlation methods can be combined with coherent scattering effects to amplify the x-ray scattering signal from a weak sample (refer to XAmp).

# Fluctuation Scattering

**Fluctuation Scattering**: TBD

# Variance Scattering

The term **Variance Scattering** has been used to describe methods that intentionally emphasize, and analyze, variations in x-ray scattering signals.

- Ring graininess analysis (to determine grain count, grain size and size-distribution, crystallinity, etc.)
- Yager, K.G.; Majewski, P.W. Metrics of graininess: robust quantification of grain count from the non-uniformity of scattering rings
*Journal of Applied Crystallography***2014**, 47, 1855–1865. doi: 10.1107/S1600576714020822

- Yager, K.G.; Majewski, P.W. Metrics of graininess: robust quantification of grain count from the non-uniformity of scattering rings
- Heterogeneity
- C. J. Gommes Small-angle scattering and scale-dependent heterogeneity
*J. Appl. Cryst.***2016**, 49, 1162-1176. doi: 10.1107/S1600576716007810

- C. J. Gommes Small-angle scattering and scale-dependent heterogeneity

# XPCS

**X-ray Photon Correlation Spectroscopy (XPCS)** measures the temporal fluctuation of coherent speckle. From the reconstructed time correlation function, one can infer system dynamics.

# References

### XCCA

- Peter Wochner, Christian Gutt, Tina Autenrieth, Thomas Demmer, Volodymyr Bugaev, Alejandro Díaz Ortiz, Agnès Duri, Federico Zontone, Gerhard Grübel and Helmut Dosch X-ray cross correlation analysis uncovers hidden local symmetries in disordered matter
*Proceedings of the National Academy of Sciences***2009**, 106 (28), 11511–11514. doi: 10.1073/pnas.0905337106 - M. Altarelli, R. P. Kurta, and I. A. Vartanyants X-ray cross-correlation analysis and local symmetries of disordered systems: General theory
*Phys. Rev. B.***2010**, 82, 104207. doi: 10.1103/PhysRevB.82.104207 - R. P. Kurta, M. Altarelli, E. Weckert, and I. A. Vartanyants X-ray cross-correlation analysis applied to disordered two-dimensional systems
*Phys. Rev. B***2012**, 85, 184204. doi: 10.1103/PhysRevB.85.184204 - R P Kurta, R Dronyak, M Altarelli, E Weckert and I A Vartanyants Solution of the phase problem for coherent scattering from a disordered system of identical particles
*New Journal of Physics***2013**, 15. doi: 10.1088/1367-2630/15/1/013059 - F. Lehmkühler, G. Grübel and C. Gutt Detecting orientational order in model systems by X-ray cross-correlation methods
*J. Appl. Cryst.***2014**, 47, 1315-1323. doi: 10.1107/S1600576714012424 - Lehmkühler, F.; Fischer, B.; Müller, L.; Ruta B.; Grübel, G. Structure beyond pair correlations: X-ray cross-correlation from colloidal crystals
*Journal of Applied Crystallography***2016**, 49, doi: 10.1107/S1600576716017313 - Julien R. Lhermitte, Cheng Tian, Aaron Stein, Atikur Rahman, Yugang Zhang, Lutz Wiegart, Andrei Fluerasu, Oleg Gang, and Kevin G. Yager Robust X-ray angular correlations for the study of meso-structures
*J. Appl. Cryst.***2017**, 50(3). doi: 10.1107/S1600576717003946- Julien R. Lhermitte, Aaron Stein, Cheng Tian, Yugang Zhang, Lutz Wiegart, Andrei Fluerasu, Oleg Gang, and Kevin G. Yager Coherent Amplification of X-ray Scattering from Meso-structures
*IUCrJ***2017**4(5) doi: 10.1107/S2052252517008107

- Julien R. Lhermitte, Aaron Stein, Cheng Tian, Yugang Zhang, Lutz Wiegart, Andrei Fluerasu, Oleg Gang, and Kevin G. Yager Coherent Amplification of X-ray Scattering from Meso-structures
- Felix Lehmkühler, Florian Schulz, Martin A. Schroer, Lara Frenzel, Holger Langeb, and Gerhard Grübel Heterogeneous local order in self-assembled nanoparticle films revealed by X-ray cross-correlations
*IUCrJ***2018**, 5 (3), 354-360. doi: 10.1107/S2052252518005407

### Reconstruction

- Zvi Kam The Reconstruction of Structure from Electron Micrographs of Randomly Oriented Particles ;;Journal of Theoretical Biology
**1980**, 82 (1), 15-39. doi: 10.1016/0022-5193(80)90088-0 - Richard A. Kirian, Kevin E. Schmidt, Xiaoyu Wang, R. Bruce Doak, and John C. H. Spence Signal, noise, and resolution in correlated fluctuations from snapshot small-angle x-ray scattering
*Phys. Rev. E***2011**, 84, 011921. doi: 10.1103/PhysRevE.84.011921 - G. Chen, M. A. Modestino, B. K. Poon, A. Schirotzek, S. Marchesini, R. A. Segalman, A. Hexemer and P. H. Zwart Structure determination of Pt-coated Au dumbbells via fluctuation X-ray scattering
*J. Synchrotron Radiation***2012**, 19, 695-700. doi: 10.1107/S0909049512023801

### Sparse Data

- K Ayyer, HT Philipp, MW Tate, JL Wierman, V Elser, SM Gruner Determination of crystallographic intensities from sparse data
*IUCrJ***2015**, 2 (1), 29-34. doi: 10.1107/S2052252514022313 - Wierman JL, Lan TY, Tate MW, Philipp HT, Elser V, Gruner SM Protein crystal structure from non-oriented, single-axis sparse X-ray data
*IUCrJ***2016**, 3 (1), 43-50. doi: 10.1107/S2052252515018795

### XFEL

- Derek Mendez, Herschel Watkins, Shenglan Qiao, Kevin S. Raines, Thomas J. Lane, Gundolf Schenk, Garrett Nelson, Ganesh Subramanian, Kensuke Tono, Yasumasa Joti, Makina Yabashi, Daniel Ratner and Sebastian Doniach Angular correlations of photons from solution diffraction at a free-electron laser encode molecular structure
*IUCrJ***2016**, 3(6), 420-429. doi: 10.1107/S2052252516013956

### Fluctuation Scattering

- G. Chen, M. A. Modestino, B. K. Poon, A. Schirotzek, S. Marchesini, R. A. Segalman, A. Hexemer and P. H. Zwart Structure determination of Pt-coated Au dumbbells via fluctuation X-ray scattering
*J. Synchrotron Rad.***2012**, 19, 695-700. doi: 10.1107/S0909049512023801 - Erik Malmerberg, Cheryl A. Kerfeld and Petrus H. Zwart Operational properties of fluctuation X-ray scattering data
*IUCrJ***2015**, 2(3), 309-316. doi: 10.1107/S2052252515002535 - Andrew V. Martin Orientational order of liquids and glasses via fluctuation diffraction
*IUCrJ***2016**, 4 (1). doi: 10.1107/S2052252516016730

- Derek Mendez, Thomas J. Lane, Jongmin Sung, Jonas Sellberg, Clément Levard, Herschel Watkins, Aina E. Cohen, Michael Soltis, Shirley Sutton, James Spudich, Vijay Pande, Daniel Ratner, Sebastian Doniach Observation of correlated X-ray scattering at atomic resolution doi: 10.1098/rstb.2013.0315
- Dataset: Mendez, Derek; Thomas J. Lane; Daniel Ratner; Sebastian Doniach Correlated x-ray scattering dataset, silver nanoparticles
*Harvard Dataverse***2013**, V2. doi: 10.7910/DVN/23244

- Dataset: Mendez, Derek; Thomas J. Lane; Daniel Ratner; Sebastian Doniach Correlated x-ray scattering dataset, silver nanoparticles