# Correlation methods

**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).

## 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

### 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

## See Also

- X-ray Photon Correlation Spectroscopy (XPCS)
- Fluctuation X-ray Scattering (FXS)
- Variance Scattering (VS)
- 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

- Ring graininess analysis (to determine grain count, grain size and size-distribution, crystallinity, etc.)