X-ray cross-correlation analysis and local symmetries of disordered systems: General theory

In a recent paper [P. Wochner , Proc. Natl. Acad. Sci. U.S.A. 106, 11511 (2009)10.1073/pnas.0905337106] x-ray scattering intensity correlations around a ring, in the speckle diffraction pattern of a colloidal glass, were shown to display a remarkable ∼cos(nφ) dependence on the angular coordinate φ around the ring, with integer index n depending on the magnitude of the scattering wave vector. With an analytical derivation that preserves full generality in the Fraunhofer diffraction limit, we clarify the relationship between this result and previous x-ray studies of bond-orientation order, and provide a sound basis to the statement that the angular intensity correlations deliver information on local bond arrangements in a disordered (or partially ordered) system. We present a detailed analysis of the angular cross-correlation function and show its applicability for studies of a wide range of structural properties of disordered systems, from local structure to spatial correlations between distant structural elements.