We present an x-ray study of liquid crystal membranes in the vicinity of hexatic-smectic phase transition by means of angular x-ray cross-correlation analysis (XCCA). By applying two-point angular intensity cross-correlation functions to the measured
series of diffraction patterns the parameters of bond-orientational (BO) order in hexatic phase were directly determined. The temperature dependence of the positional correlation lengths was analyzed as well. The obtained correlation lengths show larger values for the higher-order Fourier components of BO order. These findings indicate a strong coupling between BO and positional order that has not been studied in detail up to now.
Angular x-ray cross-correlation analysis (XCCA) is an approach to study the structure of disordered systems using the results of x-ray scattering experiments. In this paper we summarize recent theoretical developments related to the Fourier analysis
of the cross-correlation functions. Results of our simulations demonstrate the application of XCCA to two- and three-dimensional (2D and 3D) disordered systems of particles. We show that the structure of a single particle can be recovered using x-ray data collected from a 2D disordered system of identical particles. We also demonstrate that valuable structural information about the local structure of 3D systems, inaccessible from a standard small-angle x-ray scattering experiment, can be resolved using XCCA.
Angular x-ray cross-correlation analysis (XCCA) is an approach to study the structure of disordered systems using the results of coherent x-ray scattering experiments. Here, we present the results of simulations that validate our theoretical findings
for XCCA obtained in a previous paper [M. Altarelli et al., Phys. Rev. B 82, 104207 (2010)]. We consider as a model two-dimensional (2D) disordered systems composed of non-interacting colloidal clusters with fivefold symmetry and with orientational and positional disorder. We simulate a coherent x-ray scattering in the far field from such disordered systems and perform the angular cross-correlation analysis of calculated diffraction data. The results of our simulations show the relation between the Fourier series representation of the cross-correlation functions (CCFs) and different types of correlations in disordered systems. The dependence of structural information extracted by XCCA on the density of disordered systems and the degree of orientational disorder of clusters is investigated. The statistical nature of the fluctuations of the CCFs in the model `single-shot experiments is demonstrated and the potential of extracting structural information from the analysis of CCFs averaged over a set of diffraction patterns is discussed. We also demonstrate the effect of partial coherence of x-rays on the results of XCCA.
In a recent article (P.Wochner et al., PNAS (2009)) x-ray scattering intensity correlations around a ring, in the speckle diffraction pattern of a colloidal glass, were shown to display a remarkable ~ cos(n $phi$) dependence on the angular coordinate
$phi$ around the ring, with integer index n depending on the magnitude of the scattering wavevector. With an analytical derivation that preserves full generality in the Fraunhofer diffraction limit, we clarify the relationship of this result to 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 wide range of structural properties of disordered systems, from local structure to spatial correlations between distant structural elements.
