Characteristics of angular cross correlations studied by light scattering from two-dimensional microsphere films

Abstract

Recently the analysis of scattering patterns by angular cross-correlation analysis (CCA) was introduced to reveal the orientational order in disordered samples with special focus to future applications on x-ray free-electron laser facilities. We apply this CCA approach to ultra-small-angle light-scattering data obtained from two-dimensional monolayers of microspheres. The films were studied in addition by optical microscopy. This combined approach allows to calculate the cross-correlations of the scattering patterns, characterized by the orientational correlation function l (q), as well as to obtain the real-space structure of the monolayers. We show that CCA is sensitive to the orientational order of monolayers formed by the microspheres which are not directly visible from the scattering patterns. By mixing microspheres of different radii the sizes of ordered monolayer domains is reduced. For these samples it is shown that l (q) quantitatively describes the degree of hexagonal order of the two-dimensional films. The experimental CCA results are compared with calculations based on the microscopy images. Both techniques show qualitatively similar features. Differences can be attributed to the wave-front distortion of the laser beam in the experiment. This effect is discussed by investigating the effect of different wave fronts on the cross-correlation analysis results. The so-determined characteristics of the cross-correlation analysis will be also relevant for future x-ray-based studies.

Colloidal crystallite suspensions studied by high pressure small angle x-ray scattering

“…55,56 For a less characteristic shape of C(Q, ∆), as in case of amorphous systems, the orientational order can be determined by studying the angular Fourier transform of the cross correlation function. [57][58][59] For the present case of colloidal crystallite suspensions, the cross correlation function can be well described using the single cosine approach.…”

Structure beyond pair correlations: X-ray cross-correlation from colloidal crystals

“…A more detailed insight into the sample structure is typically obtained by a Fourier analysis of the cross-correlation functions, as done in recent XCCA work (Lehmkü hler et al, 2014;Schroer et al, 2014Schroer et al, , 2015Zaluzhnyy et al, 2015;Kurta et al, 2015). An overview of the analysis is given by Lehmkü hler et al (2014).…”

Operational properties of fluctuation X-ray scattering data

“…The diffraction pattern of an ensemble of molecules frozen in space and time will contain the signature of many particles, combining effects from the shape and internal structure of the particles, the so-called form factor, and their mutual arrangement in space, the structure factor. In the case of an ideal dilute solution, one can show that the mean angular intensity correlation function, C 2 (q, Á'), averaged over a large number of independent multiple-particle shots, is equivalent to that obtained from single-particle data (Kam, 1977;Saldin et al, 2009), assuming no interparticle interactions (Kam, 1977;Saldin et al, 2009;Kirian et al, 2011;Altarelli et al, 2010) and the presence of a flat X-ray wavefront during the scattering process (Lehmkü hler et al, 2014;Schroer et al, 2014). The potential effects of the coherence properties of the X-ray beam on the resulting angular correlations will be discussed elsewhere.…”