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

In a recent paper [P. Wochner et al., Proc. Natl. Acad. Sci. U.S.A. 106, 11511 (2009)] x-ray scattering intensity correlations around a ring, in the speckle diffraction pattern of a colloidal glass, were shown to display a remarkable $\sim \text{cos}\left(n\phi \right)$ dependence on the angular coordinate $\phi$ 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.

Figure 1

(Color online) Geometry of the diffraction experiment. A coherent x-ray beam illuminates a disordered sample and produces a speckle diffraction pattern on a detector. The speckle features are defined by the finite size of the beam or the finite size of the sample and its microscopic configuration. The direction of the incident beam is defined along the $z$ axis of the coordinate system.

Figure 2

(Color online) Disordered sample consisting of clusters randomly oriented and distributed in 3D space. The radius vector ${\mathbf{R}}_{k_2,k_1}={\mathbf{R}}_{k_2}-{\mathbf{R}}_{k_1}$ connects the centers ${\mathbf{R}}_{k_1}$ and ${\mathbf{R}}_{k_2}$ of two different clusters $k_1$ and $k_2$, the vectors ${\mathbf{r}}_1$ and ${\mathbf{r}}_2$ define the positions of particles inside each cluster with the origin of a local coordinate system in each cluster $k_i$ positioned at its center.

Figure 3

(Color online) (a) In the general case, the cross-correlation function can be defined as an angular average over two intensity rings with different magnitudes of the scattering vectors $q_1^{\perp }\ne q_2^{\perp }$. The perpendicular components of the scattering vectors ${\mathbf{q}}_1^{\perp }$ and ${\mathbf{q}}_2^{\perp }$ are defined in the plane $\left(k_x,k_y\right)$ in the polar coordinate system as ${\mathbf{q}}_1^{\perp }=\left(q_1^{\perp },\phi \right)$ and ${\mathbf{q}}_2^{\perp }=\left(q_2^{\perp },\phi +\Delta \right)$. (b) Scattering geometry in reciprocal space. Here ${\mathbf{k}}^{\text{in}}$ is the wave vector of the incident beam directed along the $z$ axis, ${\mathbf{k}}_1^{\text{out}}$ and ${\mathbf{k}}_2^{\text{out}}$ are the wave vectors of two scattered waves with the scattering angles $2{\alpha }_1$ and $2{\alpha }_2$. The scattering vectors ${\mathbf{q}}_1=\left({\mathbf{q}}_1^{\perp },q_1^z\right)$ and ${\mathbf{q}}_2=\left({\mathbf{q}}_2^{\perp },q_2^z\right)$ are decomposed into two components: ${\mathbf{q}}_j^{\perp }$ that is perpendicular and $q_j^z$ that is parallel to the direction of the incident beam.

Figure 4

(Color online) 2D particles with simple geometrical shapes exhibiting different rotational symmetries (rotational axes are perpendicular to the plane of the figure): (a) twofold, (b) threefold, (c) fourfold, (d) fivefold, and (e) sixfold.

Figure 5

(Color online) Schematic illustration of different types of correlations contributing to the Fourier components $C_{q_1,q_2}^n$ of the angular CCF corresponding to four different terms in Eq. (38) (see text).

Figure 6

(Color online) Normalized contributions of different terms to the Fourier coefficients $C_q^n$ at $q=0.037{\text{nm}}^{-1}$. A 2D disordered system consisting of pentagonal clusters was considered. (a) Gaussian distribution of the in-plane angular orientations of the pentagonal clusters (with a standard deviation $\sigma =0.2\times 2\pi /5$). Two bold lines bound a central angle $\varphi =2\pi /5$. (b) The case of a dilute system $\left(D/d=150\right)$. (c) The case of a close-packed system $\left(D/d=1.5\right)$.