STRUCTURAL ANALYSIS BY X-RAY INTENSITY ANGULAR CROSS CORRELATIONS

RUSLAN P. KURTA1, MASSIMO ALTARELLI1, and IVAN A. VARTANYANTS2,3

1European XFEL GmbH, Schenefeld, Germany

2Deutsches Elektronen-Synchrotron, DESY, Hamburg, Germany

3National Research Nuclear University 'MEPhI' (Moscow Engineering Physics Institute), Moscow, Russia

CONTENTS

1. Introduction
2. Theory
   1. Scattering from a Disordered System of Reproducible Particles
   2. 2D Disordered Systems
      1. Dilute Systems
      2. Dense Systems
   3. 3D Disordered Systems
   4. Two- and Three-Point Angular CCFs and Their Fourier Decomposition
      1. General Definitions
      2. Analysis of Disordered Systems by Angular CCFs
3. Applications
   1. Single-Particle Structure Recovery from FXS
      1. 2D Structure Determination
      2. 3D Structure Determination
   2. Correlations in Disordered and Partially Ordered Phases
      1. Local Structure of Colloidal Systems
      2. BO Order in Liquid Crystals
      3. Structural Inhomogeneities in Semicrystalline Polymers
      4. Short-Range and Medium-Range Order in Metallic Glasses
      5. Emergent Rotational Symmetries and Domain Memory in Magnetic Multilayers
4. Conclusions and Outlook

Acknowledgments

References

I. INTRODUCTION

The aim of this chapter is to review the topic of angular intensity correlations in X-ray diffraction. This topic has a history going back by almost 40 years, and is intertwined with developments in the related areas of intensity correlations in optical laser scattering and in electron scattering; but it has recently known a revival, partly related to the progress in X-ray sources and in instrumentation.

In the early literature, scattering experiments performed on randomly oriented objects in a solution were mostly discussed. Correlations between scattered intensity I(q,?t) in different directions (for different scattering vectors q) at the same or at different times t were considered,

where the brackets indicate an average over many measurements. In the case of light scattering [1,2], a laser source was generally used and its full transverse coherence always implies an interference between the scattering by different particles inside the scattering volume. Also, rather large particles of the order of hundreds of nanometers, matching the wavelength of light, were investigated, in a concentration such that the average number of particles within the illuminated volume (defined by apertures) was rather small. The large particle size and the intensity of laser light combine to achieve an exposure time shorter than the characteristic orientational relaxation time of the objects. In the pioneering work by Kam [3], the intensity correlations between scattering of X-rays or neutrons from macromolecules in solution were addressed, also in the limit in which the data could be acquired in a time shorter than the characteristic molecular reorientation time. The possibility of this correlation analysis to obtain structural information without crystallization was proposed. Similar concepts were applied to electron microscopy; see, for example, Refs. 4 and 5.

The conventional X-ray scattering pattern of a disordered system, for example, a liquid, molecules in solution, or an amorphous solid is isotropic (Debye-Scherrer rings) when recorded with a weak, and low coherence, source [6,7]. The weak source, in contrast to the previously discussed examples, means that the exposure time required to collect a sufficient signal, in the case of a liquid or a solution, is long compared to characteristic relaxation times of the rotational and translational agitation. If, on the other hand, the signal can be acquired in a short time with a brilliant X-ray source with a high degree of coherence, such as available with a third-generation synchrotron source or X-ray free-electron laser, the recorded pattern is not isotropic, but is an apparently random collection of speckles. These speckles are in fact encoding the instantaneous positions and orientations of the molecules. In an amorphous solid, in a random alloy or in a glass, with slow dynamics, on the other hand, the duration of the exposure is not so relevant, but a source with a high degree of partial coherence can here also reveal a speckle pattern encoding local fluctuations in orientation or ordering.

Due to high penetration of X-rays, multiple scattering effects on a disordered sample of few microns size can be safely neglected and kinematical scattering will be assumed to be valid. This is a very important simplification of the theory that is valid only for a limited number of samples studied by visible light or by electrons, where multiple scattering effects can seldom be neglected. This makes X-rays especially attractive for the study of disordered systems. A low-noise, high-dynamic range detector, with sufficient angular resolution, is also needed to record meaningful angular anisotropies. The very recent emergence of X-ray free-electron lasers (XFELs) [8-10], with ultrabright pulses of few femtoseconds duration and a high degree of transverse coherence, is opening up the promise of a completely new set of experimental conditions and provides further motivation for exploring the potential benefits of correlation analysis.

The revival of angular correlation studies was recently prompted by the work of Wochner et al. [11], which reported angular correlations with pronounced periodic character in a colloidal suspension of polymethylmethacrylate (PMMA) spheres, expected to form icosahedral clusters near the glass formation concentration. They considered angular averages in the form of a cross-correlation function (CCF) calculated on the same scattering ring ( ) and at the same time (the scattering vector q, in the plane normal to the incoming beam, being expressed as )

where is the angular coordinate, is the intensity fluctuation, function, and denotes the average over the angle f. This work stimulated further theoretical [12-21] and experimental [12-21] exploration of the CCFs in the studies of disordered materials by X-ray scattering, as well as light [32] and electron scattering [33].

There are two main scientific drivers for the investigation of the angular correlations of X-ray scattering patterns. On the one hand, the angular correlations in scattering experiments are investigated as a possible tool to solve structures of molecules in solutions or, more generally, in noncrystalline systems (see Fig. 1). This line of thought, as we saw, goes back to the work of Kam [3], almost 40 years ago; his seminal (although so far not yet implemented in full) idea, was that the intensity fluctuations contain additional information, with respect to the average around the scattering intensity rings. This could allow to go beyond the quantities traditionally extracted from the isotropic patterns (average pair correlation functions in a liquid, or radius of gyrations for molecules in solutions, etc.), possibly all the way to the high-resolution molecular structure. In the more recent applications, the progress in instrumentation opens the door to a rapid acquisition of many scattering patterns; this makes acquisition of angular correlations not only in each diffraction pattern but also over an ensemble of many diffraction patterns possible [3,34].

Figure 1. Different types of structural disorder in a system of particles recorded by coherent X-ray snapshots. (a) A single benzene-like molecule and the corresponding simulated coherent diffraction pattern (e). (b) Oriented system of molecules, where particles have random positions but the same orientation as the molecule shown in (a). Corresponding scattering pattern (f) encodes information about a single-particle structure (compare with (e)), modulated by coherent superposition of waves scattered from molecules in different positions. (c) Aligned system of molecules, where in addition to positional disorder particles have random orientations about z -axis. Only the central part of the respective scattering pattern (g) reminds about the single-molecule diffraction pattern (a). (d) Completely disordered system of molecules, where particles have random positions and orientations. Scattering pattern simulated for this system can not be directly associated with the single molecule. In all simulations direction of the incoming beam is assumed along z-axis.

On the other hand, an alternative application of angular correlations could be very important for the physics of disordered or partially ordered systems: it is the unveiling of hidden symmetries and of partial order. This includes systems displaying short-range order (SRO) [35,36], as well as complicated dynamics, aging, dynamical heterogeneity, and medium-range order (MRO) in a large class of glass-forming liquids [37-41]. In such systems, a relevant question is, for example, if one can recognize and identify an n -fold symmetry axis of an...