Diffraction with a coherent X-ray beam: dynamics and imaging

Monte Carlo static and dynamic simulations of a three-dimensional Ising critical model

The critical dynamics of 'model A" of Hohenberg and Halperin has been studied by the Monte Carlo method. Simulations have been carried out in the three-dimensional (3d) simple cubic Ising model for lattices of sizes L=16 to L=512. Using Wolff's cluster algorithm, the critical temperature is precisely found as β_{c}=0.22165468(5). By Fourier transform of the lattice configurations, the critical scattering intensities I(q[over ⃗]) can be obtained. After circular averaging, the static simulations with L=256 and L=512 provide an estimate of the critical exponent γ/ν=2-η=1.9640(5). The |q[over ⃗]|-dependent distribution of I(q[over ⃗]) showed an exponential distribution, corresponding to a Gaussian distribution of the scattering amplitudes for a large q domain. The time-dependent intensities were then used for the study of the critical dynamics of 3d lattices at the critical point. To simulate results of an x-ray photon correlation spectroscopy experiment, the time-dependent correlation function of the intensities was studied for each |q[over ⃗]|-value. In the q region where I(q[over ⃗]) had an exponential distribution, the time correlations can be fit to a stretched exponential, where the exponent μ=γ/νz≃0.975 provides an estimate of the dynamic exponent z. This corresponds to z=2.0145, in agreement with the observed variations of the characteristic fluctuation time of the intensity: τ(q)∝q^{-z}, which gives z=2.015. These results agree with the ε expansion of field-theoretical methods (2.017). In this paper, the need to take account of the anomalous time behavior (μ<1) in the dynamics is exemplified. This dynamics reflects a nonlinear time behavior of model A, and its large time extension is discussed in detail.

In situ coherent X-ray diffraction imaging of radiation-induced mass loss in metal-polymer composite spheres

A major limitation to the use of coherent X-ray diffraction imaging (CXDI) for imaging soft materials like polymers and biological tissue is that the radiation can cause extensive damage to the sample under investigation. In this study, CXDI has been used to monitor radiation-induced structural changes in metal-coated poly(methyl methacrylate) microspheres. Using a coherent undulator X-ray beam with 8.10 keV photon energy, 14 tomograms at a resolution of ∼30 nm were measured consecutively, which resulted in an accumulated dose of 30 GGy. The three-dimensional images confirmed that the polymer core was strongly affected by the absorbed dose, giving pronounced mass loss. Specifically, as the metal-polymer composite was exposed to the X-ray beam, a bubble-like region of reduced density grew within the composite, almost filling the entire volume within the thin metallic shell in the last tomogram. The bubble seemed to have its initiation point at a hole in the metal coating, emphasizing that the free polymer surface plays an important role in the degradation process. The irradiation of an uncoated polystyrene microsphere gave further evidence for mass loss at the free surface as the radius decreased with increased dose. The CXDI study was complemented by X-ray photon correlation spectroscopy, which proved efficient in establishing exposure dose limits. Our results demonstrate that radiation-induced structural changes at the tens of nanometer scale in soft materials can be followed as a function of dose, which is important for the further development of soft-matter technology.

Deterministic Bragg Coherent Diffraction Imaging

A deterministic variant of Bragg Coherent Diffraction Imaging is introduced in its kinematical approximation, for X-ray scattering from an imperfect crystal whose imperfections span no more than half of the volume of the crystal. This approach provides a unique analytical reconstruction of the object's structure factor and displacement fields from the 3D diffracted intensity distribution centred around any particular reciprocal lattice vector. The simple closed-form reconstruction algorithm, which requires only one multiplication and one Fourier transformation, is not restricted by assumptions of smallness of the displacement field. The algorithm performs well in simulations incorporating a variety of conditions, including both realistic levels of noise and departures from ideality in the reference (i.e. imperfection-free) part of the crystal.

PyNX.Ptycho: a computing library for X-ray coherent diffraction imaging of nanostructures

X-ray imaging techniques have undergone a remarkable development during the past decade, taking advantage of coherent X-ray sources. Among these techniques, ptychography allows reconstruction of the specimen and the illumination probe from a series of diffraction patterns without any prior knowledge about the sample. However, the reconstruction of the ptychographic data remains challenging and the reconstruction software is often not publicly available. Presented here is an open-source library for the reconstruction of two-dimensional ptychographic data, written in Python. This library implements existing algorithms, with examples of data reconstruction on both simulated and experimental (Bragg ptychography on heterogeneous strained InAs/GaAs nanowires) data sets. It can be used for educational (simulation) purposes or experimental data analysis, and also features an OpenCL version of the ptychography algorithm for high-performance computing.

Signature of dislocations and stacking faults of face-centred cubic nanocrystals in coherent X-ray diffraction patterns: a numerical study

Crystal defects induce strong distortions in diffraction patterns. A single defect alone can yield strong and fine features that are observed in high-resolution diffraction experiments such as coherent X-ray diffraction. The case of face-centred cubic nanocrystals is studied numerically and the signatures of typical defects close to Bragg positions are identified. Crystals of a few tens of nanometres are modelled with realistic atomic potentials and 'relaxed' after introduction of well defined defects such as pure screw or edge dislocations, or Frank or prismatic loops. Diffraction patterns calculated in the kinematic approximation reveal various signatures of the defects depending on the Miller indices. They are strongly modified by the dissociation of the dislocations. Selection rules on the Miller indices are provided, to observe the maximum effect of given crystal defects in the initial and relaxed configurations. The effect of several physical and geometrical parameters such as stacking fault energy, crystal shape and defect position are discussed. The method is illustrated on a complex structure resulting from the simulated nanoindentation of a gold nanocrystal.

X-ray photon correlation spectroscopy

In recent years, X-ray photon correlation spectroscopy (XPCS) has emerged as one of the key probes of slow nanoscale fluctuations, applicable to a wide range of condensed matter and materials systems. This article briefly reviews the basic principles of XPCS as well as some of its recent applications, and discusses some novel approaches to XPCS analysis. It concludes with a discussion of the future impact of diffraction-limited storage rings on new types of XPCS experiments, pushing the temporal resolution to nanosecond and possibly even picosecond time scales.

On the estimation of statistical uncertainties on powder diffraction and small-angle scattering data from two-dimensional X-ray detectors

Optimal methods are explored for obtaining one-dimensional powder pattern intensities from two-dimensional planar detectors with good estimates of their standard deviations. Methods are described to estimate uncertainties when the same image is measured in multiple frames as well as from a single frame. The importance of considering the correlation of diffraction points during the integration and the resampling process of data analysis is shown. It is found that correlations between adjacent pixels in the image can lead to seriously overestimated uncertainties if such correlations are neglected in the integration process. Off-diagonal entries in the variance–covariance (VC) matrix are problematic as virtually all data processing and modeling programs cannot handle the full VC matrix. It is shown that the off-diagonal terms come mainly from the pixel-splitting algorithm used as the default integration algorithm in many popular two-dimensional integration programs, as well as from rebinning and resampling steps later in the processing. When the full VC matrix can be propagated during the data reduction, it is possible to get accurate refined parameters and their uncertainties at the cost of increasing computational complexity. However, as this is not normally possible, the best approximate methods for data processing in order to estimate uncertainties on refined parameters with the greatest accuracy from just the diagonal variance terms in the VC matrix is explored.

Coherent Bragg nanodiffraction at the hard X-ray Nanoprobe beamline

Bragg coherent diffraction with nanofocused hard X-ray beams provides unique opportunities for quantitative in situ studies of crystalline structure in nanoscale regions of complex materials and devices by a variety of diffraction-based techniques. In the case of coherent diffraction imaging, a major experimental challenge in using nanoscale coherent beams is maintaining a constant scattering volume such that coherent fringe visibility is maximized and maintained over the course of an exposure lasting several seconds. Here, we present coherent Bragg diffraction patterns measured from different nanostructured thin films at the Sector 26 Nanoprobe beamline at the Advanced Photon Source and demonstrate that with nanoscale positional control, coherent diffraction patterns can be measured with source-limited fringe visibilities more than 50% suitable for imaging by coherent Bragg ptychography techniques.

Homometry in the light of coherent beams

Two systems are homometric if they are indistinguishable by diffraction. A distinction is first made between Bragg and diffuse scattering homometry, and it is shown that in the last case coherent diffraction can allow the diffraction diagrams to be differentiated. The study of the Rudin–Shapiro sequence, homometric to random sequences, allows one to manipulate independently two-point and four-point correlation functions, and to show their effect on the statistics of speckle patterns. This study provides evidence that long-range order in high-order correlation functions has a measurable effect on the speckle statistics.

Everything SAXS: small-angle scattering pattern collection and correction

For obtaining reliable nanostructural details of large amounts of sample--and if it is applicable--small-angle scattering (SAS) is a prime technique to use. It promises to obtain bulk-scale, statistically sound information on the morphological details of the nanostructure, and has thus led to many a researcher investing their time in it over the last eight decades of development. Due to pressure from scientists requesting more details on increasingly complex nanostructures, as well as the ever improving instrumentation leaving less margin for ambiguity, small-angle scattering methodologies have been evolving at a high pace over the past few decades. As the quality of any results can only be as good as the data that go into these methodologies, the improvements in data collection and all imaginable data correction steps are reviewed here. This work is intended to provide a comprehensive overview of all data corrections, to aid the small-angle scatterer to decide which are relevant for their measurement and how these corrections are performed. Clear mathematical descriptions of the corrections are provided where feasible. Furthermore, as no quality data exist without a decent estimate of their precision, the error estimation and propagation through all these steps are provided alongside the corrections. With these data corrections, the collected small-angle scattering pattern can be made of the highest standard, allowing for authoritative nanostructural characterization through its analysis. A brief background of small-angle scattering, the instrumentation developments over the years, and pitfalls that may be encountered upon data interpretation are provided as well.

The use of a mini-κ goniometer head in macromolecular crystallography diffraction experiments

Most macromolecular crystallography (MX) diffraction experiments at synchrotrons use a single-axis goniometer. This markedly contrasts with small-molecule crystallography, in which the majority of the diffraction data are collected using multi-axis goniometers. A novel miniaturized κ-goniometer head, the MK3, has been developed to allow macromolecular crystals to be aligned. It is available on the majority of the structural biology beamlines at the ESRF, as well as elsewhere. In addition, the Strategy for the Alignment of Crystals (STAC) software package has been developed to facilitate the use of the MK3 and other similar devices. Use of the MK3 and STAC is streamlined by their incorporation into online analysis tools such as EDNA. The current use of STAC and MK3 on the MX beamlines at the ESRF is discussed. It is shown that the alignment of macromolecular crystals can result in improved diffraction data quality compared with data obtained from randomly aligned crystals.

Structural characterization of the phospholipid stabilizer layer at the solid-liquid interface of dispersed triglyceride nanocrystals with small-angle x-ray and neutron scattering

Dispersions of crystalline nanoparticles with at least one sufficiently large unit cell dimension can give rise to Bragg reflections in the small-angle scattering range. If the nanocrystals possess only a small number of unit cells along these particular crystallographic directions, the corresponding Bragg reflections will be broadened. In a previous study of phospholipid stabilized dispersions of β-tripalmitin platelets [Unruh, J. Appl. Crystallogr. 40, 1008 (2007)], the x-ray powder pattern simulation analysis (XPPSA) was developed. The XPPSA method facilitates the interpretation of the rather complicated small-angle x-ray scattering (SAXS) curves of such dispersions of nanocrystals. The XPPSA method yields the distribution function of the platelet thicknesses and facilitates a structural characterization of the phospholipid stabilizer layer at the solid-liquid interface between the nanocrystals and the dispersion medium from the shape of the broadened 001 Bragg reflection. In this contribution an improved and extended version of the XPPSA method is presented. The SAXS and small-angle neutron scattering patterns of dilute phospholipid stabilized tripalmitin dispersions can be reproduced on the basis of a consistent simulation model for the particles and their phospholipid stabilizer layer on an absolute scale. The results indicate a surprisingly flat arrangement of the phospholipid molecules in the stabilizer layer with a total thickness of only 12 Å. The stabilizer layer can be modeled by an inner shell for the fatty acid chains and an outer shell including the head groups and additional water. The experiments support a dense packing of the phospholipid molecules on the nanocrystal surfaces rather than isolated phospholipid domains.

High contrast x-ray speckle from atomic-scale order in liquids and glasses

The availability of ultrafast pulses of coherent hard x rays from the Linac Coherent Light Source opens new opportunities for studies of atomic-scale dynamics in amorphous materials. Here, we show that single ultrafast coherent x-ray pulses can be used to observe the speckle contrast in the high-angle diffraction from liquid Ga and glassy Ni(2)Pd(2)P and B(2)O(3). We determine the thresholds above which the x-ray pulses disturb the atomic arrangements. Furthermore, high contrast speckle is observed in scattering patterns from the glasses integrated over many pulses, demonstrating that the source and optics are sufficiently stable for x-ray photon correlation spectroscopy studies of dynamics over a wide range of time scales.

Dynamics in dense hard-sphere colloidal suspensions

The dynamic behavior of a hard-sphere colloidal suspension was studied by x-ray photon correlation spectroscopy and small-angle x-ray scattering over a wide range of particle volume fractions. The short-time mobility of the particles was found to be smaller than that of free particles even at relatively low concentrations, showing the importance of indirect hydrodynamic interactions. Hydrodynamic functions were derived from the data, and for moderate particle volume fractions (Φ≤ 0.40) there is good agreement with earlier many-body theory calculations by Beenakker and Mazur [Physica A 120, 349 (1984)]. Important discrepancies appear at higher concentrations, above Φ≈ 0.40, where the hydrodynamic effects are overestimated by the Beenakker-Mazur theory, but predicted accurately by an accelerated Stokesian dynamics algorithm developed by Banchio and Brady [J. Chem. Phys. 118, 10323 (2003)]. For the relaxation rates, good agreement was also found between the experimental data and a scaling form predicted by the mode coupling theory. In the high concentration range, with the fluid suspensions approaching the glass transition, the long-time diffusion coefficient was compared with the short-time collective diffusion coefficient to verify a scaling relation previously proposed by Segrè and Pusey [Phys. Rev. Lett. 77, 771 (1996)]. We discuss our results in view of previous experimental attempts to validate this scaling law [L. Lurio et al., Phys. Rev. Lett. 84, 785 (2000)].

X-ray optics simulation using Gaussian superposition technique

We present an efficient method to perform x-ray optics simulation with high or partially coherent x-ray sources using Gaussian superposition technique. In a previous paper, we have demonstrated that full characterization of optical systems, diffractive and geometric, is possible by using the Fresnel Gaussian Shape Invariant (FGSI) previously reported in the literature. The complex amplitude distribution in the object plane is represented by a linear superposition of complex Gaussians wavelets and then propagated through the optical system by means of the referred Gaussian invariant. This allows ray tracing through the optical system and at the same time allows calculating with high precision the complex wave-amplitude distribution at any plane of observation. This technique can be applied in a wide spectral range where the Fresnel diffraction integral applies including visible, x-rays, acoustic waves, etc. We describe the technique and include some computer simulations as illustrative examples for x-ray optical component. We show also that this method can be used to study partial or total coherence illumination problem.

Firmware lower-level discrimination and compression applied to streaming x-ray photon correlation spectroscopy area-detector data

We present a data acquisition system to perform on-the-fly background subtraction and lower-level discrimination compression of streaming x-ray photon correlation spectroscopy data from a fast charge-coupled device (CCD) area detector. The system is built using a commercial frame grabber with an on-board field-programmable gate array. The system is capable of continuously processing at least 60 CCD frames per second each consisting of 1024 × 1024 16-bit pixels with ≲ 15,000 photon hits per frame at a maximum compression factor of ≈95%.

Bulk dislocation core dissociation probed by coherent x rays in silicon

We report on a new approach to probe bulk dislocations by using coherent x-ray diffraction. Coherent x rays are particularly suited for bulk dislocation studies because lattice phase shifts in condensed matter induce typical diffraction patterns which strongly depend on the fine structure of the dislocation cores. The strength of the method is demonstrated by performing coherent diffraction of a single dislocation loop in silicon. A dissociation of a bulk dislocation is measured and proves to be unusually large compared to surface dislocation dissociations. This work opens a route for the study of dislocation cores in the bulk in a static or dynamical regime, and under various external constraints.

Kinoform optics applied to X-ray photon correlation spectroscopy

Moderate-demagnification higher-order silicon kinoform focusing lenses have been fabricated to facilitate small-angle X-ray photon correlation spectroscopy (XPCS) experiments. The geometric properties of such lenses, their focusing performance and their applicability for XPCS measurements are described. It is concluded that one-dimensional vertical X-ray focusing via silicon kinoform lenses significantly increases the usable coherent flux from third-generation storage-ring light sources for small-angle XPCS experiments.

Longitudinal coherence function in X-ray imaging of crystals

The longitudinal coherence function at the Advanced Photon Source beamline 34-ID-C has been measured by a novel method and the coherence length (xi(L)) determined to be, xi(L) = 0.66 +/- 0.02 microm. Three dimensional Coherent X-ray Diffraction (CXD) patterns were measured for multiple Bragg reflections from two Zinc Oxide (ZnO) nanorods with differing aspect ratios. The visibility of fringes corresponding to the 002 crystal direction for each reflection were found to be different and used to map the coherence function of the incident radiation. Partial coherence was found to be associated with amplitude 'hot' spots in three dimensional reconstructions of the crystal structure.

Observation of correlations up to the micrometer scale in sliding charge-density waves

A high resolution coherent x-ray diffraction experiment has been performed on the charge-density wave (CDW) system K0.3MoO3. The 2kF satellite reflection associated with the CDW has been measured with respect to external dc currents. In the sliding regime, the 2kF satellite reflection displays secondary satellites along the chain axis which corresponds to correlations up to the micrometer scale. This super long-range order is 1500 times larger than the CDW period itself. This new type of electronic correlation seems inherent to the collective dynamics of electrons in charge-density wave systems. Several scenarios are discussed.

Soft X-ray coherent scattering: instrument and methods at ESRF ID08

An experimental setup has been developed to perform soft x-ray coherent scattering at beamline ID08 of the European Synchrotron Radiation Facility. An intense coherent beam was obtained by filtering the primary beam with the monochromator and a circular pinhole. A pinhole holder with motorized translations was installed inside the UHV chamber of the diffractometer. The scattered intensity was recorded in reflection geometry with a back-illuminated charge coupled device camera. As a demonstration we report experimental results of resonant magnetic scattering using coherent beam. The degree of coherence is evaluated, and it is shown that, while the vertical coherence is much higher than the horizontal one at the source, the situation is reversed at the diffractometer. The intensity of the coherent beam is also discussed.