General approaches for shear-correcting coordinate transformations in Bragg coherent diffraction imaging: Part 2

X-ray Bragg coherent diffraction imaging has been demonstrated as a powerful three-dimensional (3D) microscopy approach for the investigation of sub-micrometer-scale crystalline particles. It is based on the measurement of a series of coherent diffraction intensity patterns that are numerically inverted to retrieve an image of the spatial distribution of relative phase and amplitude of the Bragg structure factor of the scatterer. This 3D information, which is collected through an angular rotation of the sample, is necessarily obtained in a non-orthogonal frame in Fourier space that must be eventually reconciled. To deal with this, the currently favored approach (detailed in Part I) is to perform the entire inversion in conjugate non-orthogonal real and Fourier space frames, and to transform the 3D sample image into an orthogonal frame as a post-processing step for result analysis. In this article, a direct follow-up of Part I, we demonstrate two different transformation strategies that enable the entire inversion procedure of the measured data set to be performed in an orthogonal frame. The new approaches described here build mathematical and numerical frameworks that apply to the cases of evenly and non-evenly sampled data along the direction of sample rotation (the rocking curve). The value of these methods is that they rely on and incorporate significantly more information about the experimental geometry into the design of the phase retrieval Fourier transformation than the strategy presented in Part I. Two important outcomes are 1) that the resulting sample image is correctly interpreted in a shear-free frame, and 2) physically realistic constraints of BCDI phase retrieval that are difficult to implement with current methods are easily incorporated. Computing scripts are also given to aid readers in the implementation of the proposed formalisms

General approaches for shear-correcting coordinate transformations in Bragg coherent diffraction imaging: Part 2

X-ray Bragg coherent diffraction imaging has been demonstrated as a powerful three-dimensional (3D) microscopy approach for the investigation of sub-micrometer-scale crystalline particles. It is based on the measurement of a series of coherent diffraction intensity patterns that are numerically inverted to retrieve an image of the spatial distribution of relative phase and amplitude of the Bragg structure factor of the scatterer. This 3D information, which is collected through an angular rotation of the sample, is necessarily obtained in a non-orthogonal frame in Fourier space that must be eventually reconciled. To deal with this, the currently favored approach (detailed in Part I) is to perform the entire inversion in conjugate non-orthogonal real and Fourier space frames, and to transform the 3D sample image into an orthogonal frame as a post-processing step for result analysis. In this article, a direct follow-up of Part I, we demonstrate two different transformation strategies that enable the entire inversion procedure of the measured data set to be performed in an orthogonal frame. The new approaches described here build mathematical and numerical frameworks that apply to the cases of evenly and non-evenly sampled data along the direction of sample rotation (the rocking curve). The value of these methods is that they rely on and incorporate significantly more information about the experimental geometry into the design of the phase retrieval Fourier transformation than the strategy presented in Part I. Two important outcomes are 1) that the resulting sample image is correctly interpreted in a shear-free frame, and 2) physically realistic constraints of BCDI phase retrieval that are difficult to implement with current methods are easily incorporated. Computing scripts are also given to aid readers in the implementation of the proposed formalisms

General approaches for shear-correcting coordinate transformations in Bragg coherent diffraction imaging. Part II

International audienceX-ray Bragg coherent diffraction imaging (BCDI) has been demonstrated as a powerful 3D microscopy approach for the investigation of sub-micrometre-scale crystalline particles. The approach is based on the measurement of a series of coherent Bragg diffraction intensity patterns that are numerically inverted to retrieve an image of the spatial distribution of the relative phase and amplitude of the Bragg structure factor of the diffracting sample. This 3D information, which is collected through an angular rotation of the sample, is necessarily obtained in a non-orthogonal frame in Fourier space that must be eventually reconciled. To deal with this, the approach currently favored by practitioners (detailed in Part I) is to perform the entire inversion in conjugate non-orthogonal real- and Fourier-space frames, and to transform the 3D sample image into an orthogonal frame as a post-processing step for result analysis. In this article, which is a direct follow-up of Part I, two different transformation strategies are demonstrated, which enable the entire inversion procedure of the measured data set to be performed in an orthogonal frame. The new approaches described here build mathematical and numerical frameworks that apply to the cases of evenly and non-evenly sampled data along the direction of sample rotation (i.e. the rocking curve). The value of these methods is that they rely on the experimental geometry, and they incorporate significantly more information about that geometry into the design of the phase-retrieval Fourier transformation than the strategy presented in Part I. Two important outcomes are (1) that the resulting sample image is correctly interpreted in a shear-free frame and (2) physically realistic constraints of BCDI phase retrieval that are difficult to implement with current methods are easily incorporated. Computing scripts are also given to aid readers in the implementation of the proposed formalisms

General approaches for shear-correcting coordinate transformations in Bragg coherent diffraction imaging: Part 1

In this two-part article series we provide a generalized description of the scattering geometry of Bragg coherent diffraction imaging (BCDI) experiments, the shear distortion effects inherent to the resulting three-dimensional (3D) image from current phase retrieval methods and strategies to mitigate this distortion. In this Part I, we derive in general terms the real-space coordinate transformation to correct this shear, which originates in the more fundamental relationship between the representations of mutually conjugate 3D spaces. Such a transformation, applied as a final post-processing step following phase retrieval, is crucial for arriving at an un-distorted and physically meaningful image of the 3D scatterer. As the relevance of BCDI grows in the field of materials characterization, we take this opportunity to generalize the available sparse literature that addresses the geometric theory of BCDI and the subsequent analysis methods. This aspect, specific to coherent Bragg diffraction and absent in two-dimensional transmission CDI experiments, gains particular importance concerning spatially-resolved characterization of 3D crystalline materials in a realiable, non-destructive manner. These articles describe this theory, from the diffraction in Bragg geometry, to the corrections needed to obtain a properly rendered digital image of the 3D scatterer. Part I provides the experimental BCDI communitcy with the theoretical underpinnings of the 3D real-space distortions in the phase-retrieved object, along with the necessary post-retrieval correction method. Part II builds upon the geometric theory developed in Part I with the formalism to correct the shear distortions directly on an orthogonal grid within the phase retrieval algorithm itself, allowing more physically realistic constraints to be applied

Structural Dynamics of Materials Probed by X-Ray Photon Correlation Spectroscopy

In this chapter we discuss coherent X-ray scattering, photon statistics of speckle patterns, and X-ray photon correlation spectroscopy (XPCS). XPCS is a coherent X-ray scattering technique used to characterize dynamic properties of condensed matter by recording a fluctuating speckle pattern. In the experiments, the time correlation function of the scattered intensity is calculated at different momentum transfers Q and thereby detailed information about the dynamics is obtained. Recently, XPCS applications have broadened to include the study of nonequilibrium and heterogeneous dynamics, e.g., in systems close to jamming or at the glass transition. This is enabled through multi-speckle techniques where a 2D area detector (CCDs or pixel detectors) is employed, and the correlation function is evaluated by averaging over subsets of equivalent pixels (same Q ). In this manner time averaging can be avoided, and the time-dependent dynamics is quantified by the so-called two-times correlation functions. Higher-order correlation functions may also be calculated to investigate questions related to non-Gaussian dynamics and dynamical heterogeneity. We discuss recent forefront applications of XPCS in the study of soft and hard condensed matter dynamics, including phase-separation dynamics of colloid-polymer mixtures, motion of Au nanoparticles at the air-water interface, dynamics of atoms in metallic crystals and glasses, and domain coarsening in phase-ordering binary alloys