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Register a new userPhase retrieval for Bragg coherent diffraction imaging at high X-ray energies
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Coherent X-ray beams with energies $geq 50$ keV can potentially enable three-dimensional imaging of atomic lattice distortion fields within individual crystallites in bulk polycrystalline materials through Bragg coherent diffraction imaging (BCDI). However, the undersampling of the diffraction signal due to Fourier space compression at high X-ray energies renders conventional phase retrieval algorithms unsuitable for three-dimensional reconstruction. To address this problem we utilize a phase retrieval method with a Fourier constraint specifically tailored for undersampled diffraction data measured with coarse-pitched detector pixels that bin the underlying signal. With our approach, we show that it is possible to reconstruct three-dimensional strained crystallites from an undersampled Bragg diffraction data set subject to pixel-area integration without having to physically upsample the diffraction signal. Using simulations and experimental results, we demonstrate that explicit modeling of Fourier space compression during phase retrieval provides a viable means by which to invert high-energy BCDI data, which is otherwise intractable.
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Unique intensity features arising from dynamical diffraction arise in coherent x-ray nanobeam diffraction patterns of crystals having thicknesses larger than the x-ray extinction depth or exhibiting combinations of nanoscale and mesoscale features. We demonstrate that dynamical scattering effects can be accurately predicted using an optical model combined with the Darwin theory of dynamical x-ray diffraction. The model includes the highly divergent coherent x-ray nanobeams produced by Fresnel zone plate focusing optics and accounts for primary extinction, multiple scattering, and absorption. The simulation accurately reproduces the dynamical scattering features of experimental diffraction patterns acquired from a GaAs/AlGaAs epitaxial heterostructure on a GaAs (001) substrate.
Coherent X-ray photons with energies higher than 50 keV offer new possibilities for imaging nanoscale lattice distortions in bulk crystalline materials using Bragg peak phase retrieval methods. However, the compression of reciprocal space at high energies typically results in poorly resolved fringes on an area detector, rendering the diffraction data unsuitable for the three-dimensional reconstruction of compact crystals. To address this problem, we propose a method by which to recover fine fringe detail in the scattered intensity. This recovery is achieved in two steps: multiple undersampled measurements are made by in-plane sub-pixel motion of the area detector, then this data set is passed to a sparsity-based numerical solver that recovers fringe detail suitable for standard Bragg coherent diffraction imaging (BCDI) reconstruction methods of compact single crystals. The key insight of this paper is that sparsity in a BCDI data set can be enforced by recognising that the signal in the detector, though poorly resolved, is band-limited. This requires fewer in-plane detector translations for complete signal recovery, while adhering to information theory limits. We use simulated BCDI data sets to demonstrate the approach, outline our sparse recovery strategy, and comment on future opportunities.
By using phase retrieval, Bragg Coherent Diffractive Imaging (BCDI) allows tracking of three-dimensional displacement fields inside individual nanocrystals. Nevertheless, in the presence of significant (1% and higher) strains, such as in the process of a structural phase transformation, fails due to the Bragg peak distortions. Here we present an advanced BCDI algorithm enabling imaging three-dimensional strain fields in highly strained crystals. We test the algorithm on particles simulated to undergo a structural phase transformation. While the conventional algorithm fails in unambiguously reconstructing the phase morphology, our algorithm correctly retrieves the morphology of coexistent phases with a strain difference of 1%. The key novelty is the simultaneous reconstruction of multiple scans of the same nanoparticle at snapshots through the phase transformations. The algorithm enables visualizing phase transformations in nanoparticles of lithium-ion, sodium-ion nanoparticles, and other nanoparticulate materials in working conditions (operando).
Focused ion beam (FIB) techniques are commonly used to machine, analyse and image materials at the micro- and nanoscale. However, FIB modifies the integrity of the sample by creating defects that cause lattice distortions. Methods have been developed to reduce FIB-induced strain, however these protocols need to be evaluated for their effectiveness. Here we use non-destructive Bragg coherent X-ray diffraction imaging to study the in situ annealing of FIB-milled gold microcrystals. We simultaneously measure two non-collinear reflections for two different crystals during a single annealing cycle, demonstrating the ability to reliably track the location of multiple Bragg peaks during thermal annealing. The thermal lattice expansion of each crystal is used to calculate the local temperature. This is compared to thermocouple readings, which are shown to be substantially affected by thermal resistance. To evaluate the annealing process, we analyse each reflection by considering facet area evolution, cross-correlation maps of displacement field and binarised morphology, and average strain plots. The crystals strain and morphology evolve with increasing temperature, which is likely to be caused by the diffusion of gallium in gold below ~280{deg}C and the self-diffusion of gold above ~280{deg}C. The majority of FIB-induced strains are removed by 380-410degree C, depending on which reflection is being considered. Our observations highlight the importance of measuring multiple reflections to unambiguously interpret material behaviour.
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.
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