Modeling and experimental validation of dynamical effects in Bragg coherent x-ray diffractive imaging of finite crystals

Yuan Gao, Xiaojing Huang, Ross Harder, Wonsuk Cha, Garth J. Williams, and Hanfei Yan

Phys. Rev. B 106, 184111 – Published 30 November 2022

Abstract

Bragg coherent diffractive imaging (BCDI) is a noninvasive microscopy technique that can visualize the shape and internal lattice deviations of crystals with nanoscale spatial resolution and picometer deformation sensitivity. Its strain imaging capability relies on Fourier transform–based iterative phase retrieval algorithms, which are mostly developed under the kinematical approximation. Such approximation prohibits the application of BCDI on larger crystals, which are commonly seen in most emerging functional materials. Understanding the dynamical effect in BCDI, as well as developing a validated method for modeling BCDI at the dynamical diffraction limit, is crucial for applying BCDI to hierarchical systems that contain micron-sized crystals and grains. Here we report a comparative study on the impact of dynamical diffraction effects by comparing the reconstruction results from two measurements of the same crystal. Forward simulation is implemented to show subtle changes of interference fringes in the diffraction pattern due to the dynamical diffraction, and is compared directly with the experimental data.

Received 6 September 2022
Revised 8 November 2022
Accepted 15 November 2022

DOI:https://doi.org/10.1103/PhysRevB.106.184111

Physics Subject Headings (PhySH)

Crystal structures

Coherent X-ray scattering X-ray diffraction X-ray imaging

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Yuan Gao ^1,*, Xiaojing Huang¹, Ross Harder², Wonsuk Cha², Garth J. Williams^1,†, and Hanfei Yan^1,‡

¹National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, New York 11973, USA
²Advanced Photon Source, Argonne National Laboratory, Lemont, Illinois 60439, USA

^*yuangao@bnl.gov
^†gwilliams@bnl.gov
^‡hyan@bnl.gov

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Vol. 106, Iss. 18 — 1 November 2022

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Images

Figure 1
Reconstruction from the [002] peak at 7.5 keV. The diffraction data were first inverted using an intentionally loosened support (a) and then using a fixed support (b). The scale bar is 250 nm. For both (a) and (b), slices of amplitude (top row) and phase (bottom row) are plotted, where the left, middle, and right columns are the slices along x-y, x-z, and y-z planes, respectively. The definition of diffraction geometry is shown in (c). $k_{i}$ and $k_{f}$ are the wavevectors of incident and diffracted x-ray photons, respectively. The laboratory coordinate is right-handed, where $y$ is upward, and $z$ is the propagation direction of the incident x-ray beam. δ and γ are the detector angles. (d) Error metrics during the iterative phase retrieval. $χ^{2}$ of the first (blue) and second (amber) trials, as well as the corresponding $η^{2}$ (green and red, respectively), are plotted in logarithmic scale.
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Figure 2
Reconstruction from the [004] peak at 15 keV. The diffraction data were inverted following the same two-step procedure. The result using a fixed support is shown in (a). The scale bar is 250 nm. Slices of amplitude (top row) and phase (bottom row) are demonstrated, where the left, middle, and right columns are the slices along x-y, x-z, and y-z planes, respectively. Error metrics— $χ^{2}$ (blue) and $η^{2}$ (amber)—during the phase retrieval with fixed support are plotted in logarithmic scale in (b).
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Figure 3
Forward simulation of the [004] peak at 15 keV. (a) Schematic of x-ray diffraction from an arbitrary crystal. $s_{0}, s_{h}$ represent the directions of transmitted and diffracted waves, respectively. Boundary conditions must be satisfied on $Γ$ (blue) for the transmitted wave and on $Ω$ (purple) for the diffracted wave. The wavefield at an arbitrary voxel P inside the crystal is integrated from all upstream voxels, as marked by red and green arrows. (b) 3D diffraction intensity of experimental data (left) and forward simulation from the DM (right), plotted in the detector frame. (c) Line intensity variations across the center of 3D diffraction intensity: along the $y$ -axis (top) and $x$ -axis (middle) of the detector, and the rocking axis (bottom). Simulation results from the DM (amber), AR (green), and KA (red) are normalized to the experimental data (blue) by integrated intensity, and then aligned using cross-correlation. Black error bars represent the Poisson noise of the DM simulation at $\pm 50$ and $\pm 70 μ m^{- 1}$ , respectively.
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Figure 4
Forward simulation of the [002] peak at 7.5 keV. Line intensity variations from experimental data (blue), DM (amber), AR (green), and KA (red) are plotted across the center of 3D diffraction intensity, along (a) $y$ -axis and (b) $x$ -axis of the detector, and (c) the rocking axis. Black error bars represent the Poisson noise of the DM simulation at $\pm 50$ and $\pm 70 μ m^{- 1}$ , respectively. (d) Detail of the first few orders of side lobes on the $- Δ q$ side from data shown in (a). Black error bars represent the Poisson noise of the DM simulation at −20, −30, and $- 40 μ m^{- 1}$ , respectively.
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Figure 5
Reconstructions from diffraction data simulated using different models. (a) and (d) are retrieved from KA, (b) and (e) are from AR, and (c) and (f) are from DM. The traditional shrink-wrap algorithm was used when inverting (a), (b), and (c), while (d), (e), and (f) were inverted using the two-step approach described in Sec. 3b.
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