Three Dimensional Variable-Wavelength X-Ray Bragg Coherent Diffraction Imaging

W. Cha, A. Ulvestad, M. Allain, V. Chamard, R. Harder, S. J. Leake, J. Maser, P. H. Fuoss, and S. O. Hruszkewycz

Phys. Rev. Lett. 117, 225501 – Published 23 November 2016

Abstract

We present and demonstrate a formalism by which three-dimensional (3D) Bragg x-ray coherent diffraction imaging (BCDI) can be implemented without moving the sample by scanning the energy of the incident x-ray beam. This capability is made possible by introducing a 3D Fourier transform that accounts for x-ray wavelength variability. We demonstrate the approach by inverting coherent Bragg diffraction patterns from a gold nanocrystal measured with an x-ray energy scan. Variable-wavelength BCDI will expand the breadth of feasible in situ 3D strain imaging experiments towards more diverse materials environments, especially where sample manipulation is difficult.

Received 15 September 2016

DOI:https://doi.org/10.1103/PhysRevLett.117.225501

Physics Subject Headings (PhySH)

Crystal defects Imaging & optical processing Nanocrystals X-ray beams & optics

Condensed Matter, Materials & Applied PhysicsAtomic, Molecular & OpticalInterdisciplinary Physics

Authors & Affiliations

W. Cha¹, A. Ulvestad¹, M. Allain², V. Chamard², R. Harder³, S. J. Leake⁴, J. Maser³, P. H. Fuoss¹, and S. O. Hruszkewycz^1,*

¹Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
²Aix-Marseille University, CNRS, Centrale Marseille, Institut Fresnel, 13013 Marseille, France
³X-ray Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA
⁴ESRF—The European Synchrotron, CS 40220, 38043 Grenoble Cedex 9, France

^*Corresponding author. shrus@anl.gov

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Vol. 117, Iss. 22 — 25 November 2016

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Images

Figure 1
This schematic depicts an isolated nanocrystal illuminated with a coherent beam and oriented such that the incident ( $k_{i}$ ) and exit ( $k_{f}$ ) beam wave vectors satisfy a Bragg condition for the HKL reflection (denoted by the reciprocal space vector $G_{H K L}$ ). In such an experiment, the area detector accesses a 2D slice through the 3D reciprocal space intensity pattern. To measure various components of the 3D Bragg peak intensity distribution, the scattering condition $q = k_{f} - k_{i}$ must be changed relative to $G_{H K L}$ , thus changing $Q = q - G_{H K L}$ . In a monochromatic experiment [BCDI rocking curve, panels (a) and (b)], this is done by changing the angle of the sample at a fixed $| q |$ . Alternatively, with a fixed sample position [vwBCDI scan, panels (c) and (d)], the reciprocal space volume about the Bragg peak can be sampled by changing the wavelength of the x-ray beam.
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Figure 2
Schematic of a slice-by-slice calculation of vwBCDI diffraction patterns. (a) The nanocrystal $ρ$ is multiplied by a phase factor, resulting (b) in $Q_{j} ρ$ . (c) The 3D quantity $Q_{j} ρ$ at a given $λ$ is projected onto the ( $r_{x}$ , $r_{y}$ ) plane via the projection operator $R$ . In order to properly scale the diffraction pattern for this $λ$ , the operator $F_{λ}$ is invoked, defined as $S_{λ}^{- 1} F S_{λ}$ , shown in (d)–(f). (d) $S_{λ}$ changes the number of pixels in the image to $N_{pix} (λ_{j})$ by padding with zeros. (e) A 2D Fourier transform $F$ of the padded projection array is applied. (f) $S_{λ}^{- 1}$ resizes the resulting array back to a fixed pixel size, in this case $N_{\det} \times N_{\det}$ . (g) In this manner, each slice (gray plane) through a 3D Bragg peak intensity distribution (yellow isosurface) is calculated resulting in the set ${ψ_{1}, \dots, ψ_{J}}$ that mimics a vwBCDI measurement. The inverse process of reconstructing $ρ$ from ${ψ_{1}, \dots, ψ_{J}}$ involves inverting the above operators (including $F_{λ}^{- 1}$ ), and is demonstrated graphically in Supplemental Fig. S1 [23].
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Figure 3
(a) A density isosurface of a Au nanocrystal reconstructed using vwBCDI. Coloring corresponds to near-surface lattice displacements. ${111}$ facets are labeled, and the arrow $q_{111}$ indicates the scattering vector direction of the measured Bragg peak. (b) The lattice displacement within the nanocrystal along the gray plane in (a). (c),(d) Corresponding images of the same crystal reconstructed from rocking-curve-based BCDI data. (e) Comparison of lattice displacement lineouts along the dotted lines in (b) and (d). Error metrics from vwBCDI and standard rocking-curve phase retrieval are shown in (e).
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