The problem is whether the defect content of a face-centred cubic (f.c.c.) nanocrystal can be determined quantitatively from its three-dimensional Bragg coherent X-ray diffraction (CXD) patterns once defects are no longer single and isolated. A 2015 numerical study catalogued the reciprocal-space signatures of individual dislocations, stacking faults and dislocation loops in atomistically relaxed f.c.c. crystals and proposed a selection-rule procedure to identify a defect known to be alone in a strain-free crystal; its Discussion explicitly left open the case of multiple interacting defects at high density, superposition on residual or epitaxial/interface strain, transfer to other lattices, and experimental validation. The open core is a general, quantitatively validated inversion covering the many-defect, strain-bearing regime.

Background

Coherent X-ray diffraction (CXD) is a lensless structural probe in which a nanometre-scale crystal is illuminated by a coherent X-ray beam and the intensity in reciprocal space around a Bragg peak — a reciprocal-lattice point, labelled by Miller indices (h,k,l)(h,k,l), where constructive interference concentrates intensity — is recorded. In the kinematic (single-scattering) approximation the intensity equals the squared modulus of the Fourier transform of the crystal's complex electron density, in which displacements of atoms from perfect periodic positions enter as phase factors; the measurable span of intensities from peak to background is the dynamical range. In Bragg coherent diffractive imaging (BCDI), a three-dimensional intensity map around one Bragg reflection is inverted by iterative phase-retrieval algorithms to image the crystal's shape and displacement (strain) field. In face-centred cubic (f.c.c.) metals, whose close-packed planes are the {111}\{111\} planes, plastic deformation is carried by dislocations — line defects characterized topologically by a Burgers vector b\mathbf{b}, equal to 12⟨110⟩\tfrac{1}{2}\langle110\rangle for perfect dislocations in f.c.c. In f.c.c. metals a perfect dislocation usually dissociates into two Shockley partial dislocations (partials with b=16⟨112⟩\mathbf{b}=\tfrac{1}{6}\langle112\rangle) bounding a stacking fault, a ribbon where the ABC stacking sequence of close-packed planes is locally violated. Closed faulted structures include Frank loops (faulted loops with b=13⟨111⟩\mathbf{b}=\tfrac{1}{3}\langle111\rangle) and prismatic loops. Two contrast rules mirror transmission-electron-microscopy conventions: a stacking fault produces a phase jump of ±2π/3\pm 2\pi/3 between the crystal parts on either side of the fault, visible at reflections with h+k+l≠3nh+k+l\ne 3n, and a dislocation is invisible at diffraction vector g\mathbf{g} when g⋅b=0\mathbf{g}\cdot\mathbf{b}=0 (invisibility criteria). Dupraz, Beutier, Rodney, Mordehai and Verdier (2015) computed kinematic CXD patterns near Bragg peaks for crystals of a few tens of nanometres containing single well defined defects, using atomic positions relaxed with embedded-atom-method (EAM) empirical interatomic potentials. They catalogued how each defect's signature depends on the reflection chosen, gave selection rules maximizing sensitivity to a given defect, proposed an experimental procedure — measure a few chosen reflections and use invisibility and sign rules — to identify a defect known to be alone, and showed that the few defects nucleated in a simulated nanoindentation of a gold crystal can in principle be identified. Their Discussion states explicitly what the study does not do: it does not address multiple defects as encountered in real samples; it discards residual strain (from growth or epitaxy) and is not suited to interface dislocations; transfer of the f.c.c. results to hexagonal or body-centred cubic lattices is unverified, with dissociated-dislocation results expected not to carry over; and the procedure is validated only in simulation. The unresolved target is thus the passage from catalog-based identification of single or few, non-interacting defects in strain-free free-standing f.c.c. crystals to a general, quantitative and experimentally validated inversion of Bragg CXD patterns of densely defected crystals.

Problem Statement

Develop and validate a general quantitative analysis (inversion) framework that determines the defect content of a face-centred cubic nanocrystal from its three-dimensional Bragg coherent X-ray diffraction patterns in the regime not covered by the 2015 catalog: multiple, interacting defects at high density, optionally superposed on inhomogeneous (residual or epitaxial/interface) strain. Given measured or simulated CXD patterns at a suitable selection of Bragg reflections, the framework should recover the defect content — types (perfect or dissociated dislocations and their character, stacking faults, Frank or prismatic loops), Burgers vectors, fault orientations and characters, positions or spatial distributions, and number densities — together with the strain contribution, within a quantified validity domain (defect-density range, crystal size and shape, noise level and reciprocal-space sampling). The related extension named in the same source — analogous validated catalogs for body-centred cubic and hexagonal close-packed lattices — is a natural follow-on within such a framework but does not by itself answer the question.

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