Experimental demonstration of coupled multi-peak Bragg coherent diffraction imaging with genetic algorithms

Bragg coherent diffraction imaging has the potential to provide significant insight into the structure-properties relationship for crystalline materials by imaging, with nanoscale resolution, three-dimensional strain fields within individual grains and nanoparticles. The capability of present-day synchrotrons to locate and measure a multiplicity of Bragg reflections from a single grain makes it possible to recover the full strain tensor with nanometer resolution. Recent methods for coupling reconstructions from several peaks to determine the strain tensor have been developed and applied to synthetic data, but have not been applied to experimental data. Here, using a coupled genetic reconstruction algorithm, we reconstruct an experimental data set and demonstrate improvements in the ability to resolve vector-valued displacement fields internal to the particle as compared to what is achieved with a noncoupled approach. The coupled approach developed in this work was also validated on simulated data sets. In both simulated and experimental data, reconstructions from our coupled Bragg peak algorithm show improvements over the noncoupled independent reconstruction method of $5\%$ in terms of accuracy and $53\%$ in terms of consistency.

Figure 1

Schematic of (a) the CPR routine and (b) CPR-GA. Each peak is put through a phase retrieval recipe including iterations of ER and HIO. At designated iterations during the recipe, the universal values $\mathbf{u}\left(\mathbf{x}\right)$ and $\overline{\rho }\left(\mathbf{x}\right)$ are computed from the most recent phases and amplitudes from each constituent. The new $\mathbf{u}\left(\mathbf{x}\right)$ is projected along each $G_{j,hkl}$ vector to update the phase guess for each constituent ${{\varphi }_j}^{\prime }\left(\mathbf{x}\right)$, and the new $\overline{\rho }\left(\mathbf{x}\right)$ is used to update the amplitude guess ${{\rho }_j}^{\prime }\left(\mathbf{x}\right)$. At specified iterations, shrinkwrap is applied to the universal amplitude to update the universal support $\mathbf{S}$, which is used for all constituents. The CPR-GA depicted consists of three generations of three individuals. Individuals are instances of the CPR routine in (a). Individuals are bred with the fittest individual after each generation by averaging their $\mathbf{u}\left(\mathbf{x}\right)$ and $\overline{\rho }\left(\mathbf{x}\right)$ values.

Figure 2

Comparison of the average $\frac{{\chi }^2}{{\chi }_{\text{tot}}^2}$ and $\frac{L}{L_{\text{tot}}}$ over 40 reconstructions of the simulated (top) and experimental (bottom) data sets using the independent and coupled routines. It is clear that the coupled routine performs favourably compared to the independent routine for both data sets. Spurious errors for the $\frac{L}{L_{\text{tot}}}$ values for the independent reconstruction reflect the fact that $\frac{L}{L_{\text{tot}}}$ is unconstrained in this method.

Figure 3

$L\left(\mathbf{x}\right)$ map for IPR-GA (top) and CPR-GA (bottom) for the simulated data set. The best reconstructions were chosen for each method. Cross sections in the $x\text{−}y$, $x\text{−}z$, and $y\text{−}z$ planes are shown. The CPR-GA reconstruction shows lower $L$ values at nearly every voxel, suggesting good agreement between its constituents. $L$ values have been capped at ${10}^{-6}$ to visualize the values in the coupled reconstruction.

Figure 4

CPR-GA reconstruction of experimental Au nanocrystal on Si substrate. The strong facets and existence of a flat surface on the $x\text{−}z$ plane agree with expectations.

Figure 5

$L\left(\mathbf{x}\right)$ map for IPR-GA (top) and CPR-GA (bottom) for the experimental data set. The best reconstructions were chosen for each method. Cross sections in the $x\text{−}y$, $x\text{−}z$, and $y\text{−}z$ planes are shown. The CPR-GA reconstruction shows lower $L$ values than the IPR-GA, suggesting good agreement between its constituents. $L$ values have been capped at $4\times {10}^{-5}$ to compare smaller values in the interior of the crystal.