Three-Dimensional Coherent Bragg Imaging of Rotating Nanoparticles

Bragg coherent diffraction imaging is a powerful strain imaging tool, often limited by beam-induced sample instability for small particles and high power densities. Here, we devise and validate an adapted diffraction volume assembly algorithm, capable of recovering three-dimensional datasets from particles undergoing uncontrolled and unknown rotations. We apply the method to gold nanoparticles which rotate under the influence of a focused coherent x-ray beam, retrieving their three-dimensional shapes and strain fields. The results show that the sample instability problem can be overcome, enabling the use of fourth generation synchrotron sources for Bragg coherent diffraction imaging to their full potential.

Figure 1

(a) Geometrical description of the BCDI experiment. An example model particle (bottom) and its corresponding diffraction volume (top), which is sampled across the detector plane. Measured intensities are sensitive to small variations of the rocking ($\theta$) and roll ($\omega$) angles, but less sensitive to similar variations of the azimuth ($\phi$). (b) Diffraction frames recorded as a real particle rotates through the Bragg condition, with 48 ms exposure time. They correspond to cuts through the diffraction volume in (a) at unknown $\theta$ and $\omega$. The white vertical line is a gap between two detector modules. (c) The total diffracted intensity of a particle undergoing an uncontrolled rocking curve, recorded at 100 Hz.

Figure 2

A numerical experiment where data are simulated for arbitrary $(\theta ,\omega )$ trajectories, assembled, and phase retrieved. (a) Input model particle [cf. Fig. 1], described using slices of its phase and amplitude. (b) Selected frames illustrating the noise level, on an arbitrary and logarithmic scale. (c) The rocking and roll angles used as simulation input. (d) The matrices $P_{jk}={\sum }_l{P}_{jlk}$ (left) and $P_{lk}={\sum }_j{P}_{jlk}$ (right) after 100 iterations of Eqs. (1)–(4). Both matrices are shown on a linear scale from 0 (navy) to 1 (yellow). (e) Reconstruction of the original particle by phase retrieval of the assembled diffraction volume, as in (a).

Figure 3

Experimental results for a selected particle. (a) Assembly results presented as in Fig. 2. (b) Amplitude isosurface colored according to the (111) strain component on the surface. (c),(d) Cuts through the particle displaying the internal strain. (e) Orientation of the reconstructed particle as compared to the expected shape. (f) Fourier shell correlation analysis of the two half datasets, estimating the resolution of the final reconstruction. The dashed line shows the half-bit criterion [Eq. (17) of Ref. [33] ]. The criterion is corrected for the support by scaling the number of voxels in each shell by $(D/L{)}^2$, where $D$ and $L$ are the linear dimensions of support and phased volume, respectively [33].