Procedure for 3D atomic resolution reconstructions using atom-counting and a Bayesian genetic algorithm

Abstract text (incl. figure legends and references)

To understand the catalytic properties of metallic nanoparticles (NPs), a 3D characterization at high resolution is often required. Although electron tomography is a valuable tool to get insight in the 3D shape of NPs, this approach is often not feasible when investigating small beam-sensitive catalysts or dynamical processes. Therefore, alternative methods have been developed where 3D atomic models are reconstructed from a single ADF STEM projection [1-4]. For this purpose, atom counts are used to create an initial atomic model which serves as an input for an energy minimization to obtain a relaxed 3D reconstruction.

The existing methods often show some limitations. Either the finite atom-counting precision is ignored, or the final reconstruction ends up in a global energy minimum while the NP"s structure deviates from a ground state configuration. To overcome these limitations, we developed a Bayesian genetic algorithm. Bayesian methods are powerful tools to rationally combine a priori information with observed data and genetic algorithms are typically used for solving large optimization problems [4,5]. Our novel method enables to minimize the NP"s energy while utilizing a priori information about the finite precision of the atom-counting results and neighbour-mass relations.

In an extensive simulation study, the quality of the obtained reconstructions is quantitatively evaluated in terms of the surface atoms, which are of general interest for catalysis. Fig. 1 shows the fraction of correctly reconstructed surface atoms of a simulated Pt NP. As a reference, the results for the reconstructions without prior knowledge are also included. We also applied the method to study an experimental time-series of a catalyst Pt NP. To reliably count the number of atoms from the time series, we used a hidden Markov model which explicitly describes the possibility of structural changes over time [6]. The reconstructed models are schematically represented in Fig. 2.

In summary, a significant improvement is observed when including more relevant prior knowledge, especially at lower doses. Therefore, the method is very promising for obtaining reliable reconstructions of beam-sensitive NPs during dynamical processes from images acquired with lower incident electron doses [7].

References

[1] S. Bals et al., Nat. Comm. 3 (2012) 897

[2] L. Jones et al., Nano Lett. 14 (2014) 6336

[3] A. De Backer et al., Nanoscale 9 (2017) 8791

[4] M. Yu et al., ACS Nano 10 (2016) 4031

[5] Aarons et al., Nano Lett. 17 (2017) 4003

[6] De wael et al., Phys. Rev. Lett. 124 (2020) 106105

[7] This work was supported by the European Research Council (Grant 770887 PICOMETRICS to SVA and Grant 823717 ESTEEM3). The authors acknowledge financial support from the Research Foundation Flanders (FWO Belgium) through project fundings and a postdoctoral grant to ADB. LJ acknowledges Science Foundation Ireland (SFI), the Royal Society, and the AMBER Centre.

Figure 1 Fraction of the correctly reconstructed surface atoms l as a function of the incident electron dose. The inset shows the ground truth reconstruction of the Pt NP where the colouring of the Pt atoms indicates the nearest-neighbour coordination.

Figure 2 a) ADF STEM time series. b) corresponding reconstructed 3D atomic models for the time sequence viewed along the beam direction. c) Rotated models to show the dominant surface facets. The colouring of the atoms corresponds to the nearest-neighbour coordination.