Abstract
Three-dimensional atomic structure is routinely determined for periodic crystals. However, extending such analysis to amorphous materials remains a substantial challenge, despite the scientific and technological importance1,2. In this context, a recent report describing the three-dimensional structure determination of an amorphous solid using atomic-resolution electron tomography (AET) is truly remarkable3. If validated, such an analysis would be groundbreaking. Here we address this issue and investigate whether and when AET can identify all or most atoms in an amorphous nanoparticle. By simulating AET, we reveal limitations on the structural and chemical information AET can determine from noisy electron images. For monoatomic nanoparticles, the structure can be determined with an atomic-position accuracy of tens of picometres under stringent fluence, sampling and projection requirements. For multi-element amorphous nanoparticles, chemical identification resolution is determined by noise and experimental sampling. Heavier atoms are more easily resolved than lighter ones, and large chemical analysis uncertainties emerge when atomic peak and background intensities overlap. Using these insights, we delineate nanoparticle size, composition, electron fluence and image sampling requirements for AET. The results serve as a benchmark for future experiment design and demonstrate a viable approach for amorphous structure determination validation using AET.
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Data availability
The source data, including atomic models and simulated projection images, are available via Zenodo at https://zenodo.org/records/10850980 (ref. 34) upon publication.
Code availability
The codes supporting this study’s findings are publicly available from sources cited in relevant references. The ZMULT program, which we used to simulate HAADF STEM images, is available at https://github.com/rtbusch/AET_CodeModRepo upon publication.
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Acknowledgements
This work was primarily supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences, Division of Materials Sciences and Engineering under award number DE-SC0024064. J.-M.Z. and R.B. acknowledge additional support from two NSF grants (DMR 2139185 and DMR 2226495) and Microscopy Australia at Monash University.
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J.-M.Z., P.R. and M.M.J.T. conceived of the idea, and J.-M.Z. directed the project. R.B. developed the tomographic reconstruction flow and conducted the simulation experiments and data analysis. M.M.J.T. provided the atomic models. J.-M.Z. developed the image simulation flow. All authors contributed to the writing of the paper.
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Nature thanks Robert Hovden, Jianwei (John) Miao and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.
Extended data figures and tables
Extended Data Fig. 1 Impact of background noise on the reconstruction of a 208 atoms amorphous silicon nanoparticle.
Two examples of (a) without and (b) with the background are shown using simulated and reconstructed projection images and the reconstructed tomograms (in the form of iso-surfaces). The amount of background noise is equivalent to the Poisson noise of 1 electron. The incident fluence is 1.6 × 104 e/Å2.
Extended Data Fig. 2 Simulation of the AET experiment reported by Yang et al.3 using the reported atomic structure and Co, Pd, and Pt as representative atoms.
(a) A simulated projection image (300 × 300 pixels) at the sampling of 0.33 Å/pixel and D = 5.6 × 104 e/Å2, using a 200 kV electron beam of 25 mrad convergence angle and a background noise of 1 e. The simulated images were centered using atoms at the surface extremities (Supplementary Note 1). (b) The preprocessed image of (a) with up-sampling of 3 times and noise smoothing, and (c) the corresponding projection obtained from the reconstructed tomogram using 55 projections from −70° to 70°. (b) The input atomic model with Pt, Pd, and Ni for the heavy, medium, and light atoms and plots showing three types of atoms separately. (c) The determined atomic structure from the simulated AET experiment as shown in the same way as (b). (d) The measured integrated atomic peak intensity histograms for three electron fluences. The atomic peaks were identified using a background threshold (1% of the maximum tomogram intensity). (e) The assignment of atomic types using the k-means clustering algorithm in ref. 3 and the CT obtained at D = 5.6 × 104 e/Å2. The low success rates for atomic identification are caused by the overlap of atomic peak intensities, which is also seen in the uploaded experimental data in ref. 3.
Extended Data Fig. 3 The impact of depth-of-focus on tomographic reconstruction.
(a) The simulated electron probe intensity distribution along the optical axis. (b) The simulated projection image and the CT in a 1 nm depth slice at the nanoparticle center. (c) The distribution of atomic peak intensity at the located atomic positions for the probe convergence angle of θ = 10 and 20 mrad, respectively. The nanoparticle simulated is 8 nm in diameter. The iso-surfaces in (b) are displayed at the value of 0.05 in normalized intensity. Further details are provided in Supplementary Note 2.
Extended Data Fig. 4 Impact of the missing wedge on AET.
The panel shows the reconstructed tomograms for the missing wedge of 10°, 20°, and 30° and the rotation step size of 3° and 6°, respectively. The tomograms are displayed as the center slice of 1 nm thickness, looking down the rotation axis. All simulations were performed at the electron fluence of 1.6 × 105 e/Å2. The red sections in the illustrated diagrams on the right mark the missing wedges. The elongation of atoms is seen along the middle section of the missing wedge (the z direction).
Extended Data Fig. 5 Impact of depth section and random defocus on the reconstructed tomograms of an 8 nm amorphous Si nanoparticle.
(a) Depth sectioning, the projection image here is a sum of ten simulated images with 1 nm increment in focus at each rotation. (b) Random focus, a random focus is selected between 0 to 10 nm for the projection image at each rotation. An example of the simulated projection image after preprocessing is shown together with the reconstruction image and a 1 nm slice of the computed tomogram using the SIRT algorithm. The tomograms show the recovery of nearly uniform and isotropic peaks.
Extended Data Fig. 6 Comparison between the simulated projection images with and without dynamical diffraction effect.
(a) A multislice simulation of a section of an 8 nm amorphous Si nanoparticle using the method described by Ishizuka (ref. 25), which is implemented in the ZMULT program (ref. 33). (b) The same section simulated using the projection approximation as described in the Method section. (c) The intensity profiles of the two simulations Taken along the marked lines in (a) and (b). The images in (a, b) are 32 pixels in height, each pixel is 0.195 Å.
Extended Data Fig. 7
Comparison between tomographic reconstructions using the SIRT and RESIRE algorithms for a 5 nm diameter amorphous silicon nanoparticle as shown in (a). The over-sampling ratio of 3 times is used for RESIRE reconstructions, while the input images are up-sampled by 3 times using interpolation for the SIRT reconstructions to match with the RESIRE algorithm. (b,c,d) The reconstructed tomograms using the SIRT and RESIRE algorithms are viewed along the z, x, and y axes. The tomograms appear nearly identical. The atomic identification yielded the same number of atoms, and the average deviation distance \(\overline{\Delta d}\) = 0.1 Å between the two reconstructions.
Supplementary information
Supplementary Information
This file contains Supplementary Notes 1–4 and Figs. 1–4.
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Busch, R., Rez, P., Treacy, M.M.J. et al. Limit of atomic-resolution-tomography reconstruction of amorphous nanoparticles. Nature 649, 1119–1122 (2026). https://doi.org/10.1038/s41586-025-09924-w
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DOI: https://doi.org/10.1038/s41586-025-09924-w
