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Determining the three-dimensional atomic structure of an amorphous solid


Amorphous solids such as glass, plastics and amorphous thin films are ubiquitous in our daily life and have broad applications ranging from telecommunications to electronics and solar cells1,2,3,4. However, owing to the lack of long-range order, the three-dimensional (3D) atomic structure of amorphous solids has so far eluded direct experimental determination5,6,7,8,9,10,11,12,13,14,15. Here we develop an atomic electron tomography reconstruction method to experimentally determine the 3D atomic positions of an amorphous solid. Using a multi-component glass-forming alloy as proof of principle, we quantitatively characterize the short- and medium-range order of the 3D atomic arrangement. We observe that, although the 3D atomic packing of the short-range order is geometrically disordered, some short-range-order structures connect with each other to form crystal-like superclusters and give rise to medium-range order. We identify four types of crystal-like medium-range order—face-centred cubic, hexagonal close-packed, body-centred cubic and simple cubic—coexisting in the amorphous sample, showing translational but not orientational order. These observations provide direct experimental evidence to support the general framework of the efficient cluster packing model for metallic glasses10,12,13,14,16. We expect that this work will pave the way for the determination of the 3D structure of a wide range of amorphous solids, which could transform our fundamental understanding of non-crystalline materials and related phenomena.

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Fig. 1: Determining the 3D atomic structure of a multi-component glass-forming nanoparticle with AET.
Fig. 2: SRO of the glass-forming nanoparticle.
Fig. 3: Connectivity and distribution of MROs in the glass-forming nanoparticle.
Fig. 4: Quantitative characterization of MROs.
Fig. 5: 3D atomic packing of four representative MROs.

Data availability

The raw and processed experimental data are available at The 3D atomic coordinates of the glass-forming nanoparticle have been deposited in the Materials Data Bank ( with MDB ID NiRh00001.

Code availability

The MATLAB source codes for the RESIRE reconstruction and data analysis used in this work are available at


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We thank J. Ding for discussions, H. B. Yu for providing a molecular-dynamics-simulated atomic model of the Cu65Zr35 metallic glass, and D. J. Kline and M. R. Zachariah for assistance with the temperature measurements. This work was primarily supported by STROBE: A National Science Foundation Science & Technology Center under grant number DMR 1548924. This work was also supported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences, Division of Materials Sciences and Engineering under award number DE-SC0010378 (data acquisition and 3D image reconstruction) and the NSF DMREF programme under award number DMR-1437263. J.M. acknowledges partial support from an Army Research Office MURI grant on Ab-Initio Solid-State Quantum Materials: Design, Production and Characterization at the Atomic Scale. Y. Yao and L.H. are supported by the National Science Foundation (award number 1635221).The ADF-STEM imaging with TEAM 0.5 was performed at the Molecular Foundry, which is supported by the Office of Science, Office of Basic Energy Sciences of the US DOE under contract number DE-AC02—05CH11231.

Author information

Authors and Affiliations



J.M. conceived the idea and directed the project; Y. Yao and L.H. synthesized the samples; J.Z., P.E., A.K.S. and J.M. discussed and/or carried out the experiments; M.P., Y. Yuan, A.R., S.J.O. and J.M. developed the RESIRE algorithm. Y. Yang, F.Z., Y. Yuan, D.J.C., J.Z., D.S.K., X.T. and J.M. performed image reconstruction, atom tracing and classification, analysed the data and/or interpreted the results; J.M., Y. Yang, J.Z. and F.Z. wrote the manuscript. All authors commented on the manuscript.

Corresponding author

Correspondence to Jianwei Miao.

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The authors declare no competing interests.

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Peer review information Nature thanks Simon Billinge, Paul Voyles, Yong Yang and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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Extended data figures and tables

Extended Data Fig. 1 Cooling rate measurement, EDX and EELS maps of the nanoparticles.

a, The cooling rate for the average (Taverage) and maximum (Tmax) temperature curves was measured to be 51,000 K s−1 (slope of the red line) and 69,000 K s−1 (slope of the green line), respectively. b, Low-resolution ADF-STEM image of the nanoparticles. cj, EDX maps showing the distribution of Ni (c), Co (d), Ru (e), Rh (f), Pd (g), Ag (h), Ir (i) and Pt (j). k, EDX spectrum of all the elements shown in cj; cps, counts per second. l, Low-resolution ADF-STEM image of a large area, with the white square indicating the aggregation of several nanoparticles used for the EELS measurement. m, ADF-STEM image of the region in the white square in l. np, EELS maps show the distribution of Co (n), Ni (o) and O (p) in the same region. q, EELS spectrum obtained from np. No oxygen signal was detected in the EELS map or spectrum. Scale bars, 20 nm (b); 100 nm (l); and 10 nm (o).

Extended Data Fig. 2 Analysis of seven multi-component metallic nanoparticles.

ag, Representative ADF-STEM images of particles 1–7, respectively. Scale bar, 2 nm. hn, Local BOO parameters for all atoms in particles 1–7, where the dashed red curves correspond to a normalized BOO parameter of 0.5. The percentage at the top of each panel shows the fraction of disordered atoms in each particle. o, Local BOO parameters of a 3D atomic model cropped from a molecular-dynamics-simulated Cu65Zr35 metallic glass67 used as a reference, from which a normalized BOO parameter of 0.5 (dashed red curve) was chosen as a cut-off to separate crystal nuclei from amorphous structure. For a fair comparison, the 3D atomic model was cropped to have a similar 3D shape and size to the experimental nanoparticle (particle 1). pv, PDFs of all atoms in particles 1–7, respectively. With decreasing fraction of disordered atoms in the nanoparticles, the peaks in the PDFs become narrower and new peaks corresponding to different crystal planes appear.

Extended Data Fig. 3 Experimental tomographic tilt series of a multi-component glass-forming nanoparticle (particle 1).

55 raw ADF-STEM images of the nanoparticle with a tilt range from −69.4° to +72.6°. The 2D power spectra of the images are shown in the insets, where the amorphous halo is visible. Some crystalline features are visible in several experimental images and the 2D power spectra. Scale bar, 2 nm.

Extended Data Fig. 4 Angular errors in the experimental images and verification of the experimental 3D atomic model using multi-slice simulations.

a, Angular errors in the experimental images determined by an angular refinement procedure (Methods), where the colour dots and lines represent the deviation of the three Euler angles (ϕ, θ and φ) from the correct ones (0°) at each tilt angle. These angular errors were taken into account in the multi-slice simulations. b, The angular errors were correctly refined in the 3D reconstruction of the 55 multi-slice images using RESIRE (Methods). After the angular refinement, the largest error is only 0.2°. c, d, Comparison between a representative experimental (after denoising) (c) and a multi-slice (d) image at 0°. To account for the source size and incoherence effects, each multi-slice image was convolved with a Gaussian function (Methods). e, Histogram of the deviation of the atomic positions between the experimental atomic model and that obtained from 55 multi-slice images. The peak, mean and root-mean-square deviation of the histogram are 6 nm, 15 nm and 21 pm, respectively. Scale bar, 2 nm.

Extended Data Fig. 5 3D distribution of the crystal nuclei in the glass-forming nanoparticle, partial coordination numbers and Voronoi polyhedra of the solute-centre clusters.

a, 3D distribution of the atoms with a normalized BOO parameter ≥0.5, revealing that 15.46% of the total atoms form crystal nuclei in the nanoparticle. b, Normalized partial coordination numbers of type-1, -2 and -3 atoms. c, 3D distribution of the 2,682 solute centres (red dots) that are between the first (3.78 Å) and the second (6.09 Å) minimum of the PDF curve (Fig. 1g). d, Ten most abundant Voronoi polyhedra of the solute-centre clusters.

Extended Data Fig. 6 Identification of MROs with a 1-Å radius cut-off.

a, Histogram of the four types of MRO—fcc- (blue), hcp- (red), bcc- (green) and sc-like (purple)—as a function of size (that is, the number of solute centres). b, Population of the solute-centre atoms for the four types of MRO. cj, Representative fcc- (c), hcp- (e), bcc- (g) and sc-like (i) MROs, containing 23, 18, 10 and 27 solute centres (large red spheres), respectively. The solute centres are orientated along the fcc (d), hcp (f), bcc (h) and sc (j) zone axes.

Extended Data Fig. 7 Identification of MROs with a 0.5-Å radius cut-off.

a, Histogram of the four types of MRO—fcc- (blue), hcp- (red), bcc- (green) and sc-like (purple)—as a function of size. b, Population of the solute-centre atoms for the four types of MRO. cj, Representative fcc- (c), hcp- (e), bcc- (g) and sc-like (i) MROs, containing 15, 10, 8 and 8 solute centres (large red spheres), respectively. The solute centres are orientated along the fcc (d), hcp (f), bcc (h) and sc (j) zone axes.

Extended Data Fig. 8 Tomographic tilt series of an amorphous CuTa thin film.

ADF-STEM images of a portion of the CuTa thin film. The insets show the 2D power spectra of the experimental images, in which the amorphous halo are clearly visible. Scale bar, 2 nm.

Extended Data Fig. 9 Determination of the 3D atomic structure of the amorphous CuTa thin film.

a, Large-field-of-view image of amorphous CuTa. b, Magnified image of the region in the white square in a. c, Average 2D power spectrum of all the experimental images. d, 3D atomic model of a portion of the CuTa thin film with a total of 1,808 Cu (gold) and 12,774 Ta (blue) atoms, determined from the tilt series shown in Extended Data Fig. 8 (Methods). Because the CuTa film is thinner than ~6 nm, 40 experimental images are sufficient to produce a good 3D reconstruction. e, A 2-Å-thick internal slice of the 3D reconstruction of the amorphous CuTa thin film, showing the disordered atomic structure. f, Local BOO parameters of the 3D atomic model, where only 0.47% of the total atoms with a normalized BOO parameter ≥0.5 form crystal nuclei. g, PDF of the disordered atoms with a normalized BOO parameter <0.5. Scale bars, 30 nm (a) and 2 nm (b, e).

Extended Data Table 1 AET data collection, processing, reconstruction, refinement and statistics

Supplementary information

Supplementary Information

This file contains details regarding the Data and Source Code Package and Supplementary Figures 1-4.

Video 1

Temperature measurement of the carbothermal shock process by a high-speed Phantom Miro M110 camera. The movie shows the colormap of the temperature distribution with an interval of 55 ms and a pixel size of 25 μm, where the inhomogeneity is due to the difference of the local heat transfer.

Video 2

Progressive slices of the 3D reconstruction of the multi-component glass-forming nanoparticle, showing the disordered nature of the nanoparticle and that the majority of type 3 atoms (red blobs) are distributed in the second coordination shell. Each slice corresponds to 0.347 Å thick.

Video 3

Experimental 3D atomic model of the multi-component glass-forming nanoparticle with type 1, 2 and 3 atoms in green, blue and red, respectively, which exhibits disordered atomic structure. Compared with type 1 and 2 atoms, type 3 atoms are more uniformly distributed in the 9-nm-diameter nanoparticle.

Video 4

3D distribution of the four types of the MROs in the multi-component glass-forming nanoparticle, where fcc-, hcp-, bcc- and sc-like MROs are in blue, red, green and purple, respectively. To better visualize the networks, only those with eight solute centre atoms or more are shown.

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Yang, Y., Zhou, J., Zhu, F. et al. Determining the three-dimensional atomic structure of an amorphous solid. Nature 592, 60–64 (2021).

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