Determining the three-dimensional atomic structure of a 1 metallic glass 2

(3D) experimental without model fitting using a multi- component metallic glass as a proof-of-principle, we advance atomic electron tomography to determine the 3D atomic positions in an amorphous solid for the first time. We quantitatively characterize the short-range order (SRO) and medium- range order (MRO) of the 3D atomic arrangement. We find that although the 3D atomic packing of the SRO is geometrically disordered, some SRO connect with each other to form crystal-like networks and give rise to MRO. We identify four crystal- networks - face-centred cubic, hexagonal close-packed, body-centered

. Looking forward, we anticipate this experiment will open the door to 28 determining the 3D atomic coordinates of various amorphous solids, whose impact 29 on non-crystalline solids may be comparable to the first 3D crystal structure solved 30 by x-ray crystallography over a century ago 14 . 31 Since the first discovery in 1960 15 , metallic glasses have been actively studied for 32 fundamental interest and practical applications 7-12,16-20 . However, due to their disordered 33 structure, the 3D atomic arrangement of metallic glasses cannot be determined by 34 crystallography 21 . Over the years, a number of experimental and computational methods 35 have been used to study the metallic glass structure, such as x-ray/neutron diffraction 22,23 , 36 x-ray absorption fine structure 9 , high-resolution transmission electron microscopy 24 , 37 fluctuation electron microscopy 25 , angstrom-and nano-beam electron diffraction 13,26,27 , 38 nuclear magnetic resonance 28 , density functional theory 29 , molecular dynamics 39 simulations 30-33 and reverse Monte Carlo modelling 9,25 . Despite all these developments, 40 however, there was no experimental method available to directly determine all the 3D 41 atomic positions in metallic glass samples. One experimental method that can potentially 42 solve this long-standing problem is atomic electron tomography (AET) 34,35 . AET 43 combines high-resolution tomographic tilt series with advanced iterative algorithms to 44 resolve the 3D atomic structure of materials without assuming crystallinity, which has 45 been applied to image grain boundaries, anti-phase boundaries, stacking faults, 46 dislocations, point defects, chemical order/disorder, atomic-scale ripples, bond distortion 47 and strain tensors with unprecedented 3D detail [36][37][38][39][40][41] . More recently, 4D (3D + time) AET 48 3 has been developed to observe crystal nucleation at atomic resolution, showing that early 49 stage nucleation results are inconsistent with classical nucleation theory 42 . Here, we use 50 a multi-component metallic glass as a model and advance AET to determine its 3D atomic 51 positions with a precision of 21 picometers. 52 Determining the 3D atomic positions in a multi-component metallic glass 53 The samples were synthesized by a carbothermal shock technique with a high cooling 54 rate (Extended Data Fig. 1a, Supplementary video 1 and Methods), which created high 55 entropy alloy nanoparticles with multi-metal components 43 . The energy-dispersive X-ray 56 spectroscopy data show the nanoparticles are composed of eight elements: Co, Ni, Ru, 57 Rh, Pd, Ag, Ir and Pt (Extended Data Fig. 1b-k). Tomographic tilt series were acquired 58 from seven nanoparticles using a scanning transmission electron microscope with an 59 annular dark-field detector (Extended Data Table 1). While most of the nanoparticles are 60 crystalline or polycrystalline, particles 1 and 2 have disordered structure (Extended Data 61 Fig. 2). In this study, we focus on the more disordered nanoparticle (particle 1), from 62 which a tilt series of 55 images were measured ( Fig. 1a and Extended Data Fig. 3a). 63 Although some crystalline features are present in several images, the diffraction patterns 64 calculated from the images show the amorphous halo. From the average diffraction 65 pattern (Fig. 1b), we derived the radial distribution function (RDF) (Extended Data Fig.   66 3c), exhibiting the amorphous structure of the nanoparticle. After pre-processing and 67 image denoising, the tilt series was reconstructed and the 3D atomic positions were traced 68 and classified (Fig. 1c, atomic model of the nanoparticle, consisting of 8322, 6896 and 3138   73   atoms for type 1, 2 and 3, respectively. To verify the reconstruction, atom tracing and   74 classification procedure, we calculated 55 images from the experimental atomic model   Figure 1g shows the RDF of the amorphous structure of the 3D atomic model 97 (Methods), where the weak second-peak splitting is consistent with previous observation 98 in high entropy bulk metallic glasses 44 . The ratios of the second, third, fourth and fifth to 99 the first peak position are 1.74, 1.99, 2.64 and 3.51, respectively, which agree with those 100 of metallic glasses 45,46 . The partial pair distribution functions (PDFs) between type 1, 2 101 and 3 atoms are shown in Fig. 1h. By fitting a Gaussian to the first peaks in the partial 102 PDFs, we determined the type 11, 12, 13, 22, 23 and 33 bond lengths to be 2.59, 2.71, 103 2.78, 2.72, 2.75 and 2.9 Å, respectively. In particular, the partial PDF for the type 33 104 atoms (the yellow curve) exhibits a unique feature with the second peak higher than the 105 first peak, indicating that the majority of type 3 atoms are distributed beyond the SRO.

106
The short-range order 107 To determine the SRO in the metallic glass sample, we used the Voronoi tessellation to 108 characterize the local atomic arrangement 6 . This method identifies the nearest neighbour 109 atoms around each central atom to form a Voronoi polyhedron, which is designated by a 110 Voronoi index <n3, n4, n5, n6> with ni denoting the number of i-edged faces. Figure 2a   111 shows the ten most abundant Voronoi polyhedra in the nanoparticle with a fraction 112 ranging from 5.02% to 1.72%, most of which are geometrically disordered and commonly 113 observed in model metallic glasses 11 such as <0,4,4,3>, <0,3,6,3>, <0,4,4,2> and 114 <0,3,6,2> (Fig. 2b). The small fractions of the Voronoi polyhedra suggest that the sample 115 is not a well relaxed metallic glass due to its poor glass forming ability 18 . Figure 2c   The medium-range order 130 From the partial PDF of type 33 atoms (Fig. 1h, the yellow curve), we observed that the 131 highest peak is located at 4.77 Å and 1.49 times higher than the nearest neighbour peak. 132 This result indicates that the majority of type 3 atoms are distributed in the second 133 coordination shell, which is between the first (3.86 Å) and the second minimum (6.08 Å) 134 of the RDF curve (Fig. 1g). According to the efficient cluster packing model 8,10-12,20 , 135 solute atoms are surrounded by solvent atoms to form solute-centred clusters. These

400
The homogenously mixed precursor solution was loaded onto the carbon substrates 51 (reduced graphene 401 oxide) and heated to a temperature as high as 1,763 K for 55 milliseconds (Extended Data Fig. 1a) Table 1). To minimize sample drift, four sequential images per tilt angle were measured with a dwell 417 time of 3 µs. To monitor any potential damage induced by the electron beam, we took 0° images before, 418 during and after the acquisition of each tilt series and ensured that no noticeable structural change was 419 observed for the seven nanoparticles. The total electron dose of each tilt series was estimated to be between 420 7´10 5 e -/Å 2 and 9.5´10 5 e -/Å 2 (Extended Data Table 1

496
Determination of 3D atomic coordinates and species. From each final 3D reconstruction, the atomic 497 coordinates and species were identified using the following procedure 40,42 . 498 i) Each 3D reconstruction was upsampled by a factor of 3 using the spline interpolation, from which all the 499 local maxima were identified. Starting from the highest intensity peak, polynomial fitting 60 was performed 500 on a 0.8×0.8×0.8 Å 3 (7×7×7 voxel) volume around each local maximum to locate the peak position. If the 501 distance between the fitted peak position and existing potential atom positions is larger than or equal to 2 502 Å, it was listed as a potential atom. After repeating this step for all the local maxima, a list of potential atom 503 positions was obtained. This method to trace the positions of potential atoms has previously been rigorously 504 tested by using two independent experimental tilt series acquired from the same sample 42 . 505 ii) A 3D difference map was generated by taking the difference between the 3D reconstruction and the list 506 of the potential atoms. Based on the difference map, we manually adjusted a very small fraction of the 507 atoms (167 out of 18356), which has been routinely used in protein crystallography 61 . 508 23 iii) A K-mean clustering method 40,42,62 was used to classify three types of atoms and non-atoms (Co and Ni 509 as type 1, Ru, Rh, Pd and Ag as type 2, and Ir and Pt as type 3) based on the integrated intensity of a 0.8 Å 510 × 0.8 Å × 0.8 Å volume around each potential atom position. An initial atomic model with 3D atomic 511 coordinates was determined from each 3D reconstruction. 512 iv) Due to the missing wedge problem and noise in the experimental images, there is local intensity 513 variation in each 3D reconstruction. A local reclassification was iteratively performed to refine the type 1, 514 2 and 3 atoms. Each atom was defined as the centre of a 10-Å-radius sphere. The average intensity 515 distribution of type 1, 2 and 3 atoms was computed within the sphere. The L2 norm of the intensity 516 distribution between the centre atom and the average type 1, 2 and 3 atom was calculated. The centre atom 517 was assigned to the type with the smallest L2 norm. The procedure was iteratively repeated until there were

533
Where ∇ Y is the spatial gradient operator with respect to the atomic position ( Y , Y , Y ). The iterative 534 refinement process was terminated when the L2 norm error could not be further reduced.

535
The local bond orientational order (BOO) parameters. The local BOO parameters (Q4 and Q6) were 536 calculated from the 3D atomic model of each nanoparticle using a method described elsewhere 63,64 . The Q4 537 and Q6 order parameters were computed up to the second shell with a shell radius set by the first valley in 538 the RDF curve of the 3D atomic model. Figure 1f and Extended Data Fig. 2h-n show the distribution of the 539 local BOO parameters of all the atoms in particles 1-7. To separate the amorphous structure from the crystal 540 nuclei, we calculated the normalized local BOO parameter, defined as nQ p 5 + Q q 5 /nQ p stt 5 + Q q stt 5 ,