Abstract
Amorphous solids such as glass are ubiquitous in our daily life and have found broad applications ranging from window glass and solar cells to telecommunications and transformer cores 1,2 . However, due to the lack of long-range order, the threedimensional (3D) atomic structure of amorphous solids have thus far defied any direct experimental determination 3-12 . Here, using a multi-component metallic glass as a model, we advance atomic electron tomography to determine its 3D atomic positions and chemical species with a precision of 21 picometer. We quantify 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 SROs connect with each other to form crystal-like 2 networks and give rise to MROs, which exhibit translational but no orientational order. We identify four crystal-like MROs  face-centred cubic, hexagonal closepacked, body-centered cubic and simple cubic  coexisting in the sample, which significantly deviate from the ideal crystal structures. We quantify the size, shape, volume, and structural distortion of these MROs with unprecedented detail.
Abstract
Looking forward, we anticipate this experiment will open the door to determining the 3D atomic coordinates of various amorphous solids, whose impact on noncrystalline solids may be comparable to the first 3D crystal structure solved by xray crystallography over a century ago 13 .
Abstract
Since the first discovery in 1960 14 , metallic glasses have been actively studied for fundamental interest and practical applications 7-12,15-19 . However, due to their disordered internal structure, the 3D atomic arrangement of metallic glasses cannot be determined by crystallography 20 . Over the years, a number of experimental and computational methods have been used to study the metallic glass structure, such as x-ray/neutron diffraction 21,22 , x-ray absorption fine structure 9 , high-resolution transmission electron microscopy 23 , fluctuation electron microscopy 24 , nanobeam electron diffraction 25,26 , nuclear magnetic resonance 27 , density functional theory 28 , molecular dynamics simulations 29-31 and reverse Monte Carlo modelling 9,24 . Despite all these developments, however, there is currently no experimental method available to directly resolve the 3D atomic structure of metallic glasses. One experimental method that can potentially solve this long-standing problem is atomic electron tomography (AET) 32,33 . AET combines high-resolution tomographic tilt series with advanced iterative algorithms to determine the 3D atomic positions in materials without assuming crystallinity, which has been applied to image grain boundaries, anti-phase boundaries, stacking faults, dislocations,

point defects, chemical order/disorder, atomic-scale ripples, bond distortion and strain tensors with high precision [34] [35] [36] [37] [38] [39] [40] . More recently, 4D (3D + time) AET has been developed to observe crystal nucleation at atomic resolution, showing early stage nucleation results are not consistent with classical nucleation theory 41 . Here, we use a multi-component metallic glass as a model and advance AET to determine the 3D atomic positions in an amorphous solid for the first time.
Determining the 3D atomic positions in a multi-component metallic glass
The samples were synthesized by a carbothermal shock technique with a cooling rate as high as 69000 K/s ( Supplementary Fig. 1a and Methods), which created high entropy alloy nanoparticles with multi-metal components 42 . The energy-dispersive X-ray spectroscopy data show the nanoparticles are composed of eight elements: Co, Ni, Ru, Rh, Pd, Ag, Ir and Pt ( Supplementary Fig. 2 ). Tomographic tilt series were acquired from six nanoparticles using a scanning transmission electron microscope with an annular dark-field detector ( Supplementary Table 1 ). While most of the nanoparticles show the crystalline structure, particles 1 and 2 exhibit more disordered structures ( Supplementary   Fig. 3 ). In this study, we focus on the most disordered nanoparticle (particle 1) among the six, from which a tilt series of 51 high-resolution images was acquired ( Supplementary   Fig. 4 ). After pre-processing and image denoising, the tilt series was reconstructed by a newly developed real space iterative algorithm and the 3D atomic positions and chemical species were traced and classified ( Supplementary Fig. 5, Methods) . Since the image contrast in the 3D reconstruction is based on the atomic number 39, 41 , presently AET is only sensitive enough to classify the eight elements into three different types: Co and Ni as type 1, Ru, Rh, Pd and Ag as type 2, and Ir and Pt as type 3, as the atomic number of the elements in each type is close to each other. After atom classification, we obtained the experimental 3D model of the disordered nanoparticle, consisting of 8794, 8587 and 3472 atoms for type 1, 2 and 3, respectively. To validate the reconstruction, atom tracing and classification procedure, we calculated 51 projections from the experimental atomic model using multislice simulations 43 . Supplementary Fig. 6a and b shows the consistency between the experimental and calculated projections. We then applied the same reconstruction, atom tracing and classification procedure to determine a new 3D atomic model from the 51 multislice projections. By comparing the two models, we confirmed that 96.5% of atoms were correctly identified with a 3D precision of 21 pm ( Supplementary Fig. 6c ). Figure 1a and Supplementary video 1 show the experimental 3D atomic model of the nanoparticle with type 1, 2 and 3 atoms in green, blue and red, respectively, which exhibits disordered atomic structure. To quantify the disorder, we calculated the local bond orientational order (BOO) parameter of the atoms 44 (Methods) . Figure 1b shows the BOO distribution of all the atoms in the nanoparticle, indicating the majority of atoms significantly deviate from the ideal crystal structures. For a comparison, the BOO distribution of the other five nanoparticles is shown in Supplementary Fig. 3h -l. By using a threshold of BOO  0.5 ( Supplementary Fig. 3m ), we found 884 atoms (i.e. 4.2% of the total atoms, Fig. 1c ) satisfy this criterion. 500 of these atoms formed 19 crystal nuclei, each of which was defined to have a minimum of 13 atoms 41 . 18 nuclei are situated on or near the surface and only one nucleus is located near the centre of the nanoparticle. To separate the amorphous structure from the crystal nuclei, we focus on the analysis of the atoms with BOO < 0.5 in the following sessions. Figure 1d shows the radial distribution function (RDF) of the amorphous structure, where the splitting of the 2 nd and 3 rd peak was observed with the Gaussian fit ( Fig. 1d , inset). The ratios of the 2 nd , 3 rd , 4 th and 5 th to the 1 st peak position were determined to be 1.82, 2.13, 2.74 and 3.62, respectively, which are consistent with those of metallic glasses 45 . The pair distribution functions (PDFs) between type 1, 2 and 3 atoms are shown in Fig. 1e , which consist of 6 pairs -type 11, 12, 13, 22, 23 and 33 atoms. By fitting the Gaussian to the 1 st peaks in the PDFs, we measured the bond lengths of type 11, 22 and 33 atoms to be 2.60, 2.67 and 2.66 Å, respectively, which are consistent with our estimation that the average size of type 2 and 3 atoms is comparable to each other and both are slightly larger than that of type 1 atoms (Methods). While the bond lengths of type 12 (2.65 Å) and type 13 (2.66 Å) agree with the average values of the corresponding atom types, the bond length of type 23 (2.60 Å) is 2.4% shorter than the expected value (Methods), indicating a bond shortening between type 2 and 3 atoms. Furthermore, unlike other five PDFs, the PDF for the type 33 atoms (the yellow curve) exhibits a unique feature with a higher 2 nd peak than the 1 st peak, suggesting that the majority of type 3 atoms are distributed beyond the SRO.
The short-range order
To determine the SRO in the metallic glass sample, we used the Voronoi tessellation method to characterize the local atomic arrangement 6 . This method identifies the nearest neighbour atoms around each central atom to form a Voronoi polyhedron, which is designated by a Voronoi index <n3, n4, n5, n6> with ni denoting the number of i-edge faces. Figure 2a shows the ten most abundant Voronoi polyhedra in the nanoparticle with a fraction ranging from 7.3% to 3.1%, most of which are geometrically disordered and commonly observed in model metallic glasses 11 such as <0,4,4,3>, <0,4,4,2>, <0,2,8,1> and <0,3,6,3> (Fig. 2b ). Although the Voronoi index of <0,4,4,4> can be considered a distorted fcc structure, the 3D atomic positions in these clusters severely deviate from an idea fcc lattice and exhibit a disordered structure ( Fig. 2b ). Figure 2c shows the local symmetry distribution for all the faces of the Voronoi polyhedra. The 3-, 4-, 5-and 6-fold faces account for 3.2%, 31%, 45.3% and 20.5%, respectively, revealing 5-fold faces are most abundant in the SRO. But only 9.2% of all the Voronoi polyhedra are distorted icosahedra, which include Voronoi indices <0,0,12,0>, <0,1,10,2>, <0,2,8,2> and <0,2,8,1>. This result indicates that most 5-fold faces do not form distorted icosahedra in this metallic glass nanoparticle. From the Voronoi tessellation, we also calculated the coordination number (CN) distribution for the atoms (Fig. 2d ), where the average CNs for types 1, 2 and 3 are 10.9, 10.9 and 10.7, respectively. The smaller average CN of type 3 relative to the other two types of atoms can be attributed to the shortening of the type 23 bond (Fig. 1e, Methods) , which reduces the space for packing solvent atoms around the type 3 atoms. The bond shortening also implies a stronger interaction or chemical SRO between type 2 and 3 atoms. As a verification, we quantified the chemical SRO using the Warren-Cowley parameter (Methods), confirming that the type 11 and 23 bond are favoured in the metallic glass sample ( Supplementary Fig. 7a ).
The medium-range order
From the PDF of type 33 atoms ( Fig. 1e , the yellow curve), we observed that the highest peak is matthew55@example.net Å and is 1.22 times higher than the nearest neighbour peak. This result indicates that the majority of type 3 atoms are distributed in the second coordination shell, which is between the first (3.78 Å) and the second minimum (6.09 Å) of the RDF curve ( Fig. 1d ). According to the efficient cluster packing model 8, [10] [11] [12] 19 , solute atoms are surrounded by solvent atoms to form solute-centred clusters. These solute-centred clusters act as the basic building blocks and are densely packed in 3D space to constitute the MRO of metallic glasses. To quantitatively test this model with experimental data, we first examined the distribution of the type 3 atoms in the second coordination shell. Supplementary Fig. 7b and c shows that 79.8% of type 3 atoms (2769) satisfy this criterion and are fairly uniformly distributed inside the nanoparticle. These type 3 atoms act as solute centres and are surrounded mainly by type 1 and 2 solvent atoms to form atomic clusters. Supplementary Fig. 7d shows the ten most abundant Voronoi polyhedra of the solute-centred clusters. The solute-centred clusters connect with each other by sharing one (a vertex), two (an edge) and three atoms (a face) as well as protrude into each other by sharing four and five atoms ( Fig. 3a-e ). Figure 3f shows the statistical distribution of the number of the solute-centred cluster pairs, which share from one to five atoms.
The medium-range order
To locate the MRO, we implemented a breadth-first search algorithm to search for the fcc-, hcp-, bcc-, simple cubic (sc-) and icosahedral-like networks of the solute centres in the metallic glass nanoparticle (Methods). This algorithm enables us to globally search for MROs with a maximum number of solute centres. Each MRO is defined to have five or more solute centres with each solute centre falling within a 0.75 Å radius relative to the perfect fcc, hcp, bcc, sc lattice or icosahedral vertices. We found there are four types of MROs (fcc-, hcp-, bcc-and sc-like), but no icosahedral-like MROs in the nanoparticle (Methods). Figure 3g shows the histogram of the four types of MROs as a function of the size (i.e. the number of solute centres), where the inset illustrates the fraction of the four MRO solute centre atoms. Figure 3h and Supplementary Video 2 show the 3D distribution of the four types of MROs with each having ten solute centres or more.
The medium-range order
To validate our analysis, we also searched for the MROs with a 1 Å and 0.5 Å radius cutoff. While the total number of the MROs changes as the cut-off radius, the fraction of the four MRO solute centre atoms is consistent among the three different cut-off radii ( Supplementary Figs. 8 and 9 ).
The medium-range order
Next, we quantified the MROs with a 0.75 Å radius cut-off. Figure 4a the other three partial RDFs, the partial RDF of the sc-like MRO has two peaks and the ratio of the 2 nd to the 1 st peak position is about √2 (Fig. 4c , the purple curve), which corresponds to the ratio of the diagonal to the side length of a square. The shorter nearest neighbour distance of the sc-like MROs relative to the other three crystal-like networks and the appearance of the two peaks in the partial RDF indicate that the sc-like solutecentred clusters are more closely connected with their neighbours. Figure 4d shows the distribution of sharing one, two, three, four and five atoms between neighbouring solutecentred clusters for the four types of the MRO, confirming that the solute-centred clusters in sc-like MROs tend to share more atoms with their neighbours than those in other types of MROs.
The medium-range order
Our quantitative analysis of the SRO and MRO in a multi-component metallic glass confirm the general framework of the efficient cluster packing model 8, [10] [11] [12] 19 , that is, solute-centred clusters are densely packed in 3D space to give rise to the MRO.
The medium-range order
However, we observed that the shortening of a solute and solvent bond contributes to the formation of the solute-centred clusters. We revealed that fcc-, hcp-, bcc-and sc-like MROs co-exist in the multi-component metallic glass and there are no icosahedral-like MROs. By quantifying their length, volume and 3D structure, we found that the MROs not only have a large variation in length and volume, but also significantly deviate from the ideal crystal structures (Fig. 4c ). These results indicate that the MROs in real metallic glasses are more complicated than previously thought. Furthermore, as the size of MROs is comparable to that of shear transformation zones in metallic glasses 11, 19, 46, 47 , AET could also be applied to determine the 3D atomic structure of shear transformation zones and link the structure and properties of metallic glasses 11 .
Outlook
Over the last century, crystallography has been broadly applied to determine the 3D atomic structure of crystalline samples 20 . The quantitative 3D structural information has been fundamental to the development of many scientific fields. However, for amorphous solids, their 3D structure has been primarily inferred from experimental data, where the statistical structural information can be obtained but the 3D local structure is lost [8] [9] [10] [11] [12] . This qualitative approach has hindered our fundamental understanding of the 3D structure of amorphous solids and related phenomena such as the crystal-amorphous phase transition and the glass transition 48, 49 . Here, we demonstrate the ability to directly determine the 3D atomic structure of an amorphous solid using AET, which enables us to quantitatively analyse the SRO and MRO at the single-atom level. Although we focus on a metallic glass nanoparticle in this study, this method is generally applicable to different sample geometry such as thin films and extended objects (Methods). Furthermore computer simulations have indicated that AET can also be used to resolve the 3D atomic positions of silicate glasses 50 . Thus, we anticipate that this work will open the door to determining the 3D structure of a wide range of amorphous solids. Furthermore, by annealing amorphous solid samples at different times and temperatures (below and above the transition temperature), 4D AET can be applied to reveal their atomic structure as a The inset shows the splitting of the 2 nd and 3 rd peaks with the Gaussian fit. e, The PDFs between type 1, 2 and 3 atoms, consisting of 6 pairs -type 11, 12, 13, 22, 23 and 33 atoms.
Figures and Figure legends
The PDF for the type 33 atoms (the yellow curve) shows a unique feature with a higher 2 nd peak than the 1 st peak. The homogenously mixed precursor solution was loaded onto the carbon substrates 51 (reduced graphene oxide) and heated to a temperature as high as 1,763 K for 55 milliseconds ( Supplementary Fig. 1 ). The sample was suspended on a trench and connected with copper electrodes by silver paste for both heating and effective cooling as a giant heat sink. The thermal shock synthesis was triggered by electric Joule heating in an argon-filled glovebox using a Keithley 2425 SourceMeter where the high temperature and duration can be effectively controlled by tuning the input power and duration. The temperature of this process was monitored by a time-resolved spectrometer. The max cooling rate was estimated to be 69,000 K/s at the cooling stage ( Supplementary Fig. 1 ). The resulting nanoparticles on reduced graphene oxide were dispersed in ethanol with sonication. After deposited on to 5-nm-thick silicon nitride membranes, the nanoparticles were baked at 100 °C for 12 hours in vacuum to eliminate any hydrocarbon contamination.
Data acquisition.
A set of tomographic tilt series were acquired from six nanoparticles using the TEAM 0.5 microscope with the TEAM stage 52 . Images were collected at 200 kV in ADF-STEM mode (Supplementary Table 1 ). To minimize sample drift, four sequential images per tilt angle were measured with a dwell time of 3 μs. To monitor any potential damage induced by the electron beam, we took 0° projection images before, during and after the acquisition of each tilt series and ensured that no noticeable structural change was observed for the six particles. The total electron dose of each tilt series was estimated to be between 710 5 e -/Å 2 and 9.510 5 e -/Å 2 ( Supplementary Table 1 ).
Data acquisition.
Image pre-processing and denoising. For each experimental tilt series, we performed the following procedure for image post-processing and denoising. i) Image registration. At each tilt angle, we used the first image as a reference and calculated normalized cross-correlation between the reference and the other three images using a step size of 0.1 pixel 53 . These four images were aligned and summed to form an experimental image at that tilt angle.
Data acquisition.
ii) Scan distortion correction 36 . Two steps were used to correct the scan distortion for the experimental images. First, a set of low-magnification images were taken from nanoparticles and their positions were fitted with a Gaussian. Based on the geometric relation of the nanoparticles at different angles, the scan coil directions were calibrated to be perpendicular and equal in strength. Second, six high-magnification images were taken from a multi-component metallic nanoparticle and scan distortion parameters were estimated by minimizing the mean squared error of the common line of the six images. These scan distortion parameters were applied to the experimental images.
Data acquisition.
iii) Image denoising. The experimental images contain mixed Poisson and Gaussian noise and were denoised by the block-matching and 3D filtering (BM3D) algorithm 54 , which has been demonstrated to be effective in reducing noise in AET 36, 39, 41 . The BM3D denoising parameters were optimized by the following three steps. First, Poisson and Gaussian noise level were estimated from the experimental tilt series. Second, several images were simulated based on a model nanoparticle, which has a similar size and elemental distribution as those of an experimental image. The same level of Poisson and Gaussian noise was added to the simulated images. Third, these noisy images were denoised by BM3D with different parameters. The denoising parameters corresponding to the largest cross-correlation coefficient between the denoised and the original images were chosen and applied to denoise the experimental images. iv) Background subtraction and alignment. After denoising, a 2D mask was defined from each experimental image, which is slightly larger than the size of the nanoparticle. The background inside the mask was estimated by the discrete Laplacian in Matlab. After background subtraction, each tilt series of the experimental images were scaled and aligned by the centre of mass and common line methods 32, 34 .
Data acquisition.
The REal Space Iterative REconstruction (RESIRE) algorithm. After post-processing and denoising, the experimental images were reconstructed by the RESIRE algorithm. The algorithm iteratively minimizes an error function defined by,
Data acquisition.
where ( ) is an error function of a 3D object ( ) at tilt angle , Π ( ) projects to generate a 2D image at angle , is the experimental image at angle , and { , } is the coordinates. The minimization is solved via the gradient descent,
Data acquisition.
where ∇ represents the gradient and is the rotation matrix at tilt angle θ, which transforms coordinates { , , } to { , , }. Supplementary Fig. 5 shows the schematic diagram of the iterative process, where the j th iteration consists of the following four steps.
Data acquisition.
i) A set of images are calculated from the 3D object of the j th iteration using a Fourier method. The 3D object is first padded with zeros by properly choosing an oversampling ratio 55 . Applying the fast Fourier transform to the zero-padded object generates a 3D array in reciprocal space, from which a series of 2D Fourier slices are obtained at different tilt angles. These 2D Fourier slices are inverted to a set of images via the inverse Fourier transform.
Data acquisition.
ii) The error function defined in equation 1 is calculated between the computed and experimental images.
Data acquisition.
iii) The gradient of the error function is computed for every voxel using equation 2.
Data acquisition.
iv) The 3D object of the (j+1) th iteration is updated by,
Data acquisition.
where ∆ is the step size (∆ = 2 was chosen for the reconstruction of our experimental data), n is the number of images and N is the dimension of each image (N × N). +1 { , , } is used as an input for the (j+1) th iteration.
Data acquisition.
The convergence of the algorithm is monitored by the R-factor,
Data acquisition.
Usually, after several hundreds of iterations, the algorithm converges to a high-quality 3D reconstruction from a limited number of images. Both our numerical simulation and experimental results have indicated that RESIRE outperforms other iterative tomographic algorithms such as generalized Fourier iterative reconstruction 56 and simultaneous iterative reconstruction technique 57 . By avoiding iterating between real and reciprocal space, RESIRE can be applied to general sample geometry such as thin films and extended objects. The details of the RESIRE algorithm will be reported in a follow-up paper.
Data acquisition.
For each aligned and scaled experimental tilt series, we first ran RESIRE for 200 iterations. From the initial 3D reconstruction, angular refinement and spatial alignment were applied iteratively until there was no further improvement. Next, the background of each experimental image was re-evaluated and resubtracted. Using the refined experimental images and tilt angles, we ran another 200 iterations of RESIRE to obtain the final 3D reconstruction of each experimental tilt series (Supplementary Table 1 ).
Determination of 3D atomic coordinates and species.
From each final 3D reconstruction, the atomic coordinates and chemical species were identified using the following procedure 39, 41 .
Determination of 3D atomic coordinates and species.
i) Each 3D reconstruction was upsampled by a factor of 3 using the spline interpolation, from which all the local maxima were identified. Starting from the highest intensity peak, polynomial fitting 58 was performed on a 0.8 Å × 0.8 Å × 0.8 Å (7 × 7 × 7 voxel) volume around each local maximum to locate the peak position.
Determination of 3D atomic coordinates and species.
If the distance between the fitted peak position and existing potential atom positions is larger than or equal to 2 Å, it was listed as a potential atom. After repeating this step for all the local maxima, a list of potential atom positions was obtained.
Determination of 3D atomic coordinates and species.
ii) From the list of the potential atoms, manual checking was performed to correct any unidentified or misidentified atoms due to the broadened local intensity peaks from multiple atoms.
Determination of 3D atomic coordinates and species.
iii) A K-mean clustering method 39, 41, 59 iv) Due to the missing wedge problem and noise in the experimental images, there is local intensity variation in each 3D reconstruction. A local reclassification was iteratively performed to refine the type 1, 2 and 3 atoms. Each atom was defined as the centre of a 10-Å-radius sphere. The average intensity distribution of type 1, 2 and 3 atoms was computed within the sphere. The L2 norm of the intensity distribution between the centre atom and the average type 1, 2 and 3 atom was calculated. The centre atom was assigned to the type with the smallest L2 norm. The procedure was iteratively repeated until there are no further changes.
Refinement of 3D atomic coordinates and species.
The 3D atomic coordinates were refined by minimizing the error between the calculated and measured images using the gradient descent 36, 39, 41 . Each atom was first fit with a 3D Gaussian function with a height and a width ′ , where and ′ were considered the same for the same type of atoms. A 3D atomic model was obtained by,
Refinement of 3D atomic coordinates and species.
where , , , and ′ are the coordinates, height and standard deviation of the i th atom, respectively, and is a cut-off size of the 3D Gaussian function. From the 3D atomic model, a set of projection images
Refinement of 3D atomic coordinates and species.
were computed at different tilt angle θ by,
Refinement of 3D atomic coordinates and species.
Substituting equation (6) into (1), an error function was calculated, from which the gradient descent method was used to search for the optimal atomic position at the (j+1) th iteration,
Refinement of 3D atomic coordinates and species.
Where ∇ is the spatial gradient operator with respect to the atomic position ( , , ). The iterative refinement process was terminated when the L2 norm error could not be further reduced.
Refinement of 3D atomic coordinates and species.
The local bond orientational order (BOO) parameter. The local BOO parameter was calculated from the 3D atomic model of each nanoparticle using a method described elsewhere 44, 60 . The Q4 and Q6 order parameters were computed up to the second shell with a shell radius set by the first valley in the RDF curve of the 3D atomic model. The local BOO parameter was normalized between 0 and 1, where 0 means Q4 = Q6 = 0 and 1 represents a perfect fcc crystal structure. Fig. 1b and Supplementary Fig. 3g -l show the distribution of the local BOO parameter of all the atoms in particles 1-6. Based on the BOO distribution of a Cu65Zr35 metallic glass structure obtained from molecular dynamics simulations 61 ( Supplementary Fig.   3m ), we chose BOO = 0.5 as a cut-off to separate crystal nuclei from amorphous structure. The histogram was scaled so that the RDF approaches one for large separations. After calculating the RDF for each nanoparticle, the first valley of the RDF was used to determine the local BOO parameter ( Fig. 1b and Supplementary Fig. 3g-l) . By choosing the atoms in the metallic glass nanoparticle (particle 1) with BOO < 0.5, we applied the above procedure to plot the RDF (Fig. 1d ). For type 1, 2 and 3 atoms, we identified six sets of atoms pairs (type 11, 12, 13, 22, 23 and 33) in the nanoparticle. For each set of atom pairs, we used the above procedure to calculate the PDF shown in Fig. 1e .
Refinement of 3D atomic coordinates and species.
Voronoi tessellation and the coordination number (CN). The analysis of Voronoi tessellation was performed by following the procedure published elsewhere 6 , where the surface atoms of the nanoparticle were excluded. To reduce the effect of the experimental and reconstruction error on Voronoi tessellation, those surfaces with areas less than 1% of the total surface area of each Voronoi polyhedron were removed 9 .
Refinement of 3D atomic coordinates and species.
From the Voronoi tessellation, each polyhedron is designated by a Voronoi index 〈 1 , 2 , 3 , 4 , ⋯ 〉 with denoting the number of i-edge faces and the CN was calculated by ∑ .
Measurement of the bond length and quantification of the chemical SRO.
By fitting a Gaussian to the 1 st peak of the PDFs (Fig. 1e) , we determined the type 11, 22, 33, 12, 13 and 23 bond length to be 2.60, 2.67, 2.66, 2.65, 2.66 and 2.60 Å, respectively. The type 11, 22 and 33 bond length are consistent with the metallic radii of type 1, 2 and 3 atoms 62 . But the type 23 bond is 2.4% shorter than the average length of the type 22 and 33 bond, indicating a stronger interaction between type 2 and 3 atoms than other atom pairs.
Measurement of the bond length and quantification of the chemical SRO.
To quantify the chemical SRO, we computed the Warren-Cowley parameter 63,64 ( ),
Measurement of the bond length and quantification of the chemical SRO.
where , = 1, 2 or 3, is the partial CN of type atoms around type atoms, is the fraction of type atoms, and is the total CN around type atoms. After excluding the surface atoms, we estimated 1 , 2 and 3 to be 39.16%, 42.76% and 18.08%, respectively. Using the partial CNs ( Supplementary Fig.  7a ), we calculated 11 = -0.12, 12 = 0.06, 13 = 0.12, 21 = -0.01, 22 = 0.02, 23 = -0.03, 31 = 0.04, 32 = -0.08, and 33 = 0.11, indicating that both the type 11 and 23 bond are favoured in the metallic glass sample. The favouring of the type 23 bond is also consistent with the shortening of the type 23 bond (Fig.   1e ).
Measurement of the bond length and quantification of the chemical SRO.
Determination of the MROs. The MROs were identified using the following procedure. First, 2769 solute centres were chosen from a total of 3472 type 3 atoms based on a criterion that the solute centres must be distributed within the second coordination shell, which is between the first (3.78 Å) and the second minimum (6.09 Å) of the RDF curve ( Fig. 1d ). Second, a breadth-first-search algorithm 65, 66 was used to search for the possible fcc-, hcp-, bcc-or sc-like MROs from the 2769 solute centres. Each possible MRO must satisfy the following three criteria: i) there are at least five or more solute centres; ii) every solute centre must fall within a 0.75 Å radius from its fitted lattice vector; and iii) every solute centre must be within a medium range order distance (between 3.74 Å and 6.05 Å) to at least one other solute centre. Third, after identifying all the possible MROs, they were sorted by number of solute centres to generate a possible network queue. If there was a tie, the average error of fitting the solute centres into the lattice vectors was used to break the tie. Fourth, starting from the largest possible network, it was classified as an MRO if none of the solute centres in the network was already occupied by another MRO. If some solute centres were already occupied, then those solute centres were removed from the possible network, refitted while also accounting for connectivity, and then added back into the queue of possible networks. This process was repeated until the queue was depleted and every solute centre belonged to one and only one MRO. All the classified network with five or more solute centres were counted as the final MROs (Figs. 3g, h and 5).
Measurement of the bond length and quantification of the chemical SRO.
Finally, to corroborate our analysis, we repeated the above steps with a 1 Å and 0.5 Å radius cut-off and the corresponding final MROs are shown in Supplementary Figs. 8 and 9 , respectively.
Measurement of the bond length and quantification of the chemical SRO.
An attempt was also made in searching for icosahedral-like MROs. The breadth-first-search algorithm 65,66 was used to find the possible MROs that fall within a 0.75 Å radius from the 12 vertices of an icosahedron. Because the icosahedron cannot be periodically packed in three dimensions, only the nearest neighbour vertices were searched, making the largest possible MRO have 13 solute centres (1 central solute centre plus 12 nearest neighbours). After performing the search, the resulting possible MROs have a mean value of 3.9, meaning on average each solute centre is connected to only 3 others when constrained to an icosahedron within the second coordination shell. Furthermore, although the largest possible MROs have 7 solute centres, none of these solute centres form five-fold symmetry. We also repeated this analysis with a 1 Å radius cut-off. The mean value of solute centres becomes 4.5, the largest Supplementary Table and Figures   Supplementary Table 1 | AET data collection, processing, reconstruction, refinement and statistics. Particle Supplementary Fig. 1 | Temperature change during the rapid thermal shock heating process. The average and maximum temperatures (black and blue dots, respectively) were measured by a high-speed Phantom Miro M110 camera (Vision Research). The cooling rate for the average and maximum temperature curves was determined to be 51000 K/s (i.e. the slope of the red line) and 69000 K/s (i.e. the slope of the green line), respectively.
Supplementary Fig. 3 | Analysis of six multi-component metallic nanoparticles. a-f,
Representative ADF-STEM images of particles 1-6, respectively. Scale bar, 2 nm. g-l, The BOO distribution of the atoms in particles 1-6, showing the structural transition of the nanoparticles from disorder to order, where the red lines correspond to BOO = 0.5. According to the BOO, particles 1 and 2 are more disordered than other four particles. m, The BOO distribution of a Cu65Zr35 metallic glass obtained from molecular dynamics simulations 61 as a reference, from which we chose BOO = 0.5 as a cut-off to separate crystal nuclei from amorphous structure. Supplementary Fig. 4 | Tomographic tilt series from the most disordered nanoparticle (particle 1) among the six. 51 ADF-STEM images of the nanoparticle with a tilt range from −69.4° to +69.4° with each tilt angle shown at top right of each panel. The images were denoised by BM3D (Methods). The total electron dose of the tilt series was estimated to be 9.5 × 10 5 e/Å 2 . Scale bar, 2 nm. Supplementary Fig. 5 | Schematic diagram of the RESIRE algorithm. From the experimental projections, RESIRE minimizes the L2-norm error metric using gradient descent. The j th iteration of the algorithm consists of the following steps. i) RESIRE computes a set of images from the 3D object of the (j-1) th iteration (middle left). As the algorithm is not sensitive to the initial input, a random or empty 3D object can be used as the input of the 1 st iteration. ii) The difference between the computed and corresponding experimental images is calculated (middle right), from which an error metric is defined to monitor the convergence of the algorithm. iii) The difference is back projected to real space, yielding the gradient of the 3D reconstruction (bottom right). iv) The 3D reconstruction of the j th iteration is updated by combining the gradient with the reconstruction of the (j-1) th iteration (bottom left), where constraints such as positivity and support can be enforced. Usually, after several hundreds of iterations, RESIRE converges to a 3D reconstruction from a limited number of images. Supplementary Fig. 6 | Validation of the experimental 3D atomic model using multislice simulations. Comparison between a representative experimental (a) and a multislice ADF-STEM image (b). To account for the source size and incoherent effects, the multislice image was convolved with a Gaussian function. c, Histogram of the root mean square deviation of the atomic positions between the experimental atomic model and that obtained from 51 multislice images. 96.5% of the atoms are identical between the two models with a root-mean-square deviation of 21 pm. Scale bar, 2 nm. 

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