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[Akihiko Hirata](https://orcid.org/0000-0002-1770-8780), Shuya Sato, [Motoki Shiga](https://orcid.org/0000-0003-2434-4716), [Yohei Onodera](https://orcid.org/0000-0002-3080-6991), [Koji Kimoto](https://orcid.org/0000-0002-3927-0492), [Shinji Kohara](https://orcid.org/0000-0001-9596-2680)

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[Direct observation of the atomic density fluctuation originating from the first sharp diffraction peak in SiO2 glass](https://mdr.nims.go.jp/datasets/ab0b10fa-165f-4042-8898-0def38b62371)

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Direct observation of the atomic density fluctuation originating from the first sharp diffraction peak in SiO2 glassHirata et al. NPG Asia Materials           (2024) 16:25 https://doi.org/10.1038/s41427-024-00544-w NPG Asia MaterialsART ICLE Open Ac ce s sDirect observation of the atomic densityfluctuation originating from the first sharpdiffraction peak in SiO2 glassAkihiko Hirata 1,2,3,4,5, Shuya Sato6, Motoki Shiga 7,8,9, Yohei Onodera5,10, Koji Kimoto 5 and Shinji Kohara 5,6AbstractThe intermediate-range order of covalently bonded glasses has been extensively studied in terms of their diffractionpeaks observed at low scattering angles; these peaks are called the first sharp diffraction peaks (FSDPs). Although theatomic density fluctuations originating from the quasilattice planes are a critical scientific target, direct experimentalobservations of these fluctuations are still lacking. Here, we report the direct observation of the atomic densityfluctuations in silica glass by energy-filtered angstrom-beam electron diffraction. The correspondence between thelocal electron diffraction patterns of FSDPs and the atomic configurations constructed based on the X-ray and neutrondiffraction results revealed that the local atomic density fluctuations originated from the quasi-periodic alternatingarrangements of the columnar chain-like atomic configurations and interstitial tubular voids, as in crystals. We alsodiscovered longer-range fluctuations associated with the shoulder of the FSDP on the low-Q side. The hierarchicalfluctuations inherent in materials could aid in the elucidation of their properties and performance.IntroductionUnderstanding the intermediate-range order (IRO) inglasses beyond the nearest neighbor distances is one ofthe most intriguing topics in glass science1–6. For exam-ple, tetrahedral SiO4 units in SiO2 glass have been pro-posed to be interconnected to form several different-sizedrings with corner-sharing motifs, which is typical of net-work glass7. These ring structures could be an aspect ofthe formation of IRO on the length scale8–13. Scatteringexperiments provide a distinct signature of the IROthrough the diffraction peaks observed at low scatteringangles in reciprocal space. Indeed, a diffraction peakcalled the first sharp diffraction peak (FSDP) can beobserved in the structure factors of various covalentglasses ranging from Q= 1 to 2 Å–1 14–24. Thecharacteristics and unusual properties of FSDPs have alsobeen extensively reported25–28. Nevertheless, the IRObased on diffraction data is challenging to discuss due tothe randomness and complexity of glass structures.There are several interpretations of the origin of theFSDP. For example, Elliott proposed that FSDP originatedfrom the chemical ordering of interstitial voids around thecation-centered clusters in a glass structure15,16,23. Theordering arising from the interstitial voids contributed tothe formation of a pre-peak in the concentration‒con-centration partial structure factor. Another possibleinterpretation was provided by Gaskell and Wallis, asfollows: the FSDP arises from the quasilattice planes inglass analogous to the Bragg planes in crystals19. Theyfocused on the anisotropic scattering from local structuremodels and suggested that this scattering intensity wasrelated to the periodic fluctuations in the atomic density.The fluctuations could be interrelated with the orderingof the interstitial voids, as proposed by Elliott. Christie etal. also proposed a model of disordered crystalline latticesthat could reproduce the FSDP29. Similarly, the FSDPs© The Author(s) 2024OpenAccessThis article is licensedunder aCreativeCommonsAttribution 4.0 International License,whichpermits use, sharing, adaptation, distribution and reproductionin any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate ifchangesweremade. The images or other third partymaterial in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to thematerial. Ifmaterial is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtainpermission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.Correspondence: Akihiko Hirata (ahirata@aoni.waseda.jp) orMotoki Shiga (motoki.shiga.b4@tohoku.ac.jp) orShinji Kohara (KOHARA.Shinji@nims.go.jp)1Department of Materials Science, Waseda University, Tokyo 169-8555, Japan2Kagami Memorial Research Institute for Materials Science and Technology,Waseda University, Tokyo 169-0051, JapanFull list of author information is available at the end of the article1234567890():,;1234567890():,;1234567890():,;1234567890():,;http://orcid.org/0000-0002-1770-8780http://orcid.org/0000-0002-1770-8780http://orcid.org/0000-0002-1770-8780http://orcid.org/0000-0002-1770-8780http://orcid.org/0000-0002-1770-8780http://orcid.org/0000-0003-2434-4716http://orcid.org/0000-0003-2434-4716http://orcid.org/0000-0003-2434-4716http://orcid.org/0000-0003-2434-4716http://orcid.org/0000-0003-2434-4716http://orcid.org/0000-0002-3927-0492http://orcid.org/0000-0002-3927-0492http://orcid.org/0000-0002-3927-0492http://orcid.org/0000-0002-3927-0492http://orcid.org/0000-0002-3927-0492http://orcid.org/0000-0001-9596-2680http://orcid.org/0000-0001-9596-2680http://orcid.org/0000-0001-9596-2680http://orcid.org/0000-0001-9596-2680http://orcid.org/0000-0001-9596-2680http://creativecommons.org/licenses/by/4.0/mailto:ahirata@aoni.waseda.jpmailto:motoki.shiga.b4@tohoku.ac.jpmailto:KOHARA.Shinji@nims.go.jpobserved at Q= 1 to 2 Å–1 correspond to the periodicfluctuations of 3–6 Å according to the relationship ofD= 2π/Q, where D represents the spacing between theperiodic fluctuations19. Moreover, the scale of the FSDP inreal space exceeds that of the interatomic distances. Thus,the concept of quasilattice planes is quite helpful inunderstanding the FSDP with local anisotropy in thecontext of the IRO structures.We recently utilized the angstrom-beam electron dif-fraction (ABED) technique to observe electron diffractionpatterns from local atomic arrangements of glass struc-tures30–35. As the beam size decreased to approximately1 nm, the diffraction intensity changed from isotropic toanisotropic. In addition, the correspondence between theanisotropic diffraction intensity and the local atomicarrangement could be studied using the virtual ABEDtechnique35. Thus, these techniques represent some of themost suitable tools for determining the anisotropicintensity of FSDP and revealing the relationship betweenthe FSDP and the periodic density fluctuations in realspace. In this study, we employ both ABED experimentsand virtual ABED simulations to explore the origin of theFSDP in SiO2 glass, a representative covalent glass. Inparticular, we use an energy filter and a cold field-emission electron gun to resolve the anisotropic FSDPs atlow scattering angles (Q= 0.9 to 1.5 Å–1).ResultsAngstrom-beam electron diffraction measurementsTo obtain the anisotropic diffraction intensity of thelocal regions of SiO2 glass, we performed an ABEDexperiment using a scanning transmission electronmicroscope (STEM), as shown in Fig. 1a. A focusedelectron probe was used to illuminate and scan the thinSiO2 glass sample. Subsequently, the diffraction patternsfor each local region were recorded by a CCD camera atthe bottom. The full width at half maximum and con-vergence angle of the electron probe were 0.9 nm and 1.2mrad, respectively. The diffraction patterns were obtainedusing elastic scattering through an energy filter device. Acomparison between the filtered and unfiltered diffractionpatterns is shown in Supplementary Fig. S1.Figure 1b shows the pairs of diffraction spots observedat low scattering vectors ranging from Q= 0.9 to 1.5 Å–1.The diffraction spot at Q= 1.5 Å–1 corresponded to theFSDP of the SiO2 glass. All diffraction patterns werecollected from different local regions of the sample. Asshown in the figure, diffraction spots could be observed atthe top of the FSDP (Q= 1.5 Å–1) and at the lower scat-tering vector of Q ~ 0.9 Å–1. The positions of the peaks areindicated by the structure factors S(Q) of the high-energyX-ray and neutron diffraction, as shown in Fig. 1c.Because the slope of the S(Q) curves changes significantlyFig. 1 ABED experiments of SiO2 glass. a Schematic of the angstrom-beam electron diffraction (ABED) measurement for SiO2 (silica) glass. b Paireddiffraction spots with different Q values ranging from 1.5 to 0.9 A–1 observed in the obtained ABED patterns. c Structure factor S(Q) around FSDPobtained from high-energy X-ray diffraction (XRD) and neutron diffraction (ND). The arrows indicate the Q values corresponding to Fig. 1b. d Typicalsymmetric ABED patterns with the averaged pattern (far left). The dotted circles indicate the peak-top position of FSDP.Hirata et al. NPG Asia Materials           (2024) 16:25 Page 2 of 10    25 around Q= 1.2 Å–1, we refer to the region belowQ= 1.2 Å–1 as the shoulder of the FSDP. The shoulderwas significantly reduced in the case of densified SiO2glass36. In addition, the ABED diffraction spots were alsoobserved at the shoulder, as shown in Fig. 1b.The typical symmetric two-dimensional experimentaldiffraction patterns were similar to the zone-axis patternsin crystals, as shown in Fig. 1d, with the averaged patternon the far left (also see Supplementary Fig. S2). Theaveraged pattern consisted of the so-called halo rings,corresponding to the structure factor S(Q) in Fig. 1c. Inthe three diffraction patterns on the right, we couldobserve the symmetric patterns formed by the spots withlower Q values than the top of the FSDP (dotted circles).By analogy with crystals, these two-dimensional patternscontained abundant structural information.Extraction of the local atomic configurations by virtualangstrom-beam electron diffraction analysisThe simulation of the ABED experiment was con-ducted using a computer by following a similar processthat included beam scanning35. We prepared a large-scalestructural model of SiO2 glass with dimensions of 10 ×10 × 10 nm3 by a combined method of moleculardynamics and reverse Monte Carlo simulations. Thedetails of the modeling procedure are described in aprevious paper36. The obtained model was independentlycreated from ABED and could reproduce the high-energyX-ray and neutron diffraction experiments that providedthe isotropic intensities, including FSDPs, as shown inFig. 2a.Our interest is whether the model based on isotropicintensities produces symmetric diffraction patterns withanisotropic intensities, as observed in the ABED experi-ment. To address the above statement, we conducted avirtual ABED analysis35 for the model. While the modelwas based on isotropic intensities, it was constructed withconstraints, such as coordination numbers and bondangle distributions, to ensure physical reasonableness,especially for the short-range structures. The 10-nm-thickmodel was sliced into 2-nm-thick models and then vir-tually scanned by an electron beam with a 0.2-nm step(Fig. 2b). Note that the edge portions of the crushed glassysamples were estimated to be ~2–3 nm34. The electrondiffraction patterns calculated from each local region werestored in a database together with the coordinates of theelectron beam positions. An example of a series of dif-fraction patterns is shown in Fig. 2c. Note that the array ofthe diffraction patterns (6 × 6) in Fig. 2c coincides withthe electron beam positions (6 × 6) in real space, as shownin Fig. 2b. By carefully inspecting the diffraction patterns,symmetric patterns can be observed and are marked inred. We retrieved the clearest pattern marked in red fromthe dataset containing several similar patterns.Fig. 2 ABED simulation for the structural model of SiO2 glass. a X-ray and neutron structure factors (SX(Q) and SN(Q)) obtained from the structuralmodel of SiO2. Both experimentally obtained structure factors (SX(Q) and SN(Q)) are also shown. b Structural model of SiO2 glass obtained bycombining molecular dynamics (MD) and reverse Monte Carlo (RMC) simulations. The obtained model is sliced into thin models of 2 nm thickness forvirtual ABED analysis. The electron probe is virtually irradiated on the thin model in 0.2 nm steps. The top view of the selected cylindrical region in thethin model is also shown with 6 × 6 beam positions. c Example of a partial dataset of ABED patterns obtained by virtual electron probe scanning ofthe 6 × 6 beam positions shown in (b). d Typical simulated ABED patterns found in the dataset with the averaged pattern (far left). The dotted ringsindicate the peak position of FSDP. Note that the far-right pattern is identical to the marked pattern in (c).Hirata et al. NPG Asia Materials           (2024) 16:25 Page 3 of 10    25 Figure 2d shows three examples of symmetric diffrac-tion patterns, including the pattern shown in Fig. 2c,together with the averaged pattern on the far left. Thedotted circles indicate the top position of the FSDP. Thepatterns consisting of six spots within FSDP correspondto the shoulder in the S(Q) profile of Fig. 1c and can beclearly observed. Thus, the structural model based on theisotropic structure factor S(Q) contains the local atomicarrangements that produce the symmetric diffractionpatterns with anisotropic intensities, as observed experi-mentally. Here, we focus on the symmetric pattern con-sisting of six spots shown on the far-right side of Fig. 2d;this pattern has similar features to the experimental pat-tern shown on the far-right side of Fig. 1d.To understand the structural origin of the symmetricdiffraction patterns of FSDP and its shoulder, we extrac-ted the local atomic configurations that generated thesepatterns from the original structural model of Fig. 2b.Figure 3a shows the extracted atomic configuration ofSiO2 corresponding to the symmetric diffraction patternof the inset; this is identical to the pattern marked in redin Fig. 2c and the far-right pattern of Fig. 2d. In the cal-culation of the diffraction pattern, the electron probe wasirradiated at the center of the structural model. The fourspots with strong intensities are observed near the FSDP,and six spots can be observed inside. From this incidencedirection, the atomic density fluctuations in the structuralmodel can be observed as sparse regions (e.g., region I)and dense regions (e.g., region II).Figure 3b shows the side views of the atomic arrange-ments of region I and region II corresponding to thesparse and dense regions of the atomic density, respec-tively. In region I, a tubular atomic configuration of atleast 2 nm in length is formed where a rod-shaped void issurrounded by atoms (see Supplementary Fig. S3). Con-versely, a chain-like columnar atomic configuration isfound in region II, where the local atomic density isrelatively high. Similar features can be found in anotherFig. 3 Atomic density fluctuations for the extracted structural model. a Projection of the extracted cylindrical structural model. The diameterand height of the cylinder are both 2 nm. The red and blue circles represent O and Si atoms, respectively. The simulated ABED pattern obtained fromthe central part of the model is also shown in the inset. The electron incidence of the ABED pattern is parallel to the projection direction. b Side viewsof the atomic configurations obtained from region I and region II of (a). Note that the three atoms marked by a dotted oval circle exist slightlyoutside the column. c, d Inverse fast Fourier transform (FFT) images constructed using the marked spots of the corresponding FFT patterns shown inthe insets. The FFT pattern is obtained from the projection image of (a). The atomic configuration of (a) is superimposed on the inverse FFT images.Hirata et al. NPG Asia Materials           (2024) 16:25 Page 4 of 10    25 region of the same model, as shown in Supplementary Fig.S4. Thus, the virtual ABED technique reveals that theatomic density fluctuation with local two-dimensionalityis formed in the structural model created based on theisotropic structure factors.As mentioned in the introduction, the local periodicityin the atomic density characterized by 2π/Q could berelated to the FSDPs in reciprocal space. We employedfast Fourier transform (FFT) and inverse FFT techniques(see Supplementary Fig. S5) to understand the corre-spondence between the periodic fluctuations in atomicdensity19 and FSDPs. The FFT pattern obtained from theprojected image of the atomic configuration in the insetsof Fig. 3c, d is consistent with the calculated diffractionpattern shown in the inset of Fig. 3a. Figure 3c shows theinverse FFT image constructed from the four spotsobserved at the FSDP position in the corresponding FFTpattern, together with the projected atomic configuration.The dense regions of the structural model nearly coincidewith the dark regions of the inverse FFT image. Similarly,the inverse FFT image generated by the six spots insidethe FSDP is shown in Fig. 3d. An atomic density fluc-tuation with a longer periodicity (~ 0.52–0.57 nm) thanthat in Fig. 3c (~0.41 nm) is also observed in the samestructural model.To confirm the correlation between the contrast varia-tions in the inverse FFT images and the atomic densityfluctuations, we performed kernel density estimation witha Gaussian kernel37 for the local structural model shownin Fig. 3a. This method enables the determination of theatomic density distributions by constructing smoothsymmetric functions for each projected atomic position.Supplementary Fig. S6a shows the density map for themodel of Fig. 3a obtained by the kernel density estimation.The two different inverse FFT images shown in Fig. 3c, dare superimposed on this map, as shown in Supplemen-tary Figs. S6b and S6c, respectively. The map obtained bykernel density estimation is basically consistent with thecontrast variations in the inverse FFT images.DiscussionIn the present study, we experimentally obtained char-acteristic two-dimensional diffraction (ABED) patternsrelated to FSDP from local regions of SiO2 glass andextracted the corresponding atomic arrangements. Theperiodic fluctuations in the atomic density could beinterpreted as alternating arrangements of chain-likecolumnar atomic configurations and interstitial tubularvoids, as shown in Fig. 3b. Note that the interstitial tubularvoids are longitudinally limited to a size of approximately2 nm (see Supplementary Fig. S7). These columnar objectsdo not necessarily correspond to two-dimensional struc-tures because the atomic density fluctuation is technicallychallenging to confirm over the other orientations.Next, we discuss how the arrangements of columnarobjects relate to the concept of quasilattice planes pro-posed in a previous study19. As shown in Fig. 4a, we focuson the five chain-like columnar atomic configurationscorresponding to the dense regions. The two types ofarrangements consisting of the 1-3-5 and 4-3-2 columnsare extracted from the model, as shown in Fig. 4b.Figure 4c shows the side views of the 1-3-5 and 4-3-2arrangements and their corresponding inverse FFT ima-ges. From these orientations, pseudo-two-dimensionalatomic arrangements, which could be called quasilatticeplanes, can be observed with atomic density fluctuations.The chain-like columnar atomic configurations areformed in the dense regions (dark regions in the inverseFFT images), and each is connected by bridge atomslocated in the sparse regions (bright regions). Notably, thearrangements of the 1-3-5 columns and 4-3-2 columnsintersect, similar to the observations in crystals. Thedescription mentioned above is fairly consistent with theconcept of quasilattice planes, but we believe that ourpresent results provide a more specific description interms of their similarity to crystals.Here, the quasi-crystalline models proposed by Phillipsare discussed38,39. In this model, the structure of silicaglass consists of an aggregate of β-cristobalite paracrystalswith a size of 66 Å. The boundaries between the para-crystals are assumed to be filled with a structure similar tothe grain boundaries of crystalline materials. However, asubsequent study40 noted that this model was not con-sistent with the neutron diffraction data. According to ourstudy, even when the crystal size is less than 10 Å, thesimulated structure factor does not agree with the simu-lated structure factor. Our present observation also doesnot support the quasi-crystalline models because theexperimental ABED patterns are reasonably consistentwith those calculated from the non-crystalline model thatis based on the X-ray and neutron diffraction data. Thepresence of 5- and 7-membered rings, which are notfound in crystals41, is also indicative of the non-crystallinefeatures of our model (Supplementary Fig. S8). Notably,the term quasilattice planes is used for non-crystallinestructures, and the presence of diffraction spots in theABED patterns does not necessarily indicate the presenceof crystal structures.Furthermore, diffraction spots are distinct from theBragg reflections of crystal structures and can be gener-ated from non-crystalline structures. For example, whenconsidering the diffraction intensity of a single molecule, adistinct intensity distribution is generated in a three-dimensional reciprocal space. This intensity distribution iscaused by the presence of a correlation term in the dif-fraction intensity equation34. Similarly, the diffractionintensity from the local atomic arrangements of theglasses also has a three-dimensional intensity distribution.Hirata et al. NPG Asia Materials           (2024) 16:25 Page 5 of 10    25 As the size of the local structures increases (see Supple-mentary Fig. S9), the diffraction pattern graduallybecomes isotropic halo rings; these results are basicallyconsistent with the X-ray and neutron diffraction profiles.Specifically, the diffraction spots observed in this studycan be viewed as the building blocks of the halo rings dueto their glass structures (also see Supplementary Fig. S10).Next, we consider examples of the SiO2 crystals forcomparison with the glass structures. In the case of α-cristobalite with a density of 2.33 g cm–3, the sparse anddense portions denoted by regions A and B in the pro-jection of Fig. 5a are similar to the glass structures ofregions I and II in Fig. 3 and correspond to the columnarconfigurations along the projection direction, as shown inFig. 5c. Alternating arrangements of the columnar atomicconfigurations and interstitial tubular voids can also beobserved in the figure. Additionally, the four 101-typespots (Q= 1.56 Å–1) corresponding to the FSDP in theglass in the simulated electron diffraction pattern for α-cristobalite are closely related to the alternating arrange-ments in real space, as shown in Fig. 5b. Similar featuresare also found in the α-quartz crystal with a density of2.66 g cm–3, as shown in Supplementary Fig. S11. How-ever, in this case, the 011-type reflections potentiallycorrespond to the FSDP in the glass because the intensityof 011 is much stronger than the intensity of 100, and thedensity of α-quartz is much greater than that of glass(2.21 g cm–3). In addition, the 100-type reflections, whichform the sixfold electron diffraction pattern, more likelycorrespond to the shoulder of the FSDP. A similar patternconsisting of six spots is observed in the glass model, asshown in Fig. 3a. Thus, the local structures of glasses arenot exactly the same as those of crystals but have some-what similar characteristics. Notably, the anisotropicnature is only local in glassy structures, as opposed to themacroscopic anisotropy in crystals.To confirm the relationship between the atomic densityfluctuations and inverse FFT images of the crystals, FFTanalyses on the crystals were performed similarly to thoseperformed on the above glass samples. SupplementaryFig. S12 shows inverse FFT images for α-cristobalite, α-quartz, and coesite (see also Supplementary Fig. S13)crystals superimposed on each atomic configuration,along with their corresponding FFT patterns. Similar toglass structures, the dark regions coincided with a highatomic density, while the bright regions coincided with alow atomic density. From these crystal examples, thecontrast of light and dark regions in the inverse FFTimages corresponded to the sparse and dense regions withrespect to the atomic density. Notably, the bright regionsin the coesite corresponded to the arrangements of thesmaller 4-membered rings compared to those in the othercrystals.The structural characteristics of network glasses haveoften been investigated through ring statistics analysis12.The correlation between the sizes of the rings and theFig. 4 Side views of the quasilattice planes. a Five dense regions, labeled 1–5 in the model of Fig. 3a, corresponding to the dark areas in theinverse FFT image (identical to Fig. 3c) shown in the inset. b Extracted atomic configurations corresponding to the 1-3-5 and 4-3-2 arrangements.c Side views of the atomic configurations for the 1-3-5 and 4-3-2 arrangements on the left and right, respectively, together with the correspondinginverse FFT images. The chain-like columnar atomic arrangements, which consequently form pseudo planes, are visible in the dark (dense) portionsof the inverse FFT images.Hirata et al. NPG Asia Materials           (2024) 16:25 Page 6 of 10    25 atomic density fluctuations from our study are investi-gated. As shown in Supplementary Figs. S8b and S8e, wecalculated Guttman-type ring distributions for the thickatomic columns shown in Supplementary Fig.S14a and S14c, respectively; moreover, we calculated thering-distributions for both the local structures (Figs. S8aand S8b) and the total structure (Fig. S8f). In these atomiccolumns, the proportions of 7-membered rings, whosesizes are relatively large, are quite high compared to thoseof the total structure. In addition, the spatial distributionsof the ring centers together with the inverse FFT images ofFig. 3c, d are shown in Supplementary Fig. S15. Supple-mentary Figs. S15a and S15b show the distributions of thecentral points for the rings with five members or fewer andthe rings with six members or more, respectively; thesedistributions are superimposed in Fig. 3c. The distribu-tions with five members or fewer and six members ormore are superimposed on Fig. 3d and are also shown inSupplementary Figs. S15c and S15d. According to Sup-plementary Figs. S15a and S15b, the distributions of thepoints are relatively uniform, and numerous points arelocated in the bright regions in the inverse FFT image ofFig. 3c. Considering the directions of the normal vectorsin Supplementary Fig. S16, the rings could function asbridges connecting the denser regions, as depicted inFig. 4c. On the other hand, as shown in SupplementaryFig. S15d, the distribution of rings with six members ormore correlates well with the inverse FFT image of Fig. 3d,which has longer periods because most of the points arenear the darker regions. The directions of the normalvectors shown in Supplementary Fig. S16 indicate that therings around the darker regions presumably form theouter walls of the larger tubular voids (also see Fig. S14c).Additionally, some points with normal vectors close to thedirection of incidence can be found in the center of thebright regions; thus, the ring itself surrounds the tubularcolumn, as shown in Fig. S14a. Therefore, the larger ringsare related to the longer periods originating from theshoulder of the FSDP, similar to the case of the crystalsmentioned above.Discussing the spatial extent of the IRO is alsoimportant. Supplementary Fig. S17 shows two sequentialseries of experimental ABED patterns captured in0.25 nm steps. Identical diffraction spots indicated by thered arrows are located at the peak top of FSDP and areconsistently observed over 3 steps in both cases. Con-sidering the beam size of 0.9 nm, the spatial extent of theorder is roughly estimated to be from 0.9 to 1.4 nm.Similarly, from the simulation side, the higher contrastregion of the inverse FFT image is also extended, asshown in Supplementary Fig. S18. The extension esti-mated by ABED is consistent with that estimated byX-ray diffraction (approximately 1 nm)36. However, dif-ferent from crystals, evident interfaces between IRO doFig. 5 Analyses of α-cristobalite crystals. a Projection of the α-cristobalite crystal. The projection direction is parallel to the [010] direction of α-cristobalite. b Simulation of X-ray and electron diffraction for α-cristobalite crystals. The electron incidence is parallel to the [010] direction of α-cristobalite. The four diffraction spots marked in red in the electron diffraction pattern correspond to the 101-type reflection in the X-ray diffractionprofile. c Side views of the atomic configurations corresponding to regions A and B in (a).Hirata et al. NPG Asia Materials           (2024) 16:25 Page 7 of 10    25 not exist; therefore, the extent of the IRO is not easilydetermined experimentally.According to previous work reported by Salmon et al. 20,AX2-type network glasses possessed topological and che-mical ordering on two different length scales according toneutron diffraction measurements with an isotopic sub-stitution technique. One scale was an intermediate rangeorder manifested by an FSDP, and the other scale wasmanifested by the principal peak at a higher scatteringangle than the FSDP. Notably, the latter ordering was anextended-range order with a periodicity of 3.14 Å, wherethe oscillations in the pair distribution function were cal-culated to be 6.2 nm. In contrast, in the present study, thearrangement of the columnar objects generates hier-archical structures with different length scales corre-sponding to the FSDP and its shoulder at the low Q side(see Figs. 3c, d). We show that the diffraction spots aroundthe FSDP with different scattering angles originate fromidentical local atomic configurations, producing the widedistribution of the FSDP, including the shoulder, found inthe X-ray and neutron structure factors. Specifically, theFSDP is not a single peak but a composite of multiplepeaks. The local structures of glass do not exactly corre-spond to the crystals, as mentioned above. However, theyhave partially similar features with some periodic fluc-tuations in the atomic density. For example, the structurecorresponding to the shoulder can have similar features toα-quartz despite a shift in the Q position of α-quartzcaused by density differences between α-quartz and α-cristobalite/glass. The characteristic spatial fluctuations orinhomogeneities, which cannot be described by the aver-age values, are needed to control the properties and per-formance of disordered materials beyond the average42. Inaddition, when high pressure is applied, the longer periodin the densified SiO2 glass presumably disappears due tothe decrease in the shoulder, as well as the prominent peakshift36. Considering the stability of ordinary SiO2 glass andits densification mechanism, future studies should focuson the longer periodic fluctuations associated with theshoulder.MethodsAngstrom-beam electron diffraction (ABED) measurementsABED experiments were performed with an aberration-corrected scanning transmission electron microscope(Titan cubed, Thermo Fisher Scientific, operated at300 kV) equipped with a cold field-emission electron gun.All ABED patterns were acquired by scanning the focusedelectron probe with an exposure time of 0.01 s and a scanstep of 0.25 nm. The convergence angle of the probe andthe probe size were 1.2 mrad and 0.9 nm, respectively.Inelastic electron scattering was eliminated from theABED patterns using an electron energy loss spectro-scope. The samples for ABED observation were preparedby crushing a commercial SiO2 glass sample (T-4040,Covalent Materials Corp.).Structure modelingThe atomic configuration of the SiO2 glass was obtainedby a combination of molecular dynamics (MD) andreverse Monte Carlo (RMC) simulations. First, a randomatomic configuration containing 22,052 Si atoms and44,104 O atoms was prepared in a simulation box withdimensions of 10 × 10 × 10 nm3 and a density of2.21 g cm–3. The initial configuration was held at 4000 Kfor 1.0 × 105 steps to equilibrate to the liquid state, fol-lowed by cooling to 300 K for 5.0 × 106 steps andrelaxation at 300 K for 1.0 × 105 steps. The time step wasset to 1 fs. The MD simulation was performed by theLAMMPS code43 under the NVT ensemble, where thenumber of atoms, the volume, and the temperature werekept constant, using the Verlet algorithm. Interatomicpair potentials with short-range Born‒Mayer repulsiveand long-range Coulomb terms were employed. Theconfiguration obtained by MD was further modified to fitthe experimental X-ray and neutron structure factors bythe RMC simulation using the RMC++ code44–46. TheRMC simulation was performed by considering the con-straints on the coordination numbers, O–Si–O bondangles, and partial pair-distribution functions within thefirst coordination shell. These constraints were incorpo-rated to maintain the physically reasonable structuresgenerated by the MD simulation. The details of the dif-fraction experiments and simulations are described in aprevious paper36.Virtual angstrom-beam electron diffraction analysisElectron diffraction patterns for the 1681 (41 × 41)positions of the structural model obtained by a combi-nation of MD and RMC were calculated sequentially at0.2 nm intervals using a multislice method47. The accel-erating voltage, defocus value, and third-order sphericalaberration coefficient were 300 kV, 0 nm, and 0.005 mm,respectively. The convergence semi-angle of the electronprobe was set to 1.2 mrad. The symmetric diffractionpatterns were selected from the obtained dataset con-taining 1681 patterns. Subsequently, we extracted theatomic configurations around the position wherethe electron probe was virtually irradiated. The FFT of theprojection images of the extracted structural models andthe inverse FFT were performed using a GatanDigitalMicrograph 3.5.AcknowledgementsThis work was partially supported by a JSPS Grant-in-Aid for TransformativeResearch Areas (A) “Hyper-Ordered Structures Science” Grant No. 20H05881and Grant No. 20H05884, a JSPS Grant-in-Aid for Scientific Research (B) GrantNo. 20H04241, and a JSPS Grant-in-Aid for Challenging Research ExploratoryHirata et al. NPG Asia Materials           (2024) 16:25 Page 8 of 10    25 Grant No. 23K17837. We would like to thank Takeaki Mashiko of NIMS for histechnical assistance.Author details1Department of Materials Science, Waseda University, Tokyo 169-8555, Japan.2Kagami Memorial Research Institute for Materials Science and Technology,Waseda University, Tokyo 169-0051, Japan. 3WPI Advanced Institute forMaterials Research, Tohoku University, Sendai 980-8577, Japan. 4Mathematicsfor Advanced Materials-OIL, National Institute of Advanced Industrial Scienceand Technology, Sendai 980-8577, Japan. 5Center for Basic Research onMaterials, National Institute for Materials Science, Tsukuba 305-0047, Japan.6Faculty of Science and Technology, Tokyo University of Science, 2641Yamazaki, Noda, Chiba, Japan. 7Unprecedented-scale Data Analytics Center,Tohoku University, 468-1 Aoba, Aramaki-Aza, Aoba-ku, Sendai 980-8578, Japan.8Graduate School of Information Science, Tohoku University, 6-3-09 Aoba,Aramaki-aza Aoba-ku, Sendai 980-8579, Japan. 9RIKEN Center for AdvancedIntelligence Project, 1-4-1 Nihonbashi, Chuo-ku, Tokyo 103-0027, Japan.10Institute for Integrated Radiation and Nuclear Science, Kyoto University,2-1010 Asashiro-nishi, Kumatori-cho, Sennan-gun, Osaka, JapanAuthor contributionsA.H. and S.K. conceived this study. 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Advanced Computing in Electron Microscopy; Plenum: NewYork, NY, USA, 1998.Hirata et al. NPG Asia Materials           (2024) 16:25 Page 10 of 10    25  Direct observation of the atomic density fluctuation originating from the first sharp diffraction peak in SiO2�glass Introduction Results Angstrom-beam electron diffraction measurements Extraction of the local atomic configurations by virtual angstrom-beam electron diffraction analysis Discussion Methods Angstrom-beam electron diffraction (ABED) measurements Structure modeling Virtual angstrom-beam electron diffraction analysis Acknowledgements Acknowledgements