# Fileset

[134_25125.pdf](https://mdr.nims.go.jp/filesets/7178fd03-50a8-4153-ba6b-44afc2c2ec9c/download)

## Creator

[Shinji Kohara](https://orcid.org/0000-0001-9596-2680), Koji Kimura, [Motoki Shiga](https://orcid.org/0000-0003-2434-4716), [Yohei Onodera](https://orcid.org/0000-0002-3080-6991), Akihiko Hirata, Koichi Hayashi

## Rights

[Creative Commons BY Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0/)

## Other metadata

[Quantum beam diffraction measurement and topological analysis of tetrahedrally coordinated non-crystalline materials](https://mdr.nims.go.jp/datasets/2081771d-3841-4b14-9285-1a747bd3b01e)

## Fulltext

Quantum beam diffraction measurement and topological analysis of tetrahedrally coordinated non-crystalline materialsREVIEWQuantum beam diffraction measurement and topological analysisof tetrahedrally coordinated non-crystalline materialsShinji Kohara1,³, Koji Kimura1,2, Motoki Shiga1,3,4,5, Yohei Onodera1,Akihiko Hirata1,6,7,8 and Koichi Hayashi21Center for Basic Research on Materials, National Institute for Materials Science, Tsukuba, Ibaraki 305–0047, Japan2Department of Physical Science and Engineering, Nagoya Institute of Technology, Nagoya 466–8555, Japan3Unprecedented-scale Data Analytics Center, Tohoku University, Sendai 980–8578, Japan4Graduate School of Information Science, Tohoku University, Sendai 980–8579, Japan5RIKEN Center for Advanced Intelligence Project, Tokyo 103–0027, Japan6Department of Applied Mathematics/Department of Materials Science, Waseda University, Tokyo 169–8555, Japan7Kagami Memorial Research Institute for Materials Science and Technology, Waseda University, Tokyo 169–0051, Japan8WPI Advanced Institute for Materials Research, Tohoku University, Sendai 980–8577, JapanThe construction of large quantum beam facilities such as the synchrotron radiation facility SPring-8 and thehigh-intensity proton accelerator facility J-PARC has provided access to high-intensity, high-energy quantumbeams that are essential for structural analyses of non-crystalline materials via diffraction measurements inJapan. The developments of quantum beam diffraction techniques led to significant advancements in theresearch field. By the complementary use of X-rays, which are sensitive to heavy elements, and neutrons, whichare sensitive to light elements, along with the advances in computer simulations and topological analysistechniques, we have achieved a deep understanding of disordered structures with intermediate-range ordering.In this article, we review the recent results obtained by the complementary use of quantum beam diffraction andtopological analyses of silica polymorphs, covering silica crystals and densified silica glasses. The comparisonbetween the persistent homology analysis data and the ring size distribution has led to the classification of aseries of densified silica glasses and crystals in terms of ring persistency (ring shape) and ring entropy(topological order–disorder). This is a new concept to understand the nature of order–disorder observed in aseries of silica polymorphs without using diffraction data. We also discuss the differences among disorderedmaterials, which comprises an AA4 (A = Si) tetrahedral network (amorphous silicon), an AX4 (A = Si, X = O)tetrahedral network (glassy silica), and a non-tetrahedral network due to isolated AX4 (A = C, X = Cl) tetra-hedra (liquid carbon tetrachloride) in terms of the origin of a three-peak structure, FSDP (Q1), PP (Q2), and Q3.Key-words : Non-crystalline materials, Structure, X-ray diffraction, Neutron diffraction, Topology[Received September 13, 2025; Accepted October 19, 2025]1. IntroductionNon-crystalline materials such as glass, liquids, andamorphous materials lack the structural order found incrystalline phases, which results in broad ‘halo patterns’in quantum beam diffraction data. Since non-crystallinematerials lack structural descriptors such as lattice con-stants or space groups used in crystalline phases, it is im-possible to determine atomic positions solely from diffrac-tion data. Instead, the function best suited for describingthe disordered structure of amorphous materials is the pairdistribution function (PDF) g(r).1) g(r) represents the prob-ability of finding another atom at the distance r from asingle atom located at the origin. g(r) is obtained by aFourier transform of the normalized quantum beam dif-fraction data, the Faber–Ziman2) total structure factor S(Q).gðrÞ ¼ 1þ 12³2rμZ QmaxQminQ½SðQÞ � 1� sinðQrÞMðQÞdQð1ÞHere, μ represents the atomic number density, which isthe number of atoms per unit volume. Q [= (4³/­) sin ª,where 2ª is the scattering angle and ­ is the wavelength ofthe incident X-rays or neutrons] represents the magnitudeof the scattering vector. M(Q) is a modification functionintroduced to reduce the ripples caused by the truncationerror of S(Q) in a finite Q range in the Fourier transform ofS(Q), for which the Lorch function3) has been used.However, since diffraction data over a wide Q range cannow be measured with sufficient statistics, truncation³ Corresponding author: S. Kohara; E-mail: KOHARA.Shinji@nims.go.jp‡ Preface for this article: DOI https://doi.org/10.2109/jcersj2.134.P4-1Journal of the Ceramic Society of Japan 134 [4] 208-216 2026DOI https://doi.org/10.2109/jcersj2.25125 JCS-Japan©2026 The Ceramic Society of Japan208This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/),which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.https://doi.org/10.2109/jcersj2.134.P4-1https://doi.org/10.2109/jcersj2.134.P4-1https://doi.org/10.2109/jcersj2.25125https://creativecommons.org/licenses/by/4.0/errors have been significantly reduced. Additionally, usingsuch functions can lead to a decrease in real-space reso-lution, so there are many reports where M(Q) is not intro-duced. However, the X-ray atomic form factor, whichrepresents the scattering capability of atoms, depends onQ. Therefore, in multicomponent systems where the differ-ence in atomic number between constituent atoms is large,it is often necessary to introduce M(Q) to suppress thetruncation error of the Q-dependent atomic form factor.Regarding M(Q), new functions have been proposed,4) andthe effect of M(Q) has been carefully discussed.5) Therelationships among the pair distribution function g(r), thereduced pair distribution function G(r), the total corre-lation function T(r), and the radial distribution functionRDF(r) are expressed as follows.gðrÞ ¼ GðrÞ4³rμþ 1 ð2ÞT ðrÞ ¼ GðrÞ þ 4³rμ ¼ 4³rμgðrÞ ð3ÞRDFðrÞ ¼ rGðrÞ þ 4³rμ ¼ rT ðrÞ ð4ÞTo obtain the reliable real-space function with highresolution, it is necessary to measure diffraction patternsup to high Q, since the real-space resolution is determinedby Qmax in Eq. (1).6) Large quantum beam facilities suchas the synchrotron radiation facility SPring-8 and the high-intensity proton accelerator facility J-PARC have providedaccess to high-intensity, high-energy quantum beams thatare essential for structural analyses of non-crystallinematerials via diffraction measurements with high-Q data.We previously have reviewed combined quantum beamdiffraction measurements and computer simulations,7,8)several studies on oxide glasses and high-temperatureoxide melts,9,10) and the topology in silica polymorphs.11)In this article, we review our recent studies on non-crystalline materials by diffraction measurements and aseries of topological analysis techniques such as ring size,homology, ring shape, and tetrahedral order analyses. Inparticular, we introduce the concept of ‘ring entropy’ tounderstand the effects of ring shape and ring size distri-bution to the inter mediate-range structure in silica poly-morphs. Moreover, the origins of diffraction peaks intetrahedrally coordinated non-crystalline materials arediscussed.2. X-ray and neutron diffraction dataof silica (SiO2) glassSilica is the most important material that can form glassas a single component. The short-range ordering of thisglass is the same as that of crystals consisting of SiO4tetrahedra with oxygen atoms sharing at the corner, form-ing a network structure. The structure factor S(Q) of silicaglass obtained from X-ray diffraction (SPring-8) and neu-tron diffraction (J-PARC) data is shown in Fig. 1(a). Whenthe glass is composed of n types (n ² 2) or more of atoms,the X-ray total structure factor S(Q) is the weighted sumof the partial structure factor Sij(Q) values correspondingto the correlations between atoms of the same or differenttypes.SXðQÞ ¼Xni¼1Xnj¼1WijðQÞSijðQÞ ð5ÞhfðQÞi2 ¼Xni¼1cifiðQÞ� �2ð6ÞHere, ci and fi(Q) are the molar fraction and atomicscattering factor of atom i, respectively, and Wij(Q) isthe weighting factor defined by ci and fi(Q), Wij(Q) =cicjfi(Q)fj(Q)/© f (Q)ª2. In the case of silica glass,SSiO2ðQÞ ¼ WSi­SiðQÞSSi­SiðQÞþ 2WSi­OðQÞSSi­OðQÞþWO­OðQÞSO­OðQÞ ð7ÞWhen fi(Q) is replaced with the atomic number, i.e.,fi(Q = 0), the relationships of the X-ray structure factorSX(Q) and the neutron structure factor SN(Q) obtained byreplacing fi(Q) with the neutron scattering length bi withthe partial structure factor are expressed as follows.SXðQÞ ¼ 0:218SSi­SiðQÞ þ 0:498SSi­OðQÞþ 0:284SO­OðQÞ ð8ÞSNðQÞ ¼ 0:069SSi­SiðQÞ þ 0:388SSi­OðQÞþ 0:543SO­OðQÞ ð9ÞX-rays are more sensitive to heavy elements, whereas neu-trons are more sensitive to light elements. This differencein sensitivity is reflected in the different weights assignedto each Sij(Q). In fact, this difference in weight is mani-fested as the difference in S(Q) in Fig. 1(a).12) The firstsharp diffraction peak (FSDP, Q1)13,14) is observed at Q ³1.5¡¹1 in SX,N(Q). This FSDP is considered a signature ofthe intermediate-range ordering in glass, and its peak posi-tion (2³/Q1) and half-width (2³/¦Q1) are estimated tohave a periodicity of 4¡ and a coherence length of 10¡,respectively.15)SN(Q) shows a principal peak (PP, Q2)14) at Q ³ 3¡¹1,but no such peak is observed in SX(Q), suggesting that PPreflects the oxygen–oxygen correlation. The Sij(Q)15,16) ofsilica glass derived from MD-RMC modelling is shown inFig. 1(a). It can be seen that SSi–Si(Q) and SO–O(Q) showpositive peaks at Q2 ³ 3¡¹1, whereas SSi–O(Q) shows anegative peak. Since these positive and negative peaksare weighted by Wij(Q) in Eqs. (8) and (9), a large WO–O(= 0.543) results in a positive PP in SN(Q). However, inSX(Q), a large WSi–O (= 0.498) causes a negative PP. Theorigin of the negative peak at Q ³ 3¡¹1 in SSi–O(Q) isdiscussed in Ref. 17).The peak Q3 observed at Q ³ 5¡¹1 in SX,N(Q) reflectsthe nearest-neighbour correlation.18) Onodera et al. appliedRMC modelling to liquid mercury and confirmed its valid-ity,15) but further investigation is addressed in section 6 ofthis paper. Figure 1(b) shows the total correlation functionTX,N(r) of silica glass12) obtained by a Fourier transform ofS(Q). A peak corresponding to Si–O correlations is ob-served at 1.6¡. The calculation of the peak area yieldsa coordination number of 4, confirming the formation ofSiO4 tetrahedra. An oxygen–oxygen correlation peak isobserved at 2.6¡, but since neutrons are more sensitive tooxygen atoms than X-rays, the peak is higher in neutrondiffraction data. On the other hand, a peak correspondingJournal of the Ceramic Society of Japan 134 [4] 208-216 2026 JCS-Japan209to silicon–silicon correlations is observed at 3.1¡, but thepeak is higher in the diffraction data of X-rays, which aremore sensitive to heavy elements. Thus, even in simpleoxide glasses such as silica glass, the difference betweenX-ray and neutron diffraction data is clear.3. Structure of silica crystal and glassThe short-range ordering of silica crystals (cristobalite,tridymite, quartz, and coesite) and glass consists of SiO4tetrahedra, which are interconnected by corner-sharingoxygen atoms to form a continuous network structure. Incrystalline silica, Si atoms are surrounded by four Si atomsvia O atoms and O atoms are surrounded by six O atomsvia Si atoms (see Fig. 2). Figure 3 shows the distributionsof the numbers of Si atoms around Si atom (a) and of Oatoms around O atom (b) in silica glass obtained by MD-RMD modelling based on X-ray and neutron diffractiondata.15) The number of Si atoms around Si atom is almost4, but only 70% O atoms are surrounded by six O atoms insilica glass. This is an important structural feature in silicaglass induced by disorder. Cooper proposed the topolog-ical disorder in silica glass on the basis of ring sizedistribution, because silica glass shows a various-ring-sizedistribution.19) Onodera et al. reported that cristobalite andtridymite have only sixfold rings (consisting of six SiO4tetrahedra), whereas quartz has a large fraction of eightfoldrings in addition to sixfold rings. Coesite has a various-ring-size distribution similarly to silica glass, suggestingthat coesite is topologically disordered. This feature isrelated to the broad neutron inelastic scattering spectrumof coesite, which is very similar to that of silica glass.20)SiOFig. 2. Atomic configuration of ¢-cristobalite.86420109876543210 X-ray diffraction Neutron diffraction-5052520151050210 X-ray diffraction Neutron diffractionr (Å)TX,N(r)(b)Si OO OSi SiQ (Å 1)SX,N(Q)(a)Q1   Q2     Q3(FSDP) (PP)       Sij(Q)Si SiO OSi OFig. 1. (a) X-ray and neutron structure factors SX,N(Q)12) together with Faber–Ziman partial structure factorsSij(Q) obtained from MD-RMC model.15,16) (b) X-ray and neutron total correlation functions TX,N(r).12)Kohara et al.: Quantum beam diffraction measurement and topological analysis of tetrahedrally coordinated non-crystalline materialsJCS-Japan2104. Structure and topologyof densified silica glassThe structure of silica glass has been widely studiedfrom ambient conditions1,6–12,15,16,21,22) to high-pres-sure12,16,17,20–27) and high-temperature12,17,20,21,28,30) condi-tions by X-ray and neutron diffraction measurements. Wehave recently synthesized two silica glasses with the samedensity (2.7 g cm¹3) but different structures by hot andcold densifications.20) Figure 4 shows the X-ray totalstructure factors SX(Q) of silica glasses synthesized at1200 °C/7.7GPa (hot densified glass, HDG) and RT/20GPa (cold densified glass, CDG). The height of FSDPis maximum in HDG and minimum in CDG, suggestingthat HDG is the most ordered glass and CDG is the mostdisordered glass. Moreover, we confirmed that CDG is nota permanently densified glass and its density decreases to2.21 g cm¹3 after heat treatment at 900 °C.We applied topological analysis techniques to a seriesof densified silica glasses to understand the intermediate-range ordering. The combined use of ring size distribu-tion analysis29,30) and persistent homology analysis31–33) isvery useful for understanding the topology of densifiedsilica glass.20) Figure 5(a) shows the persistence pn [seeRef. 34)] of Si cycles in ¡-cristobalite,35) ¢-cristobalite,36)¡-quartz,37) coesite,38) and a series of densified silicaglasses calculated using a combination of the R.I.N.G.S.code,29,30) SOVA,39) and HomCloud.33) It is found that theshape of a large ring changes with increasing density andthe pn of HDG is the smallest. This behaviour is in linewith the change in the height of FSDP (Fig. S1). Intrigu-ingly, the pn values of the sixfold and eightfold rings inHDG are comparable to those of quartz, whose density isidentical to that of HDG. It is also found that an unusuallylarge pn is observed for the eightfold rings in coesite.Although the fraction of eightfold rings is small, such sym-metrical eightfold rings [see Fig. 5(b)] are observed incoesite, whose density is the highest in a series of tetra-hedral corner-sharing silica.Recently, we have reported that inter-tetrahedral oxy-gen–oxygen correlation is observed in silica glass, coesite,and siliceous zeolite MFI. Figure 6 shows the typical tri-angle O–O–O correlation32) extracted from O-centric per-sistence diagrams (PDs) for HDG and coesite, in whichinter-tetrahedral O–O correlations (green sticks) are ob-served. This correlation is the origin of the sharp PP in neu-tron diffraction data for silica glass under high pressure.24)5. Topological analysis of Si–O ringsWe propose a new analysis method to understand thering statistics in silica polymorphs. To quantify the breadthof the ring size distribution, we define the ring entropy HasH ¼ �XnFn lnðFnÞ; ð10Þwhere Fn denotes the fraction of Si–O rings with n-foldnumber of atoms in the primitive ring size distribution ofeach SiO2 glass or crystal. H is zero when all rings havethe same size, and it increases as the ring size distributionbecomes broader. Thus, H provides a useful measure of the1500100050001098765432100050007654321Number of Si atoms around Si atomCountsNumber of O atoms around O atomCounts(a)                                                                              (b)  Fig. 3. (a) Number of silicon atoms around silicon atom and (b) that of oxygen atoms around oxygen atomobtained from MD-RMC model.21020151050 1200 /7.7 GPa RT/20 GPaSX (Q)Q (Å 1)Fig. 4. X-ray total structure factors SX(Q) of two densifiedsilica glasses.20)Journal of the Ceramic Society of Japan 134 [4] 208-216 2026 JCS-Japan211breadth of the ring size distribution. As mentioned insection 3, the concept of a topologically disordered net-work was proposed in Refs. 19) and 40) on the basis ofring size distribution. As a follow-up, here we introducethe ring entropy H, which enables the degree of topo-logical disorder to be characterized quantitatively.We also define the average persistence of the Si cycles,A, asA ¼XnFnpnn; ð11Þwhere pn is the persistence of the n-fold Si cycles definedby dk–bk derived by persistent homology analysis.31–33) Acharacterizes the overall degree of symmetry of the ringsin each SiO2 compound. Smaller and larger A values cor-respond to more distorted and more symmetrical ringshapes, respectively.Figure 7 shows a two-dimensional plot of A versus Hfor a series of silica polymorphs. Because both ¡- and¢-cristobalites consist only of sixfold rings,20) their Hvalues are zero. On the other hand, the ring shapes in ¢-cristobalite are more symmetric than those in ¡-cristobal-ite,20) and therefore, ¢-cristobalite exhibits a larger A valuethan ¡-cristobalite. Unlike cristobalite, ¡-quartz containsrings of two difference sizes, as determined from the prim-itive criterion,20) i.e., six- and eightfold rings. As a result,¡-quartz shows a finite H value of 0.45. In addition, ¡-quartz exhibits a smaller A value than cristobalite, whichreasonably reflects the more distorted ring shapes in ¡-quartz than in cristobalite.20)Compared with these crystals, the H values of SiO2glasses are larger because of the broader ring size distri-bution of the glasses than of cristobalite and quartz.20) Adecreases in the following order of the glass samples syn-thesized under different conditions: RT/7.7GPa, 400 °C/7.7GPa, RT/20GPa, and 1200 °C/7.7GPa. This trend iswell correlated with the increase in density.20) This corre-lation is further supported by the fact that the glasses syn-thesized at RT/20GPa and 1200 °C/7.7GPa exhibit bothsimilar densities20) and A values. These results indicatethat the densification of SiO2 glasses induces the distortionof constituent rings.Note that coesite data points cluster the SiO2 glasses inFig. 7. In particular, the similarity in H values betweencoesite and SiO2 glasses suggests a similar degree of topo-logical disorder. This similarity may be related to a broadpeak observed in the inelastic neutron scattering data for1412108642087654  RT/7.7 GPa 400 /7.7 GPa 1200 /7.7 GPa RT/20 GPaα-cristobaliteβ-cristobaliteα-quartz coesiten fold ringPersistence p n(Å2 )(a) (b)Fig. 5. (a) Ring persistence pn of a series of silica crystals and densified silica glasses. (b) Highly symmetricaleightfold Si–Si cycle observed in coesite.3.5 Å3.2 Å2.7 ÅSiO3.3 Å3.1 Å2.6 ÅcoesiteHDG (1200 /7.7 GPa)Fig. 6. Inter-tetrahedral oxygen–oxygen correlations up to 3.5¡ derived from persistent homology analysis ofO-centric PDs. Intra- and inter-tetrahedral O–O correlations are indicated by cyan sticks and green sticks,respectively.Kohara et al.: Quantum beam diffraction measurement and topological analysis of tetrahedrally coordinated non-crystalline materialsJCS-Japan212coesite,20) which is very similar to the feature of the bosonpeak observed in SiO2 glasses as mentioned in section 3.Although the H and A values of liquid silica are iden-tical to those of silica glass, they are larger, indicating thatsilica in the liquid state is topologically more disorderedthan that in the glassy state,41) whereas its rings are overallless distorted.On the basis of the above discussion, we schematicallyillustrate two-dimensional corner-sharing networks for dif-ferent combinations of A and H values in Fig. 7. For lowA and high H, the network consists of rings with uniformsize and symmetric shapes (upper left in Fig. 7), whichcorresponds to ¢-cristobalite. As H increases, a wide vari-ety of ring sizes appear, whereas the rings generally main-tain their symmetric shapes (upper right in Fig. 7). In con-trast, as A decreases, the rings become distorted, while thenetwork still consists of rings with the same size (lowerleft in Fig. 7). When A is low and H is high, the network iscomposed of distorted rings with different sizes (lowerright in Fig. 7).It is worth mentioning that both A and H are insensitiveto the distinction between crystalline and non-crystallinestates. In fact, both the crystals and glasses can exhibitsimilar values of A and H as shown in Fig. 7 for SiO2glasses and coesite. Therefore, A and H provide a unifiedframework for evaluating topological features across bothcrystalline and non-crystalline materials. In addition, thering entropy H represents different measure of order anddisorder from that inferred from the FSDP. Figure S1presents the S(Q) in the low-Q region for SiO2 glassessynthesized at various temperatures and under a pressureof 7.7GPa. Compared with the S(Q) of the glass synthe-sized at RT/7.7GPa, which is almost identical to that ofthe pristine glass in terms of the position/height of FSDPand density, the sample prepared at 400 °C/7.7GPa exhib-its an FSDP with reduced height and a shift to a higher-Qregion. As the temperature further increases, the heightof the FSDP is significantly increased, reaching its maxi-mum at 1200 °C/7.7GPa, indicating that the hot-densifiedglass synthesized at this condition is the most orderedglass, as discussed in section 4. On the other hand, thishot-densified glass shows the highest H value among theglasses synthesized under different temperatures at 7.7GPa, as shown in Fig. 7, suggesting the strong topologi-cal disorder of the glass synthesized at 1200 °C/7.7GPa.These results highlight the conceptual difference betweentopological order/disorder proposed by Cooper and Guptaand that proposed in this section.6. Understanding the origin of Q3in non-crystalline materialsThe origin of diffraction peaks of non-crystalline mate-rials has been discussed for long time.11,13,15,18) The lengthscale of intermediate-range order manifested by diffractionpeaks in terms of peak position and peak width was dis-cussed by Salmon and Zeidler in Ref. 42). Very recently,the ring size distribution in silica glass was deduced bythe analysis of FSDP, which seems to be a challengingapproach.43) Figure 8 shows atomic configurations ofamorphous silicon,41) glassy silica,15) and liquid carbontetrachloride.44) Amorphous Si has a perfect SiSi4 tetrahe-dral network, whereas glassy silica has a SiO4 tetrahedralnetwork in which Si atom is fourfold and O atom istwofold. The short-range structure of liquid carbon tetra-chloride is a CCl4 tetrahedron, but CCl4 tetrahedra areisolated. In our previous study, we confirmed that the Q3 ofliquid Hg can be reproduced by a random atomic configu-ration with a cut-off distance of 2.6¡.15) Here, we calcu-late the S(Q) of a SiSi4 tetrahedron (first and second Si–Sicorrelations), a SiO4 tetrahedron (silicon–oxygen andHigh , High High , Low 1.51.00.50.02.52.01.51.00.50.01200 °C /7.7 GPaRing entropy HA (Å2 )β-cristobaliteα-cristobaliteα-quartzcoesiteRT/20 GPa400 °C/7.7 GPaRT/7.7 GPaLiquid (0 GPa/1700 °C)Low , Low Low , High Fig. 7. Two-dimensional plot of A versus H for a series of silica polymorphs.Journal of the Ceramic Society of Japan 134 [4] 208-216 2026 JCS-Japan213oxygen–oxygen correlations), and a CCl4 tetrahedron(carbon–chlorine and chlorine–chlorine correlations) usingthe following equations:45,46)SðQÞ ¼ ciNijfiðQÞfjðQÞhfðQÞi2 exp � 12l2ijQ2� �� sinð³Q=QmaxÞ³Q=QmaxsinðQrijÞrijð12ÞandhfðQÞi2 ¼XicifiðQÞ� �2; ð13Þwhere ci and fi(Q) are the concentration and atomic formfactor of i, respectively. rij is the atomic distance between iand j, Nij is the coordination number of j around i, and lijis a convergence factor that represents the static and ther-mal disorders of the i–j correlation.Figure 9 shows the X-ray total structure factor SX(Q)of amorphous Si,47) the neutron total structure factorsSN(Q) of glassy silica12) and liquid carbon tetrachloride,48)together with the SX(Q) of a SiSi4 tetrahedron and theSN(Q) of a SiO4 tetrahedron and a CCl4 tetrahedron calcu-lated using Eq. (12). It is confirmed that FSDP and PPcannot be reproduced by a tetrahedron for glassy silica. Ascan be seen in the figure, a CCl4 model shows a single PP,but experimental data shows two peaks due to the orien-tational correlation49,50) of CCl4 molecules. The hardsphere Monte Carlo50,51) model gives a single PP.52) Onlythe Q3 of liquid carbon tetrachloride is reproduced by aCCl4 tetrahedron probably owing to the non-tetrahedralnetwork structure (CCl4 molecules are isolated).Here, we address more the origin of PP. The origin ofFSDP has been widely discussed,11,13,15,18) while the originof PP can be understood in terms of inter-tetrahedral X–Xcorrelation in AX4 tetrahedrally coordinated system in thisarticle. Since metallic glasses do not show an FSDP nor aPP,15,18) we suggest that the origin of PP can be attributedto the correlation of the vertex of polyhedra in amorphoussilicon and liquid carbon tetrachloride.SiO2TetrahedralnetworkRed : OBlue : SiNSi O = 4NO Si = 2Nave = 2.67SparseDenseCCl4Isolated tetrahedraRed : ClBlue : CNC Cl = 4NCl C = 1Nave = 1.6SiPerfect tetrahedral networkN = 4Fig. 8. Atomic configurations of amorphous silicon,41) glassy silica,15) and liquid carbon tetrachloride44)together with coordination numbers obtained by MD-RMC models for amorphous silicon and glassy silica, andRMC-MM model for liquid carbon tetrachloride.210210210302520151050QrA XSN(Q)SN(Q)SX (Q)Q1 Q2 Q3(FSDP)   (PP)       Amorphous SiLiquid CCl4Glassy SiO2Fig. 9. X-ray total structure factors SX(Q) of amorphous sili-con47) and neutron total structure factors SN(Q) of liquid carbontetrachloride48) and glassy silica12) together with SX(Q) of SiSi4and SN(Q) of SiO4 and CCl4 calculated using Eq. (12). The scat-tering vector Q is scaled by multiplying by rA–X (distancebetween centre and corner of tetrahedron). Black curve; experi-mental data, red curve; calculated data.Kohara et al.: Quantum beam diffraction measurement and topological analysis of tetrahedrally coordinated non-crystalline materialsJCS-Japan2147. ConclusionsIn this article, we review recent results obtained by thecomplementary use of quantum beam diffraction andtopological analysis for silica polymorphs. First, we intro-duced the state-of-the-art X-ray and neutron diffractiondata of silica glass measured at SPring-8 and J-PARC,respectively.A combination of quantum beam diffraction measure-ments and computer simulations,7,8) several studies on ox-ide glasses and high-temperature oxide melts,9,10) and thetopology in silica polymorphs11) have been reviewed in ourprevious articles. In the present article, we compare atomicarrangements in silica crystals and glass, with special focuson the number of silicon atoms around silicon atom, andthat of oxygen atoms around oxygen atom. Moreover, weintroduced a series of densified silica glasses synthesizedby hot and cold densifications to understand the differ-ences in diffraction data and ring persistency. The com-parison between the persistent homology analysis data andthe ring size distribution has led to the classification of aseries of densified silica glasses and crystals in terms ofring persistency (ring shape) and ring entropy (topologicalorder–disorder) without using diffraction data. This is acrucial new concept to understand the nature of order with-in disorder53) observed in a series of silica polymorphs.Finally, we interpreted the diffraction peak in tetrahe-drally coordinated non-crystalline materials. As a result,we found differences among amorphous silicon, glassysilica, and liquid carbon tetrachloride in terms of the ori-gins of a three-peak structure, FSDP (Q1), PP (Q2), and Q3.The understanding of diffraction peaks in non-crystal-line materials is the first step toward understanding thenature of order within disorder. Moreover, the introduc-tion of topological analysis enables us to directly comparebetween crystalline and non-crystalline materials, whichdoes not depend on the presence or absence of the Braggpeak.Acknowledgements This work was partially supportedby JSPS Grants-in-Aid for Transformative Research Areas(A) “Hyper-Ordered Structures Science” (Grant Numbers20H05878, 20H05881, and 20H05884). Discussions withProfessor I. Obayashi and Dr. K. Nakashima are gratefullyappreciated. MD simulations were performed using theNumerical Materials Simulator at the National Institute forMaterials Science (NIMS).References1) K. A. Kirchner, D. R. Cassar, E. D. Zanotto, M. Ono,S. H. Kim, K. Doss, M. L. Bødker, M. M. Smedskjaer,S. Kohara, L. Tang, M. Bauchy, C. J. Wilkinson, Y.Yang, R. S. Welch, M. Mancini and J. C. Mauro, Chem.Rev. 123, 1774 (2023).2) T. E. Faber and J. M. Ziman, Philos. Mag. 11, 153(1965).3) E. Lorch, J. Phys. C Solid State 2, 229 (1969).4) A. K. Soper and E. R. Barney, J. Appl. Crystallogr. 44,714 (2011).5) J. E. Proctor, C. G. Pruteanu, B. Moss, M. A.Kuzovnikov, G. J. Ackland, C. W. Monk and S.Anzellini, J. Appl. Phys. 134, 114701 (2023).6) S. Kohara, K. Ohara, H. Tajiri, C. Song, O. Sakata, T.Usuki, Y. Benino, A. Mizuno, A. Masuno, J. T. Okada,T. Ishikawa and S. Hosokawa, Z. Phys. Chem. 230, 339(2016).7) S. Kohara and P. S. Salmon, Adv. Phys.: X 1, 640(2016).8) S. Kohara, J. Ceram. Soc. Jpn. 125, 799 (2017).9) S. Kohara, J. Ceram. Soc. Jpn. 130, 531 (2022).10) Y. Onodera, J. Ceram. Soc. Jpn. 130, 627 (2022).11) S. Kohara, J. Ceram. Soc. Jpn. 133, 488 (2025).12) S. Sato, M. Miyakawa, T. Taniguchi, Y. Onodera, K.Ohara, K. Ikeda, N. Kitamura, Y. Idemoto and S.Kohara, J. Ceram. Soc. Jpn. 132, 427 (2024).13) D. L. Price, S. C. Moss, R. Reijers, M.-L. Saboungi andS. Susman, J. Phys. C Solid State 21, L1069 (1988).14) P. S. Salmon, R. A. Martin, P. E. Mason and G. J.Cuello, Nature 435, 75 (2005).15) Y. Onodera, S. Kohara, S. Tahara, A. Masuno, H. Inoue,M. Shiga, A. Hirata, K. Tsuchiya, Y. Hiraoka, I.Obayashi, K. Ohara, A. Mizuno and O. Sakata,J. Ceram. Soc. Jpn. 127, 853 (2019).16) K. Ohara, Y. Onodera, M. Murakami and S. Kohara,J. Phys.-Condens. Mat. 33, 383001 (2021).17) S. Hayafune, T. Sakamaki, H. Ichikawa, Y. Onodera, S.Kohara and A. Suzuki, J. Ceram. Soc. Jpn. 133, 242(2025).18) P. S. Salmon and A. Zeidler, J. Stat. Mech.-Theory E2019, 114006 (2019).19) A. R. Cooper, Phys. Chem. Glasses 19, 60 (1978).20) Y. Onodera, S. Kohara, P. S. Salmon, A. Hirata, N.Nishiyama, S. Kitani, A. Zeidler, M. Shiga, A. Masuno,H. Inoue, S. Tahara, A. Polidori, H. E. Fischer, T. Mori,S. Kojima, H. Kawaji, A. I. Kolesnikov, M. B. Stone,M. G. Tucker, M. T. McDonnell, A. C. Hannon, Y.Hiraoka, I. Obayashi, T. Nakamura, J. Akola, Y. Fujii,K. Ohara, T. Taniguchi and O. Sakata, NPG Asia Mater.12, 85 (2020).21) Y. Inamura, Y. Katayama, W. Utsumi and K. Funakoshi,Phys. Rev. Lett. 93, 015501 (2004).22) T. Sato and N. Funamori, Phys. Rev. Lett. 101, 255502(2008).23) C. J. Benmore, E. Soignard, S. A. Amin, M. Guthrie,S. D. Shastri, P. L. Lee and J. L. Yarger, Phys. Rev. B 81,054105 (2010).24) A. Zeidler, K. Wezka, R. F. Rowlands, D. A. J.Whittaker, P. S. Salmon, A. Polidori, J. W. E. Drewitt,S. Klotz, H. E. Fischer, M. C. Wilding, C. L. Bull, M. G.Tucker and M. Wilson, Phys. Rev. Lett. 113, 135501(2014).25) M. Murakami, S. Kohara, N. Kitamura, J. Akola, H.Inoue, A. Hirata, Y. Hiraoka, Y. Onodera, I. Obayashi, J.Kalikka, N. Hirao, T. Musso, A. S. Foster, Y. Idemoto,O. Sakata and Y. Ohishi, Phys. Rev. B 99, 045153(2019).26) Y. Kono, K. Ohara, N. M. Kondo, H. Yamada, S Hiroi,F. Noritake, K. Nitta, O. Sekizawa, Y. Higo, Y. Tange,H. Yumoto, T. Koyama, H. Yamazaki, Y. Senba, H.Ohashi, S. Goto, I. Inoue, Y. Hayashi, K. Tamasaku, T.Osaka, J. Yamada and M. Yabashi, Nat. Commun. 13,2292 (2022).Journal of the Ceramic Society of Japan 134 [4] 208-216 2026 JCS-Japan21527) M. Guerette, M. R. Ackerson, J. Thomas, F. Yuan, E. B.Watson, D. Walker and L. Huang, Sci. Rep.-UK 5,15343 (2015).28) Q. Mei, C. J. Benmore and J. K. R. Weber, Phys. Rev.Lett. 98, 057802 (2007).29) S. L. Roux and P. Jund, Comp. Mater. Sci. 49, 70(2010).30) S. L. Roux and P. Jund, Comp. Mater. Sci. 50, 1217(2011).31) I. Obayashi, T. Nakamura and Y. Hiraoka, J. Phys. Soc.Jpn. 91, 091013 (2022).32) Y. Hiraoka, T. Nakamura, A. Hirata, E. G. Escobar, K.Matsue and Y. Nishiura, P. Natl. Acad. Sci. USA 113,7035 (2016).33) I. Obayashi, HomCloud, https://homcloud.dev/index.en.html.34) P. S. Salmon, A. Zeidler, M. Shiga, Y. Onodera and S.Kohara, Phys. Rev. B 107, 144203 (2023).35) J. J. Pluth, J. V. Smith and J. Faber, J. Appl. Phys. 57,1045 (1985).36) R. W. G. Wyckoff, Z. Krist.-Cryst. Mater. 62, 189(1926).37) K. Kihara, Eur. J. Mineral. 2, 63 (1990).38) L. Levien and C. T. Prewitt, Am. Mineral. 66, 324(1981).39) https://www.shiga-lab.org/sova.40) P. K. Gupta, J. Am. Ceram. Soc. 76, 1088 (1993).41) S. Kohara, M. Shiga, Y. Onodera, H. Masai, A. Hirata,M. Murakami, T. Morishita, K. Kimura and K. Hayashi,Sci. Rep.-UK 11, 22180 (2021).42) P. S. Salmon and A. Zeidler, Phys. Chem. Chem. Phys.15, 15286 (2013).43) Q. Zhou, Y. Shi, B. Deng, J. Neuefeind and M. Bauchy,Sci. Adv. 7, eabh1761 (2021).44) H. Morita, S. Kohara and T. Usuki, J. Mol. Liq. 147,182 (2009).45) R. L. Mozzi and B. E. Warren, J. Appl. Crystallogr. 2,164 (1969).46) Y. Onodera, Y. Takimoto, H. Hijiya, Q. Li, H. Tajiri, T.Ina and S. Kohara, NPG Asia Mater. 16, 22 (2024).47) K. Laaziri, S. Kycia, S. Roorda, M. Chicoine, J. L.Robertson, J. Wang and S. C. Moss, Phys. Rev. Lett. 82,3460 (1999).48) Y. Onodera, private communication.49) M. Misawa, J. Chem. Phys. 91, 5648 (1989).50) Sz. Pothoczki, L. Temleitner and L. Pusztai, Chem. Rev.115, 13308 (2015).51) S. Kohara and L. Pusztai, in “Computer Simulations ofGlasses: Methodologies and Applications”, Ed. by J. Duand A. N. Cormack, Wiley-American Ceramic Society,Hoboken (2022) pp. 60–88.52) Sz. Pothoczki, L. Temleitner, P. Jóvári, S. Kohara and L.Pusztai, J. Chem. Phys. 130, 064503 (2009).53) P. S. Salmon, Nat. Mater. 1, 87 (2002).Kohara et al.: Quantum beam diffraction measurement and topological analysis of tetrahedrally coordinated non-crystalline materialsJCS-Japan216https://homcloud.dev/index.en.htmlhttps://homcloud.dev/index.en.htmlhttps://www.shiga-lab.org/sova