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Shinji Kohara, Yohei Onodera, Motoki Shiga, Hirokazu Masai, Atsunobu Masuno, Koji Kimura, Koichi Hayashi

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[Bond angle distributions in silica polymorphs](https://mdr.nims.go.jp/datasets/cc025e15-a0a6-4ab6-854a-b8a66ea74621)

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Bond angle distributions in silica polymorphsEXPRESS LETTERBond angle distributions in silica polymorphsShinji Kohara1,³, Yohei Onodera1, Motoki Shiga2,3,1,4, Hirokazu Masai5,1,Atsunobu Masuno6,1, Koji Kimura7,1 and Koichi Hayashi71Center for Basic Research on Materials, National Institute for Materials Science, Tsukuba, Ibaraki 305–0047, Japan2Unprecedented-scale Data Analytics Center, Tohoku University, Sendai 980–8578, Japan3Graduate School of Information Sciences, Tohoku University, Sendai 980–8579, Japan4RIKEN Center for Advanced Intelligence Project, Chuo-ku, Tokyo 103–0027, Japan5Department of Materials and Chemistry, National Institute of Advanced Industrial Science and Technology,Ikeda, Osaka 563–8577, Japan6Graduate School of Engineering, Kyoto University, Kyoto 615–8520, Japan7Department of Physical Science and Engineering, Nagoya Institute of Technology, Gokisocho, Showaku, Nagoya 466–8555, JapanBond angle distributions (BADs) in silica crystals, silica glass, and siliceous zeolites were compared to under-stand the topology of silica polymorphs. It is found that the Si–O–Si bond angle is 180° in ¢-cristobalite, which ischaracteristic of highly symmetrical sixfold rings. In contrast, the Si–O–Si BAD of silica glass, obtained from amolecular dynamics–reverse Monte Carlo model, has an average value of 153° with a full width at half maxi-mum (FWHM) of 20°. The Si–Si–Si BAD of ¢-cristobalite exhibits a peak at 110° demonstrates that SiSi4 hypertetrahedra possess high symmetry. This feature is consistent with the presence of symmetric six-membered ringsin ¢-cristobalite. On the other hand, both coesite and silicalite-1 (MFI) shows a variety of bond angles like silicaglass, which is consistent with the variation of (Si–O)n ring size.Keywords: Silica polymorphs, Glass, Siliceous zeolite, Structure, Bond angle distribution[Received June 10, 2026; Accepted June 23, 2026; Published online July 14, 2026]Silica is one of the most important oxide materials1) thatexhibits various polymorphs and readily forms glass.2) Thefundamental structural motif of silica crystals is corner-sharing SiO4 tetrahedra, although high-pressure phasesexhibit octahedral or higher coordination numbers.3) Thiscorner-sharing tetrahedral motif is preserved in both silicaglass and siliceous zeolites. Structural modifications underhigh pressures and high temperatures have been exten-sively studied using various experimental and simulationtechniques.2)We have been investigating the topology and homologyof silica polymorphs, focusing on (Si–O)n ring size dis-tributions, cavity volumes, and ring shapes.4–6) The bondangle distribution (BAD) is a crucial structural parameterfor understanding the network structure of silica. In par-ticular, the Si–O–Si bond angle distribution in silica glassremains a subject of ongoing debate.7,8)In this report, we compare the BADs of silica crystals,siliceous zeolites, and silica glass.Calculations of BADs were performed for variouscrystal structures [¡-cristobalite,9) ¢-cristobalite,10) ¡-quartz,11) coesite,12) siliceous zeolite, faujasite (FAU),13)sodalite (SOD),14) and silicalite-1 (MFI)15)] and the atomicconfiguration of silica glass obtained from molecular dy-namics (MD)–reverse Monte Carlo (RMC) simulations.16)The continuous distribution is discretized into a histo-gram with a bin width of ¦ª. The probability density offinding an angle in the interval [ª ¹ ¦ª/2, ª + ¦ª/2] iscalculated as:PðªÞ ¼ NðªÞNtotal�ª; ð1Þwhere N(ª) is the number of triplets falling within the bincentered at ª, and Ntotal is the total number of identifiedtriplets satisfying the cutoff condition r ¼ Rc. The solid-angle-corrected probability density g(ª) for each bin iscalculated asgðªÞ ¼ PðªÞsin ª: ð2ÞThe Rc values for Si–Si, Si–O, and O–O are 3.20, 1.85,and 2.80¡ for the crystal, and 3.50, 1.90, and 3.00¡ forthe glass, respectively. The BADs were calculated withinthe range of 6.05° < ª < 173.95°.Before discussing BADs, the ring size distributions in aseries of silica polymorphs4) calculated using King17) andprimitive18,19) criteria, are shown in Fig. S1. Note that theKing criterion is based on the shortest path, whereas theprimitive criterion is suitable for detecting larger rings, as a³ Corresponding author: S. Kohara; E-mail: KOHARA.Shinji@nims.go.jpJournal of the Ceramic Society of Japan 134 [8] 555–558 2026DOI https://doi.org/10.2109/jcersj2.26055 JCS-Japan©2026 The Ceramic Society of Japan 555This 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/jcersj2.26055https://creativecommons.org/licenses/by/4.0/-1.0 -0.5 0.0 0.5 1.0-1.0 -0.5 0.0 0.5420-1.0 -0.5 0.0 0.510050050020100500500100500Si Si Siα-cristobaliteβ-cristobaliteSi O Siα-quartzcoesiteFAUSODMFIglassO Si Sicosθg(θ)-1.0 -0.5 0.0 0.5 1.0-1.0 -0.5 0.0 0.5420-1.0 -0.5 0.0 0.5402004020040200402004020040200O Si O O O Si O O Ocosθ6040200α-cristobaliteβ-cristobaliteα-quartzcoesiteFAUSODMFIglassg(θ)4020032101206005040200402001004020040200Si Si Si Si O Si O Si SiAngle (degree)P(θ)6040200120600 18012060040200P(θ)3210120600402004020040200402006040200120600 180120600O Si O O O Si O O OAngle (degree)α-cristobaliteβ-cristobaliteα-quartzcoesiteFAUSODMFIglassα-cristobaliteβ-cristobaliteα-quartzcoesiteFAUSODMFIglassFig. 1. Bond angle distributions, P(ª) (lower) and solid-angle-corrected bond angle distributions, g(ª) (upper),of silica polymorphs.Kohara et al.: Bond angle distributions in silica polymorphsJCS-Japan556primitive ring is defined as one that cannot be decomposedinto two smaller rings. As discussed in Ref. 3), cristobal-ites possess only sixfold rings consisting of six SiO4tetrahedra, whereas silica glass shows a broad ring sizedistribution ranging from three to tenfold rings, althoughsixfold rings dominate when using the primitive criterion.According to Gupta,20) this broad ring size distribution istopologically disordered. On the other hand, ¡-quartzpossesses large fraction of eightfold rings in addition tosixfold rings and coesite possesses a variety of ring sizessimilar to silica glass, which is consistent with the resultsreported in a previous study.19) Siliceous zeolites alsopossess a variety of ring sizes similar to coesite, and theprimitive criterion can detect effectively the large ringsformed by silicalite cages.Figure 1 shows the BADs, P(ª) (lower), and the solid-angle-corrected BADs, g(ª) (upper), of silica polymorphs.Note that the right and left panels show intratetrahedraland intertetrahedral correlations, respectively. All samplesshow a peak centered at ª = 109° (cos ª = ¹0.33) in theO–Si–O BADs, owing to the formation of SiO4 tetrahedra.The O–O–O BAD of ¢-cristobalite exhibits two prom-inent peaks at ª = 60° (cos ª = 0.50, intratetrahedral) and120° (cos ª = ¹0.50, intertetrahedral). The O–O–Si BADof ¢-cristobalite also exhibits two prominent peaks atª = 35° (cos ª = 0.82, intratetrahedral) and 145° (cos ª =¹0.82, intertetrahedral). Peaks at higher angles are lessprominent in other samples, suggesting that ¢-cristobalitehas extremely high symmetry, which agrees well withour previous results derived from persistent homologyanalysis.4)All samples except FAU and SOD show peaks centeredat ª = 109.4° (cos ª = ¹0.33) in the intertetrahedral Si–Si–Si BADs, owing to the formation of SiSi4 hyper-tetrahedra.4) In particular, ¢-cristobalite exhibits a verysharp profile, suggesting that the SiSi4 hypertetrahedra arehighly symmetric. This is consistent with the fact that itssixfold rings are highly symmetric and the order parameterq of the SiSi4 hypertetrahedra is high in ¢-cristobalite.4)The intertetrahedral O–Si–Si BAD of ¢-cristobaliteexhibits a sharp peak at 110° (cos ª = ¹0.34). In contrast,other samples exhibit two broad peaks at around 109°(cos ª = ¹0.33) and 16° (cos ª = 0.96). This tetrahedralbond-angle distribution arises from the high symmetry ofthe SiSi4 hypertetrahedra in ¢-cristobalite. The most strik-ing feature in the intertetrahedral Si–O–Si BADs is that thebond angle of ¢-cristobalite is 180° (enlarged figures areshown in Fig. 2). Note that this specific angle cannot beevaluated by our calculation code for g(ª) because it leadsto division by zero (sin 180° = 0, see Eq. 2). This bondangle also indicates that sixfold rings in ¢-cristobalite arehighly symmetric. Silica glass exhibits a very broad Si–O–Si bond angle distribution owing to its inherent disorder,whereas both coesite and MFI also exhibit broad distri-butions, likely due to the distribution of ring sizes asshown in Fig. S1. Consequently, we suggest that there is acorrelation between topological disorder and broad inter-tetrahedral Si–O–Si bond angle distributions. Gaussianfitting of the P(ª) data for the glass yields an average valueof 153° with a full width at half maximum (FWHM) of20°. A comparison of these results with those of previousstudies7,8,21–25) using MD simulations, machine-learningpotential MD (MLMD) simulations, X-ray diffraction(XRD), and neutron diffraction (ND) is presented inTable 1. Our results agree well with those obtained fromRMC modeling based on ND data.In this article, we present the BADs of a series of silicapolymorphs. We found that the BAD of ¢-cristobalite6040200420-1.0 -0.5 0.010050050020100500500100500g(θ)Si O Si3210180150120905040200402001004020040200P(θ)Si O Siα-cristobaliteβ-cristobaliteα-quartzcoesiteFAUSODMFIglassα-cristobaliteβ-cristobaliteα-quartzcoesiteFAUSODMFIglassFig. 2. Enlarged view of the Si–O–Si BADs for silica poly-morphs. The curve fit for the glass data is shown as a black curve.Table 1. Comparison of Si–O–Si BADs obtained using different methodsMD–RMC(this work)MD(BKS)7)MD(VSL)7)MD(VKRE)21)MLMD24)RMC(ND)8)XRD22) XRDND23) NMR25)Mean value(degree)153 152 147 142 145 151 144 147 142FWHM(degree)20 36 10 25 — 20 38 17 26Journal of the Ceramic Society of Japan 134 [8] 555–558 2026 JCS-Japan557exhibits a highly symmetrical topology, whereas the topol-ogies of both coesite and MFI are similar to that of silicaglass. This feature suggests that the dynamics of MFI isvery similar to that of coesite, which in turn resembles thatof silica glass in terms of the boson peak observed ininelastic neutron scattering data.26)Acknowledgment This work was supported by JSPSGrants-in-Aid for Transformative Research Areas (A) “Hyper-Ordered Structures Science” (Grant Numbers 20H05878,20H05880, 20H05881, 20H05882, and 20H05884).References1) P. J. Heaney, C. T. Prewitt and G. V. Gibbs, Rev.Mineral. 29, 1 (1994).2) G. N. Greaves and S. Sen, Adv. Phys. 56, 1 (2007).3) Y. Kuwayama, K. Hirose, N. 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