## Abstract

Tetrahedral amorphous materials such as SiO_{2}, GeO_{2}, Si, Ge, C, and chalcogenides are extremely important in nature and technology. It is known that covalent bonding favors local tetrahedral order in these materials. However, how to extract information on this structural order from the scattering function has remained elusive. By analyzing the structure of simulated SiO_{2} and experimental data of various tetrahedral materials, we show that the lowest wave number peak, known as the first sharp diffraction peak (FSDP), and a few higher wave number ones in the scattering functions come from the characteristic density waves of a single tetrahedral unit. FSDP is thus a direct measure of the tetrahedrality. This finding opens the door for long-awaited experimental access to the characterization of disordered amorphous structures.

## INTRODUCTION

X-ray and neutron scattering techniques have had great success in the structural determination of crystals. For liquids and amorphous states, they are less powerful than for crystals but are still very useful tools. Because of the macroscopic isotropy and homogeneity, the structure factor *S*(**k**) has no angular dependence and is thus a function of only the (scalar) wave number *k*. *S*(*k*) of disordered systems interacting with spherically symmetric potentials are usually featureless and are characterized by a broad amorphous peak around the wave number corresponding to the interatomic distance. This makes the structural analysis of liquids and glasses extremely difficult. For example, the dynamics of a liquid slow down markedly toward the glass transition temperature despite the absence of a clear signature in *S*(*k*). This apparent disconnection between structure and dynamics is widely regarded as the main source of difficulty in the understanding of slow glassy dynamics.

However, we show below that this may not be the case for so-called tetrahedral glass-formers. A wide class of materials, such as oxides (*1*–*3*), halides (*4*), chalcogenides (*4*, *5*), monatomic amorphous solids (*6*, *7*), metallic glasses (*8*, *9*), and metal-organic frameworks (*10*, *11*), exhibit a distinct peak in *S*(*k*) at a wave number lower than that corresponding to the interparticle distance. This peak is widely known as the first sharp diffraction peak (FSDP). At this moment, the origin of FSDP is a matter of debate (see below). Here, we reveal its structural origin and show that this FSDP provides us with crucial information on structural ordering in the apparently disordered liquid and glass states of tetrahedral materials.

Many models have been proposed to explain FSDP in terms of the underlying atomic structures (*12*). The existing explanations can be categorized mainly into three groups. The first scenario ascribes it to a particular type of structures, i.e., layered (*13*, *14*) or cage-like (*15*) structures, with a spacing larger than the interatomic distance. This interpretation may work for a particular material but cannot be generally applied to, e.g., tetrahedral materials, which can form a three-dimensional (3D) corner-sharing network, a pseudo 1D edge-sharing chain, or a mixture of both (*16*). The second scenario ascribes FSDP to crystal-like structures. In several types of glasses and liquids, it was found that the FSDP position *k*_{1} is close to the first peak in the corresponding crystal structure (*17*, *18*). For example, in SiO_{2}, *k*_{1} ≈ 2π/*d*_{111}, where *d*_{111} is the distance between {111} planes in β-cristobalite. This type of explanation is, however, at odds with the observation that FSDP even appears in the equilibrium liquid state above the melting point, where crystal-like structures can hardly survive.

Last, the third compelling explanation, which was proposed by Blétry (*19*) and Elliott (*20*), attributes FSDP observed in AB_{2}-type tetrahedral materials to a chemical order originating from the clustering of voids around cations in terms of concentration-concentration partial structure factors in the Bhatia-Thornton formalism (*21*). The peak wave number of FSDP was evaluated from the close packing of binary spheres with chemical orderas follows (*19*, *20*):(1)where *r* is the nearest-neighbor cation-cation distance. It was claimed that this formula reasonably predicts the FSDP position for AB_{2}-type tetrahedral materials (*19*, *20*). However, this scenario was challenged later by a first-principle molecular dynamics simulation of liquid GeSe_{2} (*22*) and classical molecular dynamics simulations of a polarizable tetrahedral liquid model (*23*), which showed that the appearance of FSDP is not directly linked to this chemical order. Moreover, neutron and x-ray scattering experiments of vitreous SiO_{2} reported that the concentration-concentration partial structure factor exhibits no peak in the FSDP position (*24*), providing strong evidence against the chemical-order scenario.

As briefly reviewed above, there has been no consensus on the origin of FSDP despite decades of intensive theoretical and experimental research studies. Moreover, it remains unclear whether there is any link between the development of FSDP and material properties. Here, we propose a novel scenario for FSDP that it originates from the electron density distribution characteristic to a locally favored tetrahedral structure (LFTS) in tetrahedral amorphous materials. This scenario, which we call the “tetrahedron model,” explains not only FSDP but also other higher wave number peaks and shoulders in the structure factor in terms of the characteristic length scales of a tetrahedral structural unit, i.e., height, length, and width. We confirm the relevance of this tetrahedron model by carefully comparing its predictions with the peak positions of numerically simulated structure factors of an archetype tetrahedral liquid SiO_{2} and experimentally measured structure factors of a wide range of tetrahedral materials. We also show, by molecular dynamics simulation of SiO_{2}, that the integrated intensity of FSDP is directly proportional to the fraction of LFTS or a measure of the degree of local tetrahedral ordering, which was independently estimated by the degree of translational order in the second shell (*25*). Because we showed (*25*) that the fraction of LFTS controls the thermodynamic properties such as the density anomaly of silica (*26*), our finding suggests a strong connection between FSDP and the properties of tetrahedral materials.

## RESULTS

The structure factor, which is the density-density correlation in reciprocal space, is given by(2)where *N* is the number of particles, 〈 ⋯ 〉 denotes the ensemble average, and ρ_{k} is the Fourier component of the number density ρ, defined by(3)where **r**_{i} is the position vector of atom *i*.

Because crystal atoms have a periodic arrangement, density wave ρ_{k} has well-defined components at particular wave vectors **k**’s, which produce sharp peaks in the structure factor. On the other hand, the atomic arrangement in a disordered noncrystalline system lacks a long-range translational order, and thus, there are no such sharp peaks unlike the case of a crystal. However, atoms are still arranged in a regular manner locally to lower the free energy. Thus, it is natural to consider that if there is a specific regular arrangement of atoms, then it can have characteristic density waves beyond the nearest interatomic distance, which may produce FSDP in the structure factor. This is our basic idea.

Tetrahedral materials, by definition, tend to locally form tetrahedral unit structures. So we first focus on the characteristics of density waves embedded in a regular tetrahedron, which we show schematically in Fig. 1A. We can see that the tetrahedral unit, whose nearest interatomic distance is *r*, can be characterized by three key length scales: height *H*, edge length *L*, and width *D*. Here, *H*, *L*, *D* > *r*. The structure factor of an isolated regular tetrahedron (see Fig. 1A) can be straightforwardly calculated from Eqs. 2 and 3 independently along the *x*, *y*, and *z* axes. The analytical solutions are given by eqs. S2 to S4, and the results are displayed in Fig. 1B. We can see that, despite small deviations due to the interference of different peaks, the structure factor *S*(*k*) of a regular tetrahedron shows four characteristic peaks in the range of 0.5 ≤ *kr*/2π ≤ 2.75, which originates from the density waves associated with the underlying characteristic lengths, *H*, *L*, and *D*, of a tetrahedron.

We can see in Fig. 1B that the FSDP located at *kr*/2π ≃ 3/4 comes mainly from the density wave along the *z* axis with *k* = 2π/*H*. The second peak at originates from the density wave along the *x* axis with *k* = 4π/*L*. The third and fourth peaks are at and , which stem from density waves along the *y* and x axes, with *k* = 6π/*D* and 8π/*L*, respectively. The separation between these two peak positions is very small. Some other signals also exist but are substantially smaller and thus may not be observed as clear peaks. We summarize the characteristic peak positions for a regular tetrahedron in Table 1. Here, it is worth noting that our tetrahedron model predicts exactly the same FSDP location at *k*_{1} = 3π/2*r*, proposed as Blétry and Elliott’s close-packed binary spheres model (see Eq. 1) (*19*, *20*). However, we emphasize that the physical mechanism is essentially different between the two models. The FSDP in our model originates from the density waves generated by a tetrahedral unit that is common to all tetrahedral materials. However, their model ascribes FSDP to the chemical order of close-packed binary spheres, whose relevance to tetrahedral materials characterized by low coordination is questionable.

Recently, by introducing a new structural descriptor *z* that characterizes the translational order in the second shell, we revealed that liquid silica can be physically treated as a dynamical mixture of two types of local structures (or two states) (*27*), which was verified by a bimodal distribution of *z* (*25*) in molecular dynamics simulations of silica with the van Beest, Kramer, and van Santen (BKS) empirical potential (*28*). At a high temperature, silica mainly consists of disordered normal-liquid structures, whereas upon cooling more and more LFTSs with high translational and tetrahedral orders are formed in the sea disordered normal-liquid structures. We showed by simulations that this two-state model can naturally explain the density anomaly of silica (*25*).

In our model, FSDP is explained by the formation of LFTS and the resulting unique density waves characteristic to a tetrahedron. On the other hand, our structural descriptor *z* also detects the same LFTS by focusing on the translational order of the second shell, which is a consequence of the formation of a regular tetrahedral structure protected from invasion of other atoms. Thus, it is quite natural to expect a close connection between FSDP and LFTS. Here, we directly check this hypothesis by looking at the Si–Si partial structure factor of a typical LFTS, SiSi_{4} (averaged over random locations and orientations) in liquid silica by simulations. As shown in Fig. 1C, the structure factor exhibits the three characteristic peaks in the range 0.5 ≤ *kr*_{SiSi}/2π ≤ 2.5, which are predicted by our tetrahedron model (Fig. 1B). Here, *r*_{SiSi} denotes the first peak position in the Si–Si radial distribution function. The third and fourth peaks for a regular tetrahedron merge into a broad peak at *kr*_{SiSi}/2π ~ 2.25 for liquid silica. Here, we note that the emergence of FSDP is linked to the high local tetrahedral order and its peak height is positively correlated with the degree of tetrahedral order (see fig. S2).

Next, we show the Si–Si structure factor of a cluster of LFTS, SiSi_{16}, including a central Si and its 16 Si neighbors up to the second shell (averaged over random positions and orientations) in Fig. 1D. Compared to an isolated LFTS (Fig. 1C), we can see that the peaks become narrower and higher, indicating the increase in the coherence length due to a coherent spatial arrangement of neighboring tetrahedra. This is supported by the increase of the coherence length, which is estimated from the width of FSDP, from ~1.8 Å in Fig. 1C to ~2.5 Å in Fig. 1D (see Supplementary Text for details). We note that this coherence mainly comes from the local development of the translational order, but not from that of the orientational order. This short coherence length is consistent with our previous finding that orientational order basically does not extend beyond the first shell in silica, which is an origin of silica’s much higher glass-forming ability compared to water despite both having similar translational orders (*25*). Moreover, we confirm that the integrated intensities (obtained by fitting three Lorentzian peaks) of FSDP in Fig. 1 (C and D) are almost the same, supporting the proportionality between the FSDP-integrated intensity and the number of atoms involved in LFTSs. Later, we will show for silica that the integrated intensity of FSDP is a direct measure of the fraction of LFTS or the degree of local structural ordering.

Tetrahedral materials of AB_{2} type can be generally categorized into two groups in terms of preferential arrangements of tetrahedrons, corner-sharing and edge-sharing groups, depending on the connectivity of neighboring AB_{4} tetrahedral units. It is known that, for the former, the 3D tetrahedral network tends to be formed, whereas for the latter, pseudo 1D chain-like structures tend to develop (*16*). SiO_{2} is known as a typical corner-sharing tetrahedral material, and as a result, two types of tetrahedra are formed locally: a small oxygen tetrahedron SiO_{4} and a large silicon tetrahedron SiSi_{4} (see the inset of Fig. 2). According to our tetrahedron model, signals from both silicon and oxygen tetrahedra should appear in the structure factor.

Figure 2 shows the partial structure factors of liquid silica at 1 bar and 4000 K. It can be seen that, beside FSDP at *kr*_{SiSi}/2π ≃ 3/4, which is observed for all the partial structure factors, a clear shoulder appears at *kr*_{SiSi}/2π ≃ 3/2 only for the O–O partial structure factor. We note that this shoulder is observed in the O–O partial structure factor at all temperatures we studied. Given the relation *r*_{SiSi} ≃ 2*r*_{SiO}, the shoulder location can be estimated as *kr*_{SiO}/2π ≃ 3/4 after scaling by the average Si–O distance *r*_{SiO}, which in our tetrahedron model corresponds to the density wave along the height *H*_{OO} of an oxygen tetrahedron SiO_{4}. We can also see this shoulder in the experimental structure factor of glassy silica obtained by neutron scattering (see Fig. 3A). This appearance of a shoulder at *kr*_{SiO}/2π ≃ 3/4 in the O–O partial structure factor further supports the relevance of our tetrahedron model for tetrahedral materials.

It is natural to expect that our tetrahedron model, based on the density waves characteristic to a single tetrahedral unit, should be valid for all tetrahedral materials. To further check the generality of our tetrahedron model, we collected experimental structure factors of various types of tetrahedral materials, including oxides, halides, chalcogenides, and monoatomic amorphous materials, measured by x-ray and neutron scatterings, and show them in Fig. 3. We categorize these materials (denoted as monatomic A and binary AB_{2}) into corner-sharing (Fig. 3A) and edge-sharing (Fig. 3B) groups (*16*). The largest tetrahedron unit is AA_{4} and AB_{4} for the corner-sharing and edge-sharing tetrahedral materials, respectively. Therefore, we scale the structure factors by the average A-A distance *r*_{AA} for the corner-sharing group and by the average A-B distance *r*_{AB} for the edge-sharing one. As seen in Fig. 3, four characteristic peaks (or shoulders, e.g., *k* = 6π/*D*_{AA} for SiO_{2} and BeF_{2}) generally appear at the locations expected for a regular tetrahedron, strongly supporting our tetrahedron model. We note that small deviations may come from distortions of tetrahedral units under thermal fluctuations. For the edge-sharing group, we can also see a distinct low *k* peak corresponding to a length scale larger than *H*_{AB}, which is observed as the first peak in Fig. 3B. We speculate that this peak stems from some other structure (e.g., chain-like structure) extending beyond *H*_{AB}, but the underlying structure behind this peak might be material specific. Further study is necessary to elucidate its structural origin.

## DISCUSSION

Until now, we have shown that the characteristic features, including FSDP, of the structure factors of a wide class of tetrahedral materials can be well explained by our simple tetrahedron model. Then, the remaining question is what kind of information on tetrahedral materials can our model provide from the behavior of the structure factor, particularly the behavior of FSDP. To answer this question, we systematically study the temperature dependence of the structure factor of liquid silica by simulations. As shown in Fig. 4A, FSDP continuously grows upon cooling, indicating a rapid increase of the number density of LFTS. This is consistent with the increase of the tetrahedral order for the SiSi_{4} unit upon cooling (see Fig. 4B).

The degree of local structural ordering in liquid silica can be described by a two-state model (*27*), where the system is effectively treated as a dynamical mixture of LFTS and disordered normal-liquid structures (*25*). The former produces FSDP, whereas the latter gives rise to a broad peak at *k* ≃ 2π/*r*_{SiSi}, or *kr*_{SiSi}/2π ≃ 1, as typically observed in simple liquids such as hard spheres or Lennard-Jones liquids. The emergence of a peak around *kr*_{SiSi}/2π ≃ 1 can be seen in the structure factors at high pressures (see fig. S1), where liquid silica predominantly consists of disordered normal-liquid structures because LFTS costs an extra volume and is thus not favored at high pressures.

On the basis of this consideration, we find that the structure factors at all the temperatures we studied can be well fitted by three Lorentzian functions related to tetrahedral structures and one Gaussian function specifically associated with disordered normal-liquid structures, whose peak position is fixed at *kr*_{SiSi}/2π = 1 (see eq. S1). The fitting result at 4000 K is shown in Fig. 4C. We note that the fittings at other temperatures (not shown) have similar quality. Here, the green peak (FSDP) originates purely from LFTS, and thus, its integrated intensity should be proportional to the fraction of LFTS. This is confirmed in Fig. 4D, where we compare the integrated intensity *A* of FSDP (green area in Fig. 4C) and the fraction *s* of LFTS as a function of temperature at 1 bar in liquid silica. The fraction *s* was independently determined by a structural descriptor *z* (*25*) that measures local translational order in SiO_{2}. The integrated intensity of FSDP is an excellent measure of the fraction of LFTS or the degree of local structural ordering. Note that tetrahedral ordering is usually associated with local translational ordering and is thus linked to local density, energy, and entropy (*27*). For example, as we showed in our previous paper (*25*), the fraction *s*, which is an order parameter in the two-state model (*27*), controls the density anomaly of liquid silica through its direct link to the volume difference between LFTS and normal-liquid structures. Therefore, we may conclude that FSDP is a key structural feature controlling the thermodynamic properties of silica, implying that FSDP may serve as a general structural descriptor in the characterization of a wide class of materials with tetrahedral coordination. We find that the temperature dependence of the local structural ordering signaled by FSDP is consistent with the two-state prediction for SiO_{2}, but its relevance to other materials should be examined carefully in the future.

To summarize, we have proposed a novel interpretation of FSDP commonly observed in tetrahedral materials: FSDP is a direct consequence of the density waves characteristic to LFTS spontaneously formed in tetrahedral materials at relatively low temperature and pressure. Our tetrahedron model, which relies on neither the crystal-like structure (cluster) nor the chemical-ordered binary spheres, explains the structural origin of not only FSDP but also the other characteristic higher wave number peaks and shoulders in the structure factors of a wide class of tetrahedral materials. We stress that FSDP is a structural indicator of local structural ordering in disordered liquids and amorphous solids. We can directly estimate the degree and range of local structural ordering quantitatively from the integrated intensity and width of FSDP measured by scattering experiments without any assumption. This allows long-awaited experimental access to a link between the structure and the physical properties of disordered noncrystalline materials including thermal, mechanical, optical, and dynamic properties. We hope that our finding will greatly advance our understanding of tetrahedral materials such as SiO_{2}, GeO_{2}, Si, Ge, C, and chalcogenides, which are of significant importance in materials science, industrial applications, and geology. Last, we argue that the structural origin of FSDP reported here may be generally relevant to disordered materials with FSDP, such as metallic systems having a tendency to form icosahedral structures.

## MATERIALS AND METHODS

### Simulation methods

Molecular dynamics simulations were carried out in a cubic box containing 3456 BKS (*28*–*30*) SiO_{2} (3456 silicon and 6912 oxygen atoms) by using the LAMMPS (*31*) package. Periodic boundary conditions were applied in all directions. Intermolecular interactions were truncated at 5.5 Å, and long-range Coulomb interactions were treated by a particle-particle particle-mesh method. All simulations were performed in the constant-temperature, constant-pressure (NPT) ensemble. The temperature and pressure were kept constant by using the Nosé-Hoover thermostat and barostat, respectively. A time step of 1 fs was used in all the simulations below 6000 K. Above 6000 K, the time step was adjusted to 0.5 fs to avoid overlap of atoms within one step. More simulation details can be found in (*25*).

### Microscopic structural descriptors

In our recent study (*25*), we proposed a new structural descriptor *z* to measure the translational order in silicon’s second shell. *z* is defined for each Si atom *i* as the difference between the distance *d*_{ji} of the fifth nearest Si neighbor *j* and the distance of the fourth nearest O neighbor . By measuring the relative distance between the first and second shells of silicon, *z* characterizes the translational order of silicon’s second shell. The distribution of *z* shows a clear bimodal feature, supporting the existence of two states in liquid SiO_{2} (*25*).

The tetrahedral order in the first shell was also evaluated for a silicon atom by using the parameter *q* that is defined by (*32*, *33*)(4)where ϕ_{ij} is the angle formed by two vectors pointing from the center silicon to two of its neighboring silicon atoms. The summation runs over all the combinations of the four nearest neighbors.

## SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/5/3/eaav3194/DC1

Supplementary Text

Fig. S1. Si–Si partial structure factors at 5000 K and various pressures.

Fig. S2. Relationship between the structure factor *S*(*k*) and the tetrahedral order parameter *q* at 1 bar and 4000 K.

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is **not** for commercial advantage and provided the original work is properly cited.

## REFERENCES AND NOTES

**Acknowledgments:**

**Funding:**This study was partly supported by Scientific Research (A) and Specially Promoted Research (KAKENHI grants nos. JP18H03675 and JP25000002, respectively) from the Japan Society for the Promotion of Science (JSPS).

**Author contributions:**H.T. proposed and supervised the study. R.S. performed research. R.S and H.T. discussed the results and wrote the manuscript.

**Competing interests:**The authors declare that they have no competing interests.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

- Copyright © 2019 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution NonCommercial License 4.0 (CC BY-NC).