Insights into the origin of the first sharp diffraction peak in amorphous silica from an analysis of chemical and radial ordering

The structural origin of the first sharp diffraction peak (FSDP) in amorphous silica is studied by analyzing chemical and radial ordering of silicon (Si) and oxygen (O) atoms in binary amorphous networks. The study shows that the chemical order involving Si–O and O–O pairs play a major role in the formation of the FSDP in amorphous silica. This is supplemented by small contributions arising from the relatively weak Si–Si correlations in the Fourier space. A shell-by-shell analysis of the radial correlations between Si–Si, Si–O, and O–O atoms in the network reveals that the position and the intensity of the FSDP are largely determined by atomic pair correlations originating from the first two/three radial shells on a length scale of about 5–8 Å, whereas the fine structure of the intensity curve in the vicinity of the FSDP is perturbatively modified by atomic correlations arising from the radial shells beyond 8 Å. The study leads to a simple mathematical relationship between the position of the radial peaks $\left(r_k\right)$ in the partial pair-correlation functions and the diffraction peaks $\left(Q_k\right)$ that can be used to obtain approximate positions of the FSDP and the principal peak. The results are complemented by numerical calculations and an accurate semi-analytical expression for the diffraction intensity obtained from the partial pair-correlation functions of amorphous silica for a given radial shell.

Figure 1

Construction of an initial configuration of $a\text{−}{\mathrm{SiO}}_2$ from a tetrahedral model of amorphous silicon (blue) by introducing oxygen atoms (red) near the center of Si–Si bonds in the network. Each oxygen atom makes an angle $\theta$ of about 10–${25}^{\circ }$ with its nearest Si–Si bond as indicated in the diagram.

Figure 2

The neutron-weighted static structure factor of $a\text{−}{\mathrm{SiO}}_2$ obtained from averaging over two 648-atom models (blue line) and experiments (red circles) [11]. The inset shows a magnified view of the FSDP and the principal peak, at 1.5 and 2.8 Å${}^{-1}$, respectively.

Figure 3

The bond-angle distributions associated with: (a) Si–O–Si angles and (b) O–Si–O angles. The average values of Si–O–Si and O–Si–O are $141.5^{\circ }$ and $109.5^{\circ }$, respectively. The corresponding standard deviations are found to be $12.5^{\circ }$ and $3.5^{\circ }$.

Figure 4

The neutron-weighted partial reduced PCFs, $G_{ij}\left(r\right)$, for: (a) Si–Si, (b) Si–O, and (c) O–O pairs. The first four shells and their radial extent are shown in different colors. The magnitude of Si–Si pair correlations can be seen to be significantly smaller than their Si–O and O–O counterparts, by a factor of about 10.

Figure 5

Neutron-weighted total $F\left(Q\right)$ (blue) and its three partial components, $F_{\text{SiSi}}\left(Q\right), F_{\text{OO}}\left(Q\right)$ and $F_{\text{SiO}}\left(Q\right)$, originating from Si–Si (green), O–O (black), and Si–O (red) radial pair correlations, respectively. The FSDP and the principal peak of $F\left(Q\right)$ are located at 1.5 and 2.8 Å${}^{-1}$, respectively.

Figure 6

The partial structure factors $F_{ij}^{\alpha }\left(Q\right)$ obtained from the first three radial shells (for $\alpha$ = 1, 2, and 3) of the Si–O correlation function.

Figure 7

The formation of the FSDP near 1.5 Å${}^{-1}$ and a minimum near 2.8 Å${}^{-1}$ in $F_{\text{SiO}}\left(Q\right)$ originating from the first two, first three, and first four radial shells. The minimum plays an important role in the formation of the principal peak in Fig. 5.

Figure 8

The shell-by-shell contributions from the first three radial shells of $G_{\text{OO}}\left(r\right)$ to $F_{\text{OO}}\left(Q\right)$ for $\alpha$ = 1, 2, 3. The maxima associated with the principal peak appear near 2.8 Å${}^{-1}$.

Figure 9

The buildup of the FSDP and the principal peak near 1.5 and 2.8 Å${}^{-1}$, respectively, from small corrections originating from the third and fourth radial shells of O–O correlations.

Figure 10

The contribution to the FSDP and the principal peak from the first three individual radial shells of Si–Si correlations. The radial correlations from the second and third shells almost cancel each other, leaving behind a small net contribution from the first three shells.

Figure 11

The evolution of the intensity of $F_{\text{SiSi}}\left(Q\right)$ in the vicinity of the FSDP and the principal peak with an increasing number of radial shells. The plots show the contributions from the first two, first three, and first four radial shells of Si–Si correlations.

Figure 12

The partial structure factor $F_{\text{SiO}}^{\alpha }\left(Q\right)$ for the second shell ($\alpha$ = 2) obtained from exact numerical calculations (red circles) and the Gaussian approximation (blue line) using Eq. (10).

Figure 13

$F_{\text{SiO}}^{\alpha }\left(Q\right)$, for $\alpha$ = 3, obtained from exact numerical calculations (red circles) and the Gaussian approximation (blue line).

Figure 14

A comparison of $F_{\text{OO}}^{\alpha }\left(Q\right)$, obtained from exact numerical calculations (red circles), with that from the Gaussian approximation (blue line) for $\alpha =2$.

Figure 15

A comparison of the partial structure factor of O–O pairs obtained from exact numerical calculations (red circles) and that from the Gaussian approximation (blue line) for the third shell ($\alpha =3$).

Figure 16

The partial structure factor $F_{\text{SiSi}}\left(Q\right)$ for the second (red) and third (blue) shells obtained from the semi-analytical Gaussian approximation (line) and exact numerical calculations (filled circle).