Intermediate range chemical ordering in amorphous and liquid water, Si, and Ge

Neutron and x-ray diffraction data for low, high, and very high density amorphous ice and liquid water, silicon, and germanium have been compared in terms of the first sharp diffraction peak in the structure factor and at the radial distribution function level. The low and high density forms of ${\mathrm{H}}_2\mathrm{O}$, Si, and Ge are shown to have very similar structures if the contributions from the hydrogen correlations in water are neglected. The very high density amorphous ice form is shown to be structurally analogous to recently reported high pressure liquid forms of Si and Ge, although there are slight differences in the way interstitial atoms or molecules are pushed into the first coordination shell.

Figure 1

Comparison of the neutron structure factor for tetrahedral glassy $\mathrm{Si}{\mathrm{O}}_2$ to the x-ray $S\left(Q\right)$'s for amorphous Si (digitized from Ref. 2) and low density amorphous (LDA) ice (Ref. 10). The data have been scaled by the nearest neighbor distance $r_1$.

Figure 2

Comparison of the wave vectors $Q_1$ of peaks in the structure factors of several glasses and liquids (which are tetrahedral at low pressure) taken from the literature, scaled by the first distance in the radial distribution function $r_1$ and the mean atomic spacing $d_S$ [where $d_S^3=(6∕\pi \rho )$ and $\rho$ is the number density]. The open star represents the limit of dense random packing (see text). $N$ denotes neutron data and $X$ represents x-ray data. $A{X}_2$ glasses $\left(N\right)$: were taken from Ref. 21, and the lightest tetrahedral glass $\mathrm{Be}{\mathrm{D}}_2$ taken from Ref. 36. Amorphous ice $\left(X\right)$: Low density amorphous (LDA) ice, high density amorphous (HDA) ice, and very high density amorphous (VHDA) ice were taken from Ref. 10. Water $\left(X\right)$: Supercooled water measured at 253 and $268\mathrm{K}$ were taken from Ref. 37, and water at 0.2 and $0.77\mathrm{GPa}$ taken from Ref. 13. Silicon: $S\left(Q\right)$ for amorphous silicon $\left(N\right)$ was taken from Ref. 2. The $S\left(Q\right)$’s for liquid silicon $\left(X\right)$ at $T=1768\mathrm{K}$ were taken from Ref. 6 and the supercooled liquid data $\left(X\right)$ at $T=1458\mathrm{K}$ taken from Ref. 30. The data for liquid Si $\left(X\right)$ at pressures of 8 and $14\mathrm{GPa}$ were taken from Ref. 8. Germanium: Amorphous Ge $\left(N\right)$ was taken from Ref. 4 and liquid Ge $\left(N\right)$ at $1273\mathrm{K}$ (Ref. 7) and Ge at $17\mathrm{GPa}\left(X\right)$ from Ref. 11.

Figure 3

Comparison of measured structure factors for very high density (VHD), high density (HD), and low density amorphous (LDA) forms of silicon ([a] solid line), germanium ([a] dashed line), amorphous ice ([b] circles), and liquid water ([b] solid line). $N$ denotes neutron data and $X$ represents x-ray data. (a) Silicon and germanium from top to bottom: VHD liquid Si $\left(X\right)$ at a pressure of $14\mathrm{GPa}$ was taken from Ref. 8 and the VHD liquid Ge $\left(X\right)$ at $17\mathrm{GPa}$ from Ref. 11. The $S\left(Q\right)$ for HD liquid Si $\left(X\right)$ at $T=1768\mathrm{K}$ was taken from Ref. 6 and HD liquid Ge $\left(N\right)$ at $1273\mathrm{K}$ from Ref. 7. The $S\left(Q\right)$ for LDA silicon $\left(N\right)$ was taken from Ref. 2. LDA Ge $\left(N\right)$ was taken from Ref. 4. (b) Amorphous ice and water $\left(X\right)$: VHDA ice, HDA ice, and LDA ice (circles) were taken from Ref. 10. These are compared to the $S\left(Q\right)$’s of VHD liquid water ($0.77\mathrm{GPa}$, top) and HD water ($0.2\mathrm{GPa}$, middle), solid lines (Ref. 13). All the curves except those for the amorphous ices were digitized from the literature.

Figure 4

Comparison of radial distribution functions for the low density amorphous forms (thick solid lines), high density liquid forms (dash-dotted lines), and very high density liquid forms (thin solid lines) for glassy and liquid water, silicon, and germanium. These curves are the Fourier transforms of those shown in Fig. 3, all rebinned with the same $Q$ spacing of $Q_{\text{step}}=0.05{\mathrm{\AA }}^{-1}$ and truncated at the same $Q_{\max }=10{\mathrm{\AA }}^{-1}$, using a Lorch modification function (Ref. 38). $r_1$ is the first peak distance in $g\left(r\right)$ for the LDA form. The $Q_{\max }$ value was restricted by the maximum momentum transfer obtained in the published high pressure diffraction data measurements.