Positive sound dispersion in vitreous ${\mathrm{GeO}}_2$ at high pressure

We determined the density fluctuations spectrum $\left[S\right(Q,\omega \left)\right]$ of vitreous and polycrystalline ${\text{GeO}}_2$ by inelastic x-ray scattering in the wave vector ($Q$) range between 3 and $13{\mathrm{nm}}^{-1}$ at pressures ($P$) between ambient and 7.1 GPa. The polycrystalline sample displays the expected $P$ evolution of the $Q$-dispersion relation of longitudinal acoustic (LA) modes. On the contrary, the dispersion relation of the glassy sample for $P\ge 1$ GPa shows an excess energy of the LA modes, while at ambient pressure follows the trend expected from ultrasonic and light scattering data. These findings point towards a pressure-induced positive sound dispersion, with an associated high-frequency sound velocity comparable to that of the polycrystalline sample at the same $P$.

Figure 1

Main panels: Representative $I^{\mathrm{expt}.}(Q,\omega )$ line shapes (dots) of p-${\mathrm{p}\text{−}\mathrm{GeO}}_2$ [panels (a)–(c)] and v-${\mathrm{p}\text{−}\mathrm{GeO}}_2$ [panels (d)–(f)] at the indicated $(P,Q)$ values. Red full lines are the best-fit curves [see Eq. (2)]; green dashed and blue solid lines are the $\delta \left(\omega \right)$ and the DHO terms accounting for the central peak and LA modes, respectively, while solid magenta and dashed black lines are the DHO terms accounting for the additional excitations observed in p-${\mathrm{GeO}}_2$ in the second pseudo-Brillouin zone. The logarithmic scale is used to emphasize the overall good agreement even in the spectral tails. The peaks located at $\approx 80$ Trad/s in panels (a) and (d) are inelastic excitations due to the phonons of the diamond anvils, which disperse out of the probed $\omega$ window for $Q>6.6{\mathrm{nm}}^{-1}$. The quantity ${\alpha }_i$ [see Eq. (1)] is reported on a linear scale below each main panel; the points highlighted in green correspond to $|{\alpha }_i|>{\sigma }_{\text{rms}}$.

Figure 2

Statistical distributions of ${\alpha }_i$ values for the data shown in Fig. 1 (panels have the same labels); the corresponding ${\sigma }_{\text{rms}}$ values are indicated in the individual panels. Red lines are Gaussian functions with variance equal to ${\sigma }_{\text{rms}}$.

Figure 3

Blue, red, and black lines are the $I(Q,\omega =0)$ patterns (vertically displaced for clarity) from p-${\mathrm{GeO}}_2$ at $P=1.0$, 3.0, and 7.1 GPa, respectively; the Miller indexes are reported close to the corresponding peaks. Inset: Compression curve; black dots are the volumes obtained from $I(Q,\omega =0)$, while red points connected by lines are literature data [33, 34].

Figure 4

$Q$-dispersion relations of ${\Omega }_{\text{LA}}$ (upper panel) and ${\Gamma }_{\text{LA}}$ (lower panel) in p-${\mathrm{p}\text{−}\mathrm{GeO}}_2$ at $P=$ 1.0 (blue triangles), 3.0 (red dots), and 7.1 (black squares) GPa. Lines through the ${\Omega }_{\text{LA}}$ dispersions are the best fits according to Eq. (3), while the lines in the lower panel are parabolic trends with coefficients equal to 0.1 Trad ${\mathrm{nm}}^2$/s and 0.2 Trad ${\mathrm{nm}}^2$/s; vertical lines indicate the boundary between the first and second pseudo-Brillouin zone (labeled as $1^{\mathrm{st}}$-PBZ and $2^{\mathrm{nd}}$-PBZ).

Figure 5

Blue, green, red, and black lines are the $I(Q,\omega =0)$ line shapes of v-${\mathrm{GeO}}_2$ at ambient pressure, $P=$ 1.0, 3.7, and 6.2 GPa, respectively.

Figure 6

Blue diamonds, green triangles, red circles, and black squares are the ${\Omega }_{\text{LA}}$ dispersions of v-${\mathrm{GeO}}_2$ at ambient pressure [panel (a)], $P=1.0$ GPa [panel (b)], 3.7 GPa [panel (c)], and 6.2 GPa [panel (d)]. Solid lines are linear fits to the data for $Q<8{\mathrm{nm}}^{-1}$, while dashed lines are the $c_{s}Q$ dispersions [43, 44, 45]. Open blue diamonds and the dash-dotted line in panel (a) are, respectively, INS data from Ref. [15] and the high-frequency dispersion ($c_{hf}Q$), estimated from the high frequency sound velocities observed at high $P$, back extrapolated to ambient $P$ (see Fig. 7). Panel (e) $Q$ dispersion of ${\Gamma }_{\text{LA}}$ in v-${\mathrm{GeO}}_2$ [symbols as in panels (a)–(d)]. Solid black line is a parabolic trend with coefficient 0.2 Trad ${\mathrm{nm}}^2$/s, while the dashed blue line is the parabolic trend (with coefficient 0.44 Trad ${\mathrm{nm}}^2$/s) observed at ambient pressure [46]. Vertical lines indicate the values of $Q_{\max }/2$ [colors as in panels (a)–(d)].

Figure 7

$P$ dependence of $c_{L,p}$ in p-${\mathrm{GeO}}_2$ (solid circles) and $c_{hf}$ for v-${\mathrm{GeO}}_2$ (open circles), as determined by data shown in Figs. 3 and 6; the line $\left[4.85\right(\mathrm{km}/\mathrm{s})+0.13(\mathrm{km}/\mathrm{s}\text{GPa})·P(\text{GPa}\left)\right]$ is a guide for the eye.