Elastic anomalies at terahertz frequencies and excess density of vibrational states in silica glass

Giacomo Baldi, Valentina M. Giordano, and Giulio Monaco

Phys. Rev. B 83, 174203 – Published 11 May 2011

Abstract

We study the temperature dependence of acousticlike excitations measured by means of inelastic x-ray scattering at terahertz frequencies in silica glass. The apparent sound velocity shows, between 300 and 1600 K, the same temperature variation measured, at lower frequencies, by Brillouin light scattering. On the contrary the vibrations at the boson peak (BP) present a much stronger temperature dependence, as indicated by neutron scattering data. The measured dispersion and damping are used to compute the contribution to the vibrational density of states (VDOS) coming from the propagating acousticlike modes. This part of the VDOS accounts only for a fraction of the BP intensity, indicating that other kinds of excitation accumulate in this frequency range. It is consequently not surprising that the BP does not follow the temperature evolution of the Debye frequency, which describes the modification of the continuum medium. Finally we present a comparison between the experimentally accessible quantities and a recently proposed model for the vibrations in glasses, based on the assumption of random spatial variations of the shear modulus [Schirmacher, Europhys. Lett. 73, 892 (2006)].

Received 28 November 2010

DOI:https://doi.org/10.1103/PhysRevB.83.174203

Authors & Affiliations

Giacomo Baldi^*

IPCF-CNR, UOS di Roma, I-00185 Roma, Italy and Dipartimento di Fisica, Università di Parma, I-43124, Parma, Italy

Valentina M. Giordano

LPMCN, Université Claude Bernard Lyon 1 et CNRS, UMR 5586, F-69622 Villeurbanne, France

Giulio Monaco

European Synchrotron Radiation Facility, Bôite Postale 220, F-38043 Grenoble, France

^*giacomo.baldi@cnr.it

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Vol. 83, Iss. 17 — 1 May 2011

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Figure 1
Selection of IXS spectra of vitreous silica at the indicated exchanged momenta $Q$ . Each panel shows the spectra at two temperatures: 300 K (blue squares) and 1620 K (red circles). The best-fitting functions are plotted as continuous lines (wine at 300 K and olive at 1620 K). The dashed black line is the elastic line, measured on a Plexiglass sample at 10 K.Reuse & Permissions
Figure 2
(a) $Q$ dependence of the apparent longitudinal sound velocity $v_{L}$ at the two indicated temperatures (the high-temperature data are taken from Ref. 23). The lines are fits to a phenomenological function describing the negative and positive dispersions. (b) Ratio between the damping $Γ$ and the square of the angular frequency as a function of the exchanged momentum $Q$ . The con-tinuous line (black) is the best-fitted curve to the room temperature points, while the dashed (olive) one is fitted to the high-temperature data ( $T = 1620$ K, from Ref. 23). The curve is Eq. (5), describing the crossover from a $Γ ~ Q^{4}$ law for $Q ≪ Q_{c}$ to a $Γ ~ Q^{2}$ law for $Q ≫ Q_{c}$ . The open circles (magenta) are data obtained with a tunneling-junction technique at $T = 1$ K, from Ref. 41. Inset: The same quantity plotted as a function of frequency in log-log scale.Reuse & Permissions
Figure 3
Longitudinal sound velocity as a function of temperature. Diamonds[42] and triangles[36] (black): BLS data, circles (magenta): IXS data at a wave vector $Q = 1.1$ nm $^{- 1}$ .Reuse & Permissions
Figure 4
Comparison between the IXS data from the present work (room temperature) and from a recent Letter ( $T = 1620$ K, Ref. 23) with previous studies from the literature. Left column: high temperature. Stars (dark yellow), $T = 1050$ K, Ref. 27; open squares (wine), $T = 1370$ K, Refs. 30, 32 and 33; points (red), $T = 1620$ K, Ref. 23. (a) Longitudinal sound velocity at high temperature. The line (red) indicates the BLS value at $T = 1620$ K. (b) Ratio between the sound attenuation and the wave vector squared at high temperature. Right column: room temperature data. Open circles (black), Ref. 28; full squares (blue), present work. (c) Longitudinal sound velocity at room temperature. The line (blue) indicates the BLS value at room temperature. (d) Ratio between the sound attenuation and the wave vector squared at room temperature.Reuse & Permissions
Figure 5
Temperature evolution of the reduced density of vibrational states $g (ω) / ω^{2}$ as measured with inelastic neutron scattering (Refs. 43 and 44). The solid and dashed lines are estimates of the acoustic contribution to the density of states, obtained by evaluating Eq. (6) as explained in the text (continuous black, 300 K; dashed olive, 1620 K). The horizontal lines are the Debye levels at the two temperatures (continuous black, 300 K; dashed olive, 1620 K).Reuse & Permissions
Figure 6
Apparent sound velocity, sound attenuation, and reduced excess of vibrational states plotted as a function of $ω / ω_{D}$ at the two probed temperatures. (a) shows the sound velocity divided by its BLS value $v_{BLS}$ . (b) presents the sound damping divided by the square of the frequency in units of the Debye frequency. In (c) the density of states is divided by the Debye density of states, $g_{D} (ω) = 3 ω^{2} / ω_{D}^{3}$ , at the studied temperature. The curves are the prediction of the model by Schirmacher and co-workers (Refs. 7 and 37) for the three indicated values of the disorder parameter $γ$ . The ratio $v_{L} / v_{T}$ is fixed to the room temperature BLS value and corresponds to a critical disorder parameter $γ_{c} = 0.1815$ .Reuse & Permissions

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