Glasses, like all condensed phases, essentially behave as elastic continua for sound waves of sufficiently low frequencies. These can be described by plane waves with well defined wave vector q, related to the angular frequency Ω by Ω = vq, where v is the phase velocity of sound. These waves propagate with an energy mean free path l. Two principal mechanisms produce a finite l in insulating glasses: the relaxation of defects and the anharmonic coupling with modes of the thermal bath, as recently reanalysed in [1]. The question of interest here is what happens at frequencies sufficiently high to probe the intrinsic structural or dynamical inhomogeneities of glasses. Can these produce such a high attenuation that acoustic excitations lose their wavelike character, as suggested by observations of low-temperature thermal conductivities?

The corresponding frequencies became accessible thanks to Brillouin X-ray scattering on ID16. Figure 3 shows Brillouin linewidths, = v/l, obtained on two oxide glasses, permanently densified silica (d-SiO2) [2] and lithium diborate (LB2), Li2O-2B2O3. In both cases, a dramatic increase of , approximately 4, is observed up to the frequencies IR shown by arrows at = O/. The latter implies that l decreased to half the sound wavelength. Thus, IR corresponds to the Ioffe-Regel (IR) crossover beyond which plane waves cease to be a useful concept. One should remark that although similar onsets of crossover are observed for these two glasses at /2 in the THz range, the linewidths behave quite differently at lower frequencies. LB2 reveals a 1 dependence in Brillouin light scattering, characteristic of the thermal relaxation of defects, while d-SiO2 is dominated by anharmonicity, leading to 2 [1].

Fig. 3: Brillouin scattering linewidths observed with visible light () and X-rays (). Note that for clarity the two ordinate scales are distinct as indicated by colours. The insets show the same X-ray data on linear scales, emphasising the rapid onset and the smallness of the error bars, where blue symbols are for d-SiO2 and red symbols for lithium diborate. The lines are explained in the text.


It is well known that most glasses also exhibit an excess of modes in their reduced vibrational density of states, Z()/2, peaking at a relatively low frequency BP, called the boson peak. It is remarkable that IR~ BP for the two glasses of Figure 3. This is presumably not an accident. It suggests that the IR-crossover indeed results from the hybridisation of acoustic excitations with excess ones, as recently described in [3]. As shown there, this hybridisation should lead to 4 below IR. However, for most glasses, BP is lower than in the experimentally favourable cases of d-SiO2 and LB2. This generally leads to scattering vectors at the IR-crossover that fall near the lower limit attainable with the current spectrometers. It is thus understandable that the onset region 4 was so far not observed in other glasses. However, there is now sufficient information in the literature to extract good estimates and error bars for IR in a number of cases. These are shown in Figure 4 as a function of BP. Values of BP are available from various spectroscopy methods, which unfortunately, do not necessarily probe the full Z(). Hence, average values of BP have been used in Figure 4. It is remarkable that Figure 4 suggests a relation between IR and BP for all these various glasses, and that those glasses having a strong to intermediate excess of modes seem to obey IR~ BP, in agreement with [3].

Fig. 4: Relation between IR and BP from literature data. The points are: (1) LB2 at 573 K; (2) lithium tetraborate Li2O-4B2O3 at 600 K; (3) densified silica at 565 K; (4) vitreous silica at 1050 K; (5) glycerol at 175 K; (6) ethanol at 86 K; (7) selenium at 295 K; (8) polybutadiene at 140 K; (9) propylene carbonate at 167 K (Tg + 7 K); (10) orthoterphenyl at 156 K. The line is a guide to the eye. See the principal publication for references to the experimental data.



[1] R. Vacher, E. Courtens, and M. Foret, Phys. Rev. B 72, 214205 (2005).
[2] B. Rufflé, M. Foret, E. Courtens, R. Vacher, and G. Monaco, Phys. Rev. Lett. 90, 095502 (2003).
[3] V.L. Gurevich, D.A. Parshin, and H.R. Schober, Phys. Rev. B 67, 094203 (2003).

Principal Publication and Authors

B. Rufflé (a), G. Guimbretière (a), E. Courtens (a), R. Vacher (a) and G. Monaco (b), Phys. Rev. Lett. 96, 045502 (2006).
(a) LCVN, Université Montpellier 2 and CNRS (France)
(b) ESRF