Brillouin sound velocity

Condition (2) is also quite common. For instance, in crystals it results in a reduced sound velocity, v q) when q approaches a boundary of the Brillouin zone [93,96], a direct result of the periodicity of a crystal lattice. In addition, interaction between modes can lead to creation of soft mode with qi O and corresponding structural transitions [97,98]. The importance of nonlocality at fluid interfaces and the corresponding softening of surface modes has been demonstrated recently, both theoretically [99] and experimentally [100]. 

Brillouin scattering provides information about the acoustic branches of the dispersion curves of the material under study. The measured frequency shift of the radiation is equal to that of the phonon under consideration (EQN (1)), and its wave vector is deduced from EQN (2), so the sound velocity may be calculated by ... 

The agreement between fee bulk modulus deduced from Brillouin scattering measurements and fee ADX results is very good. The determination of fee elastic moduli by ultrasonics was made by fee measurement of surface acoustic wave velocities on thin films [22], The second ultrasonics experiment was made on sintered powder, by measuring fee longitudinal and transverse sound velocity at ambient and under uniaxial compression. From feat, fee bulk modulus and its pressure derivative were deduced, but this result seems to be quite imprecise. The ultrasonics experiment on thin films gives rise to a very small difference in fee bulk modulus (5%), but fee ADX or Brillouin determination should be utilised for preference. 

Keywords reentrant phase transitions, Brillouin light scattering, sound velocity, sound absorption, critical phenomena. 

During a second step we carried out measurements of the Brillouin position and Brillouin line width which are related to the sound velocity and sound absorption. We use the method of Zamir, et al. ° With our current setup, the Rayleigh line provides us with the apparatus window function from where the FSR and the finesse for each spectrum are measured. During a second step we carried out measurements of the Brillouin position and Brillouin line width which are related to the sound velocity and sound absorption. We use the method of Zamir, etal. °... 

The Brillouin frequency can be connected to the sound velocity, V by the relationship ... 

In the very high frequency range, Brillouin scattering may be used to measure the sound velocities of aerogels [53]. 

Fig. 5.3.11. Dependence of the sound velocities on the polar angle from Brillouin scattering experiments on (J-methyl butyl p((p-methoxy-benzylidene)amino) cin-namate. (a) Smectic A(T = 60.7 °C), (b) smectic B (F = 48.1 C). The dashed lines are calculated from theory. The presence of a third component in (b) indicates that the shear modulus does not vanish in smectic B at these very hi frequencies. Circles, triangles and squares represent measurements at different scattering angles. (After Liao, Clark and Pershan. )...

Thus, the choice of an incident radiation Aj defines k , whereas the choice of a scattering angle (usually 90° or 180") unambiguously determines and, henceforth, the Brillouin shifi related to each of the three acoustic modes with their own velocities. Typically, for an incident visible radiation, and a material characterized by an index of refraction about 1.5 and sound velocities of a few km/s, the Brillouin shift lies in the range of 10-30 GHz (0.33-1 cm ) in backscattering geometry. Obviously, the study of Brillouin lines requires a more consequent resoiution than the one of a conventional Raman spectrometer. 

Vibration frequencies and phonon dispersion See Figs. 20 - 23. Table 13. Perpendicular vibration frequencies /zcoi and characteristics of the phonon dispersion curves for the noble gas monolayers. The sound velocities c/ and c, were obtained from the initial slope of the dispersion curves for the longitudinal (L) and shear-horizontal (SH) modes, respectively. Where complete or partial dispersion curves are available, oidy the value at the boundary of the surface Brillouin zone is indicated. Abbreviations used F, M, K high syrtunetry points of the 2D adlayer Brillouin zone (BZ) [001], [110] and [112] crystallographic directions of the substrate surface. All data were obtained using inelastic He-atom scattering. (Ad. = adsorbate) ... 

Brillouin spectroscopy can also be used to study the change of sound velocity with deformation. Anders et al. reported the longitudinal sound velocity in stretched poly (urethane) and poly (diethylsiloxane) (PDFS, Figure 60.8) networks [15]. They used the lattice-model to determine the force constants [11]. The samples showed different deformation-dependent behavior of the force constants. For the... 

FIGURE 60.8. Sound velocity of the longitudinal polarized phonon measured parallel (filled squares) and perpendicular (empty circles) to the draw direction measured by Brillouin scattering. The samples are polydiethylsiloxanes with Mn = 45,000 and 43,000 labeled as PDES-45 and PDES-3, respectively [15]. Reprinted from S.H. Anders, H.H. Krbecek, and M. Pietralla, J. Polym. Sci., Polym. Phys. Ed. 35, 1661-1676 (1997). Copyright 1997, with permission of Wiley-VCH. 

Fig. 5. Polarized Rayleigh-Brillouin spectrum of amorphous PnHMA taken with a Burleigh plane Fabry-Perot interferometer using a free spectral range of 12.4 GHz at 295 K. The two Brillouin peaks are shifted from the incident frequency by the product of the wave vector q and the sound velocity u. The line width of the Brillouin peaks is related to the attenuation of the sound waves. PnHMA.

Alvarenga et al. [66] introduced the use of Brillouin scattering to measure the sound velocity in stretched liquid water during cooling. They reported tensions beyond —100 MPa at 20°C. The pressure was calculated from the change in sound velocity before and after cavitation, assuming a linear relation based on positive pressure data. 

As discussed in Section IV.A, a step further has been recently achieved by Davitt et al. [55], who used the acoustic method to stretch water, and measured simultaneously two physical quantities of the metastable liquid the density with the FOPH, and the sound velocity with a time-resolved Brillouin scattering experiment. They were thus able to obtain the EoS down to —26 MPa at 23.3°C. They found that the EoS is compatible with the standard extrapolation of the positive pressure data [25,26]. 

In interpreting the hypersonic data of this kind one should keep in mind the connection of sound velocity to the elastic moduli of the medium. At the high frequency involved in Brillouin scattering, the frequency dependence of these moduli is important. The Brillouin shift and linewidth depend on the real and the imaginary parts of the complex longitudinal modulus M( w ) repec-tively (11). The longitudinal modulus is connected to the compres-sional modulus K(w) and to the transversal modulus G( w ) by the equation ... 

TABLE 2. Elastic constants c. y determined at room temperature for TSHD monomer and polymer using Brillouin scattering f45j and sound velocity measurements [47], ... 

Brillouin scattering from the nematic (racemic) lyotropic polymer poly-y-benzyl glutamate (PBG) shows a significant anisotropy in both the sound velocity (22%) and the Brillouin linewidth (40%) in the solutions [22], when measurements are made at different angles with respect to the nematic director. Figure 4 shows typical Brillouin spectra of PBG as a function of scattering angle. This work indicates the importance of elastic relaxations due to concentration fluctuations in these nematic polymeric materials. 

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