Splitting Brillouin

Typical Brillouin splittings are in the range 108-1010 Hz. Brillouin linewidths are in the range 166-109 Hz. Thus, a very high resolution instrument is required. Brillouin scattering is a weak effect so that an intense source of light is needed and a sensitive means of detection. All of these criteria have been met, and Brillouin scattering is now a routine tool of experimental physics (13). 

Commercial Mylar film is partially crystalline and mechanically anisotropic. The longitudinal Brillouin splitting is observed to be greater than that for quenched amorphous PET and to be different for films oriented parallel and perpendicular to the film-roll edge. Longitudinal Brillouin splittings for Mylar and for amorphous PET are plotted vs. temperature in Figure 11. The apparent value of Tg also is elevated for the Mylar film from 70° to 80 °C. 

Figure 10. Brillouin splitting Acdz vs. temperature for cyclic trimer PET crystals on the surface of Mylar film...

Figure 11. Brillouin splittings Ao)t vs. temperature for amorphous PET, Mylar film oriented parallel to the film-roll edge and perpendicular to the film edge...

Typical Brillouin splittings are in the range 10 -10 Hz. The frequency of the acoustic phonons being studied is given directly by Aw. This can be seen easily from the relation for the velocity ... 

If the density p and ratio of specific heats y are known, measurements of A(0(i) can be used to obtain /3t- The ratio of specific heats for the case where t << 10" sec can be obtained from the Rayleigh-Brillouin spectrum of the fluid. The intensity of the central peak owing to the thermal expansion divided by the intensity of the two Brillouin peaks is equal to y — 1 (6). For n-hexadecane at 120°G (shown in Figure 1), this ratio yields y = 1.227. The density is 0.7036 (7). The Brillouin splitting is measured to be 0.131 cm" The isothermal compressibility is calculated to be 1.6 X 10" cm /dyn in good agreement with the directly measured value of (7). 

Many crystallizable polymers can be prepared in the amorphous glassy state by rapid quenching as films. Measurements of Aglass transition temperature determined. Such results are shown for amorphous polyethylene terephthalate (PET) in Figure 13 (17). The Brillouin splittings change slope at 70°C. If both Aa>(i) and Awt can be measured, the Poisson ratio (T can be determined according to ... 

Figure 12, Brillouin splittings Aw d vs, temperature near the glass-rubber relaxation for PMMA, 10,000 molecular-weight PS and 2100 molecular-weight polystyrene. The arrows indicate the value of T(g) determined uHth a differential scanning calorimeter.

The basic experiment consists of measuring the spectrum of the scattered light. It consists of a strong elastic peak at one frequency with additional components whose frequency has been shifted by the inelastic scattering processes. The frequencies of these much weaker phonon peaks are measured relative to the elastic peak. From observation of the shifted Brillouin peak with respect to the central elastic peak, the longitudinal Brillouin splitting, A >i is given by... 

NiCr204 Study of magnetic hyperfine and quadrupole splitting, H(T)IH(To) = fit to Brillouin function with 5=1... 

In the left panel of Figure 8 we show the band structure calculation of graphite in the repeated zone scheme, together with a drawing of the top half of the first Brillouin zone. The band structure is for the 1 -M direction. As the dispersion is very small along the c-axis we would find a similar result if we add a constant pc component to the line along which we calculate the dispersion [17]. The main difference is that the splitting of the a 1 and % band, caused by the fact that the unit cell comprises two layers, disappears at the Brillouin zone boundary (i.e. if the plot would correspond to the A-L direction). 

Bulk silicon is a semiconductor with an indirect band structure, as schematically shown in Fig. 7.12 c. The top of the VB is located at the center of the Brillouin zone, while the CB has six minima at the equivalent (100) directions. The only allowed optical transition is a vertical transition of a photon with a subsequent electron-phonon scattering process which is needed to conserve the crystal momentum, as indicated by arrows in Fig. 7.12 c. The relevant phonon modes include transverse optical phonons (TO 56 meV), longitudinal optical phonons (LO 53.5 meV) and transverse acoustic phonons (TA 18.7 meV). At very low temperature a splitting (2.5 meV) of the main free exciton line in TO and LO replicas can be observed [Kol5]. 

Experimental data of Gibson and Sibener appears to confirm qualitatively these predictions at least for monolayers. The phonon linewidths were broadened around T up to half of the Brillouin zone. The hybridization splitting could not be resolved, but an increase of the inelastic transition probability centered around the crossing with the Rayleigh wave and extending up to 3/4 of the zone has been observed and attributed to a resonance between the adatom and substrate modes. 

Figure 14 shows the result of a Brillouin scattering experiment in the vicinity of Tc [11]. Closed circles and open circles below Tc indicate the modes split from the doubly degenerated ferroelectric soft mode. The closed circles above Tc denote the frequency of the doubly degenerated soft u mode in the paraelectric phase. The results clearly show a softening of the soft mode toward zero frequency at Tc following the Curie-Weiss law. The soft mode remains underdamped even at Tc. Generally, a soft mode is heavily damped in the vicinity of Tc, e.g., as for PbTiOs, which are typical displacive-type... 

In solid-state physics the opening of a gap at the zone boundary is usually studied in the free electron approximation, where the application of e.g., a ID weak periodic potential V, with period a [V x) = V x + a)], opens an energy gap at 7r/a (Madelung, 1978 Zangwill, 1988). E k) splits up at the Brillouin zone boundaries, where Bragg conditions are satished. Let us consider the Bloch function from Eq. (1.28) in ID expressed as a linear combination of plane waves ... 

Fig. 333 Ground-state splitting of Fe in Au determined by Mossbauer effect (Violet and Borg 1966) (a) observed (b) Brillouin function for s= 1.

Outside of a small region around the center of the Brillouin zone, (the optical region), the retarded interactions are very small. Thus the concept of coulombic exciton may be used, as well the important notions of mixure of molecular states by the crystal field and of Davydov splitting when the unit cell contains many dipoles. On the basis of coulombic excitons, we studied retarded effects in the optical region K 0, introducing the polariton, the mixed exciton-photon quasi-particle, and the transverse dielectric tensor. This allows a quantitative study of the polariton from the properties of the coulombic exciton. 

Figure 3.10. Scheme of the 2D polaritons and radiatively very unstable 2D excitons in the coupled system of an exciton K and an effective photon continuum (a) The two subsystems are not coupled, (b) The coupled system with a discrete state split off below the continuum, called the 2D polariton excitonic solutions exist only in a small segment of the Brillouin zone,... 

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