Mixing acoustic modes

The general conclusions probably are independent of the details. The two higher-frequency peaks, derived from (o and can be reasonably well described in a local mode approximation, but the lowest mode, derived from (Oy, is so mixed with the acoustical modes that it should be treated together with the acoustical modes in a more complete calculation. 

The calculations of TSM have recently been extended to include the effects of intermolecular forces by Tasumi and Shimanouchi 35). Estimates for the magnitude of intermolecular force constants for these calculations were obtained from the small splitting observed for higher-frequency modes. It was shown that intermolecular forces split every mode into two components belonging to different symmetry species. The acoustic modes vj and of TSM were also affected by intermolecular forces. For an isolated chain, these correspond to deformation and torsional vibrations respectively, but in crystals, they are mixed. Further, the zero and n phases of the acoustic modes predicted for an isolated chain correspond to zero frequency. In the crystal, non-zero values corresponding to rotary and translational lattice vibrations are obtained. 

The activity of the low frequency modes, of acoustic-like nature or mixed acoustical and optical nature, has a more subtle origin, since it is intrinsically related to the presence of electrical and mechanical disorder (Martin, 1974). It is the disordered structure that does not allow a complete destructive interference of the scattered fields, as it occurs in crystals, where the acoustical phonons do not contribute to the Raman scattering (Benassi, 1995). The electrical disorder is caused by the disordered space distribution of the polarizability, as in the case of heavy ions in a silicate glass (Benassi, 1991). Mechanical disorder is the deviation of the vibrational mode patterns from the plane wave shape ofphonons (Martin, 1974). In particular, a depolarized broad peak is present in the Raman spectra of all glasses, the boson peak, at frequencies in the range 20-60 cm . It corresponds to an excess in... 

Apart from inversions, there is another way to determine whether or not there is mixing in the Sun. Any spherically symmetric, localized sharp feature or discontinuity in the Sun s internal structure leaves a definite signature on the solar p-mode frequencies. Gough (1990) showed that changes of this type contribute a characteristic oscillatory component to the frequencies z/ / of those modes which penetrate below the localized perturbation. The amplitude of the oscillations increases with increasing severity of the discontinuity, and the wavelength of the oscillation is essentially the acoustic depth of the sharp-feature. Solar modes... 

Experimentally this effect has been found in dhcp Pr (Houmann et al. 1979) and in PrAlj (Purwins et al. 1976). The latter has cubic site symmetry and the ground and first excited states are Fj (OK) and 7 (27.4K). It orders ferromag-netically at = 33 K. At low temperatures (F T ) only three of the field-split F3-F4 magnetic excitons are seen (fig. 30) and those with 7+ polarization show a strong anti-crossing effect with a transverse acoustic phonon mode in the [001] direction. Detailed model calculations for this mixed-mode spectrum were performed by Aksenov et al. (1981) using an equation of motion approach. 

Fig. 30. Mixed-mode dispersions in ferromagnetic PrAlj at T = 4.4 K for the [001] direction. Full points represent experimental data from Purwins et al (1976), full curves are calculations by Aksenov et al. (1981), TA stands for transverse acoustic phonon, and and are exdton branches with different polarizations.

The velocities of these propagation modes are different. Longitudinal waves are the fastest with about twice the velocity of transverse waves. The result is that the sensor detects a rather complex waveform. In process analytical applications this situation is somewhat relaxed due to these measurements being made in what is termed a diffuse field . This arises for two reasons (1) it is impossible to resolve individual acoustic events and (2) acoustic emission waves mix due to reflections from interfaces. This means that within a small area there is no real difference in the measured acoustic emission signal, no matter where the acoustic emission sensor is mounted or its orientation. 

Phonon bands occur in the SBZ, similarly to the surface states discussed in Sect. 5.2.3. When the frequency of a surface mode corresponds to a gap in the bulk spectrum, the mode is localized at the surface and is called a surface phonon. If degeneracy with bulk modes exists, one speaks of surface resonances. Surface phonon modes are labeled Sj ( / = 1, 2, 3,...), and surface resonances by Rj when strong mixing with bulk modes is present, the phonon is labeled MSj. The lowest mode that is desired from the (bulk) acoustic band is often called the Rayleigh mode, after Lord Rayleigh, who first predicted (in 1887) the existence of surface modes at lower frequencies than in the bulk. 

Ducept et al., [13] studied mixed mode failure criteria for a glass/epoxy composite and an adhesively bonded composite joint. In their study, the initiation failure point detected by acoustic signal and by the non-linearity point on the load/displacement curve and found good correspondence. Magalhaes et al., [14] studied the application of acoustic emission to investigate the creep behavior of composite bonded lap shear joints. 

Fig. 54. The longitudinal acoustic phonon dispersion relations along the [100] axis of a- and y-Ce (Stassis 1.0 1988). The splitting of the dispersion curve for a-Ce is the result of mode mixing.

These harmonics can he detected hy tuning the scanner s receiver circuitry to the second harmonic at double the transmitted frequency so that the harmonics can be separated from the fundamental signals. However, tissues also produce harmonics, especially when higher acoustic powers are used, and distinguishing between them is challenging in practice in many of the simple contrast modes available, the two signals are inextricably mixed together. 

---