Transverse acoustic

Figure 9 Factor group analysis of the symmetry of the guest vibrations and the schematic plot illustrating symmetry forhidden anti-crossing hetween localized guest vibrations and framework acoustic transverse (TA) and longimdinal (LA) vibrations.

A new chemical sensor based on surface transverse device has been developed (99) (see Sensors). It resembles a surface acoustic wave sensor with the addition of a metal grating between the tranducer and a different crystal orientation. This sensor operates at 250 mH2 and is ideally suited to measurements of surface-attached mass under fluid immersion. By immohi1i2ing atra2ine to the surface of the sensor device, the detection of atra2ine in the range of 0.06 ppb to 10 ppm was demonstrated. 

There are four acoustic modes in CNT. The lowest acoustic modes are the transverse acoustic (TA) modes, which are doubly degenerate, and have, v and y... 

When air in a room is disturbed by a person speaking the molecules of the air have movements that are along the path of the wave. If you were to draw a line from the speaker s mouth to your ear, the movement of the molecules would be along this line. This type of wave, called an acoustical wave, is said to be longitudinal. The pleasant sounds of music are produced by acoustical waves. On the other hand, destruction by a bomb blast also is caused by acoustical waves. Instead of oscillating up and down, molecules in the acoustical (or compression) wave bunch together as the wave passes. It is not a transverse wave. 

Figure 8.10 Transverse acoustic and optic modes of motion in a one-dimensional diatomic chain at the same wavelength.

Figure 8.11 (a) Dispersion curve for CuCl(s) along [110] of the cubic unit cell, (b) Density of vibrational modes [3], Here L, T, A and O denote longitudinal, transverse, acoustic and optic. Reproduced by permission of B. Hennion and The Institute of Physics. 

In three dimensions, transverse and longitudinal optic and acoustic modes result. The dispersion curve for CuCl along [100] of the cubic unit cell [3] is shown in Figure 8.11(a) as an example. The number of discrete modes with frequencies in a defined interval can be displayed as a function of the frequency. This gives what is termed the density of vibrational modes or the vibrational density of states (DoS). The vibrational DoS of CuCl is given in Figure 8.11(b). 

Note that dra(t)/dt = [H,ra]=(l/ma)[pa-qaA(ra)] and, consequently, the first term in (69) represents the kinetic energy of the system of particles in the presence of the transverse electromagnetic field. Note the analogy between this representation and the dynamical solute-solvent coupling of section 2.6 where the optical phonons are equivalent to electromagnetic photons of low frequency (the acoustical phonons are related to sound waves). 

In a homogeneous isotropic elastic medium it is possible to split acoustic waves in independent longitudinal and transverse waves, each travelling at a speed cL and cT, respectively. As 2 is greater than or equal to zero, cT is lower than or equal to cl/ J2. 

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]. 

The calculated Rayleigh mode (SJ, the lowest lying phonon branch, is in good agreement with the experimental data of Harten et al. for all three metals. Due to symmetry selection rules the shear horizontal mode just below the transverse bulk band edge can not be observed by scattering methods. The mode denoted by Sg is the anomalous acoustic phonon branch discussed above. Jayanthi et al. ascribed this anomalous soft resonance to an increased Coulomb attraction at the surface, reducing the effective ion-ion repulsion of surface atoms. The Coulomb attraction term is similar for all three metals... 

Slip is not always a purely dissipative process, and some energy can be stored at the solid-liquid interface. In the case that storage and dissipation at the interface are independent processes, a two-parameter slip model can be used. This can occur for a surface oscillating in the shear direction. Such a situation involves bulk-mode acoustic wave devices operating in liquid, which is where our interest in hydrodynamic couphng effects stems from. This type of sensor, an example of which is the transverse-shear mode acoustic wave device, the oft-quoted quartz crystal microbalance (QCM), measures changes in acoustic properties, such as resonant frequency and dissipation, in response to perturbations at the surface-liquid interface of the device. 

To model this, Duncan-Hewitt and Thompson [50] developed a four-layer model for a transverse-shear mode acoustic wave sensor with one face immersed in a liquid, comprised of a solid substrate (quartz/electrode) layer, an ordered surface-adjacent layer, a thin transition layer, and the bulk liquid layer. The ordered surface-adjacent layer was assumed to be more structured than the bulk, with a greater density and viscosity. For the transition layer, based on an expansion of the analysis of Tolstoi [3] and then Blake [12], the authors developed a model based on the nucleation of vacancies in the layer caused by shear stress in the liquid. The aim of this work was to explore the concept of graded surface and liquid properties, as well as their effect on observable boundary conditions. They calculated the hrst-order rate of deformation, as the product of the rate constant of densities and the concentration of vacancies in the liquid. 

Figure 4. Theoretical trends for —(storage) and dissipation as the inner slip is varied between no slip (0) and strong slip (1) for a coated transverse shear acoustic wave device in water. The thickness of the film is 5 nm. The solid line displays the decrease in storage, and the dashed line shows the change in dissipation.

Manson (Ref a) showed that transverse acoustic oscillations of the burned gas of the lowest permitted frequencies with none or one or two (fixed) nodal meridional planes agreed reasonably well with the observed frequencies. 

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