Vibration longitudinal

Iditional importance is that the vibrational modes are dependent upon the reciprocal e vector k. As with calculations of the electronic structure of periodic lattices these cal-ions are usually performed by selecting a suitable set of points from within the Brillouin. For periodic solids it is necessary to take this periodicity into account the effect on the id-derivative matrix is that each element x] needs to be multiplied by the phase factor k-r y). A phonon dispersion curve indicates how the phonon frequencies vary over tlie luin zone, an example being shown in Figure 5.37. The phonon density of states is ariation in the number of frequencies as a function of frequency. A purely transverse ition is one where the displacement of the atoms is perpendicular to the direction of on of the wave in a pmely longitudinal vibration tlie atomic displacements are in the ition of the wave motion. Such motions can be observed in simple systems (e.g. those contain just one or two atoms per unit cell) but for general three-dimensional lattices of the vibrations are a mixture of transverse and longitudinal motions, the exceptions... 

Longitudinal vibrations of circular rod of ntass m with large mass M on end... 

The longitudinal vibrations (Au) are parallel and the transverse vibrations (E ) perpendicular to the c-direction. 

The smaller value of i found experimentally suggests that the coupling constant g is larger than we assumed. In our model of DNA we considered hole self-trapping due to interaction with longitudinal vibrations only. As discussed in Sect. 2.1, other degrees of freedom may also contribute to the coupling constant. 

Nishinari, K. 1976. Longitudinal vibrations of high elastic gels as a method for determining viscoelastic constants. Jpn. J. Appl. Phys. 15 1263-1270. 

These are quantized vibrations of conduction electrons in a metal or semiconductor 4°). The quantized energy levels of the collective longitudinal vibrations of the electron gas are quasi-particles in the sense of the definition given above and are called plasmons. The frequency of this longitudinal vibration, the plasma frequency u>P, is given by 40>... 

Equation (2.11) is the desired equation giving frequency in terms of wave length, for longitudinal vibrations of a chain of diatomic molecules. As in Eq. (1.7), we see that the frequency depends on the quantity... 

First, let us find the number of overtone vibrations in the range dv. We follow Chap. XIV in detail and for that reason can omit a great deal of calculation. In Sec. 2 of that chapter, we found the number of overtones in the range dv, in a problem of elastic vibration, in which the velocity of longitudinal waves was Vi, that of transverse waves vt. From Eqs. (2.20) and (2.21) of that chapter, the number of overtones of longitudinal vibration in the range dv, in a container of volume F, was... 

Lateral force microscopy (LFM), where the longitudinal vibrations of the cantilever are monitored. 

The sites of hydrogen-bonded chains are rigidly fixed by the side chemical bonds in various compounds. That is why the distances between the sites and, therefore, the chain rigidity are functions of external parameters related to it— that is, the thermostat functions. In that event, longitudinal vibrations of the chain sites imposed by the matrix lead to periodical oscillations of the barrier width a between the neighboring polaron wells. If the barrier becomes thinner— that is, if a —> a — 8a—the overlap integral... 

The classic treatment of the problem of shock in pipe conduits, as developed by Joukovsky, Allievi and others, assimilates in effect the condition of the water in the pipe line during the manifestation of the phenomena in question to that of a column of air in a closed organ pipe in longitudinal vibration, the reservoir end of the line corresponding to the mouth end of the pipe and the valve end of the line to the closed end of the pipe. On this basis the theory has been developed in some detail, especially by Allievi. 

The calculations show that there are two modes with the same frequency of 1569cm one of which is Raman active and the other one is IR active. The IR active mode corresponds to longitudinal vibrations. The Raman active 1569cm mode was not obtained in our simplified calculations. This mode corresponds to a vibration of the kinks, which are directed along the chemical bonds on the ends of linear fragments. The 2100 cm mode is directed along the chain axis. 

FlG. 6.6. Polariton dispersion in the Fermi resonance region, neglecting anharmonic-ity 2 (0) is the energy of the longitudinal vibration at k = 0 min and max are the minimum and maximum energy values in the band of two-particle states. 

Appendix II. Effect of Coupling Between Longitudinal and Rotational Vibrations of HB Molecules Appendix III. Applicability of Classical Theory for Harmonic Longitudinal Vibration... 

VI. Elastic Longitudinal Vibration of a Nonrigid and Elastic Reorientation of Rigid Dipoles... 

---