Elastic longitudinal vibration

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

Figure 11a Evolution of the THz dielectric loss spectra stipulated by the rise of temperature (from bottom to top). Curves 1, 2,. .7 correspond to the temperatures 7i, 72,..., T7. The left column represents the same e"(v) loss spectra as shown in Fig. 10. The right column represents the loss contributions e" and e" pertinent to elastic longitudinal vibration (solid lines) and to elastic reorientation (dashed lines) of the H-bonded molecules.

VI. ELASTIC LONGITUDINAL VIBRATION OF A NONRIGID AND ELASTIC REORIENTATION OF RIGID DIPOLES14... 

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

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... 

In view of Table II the main difference of the parameters, fitted for HW, from those, fitted for OW, concerns (i) some increase of the libration amplitude / , (ii) decrease of the form factor /, (iii) decrease of the frequency vq (the center frequency of the T-band) and increase of the moment nq, responsible for this band, and (iv) decrease of the intensity factor gj, which strongly influences the THz band. Comparison of curves 3 in Figs. 4h and 5h shows that the partial dielectric loss peak g"max of HW, located at v near 150 cm-1 and stipulated by harmonic longitudinal vibration of HB molecules, substantially exceeds such a peak of OW, since the elastic dipole moment / (D20) 8.8 D exceeds the moment / (H20) 3.5 D. 

Hence, we have gotten the case (A9b), since Abreak/ is commensurable with unity. Thus the longitudinal elastic HB vibrations, arising in water, indeed have a feature of a quantum phenomenon. 

The other time, Tq, pertinent to an elastic nonrigid dipole and comprising about 0.07 ps, decreases insignificantly with T in a wide temperature interval (see Table VI). This result may be interpreted as an evidence of its quantum nature . The fitted Tg(water) is less than the mean HB lifetime11 tHb mentioned by Teixeira [16]. We should remark that our lifetime xq actually means a period, during which longitudinal vibration remains coherent. 

Figure 16 Temperature dependences of libration-absorption maximum (a) absorption-peak frequency (c) maximum of partial loss, pertinent to elastic longitudinal (solid curve) and elastic rotational (dashed curve) vibration (b) frequencies of the corresponding loss peaks (d). Vertical lines in parts a and c refer to the temperature 27.6°C.

A permissibility of similar approximation (formulated as uncoupling of longitudinal vibration and elastic reorientation) was proved in Section VI. 

We did not differentiate between the various modes of vibration (longitudinal, transversal, acoustical, optical) for the sake of simplicity. The vibrational states in a crystal are called phonons. Figure 5-2 illustrates the collective, correlated transversal vibrational motion of a linear elastic chain of particles. 

In our discussion of elastic constants we have imagined uniform distortions of the crystal. An elastic medium can sustain vibrations, which at any instant consist of nonuniform distortions. The normal modes of an elastic continuum arc sound waves (longitudinal and transverse) propagating in the medium, and these normal modes will also exist in the crystal. Indeed, viewing the crystal as an elastic... 

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