Frequency transverse-mode

Magnetic fields introduce hydromagnetic waves, which are transverse modes of ion motion and wave propagation that do not exist in the absence of an apphed B field. The first of these are Alfven, A, waves and their frequency depends on B and p, the mass density. Such waves move parallel to the apphed field having the following velocity ... 

Three characteristics of the MRD profile change when the protein is hydrated with either H2O or D2O. Both terms of Eq. (6) are required to provide an accurate fit to the data. The second or perpendicular term dominates once the transverse modes become important. The power law for the MRD profile is retained, but the exponent takes values between 0.78 and 0.5 depending on the degree of hydration. A low frequency plateau is apparent for samples containing H2O which derives from two sources the field limitation of the local proton dipolar field as mentioned above, and from limitations in the magnetization transfer rates that may be a bottleneck in bringing the liquid spins into equilibrium with the solid spins. 

Figure 8 Frequency interval between the fundamental mode and the first transverse mode of the Fabry-Perot etalon, measured as a function of the orientation of the plane of incidence of the auxiliary He-Ne laser. Similar behaviour has been observed for the dye laser radiation... 

For wave numbers in this direction, there arc also transverse modes in which the motion is perpendicular to k. It is clear from symmetry that the two modes are of the same frequency. The calculation of these modes is more complicated. At long wavelengths these modes arc describable by the elastic constant C44 and are seen therefore to have internal displacements. Thus we must introduce displacement amplitudes in the z-direction as well as in the y-direction. This is not a major... 

In the tetrahedral structure, Np/f2 = 3(.3 )/(]6d ), We may directly obtain the Debye wave number /( and the Debye frequency coo = where is the speed of sound for longitudinal waves, which may be read from Fig. 9-2. Notice that optical modes arc included as part of the Debye spectrum. E or transverse modes, the same ki> applies, but is different and there are twice as many modes at each allowed wave number. 

For longitudinal modes, we can therefore stale that the number of modes with frequency less than a> is 2 w/waY atom-pair, or a density of modes of 6w lwy, modes per atom-pair per frequency-range. Similarly, construct the spectrum for transverse modes and plot the total on the same abscissa as in Fig. 9-6 so that comparison can be made. (That histogram did not have a normalized scale on the ordinate, so you need not worry about the ordinate.) The principal discrepancies are understandable by comparison of the Debye approximation to the spectrum shown in Fig. 9-2. 

Of central importance for understanding the fundamental properties of ferroelec-trics is dynamics of the crystal lattice, which is closely related to the phenomenon of ferroelectricity [1]. The soft-mode theory of displacive ferroelectrics [65] has established the relationship between the polar optical vibrational modes and the spontaneous polarization. The lowest-frequency transverse optical phonon, called the soft mode, involves the same atomic displacements as those responsible for the appearance of spontaneous polarization, and the soft mode instability at Curie temperature causes the ferroelectric phase transition. The soft-mode behavior is also related to such properties of ferroelectric materials as high dielectric constant, large piezoelectric coefficients, and dielectric nonlinearity, which are extremely important for technological applications. The Lyddane-Sachs-Teller (LST) relation connects the macroscopic dielectric constants of a material with its microscopic properties - optical phonon frequencies ... 

Figure 9-19 Phonon dispersion relation (angular frequency vs. relative wave vector) for the three-stripe phase of CH4 on the external surface of a bundle. LI, L2, and L3 are longitudinal branches, i.e., molecular motion parallel to the groove. The dotted curve is the result for a ID adsorbate at the same density. The remaining curves correspond to the dispersion relation of transverse modes. (Adapted from Ref. [89].)...

This is an example of LO-TO splitting of the vibrational modes according to their direction of motion at the gamma point ( 2.6.1.3, 4.2.6). For any general direction the polarisation of the waves is not strictly longitudinal or transverse and the polarisation vectors are dependent on k as well as k. Similarly, the acoustic modes split into a longitudinal and two transverse modes. These modes vary in frequency along different directions in the unit cell [16]. 

The weaker the interaction between the metal and the hydrogen the more important are the inter-hydrogen forces in determining their dynamics. This leads to dispersion, a good example of which is that found in PdH, measured as its deuteride PdDo.es [58]. This dispersion is shown in Fig. 6.21 The low frequency acoustic modes, involving the Pd vibrations, have little hydrogen displacement and show only weakly in the INS spectrum of powdered PdH however, the optic modes appear strongly, see Fig. 6.22 The relatively undispersed transverse optic modes,... 

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