Dispersion curve displacements

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... 

In Fig. 3.5 we illustrated generally that an alkalimetric or acidimetric titration curve of a hydrous oxide dispersion becomes displaced by the adsorption of a metal ion or, - in opposite direction - by the adsorption of an anion (ligand). 

Reaction Mechanisms. Bailar (2 3) studied the formation of [Co (en) 2003]" from optically active m-[Co(en)2Cl2] and found that products of opposite configurations were obtained under different conditions. He proposed a Walden type inversion for the substitution process. The configurations of the original complex and the products were related using optical rotatory dispersion curves. Dwyer (19) studied these reactions in detail and concluded that the inversion occurs through a trans displacement process involving both Ag+ and OH ... 

As frequency increases, all dispersion curves decrease to virtually zero. This indicates that EB is produced by the dipolar-orientational mechanism whereas the anisotropy of the dielectric polarizability of the macromolecules is virtually imper-ceptile in the Kerr effect. The displacement of curves towards higher frequencies with... 

Lithium hydride, LiH, is a rather simple solid there are two atoms in the primitive cell and four electrons. It has the rock salt structure and its primitive cell contains one lithium and one hydrogen atom. Fig. 4.14. There are six dispersion curves of which three are acoustic and three optical. We can compare our calculations with the measured dispersion curves from LiD (rather than LiH because the deuterium coherent scattering cross section is required to measure the relative displacements of pairs of atoms ( 2.1). 

There are three translational dispersion curves and three rotational dispersion curves. In the absence of rotation-translation coupling these vibrations are also distinctly separate. However, external modes and internal modes of the same character will become mixed, the closer they are in frequency the greater will be the mixing. Atomic displacements are no longer controlled by exclusively internal or external forces but by both. Modes of a mostly internal (or mostly external) character now involve displacements of the centre of mass (or deformations of the molecular frame). This is a general feature of all molecular crystals but it is usual to ignore such complications when the frequency of the lowest... 

Diamond has been studied by coherent INS [16]. Fig. 11.7 shows the dispersion curves calculated by periodic DFT these are in excellent with the experimental data. Fig 11.8 shows a comparison of the INS spectrum derived from the dispersion curves and the TFXA experimental spectrum. Examination of the atomic displacements shows that the spectrum can be approximately described as C-C stretching modes above 1000 cm and deformation modes below 1000 cm. The agreement is good except for the features at 154 and 331 cm. These are not a failure of the calculation but are a consequence of the use of graphite for the analysing crystal ( 3.4.2.3) and will be discussed in 11.2.2. 

Now we have two (2) phonon dispersion curves, a so-called optical branch and a lower energy acoustical branch. The standing waves are better understood in terms of the actual displacement the atoms undergo ... 

Fig. 16. Band dispersion along PM for monolayer and bilayer Xe on Al(l 11). Solid and dashed lines represent the theoretical single layer band dispersion curves [80H] displaced by layer dependent binding energy shifts [85M].

The crystal lattice vibration and the force coefficients are the subject of Chapter 12. We describe the experimental dispersion curves and conclusions that follow from their examination. The interplanar force constants are introduced. Group velocity of lattice waves is computed and discussed. It allows one to make conclusions about the interatomic bonding strength. Energy of atomic displacements during lattice vibration (that is propagation of phonons) is related to electron structure of metals. 

Figure 12.13 shows the calculated energy versus the displacement of this phonon for Mo, Nb, and Zr. The curvature agrees well with measured phonon frequencies and corresponds to the sharp dip in the phonon dispersion curves, which is a precursor to the phase transition that actually occurs in zirconium. 

The form of the REP and the polarization dispersion curve is readily derived from Eq. (142). Consider first the case of a slightly displaced totally symmetric mode. We then have Bs 1, so that we can restrict ourselves to Dj = 0 and = 1, along with v = 0 and t), = 1. Thus the excitation profile consists of four maxima corresponding to = 00, 01, 10, and 11. For small F, the 00 and 01 maxima and the 10 and 11 maxima are pairwise equal in intensity and proportional to Xg and respectively. Since... 

Fig. 39. Excitation profiles (solid curves) and depolarization dispersion curves (broken curves) of a nontotally symmetric fundamental (

The calculations underlying Fig. 51 are not based on Eq. (212) but on a slightly more elaborate equation, involving contributions of the B state as well as Franck-Condon effects due to other modes, represented by a displaced oscillator, along the lines sketched in Section VIII. These added details are needed to secure good agreement with the experimental data but do not affect the general behavior of the dispersion curves. 

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