Complex phonon frequencies

To obtain an accurate estimate of thermodynamic properties for crystalline silica polymorphs, one needs an accurate description of the phonon density of states. Given the complexity of the problem, this is tractable only with several assumptions. For example, it is often assumed that the calculated phonon spectra are not strongly dependent on temperature. Certainly this will be a satisfactory assumption in the absence of any thermal expansion, and any changes in the interatomic potentials as a function of temperature. In this case, the force constants, and consequently the dynamical properties like phonon frequencies and density of states, will be independent of temperature. 

An overview of the current lattice dynamics methods is given in Sect. 2.2 (see also Ref [27]). In practically all cases, up to now, these methods have used semi-empirical intermolecular potentials, mostly of the atom-atom type. The parameters occurring in these potentials are usually fitted to the properties of interest, such as the lattice structure, the cohesion energy and the phonon frequencies. This procedure hides the flaws which are present in the intermolecular potentials as well as in the lattice dynamics method. In studies of solid [49-53], solid [41, 54-56] and solid [57] ab initio potentials have been used, however, which contain detailed information on the anisotropy of the potential and, in the case of O, also on its spin-dependence. Illustrative results of these studies are described in Sect. 2.3. The final Sect. 2.4, shows some typical phenomena occurring in more complex molecular crystals, such as phonon-vibron mixing, the dispersion and shifts of vibron bands and the effects of isotopic substitution, i.e. changes of nuclear masses, on the lattice- and internal vibrations. These phenomena are illustrated by results obtained on solid tetra-cyano-ethene [58] and on several chlorinated-benzene crystals [59, 60]. 

Electronic polarization through a process of transition from the lower ground states (valence band, or the mid-gap impurity states) to the upper excited states in the conduction band takes the responsibility for complex dielectrics. This process is subject to the selection rule of energy and momentum conservation, which determines the optical response of semiconductors and reflects how strongly the electrons in ground states are coupling with the excited states that shift with lattice phonon frequencies [19]. Therefore, the of a semiconductor is directly related to its bandgap Eq at zero temperature, as no lattice vibration occurs at 0 K. 

Figure 5 shows the entropy Debye temperature 0s(T) obtained as in Fig. 4 but plotted here up to the melting temperature of TiC. The rather strong temperature dependence of Qs(T) below about 300 K is due to the fact that we model the vibrational entropy of a complex phonon spectrum by a function of a single parameter (i.e., 0s). At high T and if the vibrations were strictly harmonic, 0s(T) would asymptotically approach the value 0s = 0(0). However, anhar-monic effects usually cause the vibrational frequencies to decrease with T. This is the reason... 

The phonon spectrum of a solid depends on the interatomic forces as well as on the atomic masses. For an element, the mass dependence is trivial. All phonon frequencies, and hence also all frequency moments co(n), vary with the atomic mass M as In a compound with two or several different atomic masses, the vibrational frequencies depend on the interatomic forces and on the masses in a complex way, with two exceptions. In the low-frequency part of the phonon density of states, which is uniquely given by the sound velocities, the vibrational frequencies vary as p", where p is the mass density of the solid. It follows that 0c (= 0 ) in the limit of low temperatures has interatomic forces and atomic masses separated in the form of two multiplicative factors. The force-constant part is directly related to the elastic coefficients Cy. Hence the low-temperature limit of the Debye temperature gives a certain average over the interatomic forces, as it is reflected in the sound waves. We shall now introduce another average over the interatomic forces, uniquely related to the entire phonon spectrum. 

More recently. Green s function methods have been applied to evaluate analytic expressions of phonon frequencies and widths as well as of thermodynamic properties of the linear chain [5.7-9]. These studies have shown that the complex anharmonic self-energies, that is, the shifts and widths of the phonon energies, depend on q and on the applied frequency w. The results of such calculations for three-dimensional crystals will be discussed in Sect.5.5. 

The far-infrared and visible spectra of erbium oxide have been observed by Bloor et al. (1970) about the antiferromagnetic state at 3.4 K. The complex spectra can be interpreted in terms of ions on two nonequivalent sites. The changes in the visible absorption spectrum, together with changes in phonon frequencies, are attributed to the presence of the exchange fields and a magnetostrictive expansion of the crystal lattice in the ordered state. 

Thus far we have discussed the direct mechanism of dissipation, when the reaction coordinate is coupled directly to the continuous spectrum of the bath degrees of freedom. For chemical reactions this situation is rather rare, since low-frequency acoustic phonon modes have much larger wavelengths than the size of the reaction complex, and so they cannot cause a considerable relative displacement of the reactants. The direct mechanism may play an essential role in long-distance electron transfer in dielectric media, when the reorganization energy is created by displacement of equilibrium positions of low-frequency polarization phonons. Another cause of friction may be anharmonicity of solids which leads to multiphonon processes. In particular, the Raman processes may provide small energy losses. 

Nonradiative transitions can also occur between 4/ rare-earth levels. Orbit-lattice interaction may induce these between two stark-split components of the same 4fN term or between stark-split components of different 4fN terms. One assumes that the crystal field the ion sees is modulated by the vibrations of the surrounding ions. If the spacing between the two 4fN levels is less than the Debye cut-off frequency, there will be acoustical phonons capable of inducing direct transitions between the levels. The theoretical treatment of this problem is quite complex (36). 

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