Lattice vibrations dynamical models

Dynamic models for ionic lattices recognize explicitly the force constants between ions and their polarization. In shell models, the ions are represented as a shell and a core, coupled by a spring (see Refs. 57-59), and parameters are evaluated by matching bulk elastic and dielectric properties. Application of these models to the surface region has allowed calculation of surface vibrational modes [60] and LEED patterns [61-63] (see Section VIII-2). 

At the same time, many lattice dynamics models have been constructed from force-constant models or ab-initio methods. Recently, the technique of molecular dynamics (MD) simulation has been widely used" " to study vibrations, surface melting, roughening and disordering. In particular, it has been demonstrated " " " that the presence of adatoms modifies drastically the vibrational properties of surfaces. Lately, the dynamical properties of Cu adatoms on Cu(lOO) " and Cu(lll) faces have been calculated using MD simulations and a many-body potential based on the tight-binding (TB) second-moment aproximation (SMA). " ... 

Often the electronic spin states are not stationary with respect to the Mossbauer time scale but fluctuate and show transitions due to coupling to the vibrational states of the chemical environment (the lattice vibrations or phonons). The rate l/Tj of this spin-lattice relaxation depends among other variables on temperature and energy splitting (see also Appendix H). Alternatively, spin transitions can be caused by spin-spin interactions with rates 1/T2 that depend on the distance between the paramagnetic centers. In densely packed solids of inorganic compounds or concentrated solutions, the spin-spin relaxation may dominate the total spin relaxation 1/r = l/Ti + 1/+2 [104]. Whenever the relaxation time is comparable to the nuclear Larmor frequency S)A/h) or the rate of the nuclear decay ( 10 s ), the stationary solutions above do not apply and a dynamic model has to be invoked... 

In the previous sections, we have considered that the optical center is embedded in a static lattice. In our reference model center ABe (see Figure 5.1), this means that the A and B ions are fixed at equilibrium positions. However, in a real crystal, our center is part of a vibrating lattice and so the environment of A is not static but dynamic. Moreover, the A ion can participate in the possible collective modes of lattice vibrations. 

Fig. 8. Difference in the inelastic neutron scattering data between LaFe4Sb 2 and CeFe4Sb 2 vs. energy loss (Keppens et al., 1998). CeFe4Sbi2 was used as a reference compound since the neutron scattering cross section of Ce is much smaller than that of La. The difference spectra therefore reflect the vibrational density of states (DOS) associated with the La atoms. The peak at 7 meV (78 K) corresponds to the quasi-localized La mode. The second broader peak at about 15 meV corresponds to the hybridization of La and Sb vibrational modes. Both peaks can be accounted for using lattice dynamic models based on first-principles calculations (Feldman et al., 2000).

As we have seen, the expressions for the rate constant obtained for different models describing the lattice vibrations of a solid are considerably different. At the same time in a real situation the reaction rate is affected by different vibration types. In low-temperature solid-state chemical reactions one of the reactants, as a rule, differs significantly from the molecules of the medium in mass and in the value of interaction with the medium. Consequently, an active particle involved in reaction behaves as a point defect (in terms of its effect on the spectrum and vibration dynamics of a crystal lattice). Such a situation occurs, for instance, in irradiated molecular crystals where radicals (defects) are formed due to irradiation. Since a defect is one of the reactants and thus lattice regularity breakdown is within the reaction zone, the defect of a solid should be accounted for even in cases where the total number of radiation (or other) defects is small. 

In 1912 Bom and von Karman [1, 2] proposed a model for the lattice dynamics of crystals which has become the standard description of vibrations in crystals. In it the atoms are depicted as bound together by harmonic springs, and their motion is treated collectively through traveling displacement waves, or lattice vibrations, rather than by individual displacements from their equilibrium lattice sites [3]. Each wave is characterized by its frequency, wavelength (or wavevector), amplitude and polarization. 

Behavior remarkably similar to that revealed by the one-dimensional model crystals is generally observed for lattice vibrations in three dimensions. Here the dynamical matrix is constructed fundamentally in the same way, based on the model used for the interatomic forces, or derivatives of the crystal s potential energy function, and the equivalent of Eq. (7) is solved for the eigenvalues and eigenvectors [2-4, 29]. Naturally, the phonon wavevector in three dimensions is a vector with three components, q = (qx, qy, qz)> and both the fiequency of the wave, co(q), and its polarization, e q), are functions... 

There have been a limited number of theoretical Investigations on phonon frequencies and eigenvectors of the La. (Ba,Sr) CuO, [29-31] and YBa2Cu.O. - [32-35] superconductors. Unscreened lattice dynamical models [30,35], yielding only the bare phonon frequencies, gave fair agreement with experimentally determined total phonon density of states, mean square atomic vibrational amplitudes and Debye temperatures. Weber [29] has shown that the effect of... 

A number of researchers have reported [2,3,5,6,40-44] results on polymeric organotin derivatives based on reticular dynamics investigated through the Mossbauer effect and the related theory of lattice vibrations according to the Debye model. 

More recently, interest in the lattice vibrations of molecular solids has centered around the elucidation of intermolecular potential functions. If a pair potential is assumed, it can be tested by calculating the observables by application of the appropriate lattice dynamics. Dows (1962) was the first to attempt a calculation of lattice vibrational frequencies from an assumed potential. He treated solid ethylene and used a model which represented the pair interaction by repulsions between hydrogen atoms on neighboring molecules. 

The ab initio potentials used in solid nitrogen are from Refs. [31] and [32]. They have been respresented by a spherical expansion, Eq. (3), with coefficients up to = 6 and Lg = 6 inclusive, which describe the anisotropic short-range repulsion, the multipole-multipole interactions and the anisotropic dispersion interactions. They have also been fitted by a site-site model potential, Eq. (5), with force centers shifted away from the atoms, optimized for each interaction contribution. In the most advanced lattice dynamics model used, the TDH or RPA model, the libra-tions are expanded in spherical harmonics up to / = 12 and the translational vibrations in harmonic oscillator functions up to = 4, inclusive. 

For the assignment of the different vibrational peaks to certain hydrogen sites a simple lattice dynamical model was used. The frequencies of localized hydrogen vibrations in metals are obtained by solving the eigenvalue problem... 

---