Lattice dynamical models

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). " ... 

Let us now come back to Badger s rule. The parameter a is expected to lie somewhere between S and 10. By fitting the experimental MS7 frequency of h(o= 19.4 meV with the lattice dynamical model described above, we find... 

A very much simplified lattice-dynamical model is that of Debye. In the Debye approximation, discussed in the following section, a single phonon branch is assumed, with frequencies proportional to the magnitude of the wavevector q. 

One of the consequences of the suppression of the phase transition is the presence of a special critical point, Tc = 0 K. This point, called the quantum displacive limit, is characterized by special critical exponents. Its presence gives rise to classical quantum crossover phenomena. Quantum suppression and the response at and near this limit, Tc = 0 K, have been extensively studied on the basis of lattice dynamic models solved within the framework of both classical and quantum statistical mechanics. Figure 8 is a log-log plot of the 6 T) results for ST018 [15]. The expectation from theory is that in the quantum regime, y = 2 at 0.7 kbar, after which y should decrease. The results in Fig. 8 quantitatively show the expected behavior however, y is < 2 at 0.70 kbar. Despite the difference in the methods to suppress Tc in ST018, the results in Fig. 4a and Fig. 8 are quite similar. As shown in the results in Fig. 3b, uniaxial pressure also can be a critical parameter S for the evolution of ferroelectricity in STO. 

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).

The actual dependence of pATsp on the temperature is rather complicated because of the dependence of the specific heat Cp on T, which is given by Debye s theory of specific heat for the reacting oxides and corresponding lattice dynamical model for crystalline solids. Simple assumptions regarding the net change in specific heats of the components involved in the dissolution reactions, however, allow one to avoid these complications [3]. 

In most cases, the crystal potential is not known a priori. The usual procedure is to introduce some model potential containing several parameters, which are subsequently found by fitting the calculated crystal properties to the observed data available. This procedure has the drawback that the empirical potential thus obtained includes the effects of the approximations made in the lattice dynamics model, which is mostly the harmonic model. It is very useful to have independent and detailed information about the potential from quantum-chemical ab initio calculations. Such information is available for nitrogen (Berns and van der Avoird, 1980) and oxygen (Wormer and van der Avoird, 1984), and we have chosen the results calculated for solid nitrogen and solid oxygen to illustrate in Sections V and VI, respectively, the lattice dynamics methods described in Sections III and IV. Nitrogen is the simplest typical molecular crystal as such it has received much attention from theorists and... 

Detailed measurement of the recoil-free fraction in germanium metal [8] has since been reinterpreted [11] using a different lattice-dynamical model. 

Because of the isomorphous structures of the four compounds and their phase instabilities, they are an interesting set of compounds for detailed lattice-dynamic calculations. However, despite their relative simplicity with respect to other metal azides, their structure, with eight atoms per primitive unit cell, presents a formidable calculational problem. With compounds of this complexity it is imperative that dispersion-curve data be available to test lattice-dynamic models, and, thus far, this has been possible only for KN3. 

Further comments should be made before discussing lattice dynamical models. Because the medium in which the phonon waves travel is discrete, there is a minimum wavelength and thus a maximum value of q in any given direction. The planes which bound that region in momentum space define the limits of q and the Brillouin zone. If the eigenfrequencies are summed over the Brillouin zone according to... 

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... 

Elastic constants measured as a function of temperature are available for most of the lanthanides in polycrystalline form (Rosen, 1967, 1968) and for Tb, Dy, Ho and Er single crystals (Palmer, 1970 Palmer and Lee, 1973 and du Plessis, 1976). For a summary of the elastic properties of the lanthanides reference can be made to Taylor and Darby (1972, section 2.4) and to ch. 8, section 9. If a suitable lattice dynamical model were devised, we should be able to calculate Cl from first principles. This was done for Gd, Dy and Er metals (Sundstrom, 1968), but at the time of these calculations, elastic constants were available only for polycrystalline samples at a few fixed temperatures. Nevertheless the results obtained did indicate that Lounasmaa s (1964a) interpolation idea was reasonable. With the elastic constant data available today it should be possible to calculate Cl for the entire region of interest, although this appears not to have attracted much attention, presumably because the uncertainty involved in separating off the contributions in experimental heat capacity results makes comparison with theory unrewarding as far as Cl is concerned. 

The recoil-free fraction / is temperature dependent, because (x ) decreases with decreasing temperature and hence forces / to increase. However,/does not reach unity even at 7=0 due to the fact that x ) 0 at 7= 0 because of the quantum-mechanical zero-point motion of the atoms (nuclei). In order to express / in terms of the usual experimental variables, the mean square amplitude (x ) is calculated using lattice dynamic models (e.g., Einstein. Debye) 16], [15],... 

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... 

It should also be mentioned that other lattice dynamical model exist which, in many respects, are equivalent to the shell model an important one is the deformation dipole model put forward by HARDY [4.47]. 

Inelastic neutron scattering data have been analyzed on the basis of a central force lattice dynamical model for CeH2 and CeHs (Vorderwisch et al., 1974). Glinka et al. (1977) have recently completed a study of the phonon dispersion relation in a single crystal of CeD2.i2. The results bear little resemblance to the predictions of the simplified model derived by Vorderwisch et al. (1974) to fit incoherent neutron scattering results. Even when the model is extended considerably, it cannot reproduce the optic mode data. Attempts are underway to derive a more satisfactory description of the dynamics of this system (Rush, 1976). 

In this section, we consider how the optical spectra of an impurity ion mixed-crystal system can be understood in terms of a lattice-dynamical model [6]. In particular, we consider the d manganese (IV) MnF ion doped in a CsjSiFg lattice. The sharp lines and detailed vibronic structure of the absorption and emission spectra of the A2g - Eg transition makes the Mn ion in cubic symmetry an ideal system for study [7]. 

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