Lattice dynamics, theory

Elastomers are solids, even if they are soft. Their atoms have distinct mean positions, which enables one to use the well-established theory of solids to make some statements about their properties in the linear portion of the stress-strain relation. For example, in the theory of solids the Debye or macroscopic theory is made compatible with lattice dynamics by equating the spectral density of states calculated from either theory in the long wavelength limit. The relation between the two macroscopic parameters, Young s modulus and Poisson s ratio, and the microscopic parameters, atomic mass and force constant, is established by this procedure. The only differences between this theory and the one which may be applied to elastomers is that (i) the elastomer does not have crystallographic symmetry, and (ii) dissipation terms must be included in the equations of motion. 

H. BOttger, Principles of the Theory of Lattice Dynamics (Academie-Verlag, Berlin, 1983). 149, 176 ... 

Polarized Raman spectra from the alkali fluorides LiF, NaF and CsF habe been observed with argon laser excitations by Evans and Fitchen 09). These spectra are of interest as an extreme test of lattice dynamics theories and polarizability models. 

Vibrations in a real crystal are described by the lattice dynamical theory, discussed in section 2.1, rather than by the atomic oscillator model. Each harmonic phonon mode with branch index k and wavevector q then has, analogous to Eq. 

Whereas the first microscopic theory of BaTiOs [1,2] was based on order-disorder behavior, later on BaTiOs was considered as a classical example of displacive soft-mode transitions [3,4] which can be described by anharmonic lattice dynamics [5] (Fig. 1). BaTiOs shows three transitions at around 408 K it undergoes a paraelectric to ferroelectric transition from the cubic Pm3m to the tetragonal P4mm structure at 278 K it becomes orthorhombic, C2mm and at 183 K a transition into the rhombohedral low-temperature Rm3 phase occurs. 

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. 

After the formulation of defect thermodynamics, it is necessary to understand the nature of rate constants and transport coefficients in order to make practical use of irreversible thermodynamics in solid state kinetics. Even the individual jump of a vacancy is a complicated many-body problem involving, in principle, the lattice dynamics of the whole crystal and the coupling with the motion of all other atomic structure elements. Predictions can be made by simulations, but the relevant methods (e.g., molecular dynamics, MD, calculations) can still be applied only in very simple situations. What are the limits of linear transport theory and under what conditions do the (local) rate constants and transport coefficients cease to be functions of state When do they begin to depend not only on local thermodynamic parameters, but on driving forces (potential gradients) as well Various relaxation processes give the answer to these questions and are treated in depth later. 

A. A. Maradudin, E. W. Montroll and G. H. Weiss, Theory of Lattice Dynamics in the Harmonic Approximation, Academic, New York, 1963. 

The structural interpretation of dielectric relaxation is a difficult problem in statistical thermodynamics. It can for many materials be approached by considering dipoles of molecular size whose orientation or magnitude fluctuates spontaneously, in thermal motion. The dielectric constant of the material as a whole is arrived at by way of these fluctuations but the theory is very difficult because of the electrostatic interaction between dipoles. In some ionic crystals the analysis in terms of dipoles is less fruitful than an analysis in terms of thermal vibrations. This also is a theoretically difficult task forming part of lattice dynamics. In still other materials relaxation is due to electrical conduction over paths of limited length. Here dielectric relaxation borders on semiconductor physics. 

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]. 

The shell model has its origin in the Born theory of lattice dynamics, used in studies of the phonon dispersion curves in crystals/ Although the Born theory includes the effects of polarization at each lattice site, it does not account for the short-range interactions between sites and, most importantly, neglects the effects of this interaction potential on the polarization behavior. The shell model, however, incorporates these short-range interactions. 

Maradudin AA, Montroll EW, Weiss GH, Ipatava P (1971) Theory of lattice dynamics in the harmonic approximation. Academic Press, New York March N (1992) Electro Density Theory of atoms and molecules. Academic Press Marcott C, Havel HA, Hedlund B, Overend J, Moscowitz A (1979) A vibrational rotational strength of extraordinary intensity. Azidomethemoglobin A. In Mason SF (cd) Optical activity and chiral discrimination, Reidel, Dordrecht, p 289... 

Hamiltonians formally similar to Eq. (2.4) are encountered not only in the central problems of lattice dynamics and electron propagation, but also in a large variety of other problems. Among them we mention the Frenkel theory of excitons, the coupled electron-lattice impurities in the entire range of coupling, the Jahn-Teller (or pseudo-Jahn-Teller) systems, interacting spins, and so on. 

Savrasov, S.Y, Linear-response theory and lattice dynamics a muffin-tin-orbital approach, Phys. Rev. B, 54, 16470, 1996. 

We treat, in this chapter, mainly solid composed of water molecules such as ices and clathrate hydrates, and show recent significant contribution of simulation studies to our understanding of thermodynamic stability of those crystals in conjunction with structural morphology. Simulation technique adopted here is not limited to molecular dynamics (MD) and Monte Carlo (MC) simulations[l] but does include other method such as lattice dynamics. Electronic state as well as nucleus motion can be solved by the density functional theory[2]. Here we focus, however, our attention on the ambient condition where electronic state and character of the chemical bonds of individual molecules remain intact. Thus, we restrict ourselves to the usual simulation with intermolecular interactions given a priori. 

At normal temperatures the lattice dynamics involves predominantly low amplitude atomic motions that are well described in a harmonic approximation. Therefore, potential models widely used in the theory of molecular vibration, such as a generalized valence force field (GVFF) model, may be of use for such studies. In a GVFF the potential energy of a system is described with a set... 

Parameters (18) include vibronic coupling constant, V, and phonon band-structure factors, (/, y K j, A). For a particular crystal, finding these factors is a laborious problem of crystal lattice dynamics. Instead, in the OOA, 7y are used as free parameters of the theory. Still consistent with the fundamental theory of the JT effect, in this form the OOA is not directly derived from the theory. In other words, in the theory of cooperative JT effect, the OOA is a phenomenological approach. 

The intensity of one-phonon scattering in the high-temperature limit is given by the standard result from the theory of lattice dynamics (Dove 1993)... 

Yamada Y, Tsuneyuki S, Matsui Y (1992) Presstrre-induced phase transitiorrs in rutile-type ciystals. In Y Syono, MH Manghnani (eds) High Pressttie Research Application to Earth and Planetary Sciences. Geophys Monogr Ser 67 441-446, Am Geophys Union, Washington DC Yamamoto A (1974) Lattice-dynamical theory of structmal phase transition in quartz. J Phys Soc Japan... 

III. Harmonic and Quasi-harmonic Theories of Lattice Dynamics. 149... 

Depending on the character of the molecular motions, one can distinguish several physical situations. In most cases, the molecules are trapped in relatively deep potential wells. Then, they perform small translational and orientational oscillations around well-defined equilibrium positions and orientations. Such motions are reasonably well described by the harmonic approximation. The collective vibrational excitations of the crystal, which are considered as a set of harmonic oscillators, are called phonons. Those phonons that represent pure angular oscillations, or libra-tions, are called librons. For some properties it turns out to be necessary to look at the effects of anharmonicities. Anharmonic corrections to the harmonic model can be made by perturbation theory or by the self-consistent phonon method. These methods, which are summarized in Section III under the name quasi-harmonic theories, can be considered to be the standard tools in lattice dynamics calculations, in addition to the harmonic model. They are only applicable in the case of fairly small amplitude motions. Only the simple harmonic approximation is widely used the calculation of anharmonic corrections is often hard in practice. For detailed descriptions of these methods, we refer the reader to the books and reviews by Maradudin et al. (1968, 1971, 1974), Cochran and Cowley (1967), Barron and Klein (1974), Birman (1974), Wallace (1972), and Cali-fano et al. (1981). 

In the perturbation theory of lattice dynamics one starts from the... 

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