Lattice dynamics defects

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. 

The theoretical study of the lattice dynamics of the defective crystals conventionally makes use of the local vibrational density of states (LVDS) gia(ro) It is related to the diagonal element of the Green s function Guia(co) by the expression ... 

The well-known shell model was used for modeling the local atomic structure and the lattice dynamics of the defective crystal. In this model, the lattice energy is written as... 

Because charge defects will polarize other ions in the lattice, ionic polarizability must be incorporated into the potential model. The shell modeP provides a simple description of such effects and has proven to be effective in simulating the dielectric and lattice dynamical properties of ceramic oxides. It should be stressed, as argued previously, that employing such a potential model does not necessarily mean that the electron distribution corresponds to a fully ionic system, and that the general validity of the model is assessed primarily by its ability to reproduce observed crystal properties. In practice, it is found that potential models based on formal charges work well even for some scmi-covalent compounds such as silicates and zeolites. 

Lattice dynamics in bulk perovskite oxide ferroelectrics has been investigated for several decades using neutron scattering [71-77], far infrared spectroscopy [78-83], and Raman scattering. Raman spectroscopy is one of the most powerful analytical techniques for studying the lattice vibrations and other elementary excitations in solids providing important information about the stmcture, composition, strain, defects, and phase transitions. This technique was successfully applied to many ferroelectric materials, such as bulk perovskite oxides barium titanate (BaTiOs), strontium titanate (SrTiOs), lead titanate (PbTiOs) [84-88], and others. 

The atom-atom potential fitted to the ab initio data gives fairly re stic results for the equilibrium structure (unit cell parameters and molecular oriratations in the cell), the cohesion energy and the phonon frequencies of the molecular crystal. The latter have been obtained via both a harmonic and a self-consistent phonon lattice dynamics calculation and they were compared with and Raman spectra. About some of the aninncal hydrocarbon atom-atom potentials which are fitted to the crystal data, we can say that they correspond reasonably well with the ab initio results (see figs. 6, 7, 8), their main defect being an underestimate of the electrostatic multipole-multipolc interactions. 

Until recently very little was understood as to the factors which determine whether point or extended defects are formed in a non-stoicheiometric phase, although interesting empirical correlations between shear-plane formation and both dielectric and lattice dynamical properties of the defective solid had been noted. Theoretical techniques have, however, provided valuable insight into this problem and into the related one of the relative stabilities of extended and point defect structures. The role of these techniques is emphasized in this article. 

Theory of Lattice Dynamics in the Harmonic Approximation by A. A. Maradudin, E. W. Montroll, G. H. Weiss and I. P. Ipatova, Academic Press, New York New York, 1971. A formidable volume with not only a full accounting for the fundamentals of vibrations in solids, but also including issues such as the effects of defects on vibrations in solids. 

Empirical potentials are only applicable with certainty over the range of interatomic distances used in the fitting procedure, which can lead to problems if the potential is used in a calculation that accesses distances outside this range. This can happen in defect calculations, molecular dynamics simulations or lattice dynamics calculations at high temperature and/or pressure. In addition experimental data is required and thus direct calculation is the only method available when there is no relevant experimental data. It may, of course, be possible to take potentials derived for one system and transfer them to another. This method has been successful with potentials derived for binary oxides (Lewis and Catlow, 1985 Bush et al., 1994) being transferred to ternary systems (Lewis and Catlow, 1985 Price et al., 1987 Cormack et al., 1988 Purton and Catlow, 1990 Bush et al., 1994). 

The basic tools for the modeling of the solid and liquid states belong to three main categories. We can mention first the Molecular Mechanics which rely on site-site or covalent potentials and which are used to study in particular defect formations, to calculate accoustic and optical phonon modes by lattice dynamics and to estimate mechanical and thermodynamical properties. The easy implementation of the Molecular Mechanics scheme supports its intensive use in the past and its success in commercial softwares. 

In this chapter we describe the consequences of electron-phonon coupling in the absence of electron-electron interactions. The celebrated model for studying this limit is the so-called Su-Schrieffer-Heeger model (Su et al. 1979, 1980), defined in Section 2.8.2. In the absence of lattice dynamics this model is known as the Peierls model. We begin by describing the predictions of this model, namely the Peierls mechanism for bond alternation in the ground state and bond defects in the excited states. Finally, we reintroduce lattice dynamics classically and briefly describe amplitude-breathers. 

In Part VII, Greg Rutledge discusses the modeling and simulation of polymer crystals. He uses this as an excellent opportunity to introduce principles and techniques of solid-state physics useful in the study of polymers. The mathematical description of polymer helices and the calculation of X-ray diffraction patterns from crystals are explained. Both optimization (energy minimization, lattice dynamics) and sampling (MC, MD) methods for the simulation of polymer crystals are then discussed. Applications are presented from the calculation of thermal expansion, elastic coefficients, and even activation energies and rate constants for defect migration by TST methods. 

G, Zerbi, "Defect in Organic Crystal, Nimierical Methods" in "Lattice Dynamics and IntermolecifLar Porces" (S, Califano ed. Academic Press U975 ... 

The remaining features in the Fig. 8 spectrum are associated with the low frequency lattice modes in which the MnF " impurity ions moves with respect to the nearest neighbor Cs ions. The low energy modes of the defect lattice should be very close in energy to those of the perfect lattice and an analysis of the vibronic features based on the lattice dynamical model of the perfect Cs2SiFj lattice should be valid. Patterson and Lynn [6] reported for Cs2SiFg at 6 K expe-... 

Lutsko J F ef a/1989 Molecular-dynamic study of lattice-defect-nucleated melting in metals using an embedded-atom-method potential Phys. Rev. B 40 2841... 

In principle, we could find the minimum-energy crystal lattice from electronic structure calculations, determine the appropriate A-body interaction potential in the presence of lattice defects, and use molecular dynamics methods to calculate ab initio dynamic macroscale material properties. Some of the problems associated with this approach are considered by Wallace [1]. Because of these problems it is useful to establish a bridge between the micro-... 

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