Ionic lattices, dynamic 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). 

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. 

The study of the vibrational modes of atoms in a crystal is called lattice dynamics. This study allows the formation of models for a variety of solids, including molecular solids, ionic solids, etc. The quantities of interest that can be obtained from such analysis are twofold, namely, spectroscopic and thermodynamic. 

To be able to speak of ionic superconductors, and there is a certain haziness on this subject, we need at the same time a high concerrtration of mobile charges (= 1022/cm ) and a small value of Ug (thus stabilized zirconia is not an ionic superconductor). The residence time of an ion on its site is comparable to the hopping time and the simple model given here can no longer be applied. The lattice dynamics of the mobile ions is very complex. 

In molecular crystals or in crystals composed of complex ions it is necessary to take into account intramolecular vibrations in addition to the vibrations of the molecules with respect to each other. If both modes are approximately independent, the former can be treated using the Einstein model. In the case of covalent molecules specifically, it is necessary to pay attention to internal rotations. The behaviour is especially complicated in the case of the compounds discussed in Section 2.2.6. The pure lattice vibrations are also more complex than has been described so far . In addition to (transverse and longitudinal) acoustical phonons, i.e. vibrations by which the constituents are moved coherently in the same direction without charge separation, there are so-called optical phonons. The name is based on the fact that the latter lattice vibrations are — in polar compounds — now associated with a change in the dipole moment and, hence, with optical effects. The inset to Fig. 3.1 illustrates a real phonon spectrum for a very simple ionic crystal. A detailed treatment of the lattice dynamics lies outside the scope of this book. The formal treatment of phonons (cf. e(k), D(e)) is very similar to that of crystal electrons. (Observe the similarity of the vibration equation to the Schrodinger equation.) However, they obey Bose rather than Fermi statistics (cf. page 119). 

The measurement of correlation times in molten salts and ionic liquids has recently been reviewed [11] (for more recent references refer to Carper et al. [12]). We have measured the spin-lattice relaxation rates l/Tj and nuclear Overhauser factors p in temperature ranges in and outside the extreme narrowing region for the neat ionic liquid [BMIM][PFg], in order to observe the temperature dependence of the spectral density. Subsequently, the models for the description of the reorientation-al dynamics introduced in the theoretical section (Section 4.5.3) were fitted to the experimental relaxation data. The nuclei of the aliphatic chains can be assumed to relax only through the dipolar mechanism. This is in contrast to the aromatic nuclei, which can also relax to some extent through the chemical-shift anisotropy mechanism. The latter mechanism has to be taken into account to fit the models to the experimental relaxation data (cf [1] or [3] for more details). Preliminary results are shown in Figures 4.5-1 and 4.5-2, together with the curves for the fitted functions. 

The first half of this chapter concentrates on the mechanisms of ion conduction. A basic model of ion transport is presented which contains the essential features necessary to describe conduction in the different classes of solid electrolyte. The model is based on the isolated hopping of the mobile ions in addition, brief mention is made of the influence of ion interactions between both the mobile ions and the immobile ions of the solid lattice (ion hopping) and between different mobile ions. The latter leads to either ion ordering or the formation of a more dynamic structure, the ion atmosphere. It is likely that in solid electrolytes, such ion interactions and cooperative ion movements are important and must be taken into account if a quantitative description of ionic conductivity is to be attempted. In this chapter, the emphasis is on presenting the basic elements of ion transport and comparing ionic conductivity in different classes of solid electrolyte which possess different gross structural features. Refinements of the basic model presented here are then described in Chapter 3. 

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