Ionic crystals theory

Pisani [169] has used the density of states from periodic FIP (see B3.2.2.4) slab calculations to describe the host in which the cluster is embedded, where the applications have been primarily to ionic crystals such as LiE. The original calculation to derive the external Coulomb and exchange fields is usually done on a finite cluster and at a low level of ab initio theory (typically minimum basis set FIP, one electron only per atom treated explicitly). 

It is perhaps not too fanciful to compare the stormy history of liquid crystals to that of colour centres in ionic crystals resolute empiricism followed by fierce strife between rival theoretical schools, until at last a systematic theoretical approach led to understanding and then to widespread practical application. In neither of these domains would it be true to say that the empirical approach sufficed to generate practical uses such uses in fact had to await the advent of good theory. 

Of the three principal classes of crystals, ionic crystals, crystals containing electron-pair bonds (covalent crystals), and metallic crystals, we feel that a good understanding of the first class has resulted from the work done in the last few years. Interionic distances can be reliably predicted with the aid of the tables of ionic radii obtained by Goldschmidt1) by the analysis of the empirical data and by Pauling2) by a treatment based on modem theories of atomic structure. The stability,... 

The coordination theory and the principles governing coordinated structures provide the foundation for an interpretation of the structure of the complex silicates and other complex ionic crystals which may ultimately lead to the understanding of the nature and the explanation of the properties of these interesting substances. This will be achieved completely only after the investigation of the structures of many crystals with x-rays. To illustrate the clarification introduced by the new conception the following by no means exhaustive examples are discussed. 

The theoretical treatment of the properties of ionic crystals and molecules has been carried farther than that of other types of atomic aggregates. The Bom theory of crystal energy permits the calculation to within... 

By making use of the coordination theory of ionic crystals, I have predicted a structure for topaz which agrees satisfactorily with the experimental data. This structure is described in the following paragraphs. 

In solution theory the specialized distribution functions of this kind should appear in the theory of ion pairs in ionic solutions, and a form of the Bjerrum-Fuoss ionic association theory adapted to a discrete lattice is generally used for the treatment of the complexes in ionic crystals mentioned above. In fact, the above equation is not used in this treatment. Comparison of the two procedures is made in Section VI-D. 

The free valencies of a crystal can form pairs, each such pair wandering through the crystal as an entity until it breaks up. Such formations are well known in the theory of the solid state. A pair of opposite valencies in an ionic crystal (electron - - hole bound by Coulomb interaction) forms what is called a Mott exciton. A pair of like valencies (election + electron or hole + hole bound by exchange interactions) forms a so-called doublon. Such formations have recently been investigated, 12, IS). 

A fourth method has been sometimes used which depends on the assumption of particularly simple general mathematical expressions for the repulsion energies in ionic crystals (4). Such expressions have however no adequate foundation in theory or in experiment 37, 38) and will not here be considered. 

I he notation 0e indicates that this is the dielectric function at frequencies low i ompared with electronic excitation frequencies. We have also replaced co0 with l (, the frequency of the transverse optical mode in an ionic crystal microscopic theory shows that only this type of traveling wave will be readily excited bv a photon. Note that co2 in (9.20) corresponds to 01 e2/me0 for the lattice vibrations (ionic oscillators) rather than for the electrons. The mass of an electron is some thousands of times less than that of an ion thus, the plasma liequency for lattice vibrations is correspondingly reduced compared with that lor electrons. 

Fuchs, R., 1975. Theory of the optical properties of ionic crystal cubes, Phys. Rev., Bll, 1732-1740. 

The concept of bond valence, which, as will be shown below, is the same as the bond flux derived in Chapter 2, grew out of attempts to refine Pauling s principles determining the structures of complex ionic crystals (Section 1.7). In this empirical evolution of Pauling s model, both the electrostatic and short-range components were developed simultaneously. Only later did it become apparent that it was also possible to derive the properties of the electrostatic component independently using the ionic theory. 

See, e.g., Mott, N. F., and Gurney, R. W., Electronic Processes in Ionic Crystals. Oxford, New York, 1948 Seitz, F., The Modern Theory of Solids. McGraw-Hill, New York, 1940. 

Uf all the different types of atomic aggregates, ionic crystals have been found to be most suited to simple theoretical treatment. The theory of the structure of ionic crystals described briefly in the following sections was developed about 40 years ago by Born, Haber, Land6, Madelung, Ewald, Fajans, and other investigators. The simplicity of the theory is due in part to the importance in the interionic interactions of the well-understood Coulomb terms and in part to the spherical symmetry of the electron distributions of the ions with noble-gas configurations. 

In the general theory of ionic crystals (such as table salt, NaCl), a key physical quantity is the cohesive energy Zsxtal of forming the solid crystal from its constituent ions. For sodium chloride, for example, this is the energy lowering in the reaction... 

In 1937, dost presented in his book on diffusion and chemical reactions in solids [W. lost (1937)] the first overview and quantitative discussion of solid state reaction kinetics based on the Frenkel-Wagner-Sehottky point defect thermodynamics and linear transport theory. Although metallic systems were included in the discussion, the main body of this monograph was concerned with ionic crystals. There was good reason for this preferential elaboration on kinetic concepts with ionic crystals. Firstly, one can exert, forces on the structure elements of ionic crystals by the application of an electrical field. Secondly, a current of 1 mA over a duration of 1 s (= 1 mC, easy to measure, at that time) corresponds to only 1(K8 moles of transported matter in the form of ions. Seen in retrospect, it is amazing how fast the understanding of diffusion and of chemical reactions in the solid state took place after the fundamental and appropriate concepts were established at about 1930, especially in metallurgy, ceramics, and related areas. 

This process can explain the detailed events with the help of the electron theory of catalysts advocated by Wolkenstein [25]. Figure 2b shows a conceptual illustration of an ionic crystal consisting of M2+ and O2 ions, e.g., a metal oxide... 

---