Lattice theory of ionic crystals

LATTICE THEORY OF IONIC CRYSTALS Lattice energy... 

The structural importance of the quantitative lattice theory of ionic crystals discussed above lies in the light which it throws on the stability of crystal structures and the conditions which determine the appearance of different structures in substances chemically closely related, on the types of binding which occur in different structures, and on questions of solubility. We may consider these several points separately. 

Historically the quantitative theory of ionic crystals was developed between 1918 and 1924 by Born 14), Bom and Lande 19, 20), Made-lung 88) and Haber. In particular Born devised formulae which permit the calculation of the lattice energy of an ionic crystal. The lattice energy of such a crystal may be defined as the increase in internal energy at ab-... 

The theory of ionic crystals discussed above in its application to lattice energy has also been applied to other physical properties of these crystals. We have already seen how the compressibility of a crystal may be deduced, and have shown that experimental values for this quantity... 

The mosaic structure of a crystal is intimately connected with its mechanical strength. If we consider the lattice theory of a simple ionic crystal, such as sodium chloride, it is easy to calculate the stress necessary to rupture the crystal by separating it into two halves against the forces of interionic attraction. Such calculations lead to estimates of the tensile strength which are hundreds or thousands of times greater than those actually observed. If, however, the crystal possesses a mosaic structure the mechanism of fracture will be different. The two halves of the crystal will not now be separated simultaneously at every point instead there will be local stress concentrations at which the crystal will fail, the stress concentrations will then be transferred to other points and ultimately the crystal will break in two. The process may be likened to the tearing of a sheet of paper it is not easy to sever a piece of paper by means of a uniformly applied stress, but if a tear is started the stress is concentrated at the end of the tear, failure at that point takes place and the tear is rapidly propagated across the sheet. 

About the turn of the century cuid shortly thereafter, certain developments in mathematical physics and in physical chemistry were realized which were to prove important in the theory of mass and charge transport in solids, later. Einsteinand Smoluchowski( ) initiated the modern theory of Brownian motion by idealizing it as a problem in random flights. Then some seventeen years or so later, Joffee A proposed that interstitial defects could form inside the lattice of ionic crystals and play a role in electrical conductivity. The first tenable model for ionic conductivity was proposed by Frenkel, who recognized that vaccin-cies and interstitials could form internally to account for ion movement. 

There is no fundamental theory for electro-crystallization, owing in part to the complexity of the process of lattice formation in the presence of solvent, srrrfactants, and ionic solutes. Investigations at the atomic level in parallel with smdies on nonelectrochemical crystallization wotrld be rewarding and may lead to a theory for predicting the evolution of metal morphologies, which range from dertse deposits to crystalline particles and powders. 

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. 

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. 

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. 

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