Crystal, defect, point theory

An extension of the kinetic theory on cases when a mechanical pressure interacts with kinetic processes inside solid volume and on interfaces has wide application interests. The elastic deformations in solid are presented from influence of external forces and from presence of internal defects of crystal structure point defects (vacancy, intersite atoms, complexes of atoms, etc.), extended defects (dislocations and inner interfaces in polycrystals), and three-dimensional defects (heterophases crystals, polycrystals). 

Since its formulation, solid state theory has been concerned also with non-strictly-periodic systems, due principally to the theoretical and technological importance of defects (point impurities, color centers, dislocations, surfaces, etc.). However, most of these theoretical studies and approaches exploit the results of the ideal periodic crystal as the basic ingredient on which to include impurity effects. 

Kantcrcvich L N 1988 An embedded-molecular-cluster method for calculating the electronic structure of point defects in ncn-metallic crystals. I. General theory J. Phys. C Solid State Phys. 21 5041... 

In the second part (applications) we discuss some recent applications of LCAO methods to calculations of various crystalline properties. We consider, as is traditional for such books the results of some recent band-structure calculations and also the ways of local properties of electronic- structure description with the use of LCAO or Wannier-type orbitals. This approach allows chemical bonds in periodic systems to be analyzed, using the well-known concepts developed for molecules (atomic charge, bond order, atomic covalency and total valency). The analysis of models used in LCAO calculations for crystals with point defects and surfaces and illustrations of their applications for actual systems demonstrate the eflSciency of LCAO approach in the solid-state theory. A brief discussion about the existing LCAO computer codes is given in Appendix C. 

Eshelby, J. D. (1954). Distortion of a Crystal by Point Imperfections. Journal of Applied Physics, Vol. 25, No. 2, (February 1954), pp. 255-261. ISSN 0021-8979 Eshelby, J. D. (1956), The continuum theory of lattice defects. Solid State Physics Advances in Research and Applications. Frederick Seitz and David Turnbull, (Ed.), Vol. 3, (1956), pp. 79-144, Elsevier, ISBN 978-0-12-374292-6, Amsterdam, the Netherlands Friedel, J. (1954). Electronic structure of primary solid solutions in metals. Advances in Physics, Vol. 3, No. 12, (October 1954), pp. 446-507, ISSN 0001-8732 Fan, G. J. Choo, H. Liaw, P. K. (2007). A new criterion for the glass-forming ability of liquids. Journal of Non-Crystalline Solids, Vol. 353, No.l, (January 2007), pp. 102-107, ISSN 0022-3093... 

Dislocation theory as a portion of the subject of solid-state physics is somewhat beyond the scope of this book, but it is desirable to examine the subject briefly in terms of its implications in surface chemistry. Perhaps the most elementary type of defect is that of an extra or interstitial atom—Frenkel defect [110]—or a missing atom or vacancy—Schottky defect [111]. Such point defects play an important role in the treatment of diffusion and electrical conductivities in solids and the solubility of a salt in the host lattice of another or different valence type [112]. Point defects have a thermodynamic basis for their existence in terms of the energy and entropy of their formation, the situation is similar to the formation of isolated holes and erratic atoms on a surface. Dislocations, on the other hand, may be viewed as an organized concentration of point defects they are lattice defects and play an important role in the mechanism of the plastic deformation of solids. Lattice defects or dislocations are not thermodynamic in the sense of the point defects their formation is intimately connected with the mechanism of nucleation and crystal growth (see Section IX-4), and they constitute an important source of surface imperfection. 

The effect of different types of comonomers on varies. VDC—MA copolymers mote closely obey Flory s melting-point depression theory than do copolymers with VC or AN. Studies have shown that, for the copolymers of VDC with MA, Flory s theory needs modification to include both lamella thickness and surface free energy (69). The VDC—VC and VDC—AN copolymers typically have severe composition drift, therefore most of the comonomer units do not belong to crystallizing chains. Hence, they neither enter the crystal as defects nor cause lamellar thickness to decrease, so the depression of the melting temperature is less than expected. 

The importance of interactions amongst point defects, at even fairly low defect concentrations, was recognized several years ago. Although one has to take into account the actual defect structure and modifications of short-range order to be able to describe the properties of solids fully, it has been found useful to represent all the processes involved in the intrinsic defect equilibria in a crystal (with a low concentration of defects), as well as its equilibrium with its external environment, by a set of coupled quasichemical reactions. These equilibrium reactions are then handled by the law of mass action. The free energy and equilibrium constants for each process can be obtained if we know the enthalpies and entropies of the reactions from theory or... 

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. 

A theoretical analysis of the experimental kinetics for Vk centres in KC1-Tl, as well as for self-trapped holes in a-Al203 and Na-salt of DNA, is presented in [55]. The fitting of theory to the experimental curves is shown in Fig. 4.4. Partial agreement of theory and experiment observed in the particular case of Vk centres was attributed to the violation of the continuous approximation in the diffusion description. This point is discussed in detail below in Section 4.3. Note in conclusion that the fact of the observation of prolonged increase in recombination intensity itself demonstrated slow mobility of defects. In the case of pure irradiated crystals, it is a strong... 

At that date, palladium hydride was regarded as a special case. Lacher s approach was subsequently developed by the author (1946) (I) and by Rees (1954) (34) into attempts to frame a general theory of the nature and existence of solid compounds. The one model starts with the idea of the crystal of a binary compound, of perfect stoichiometric composition, but with intrinsic lattice disorder —e.g., of Frenkel type. As the stoichiometry adjusts itself to higher or lower partial pressures of one or other component, by incorporating cation vacancies or interstitial cations, the relevant feature is the interaction of point defects located on adjacent sites. These interactions contribute to the partition function of the crystal and set a maximum attainable concentration of each type of defect. Conjugate with the maximum concentration of, for example, cation vacancies, Nh 9 and fixed by the intrinsic lattice disorder, is a minimum concentration of interstitials, N. The difference, Nh — Ni, measures the nonstoichiometry at the nonmetal-rich phase limit. The metal-rich limit is similarly determined by the maximum attainable concentration of interstitials. With the maximum concentrations of defects, so defined, may be compared the intrinsic disorder in the stoichiometric crystals, and from the several energies concerned there can be specified the conditions under which the stoichiometric crystal lies outside the stability limits. 

Up to now, our equations have been continuum-level descriptions of mass flow. As with the other transport properties discussed in this chapter, however, the primary objective here is to examine the microscopic, or atomistic, descriptions, a topic that is now taken up. The transport of matter through a solid is a good example of a phenomenon mediated by point defects. Diffusion is the result of a concentration gradient of solute atoms, vacancies (unoccupied lattice, or solvent atom, sites), or interstitials (atoms residing between lattice sites). An equilibrium concentration of vacancies and interstitials are introduced into a lattice by thermal vibrations, for it is known from the theory of specific heat, atoms in a crystal oscillate around their equilibrium positions. Nonequilibrium concentrations can be introduced by materials processing (e.g. rapid quenching or irradiation treatment). 

Some of the discussion of bonding theory will concern distorted crystals or crystals with defects then description in terms of bond orbitals will be essential. Description of electronic states is relatively simple for a perfect crystalline solid, as was shown for CsCl in Chapter 2 for these, use of bond orbitals is not essential and in fact, in the end, is an inconvenience. We shall nevertheless base the formulation of energy bands in crystalline solids on bond orbitals, because this formulation will be needed in other discussions at the point where matrix elements must be dealt with, we shall use the LCAO basis. The detailed discussion of bands in Chapter 6 is done by returning to the bonding and antibonding basis. 

Abstract The surfaces of model metal oxides offer many fundamental examples where the outcome of a specific chemical reaction might be linked to the surface structure and local electronic properties. In this work the reaction of simple molecules such as ammonia, alcohols, carboxylic and amino acids is studied on two metal oxide single crystals rutile TiO CllO) and (001) and fluorite UOj(l 11). Studies are conducted with XPS, TPD, and Plane Wave Density Functional Theory (DFT). The effect of surface structure is outlined by comparing the TiOj(llO) rutile surface to those of TiOjCOOl), while the effect of surface point defects is mainly discussed in the case of stoichiometric and substoichiometric UOjClll). 

In the present chapter we focus on the optical spectra of the transitional metal ion impurities, as the point defects in insulating crystals. The reasons are because the JT effect is most often encountered in the transitional metal complexes, and very common in the octahedral complexes. We shall make use of the configuration coordinate approach [12], which enables one to apply much of the theory developed for molecules to the case of an isolated impurity in a crystal. 

The theory of "impurity" or defect absorption Intensities in semiconductors has been studied by Rashba ( 1). By use of the Fredholm method, he finds that if the absorption transition occurs at k=0 and if the discrete level associated with the impurity approaches the conduction band, the intensity of the absorption line increases. The explanation offered for this intensity behavior is that the optical excitation is not localized in the impurity but encompasses a number of neighboring lattice points of the host crystal. Hence, in the absorption process, light is absorbed by the entire region of the crystal consisting of the impurity and its surroundings. 

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