Point defects in crystals

TABLE 25.1 Kroger-Vink Notation for Point Defects in Crystals... 

Thermodynamic considerations imply that all crystals must contain a certain number of defects at nonzero temperatures (0 K). Defects are important because they are much more abundant at surfaces than in bulk, and in oxides they are usually responsible for many of the catalytic and chemical properties.15 Bulk defects may be classified either as point defects or as extended defects such as line defects and planar defects. Examples of point defects in crystals are Frenkel (vacancy plus interstitial of the same type) and Schottky (balancing pairs of vacancies) types of defects. On oxide surfaces, the point defects can be cation or anion vacancies or adatoms. Measurements of the electronic structure of a variety of oxide surfaces have shown that the predominant type of defect formed when samples are heated are oxygen vacancies.16 Hence, most of the surface models of... 

Watts, R. K. (1977). Point Defects in Crystals Wiley, New York. 

As mentioned above, the non-stoichiometric compounds originate from the existence of point defects in crystals. Let us consider a crystal consisting of mono-atoms. In ideal crystals of elements, atoms occupy the lattice points regularly. In real crystals, on the other hand, various kinds of point defects can exist in thermodynamic equilibrium. First, we shall consider vacancies , which are empty regular lattice points. Consider a crystal composed of one element which has N atoms sited on regular lattice points and vacancies,... 

In retrospect, one can understand why solid state chemists, who were familiar with crystallographic concepts, found it so difficult to imagine and visualize the mobility of the atomic structure elements of a crystal. Indeed, there is no mobility of these particles in a perfect crystal, just as there is no mobility of an individual car on a densely packed parking lot. It was only after the emergence of the concept of disorder and point defects in crystals at the turn of this century, and later in the twenties and thirties when the thermodynamics of defects was understood, that the idea... 

The analytical formalism just discussed has two shortcomings first, the usage of quite particular hop length distribution and, secondly, the restriction to the steady-state properties. The Torrey model becomes inadequate for point defects in crystals, where single hop lengths A between the nearest lattice sites takes place, p(r) = <5(r - A) in equation (4.3.4). This results in the... 

Real substances often deviate from the idealized models employed in simulation studies. For instance, many complex fluids, whether natural or synthetic in origin, comprise mixtures of similar rather than identical constituents. Similarly, crystalline phases usually exhibit a finite concentration of defects that disturb the otherwise perfect crystalline order. The presence of imperfections can significantly alter phase behavior with respect to the idealized case. If one is to realize the goal of obtaining quantitatively accurate simulation data for real substances, the effects of imperfections must be incorporated. In this section we consider the state-of-the-art in dealing with two kinds of imperfection, poly-dispersity and point defects in crystals. 

Point defects in crystal lattices can be classified into two essential types (Fig. 13.58) ... 

A finitE number of point defects (e.g. vacancies, impurities) can be found in any crystalline material as the configirrational entropy term, TAS, for a low point defect concentration, outweighs the positive formation enthalpy in the free-energy expression, AG = AH — TAS. Thus, introduction of a small number of point defects into a perfect crystal gives rise to a free energy minimum, as illustrated in Figure 2.6a. Further increases in the point defect concentration, however, will raise the free energy of the system. Point defects in crystals are discussed in Sections 3.5.1 and 6.4.1. 

Frenkel and Schottky point defects in crystals defined. The effects they have on the density of a crystal... 

The two most common point defects in crystals are Frenkel defects and Schottky defects. What are these ... 

The Kroger-Vink notation is principally used to describe point defects in crystals. 

There are also applications of quantum theory for instance in the onset of a failure in a material. The failure starts on the atomic scale when an interatomic bonding is stressed beyond its yield-stress threshold and breaks. The initiation and diffusion of point defects in crystal lattice turn out to be a starting point of many failures. These events occur in a stress field at certain temperatures. The phenomena of strain, fatigue crack initiation and propagation, wear, and high-temperature creep are of particular interest The processes of nucleation and diffusion of vacancies in the crystal lattice determines the material behavior at many operation conditions. 

Color centers are simple point defects in crystal lattices, consisting of one or more electrons trapped at an ionic... 

Radiation more effectively increases the concentration of point defects than an increase in temperature. To study the effect of point defects on mechanical properties, such as strength or hardness-related features, large amounts of point defects are preferable. Therefore, radiation is useful for studying the effects of point defects in crystals and studies on the effects of point defects are done on irradiated materials. 

See, for example, d) R. K. Watts. Point Defects in Crystals. Wiley, New York, 1976, and b) R. J. D. Tilley. Defect Crystal Chemistry and Its Applications. Chapman and Hall, New York, 1987. 

The unrestricted and restricted open-sheU Hartree-Fock Methods (UHF and ROHF) for crystals use a single-determinant wavefunction of type (4.40) introduced for molecules. The differences appearing are common with those examined for the RHF LCAO method use of Bloch functions for crystalline orbitals, the dependence of the Fock matrix elements on the lattice sums over the direct lattice and the Brillouin-zone summation in the density matrix calculation. The use of one-determinant approaches is the only possibility of the first-principles wavefunction-based calculations for crystals as the many-determinant wavefunction approach (used for molecules) is practically unrealizable for the periodic systems. The UHF LCAO method allowed calculation of the bulk properties of different transition-metal compounds (oxides, perovskites) the qrstems with open shells due to the transition-metal atom. We discuss the results of these calculations in Chap. 9. The point defects in crystals in many cases form the open-sheU systems and also are interesting objects for UHF LCAO calculations (see Chap. 10). 

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