Defects, in crystals

Early chemists believed that inorganic compounds obeyed the law of definite proportions under which they had invariable compositions determined by the valence of the constituent atoms. From the early part of the twentieth century views began to change when many compounds were found experimentally to be non-stoichiometric, and theoretical predictions by Wagner and Schottky demonstrated that exact stoichiometric compositions are the exception rather than the rule. The literature contains many treatments of the topic the text by D.M. Smyth [4] is recommended. 

The discussion of crystalline lattices has proceeded with the implicit assumption that every crystal is a perfect one. In reality, however, most crystals contain defects. As we have seen previously, whenever two or more gaseous ions combine to make a crystalline solid, the process is favored by enthalpy and a large amount of energy is released as the lattice enthalpy. At the same time, the process of crystallization is entropically unfavorable. If the enthalpy term is greater than the entropy term, the resulting lattice will approach that of a perfect crystal. However, whenever the entropy term is comparable in magnitude with the enthalpy of formation, the resulting solid will necessarily contain defects in its crystalline lattice. Because of the temperature dependence of the entropy term, the number of defects typically increases with temperature. 

Several different types of crystal defects can be present often, more than one type is present at the same time. A Schottky defect occurs when one or more atoms in the crystalline lattice are missing. For example, NaCI contains approximately I missing ion per 430,000 ions at 1000 K and the crystalline lattice of CrO consists of 8% vacancies. For an ionic solid, the charges need to be balanced. Thus, a number 

Two different examples of Schottky defects in a two-dimensional crystalline lattice a missing cation is balanced by a missing nearby anion, or a missing anion is balanced by another anion of similar size but more negative charge nearby. 

Sometimes an electron can become trapped in one of the vacant sites. This might happen in sodium chloride if the crystal is doped with some sodium metal. Some of the sodium metal will get oxidized into Na+ and an electron. When the electron fills a vacancy where there should have been a Cl ion, the result is that the compound can absorb light in the visible region. It is for this reason that this type of crystal defect is known as an F center (from the German word Farbe which means color ). 

A second common type of defect in crystals is known as a Frenkel defect Frenkel defects occur when one of the ions (usually the smaller ion) becomes displaced from its normal position and occupies an interstitial site in the crystalline lattice. This occurs more frequently when there is a large difference in size between the cations and the anions. For example, the Ag+ ions in AgBr usually sit in the octahedral holes formed by a face-centered cubic lattice of Br ions. However, every so often, one of these Ag+ ions might find itself displaced to one of the smaller tetrahedral holes in the lattice. In the zinc blende ionic lattice, where every other tetrahedral hole is 

Although several types of lattices have been described for ionic crystals and metals, it should be remembered that no crystal is perfect. The irregularities or defects in crystal structures are of two general types. The first type consists of defects that occur at specific sites in the lattice, and they are known as point defects. The second type of defect is a more general type that affects larger regions of the crystal. These are the extended defects or dislocations. Point defects will be discussed first. 

One type of point defect that cannot be entirely eliminated from a solid compound is the substituted ion or impurity defect. For example, suppose a large crystal contains 1 mole of NaCl that is 99.99 mole percent pure and that the 0.01% impurity is KBr. As a fraction, there is 0.0001 mole of both K+ and Br ions, which is 6.02 X 1019 ions of each type present in the 1 mole of NaCl Although the level of purity of the NaCl is high, there is an enormous number of impurity ions that occupy sites in the lattice. Even if the NaCl were 99.9999 mole percent pure, there would still be 6.02 X 1017 impurity cations and anions in a mole of crystal. In other words, there is a defect, known as a substituted ion or impurity defect, at each point in the crystal where some ion other than Na+ or Cl- resides. Because K+ is larger than Na+ and Br is larger than Cl-, the lattice will experience some strain and distortion at the sites where the larger cations and anions reside. These strain points are frequently reactive sites in a crystal. 

An analogous situation exists in crystals that are not ionic. For example, a highly pure metal might contain 99.9999 mole percent of one metal but still contain 0.0001 mole percent of another. There will be atoms of the metal impurity at specific sites in the lattice, which will constitute defects that alter the structure of the lattice slightly. 

A different type of defect occurs at specific sites when an atom or ion is missing from a lattice position and is transferred to the surface of the crystal. It is also possible for pairs of ions of opposite charge 

Although this is a small fraction, for 1 mole of lattice sites, this amounts to 5.6 X1018 Schottky defects. The ability of ions to move from their sites into vacancies and by so doing creating new vacancies is largely responsible for the conductivity in ionic crystals. 

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PIsanI C, Orlando R and Cora F 1992 On the problem of a suitable definition of the cluster In embedded-cluster treatments of defects In crystals J. Chem. Phys. 97 4195-204... 

Pisani C, Doves R and Nada R 1990 Ab initio Hartree-Fock perturbed-cluster treatment of local defects in crystals J. Chem. Phys. 92 7448... 

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

TABLE 1.2 Kroger-Vink Notation for 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... 

The ability to image defects in crystals which contain long-range elastic strains as each point on the specimen selects its own appropriate wavelength band for diffraction. 

Figure 4J Examples of extended defects in crystals edge dislocation (A) and screw dislocation (B).

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... 

B.7 CLASSIFICATION OF LINE DEFECTS IN CRYSTAL/CRYSTAL INTERFACES... 

Strictly speaking, it is correct in the case of complete particle recombination at the black sphere only partial particle reflection is discussed by Doktorov and Kotomin [50]. Incorporation of the back reactions into the kinetics of geminate recombination has been presented quite recently by [74, 75]. The effective radius for an elastic interaction of defects in crystals, (3.1.4), was calculated by Schroder [3], Kotomin and Fabrikant [76],... 

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... 

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