Disorder, Frenkel type

To explore the factors that determine how far nonstoichiometry may be observed in practice, we may consider the straightforward case of an oxide, MO, in which the intrinsic lattice disorder is of Frenkel type. It will be involved in simul-... 

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. 

Clearly, Eq. (268) can be approximated by a Poole-Frenkel-type function with a coefficient a > atheor. The exact value of a varies from sample to sample dependent on the type and extent of disorder—through the disorder-dependent component (Ij, Following an example of electron injection from A1 into Alq3 id = 150 nm) (see Fig. 82b), the experimental a values can be calculated from the slopes of the PF-type plots. As predicted by (204) and (265), they should be temperature... 

In addition to the examples already mentioned we shall discuss the brominatlon of silver and Ag-Cd alloys. In all cases a parabolic rate law was observed above 200°C. Contrary to the previous example, the AgBr surface layer is an ionic conductor characterized by a Frenkel-type disorder, i.e., n g. n Ag we must write t t g B Hr 0> and consequently t becomes the rate determining factor in Bq. (27). According to Frenkel ... 

The considerations presented up to this point can be easily extended to higher ionic crystals and compounds with more than two or three components [4]. Again, quite generally, the energetically favourable defects constitute the disorder type. For a binary ionic crystal without electronic majority defects there are, in principle, only four disorder types. These are the previously described Schottky and Frenkel types and their corresponding anti-types namely, cations and an equivalent number of anions in the interstices (anti-Schottky disorder), and anion vacancies with an equal number of anions in the interstices (anti-Frenkel disorder). However, for higher ionic crystals the number of possible disorder types increases considerably because of the greater number of components and sublattices. Therefore, in such crystals, it is much more difficult to uniquely determine the disorder type. 

The variables q are molar concentrations per cm. Eq. (6-1) says that Frenkel pairs are annihilated at a rate proportional to the frequency with which vacancies and interstitial ions come together, while the rate of formation of Frenkel pairs is proportional to the concentration of lattice ions AgAg. The concentrations of defects are very small, so that the number of lattice ions Ag5 g remains constant during the reaction. Furthermore, since the condition CAgj == holds for the case of Frenkel type disorder, eq. (6-1) may be rearranged to read ... 

While intrinsic disorder of the Schottky, Frenkel, or anti-Frenkel type frequently occurs in binaiy metal oxides and metal halides, i.e., Equations (5.1), (5.3), and (5.5), Schottky disorder is seldomly encountered in temaiy compounds. However, in several studies Schottky disorder has been proposed to occur in perovskite oxides. Cation and anion vacancies or interstitials can occur in ternary compounds, but such defect stractures are usually to be related with deviations from molecularity (viz. Sections II.B.2 and II.B.3), which in fact represent extrinsic disorder and not intrinsic Schottky disorder. From Figures 5.3 and 5.4 it is apparent that deviations from molecularity always influence ionic point defect concentrations, while deviations from stoichiometry always lead to combinations of ionic and electronic point defects, as can be seen from Figures 5.2 and 5.5. 

At a given ideal composition, two or more types of defects are always present in every compound. The dominant combinations of defects depend on the type of material. The most prominent examples are named after Frenkel and Schottky. Ions or atoms leave their regular lattice sites and are displaced to an interstitial site or move to the surface simultaneously with other ions or atoms, respectively, in order to balance the charge and local composition. Silver halides show dominant Frenkel disorder, whereas alkali halides show mostly Schottky defects. 

The notion of point defects in an otherwise perfect crystal dates from the classical papers by Frenkel88 and by Schottky and Wagner.75 86 The perfect lattice is thermodynamically unstable with respect to a lattice in which a certain number of atoms are removed from normal lattice sites to the surface (vacancy disorder) or in which a certain number of atoms are transferred from the surface to interstitial positions inside the crystal (interstitial disorder). These forms of disorder can occur in many elemental solids and compounds. The formation of equal numbers of vacant lattice sites in both M and X sublattices of a compound M0Xft is called Schottky disorder. In compounds in which M and X occupy different sublattices in the perfect crystal there is also the possibility of antistructure disorder in which small numbers of M and X atoms are interchanged. These three sorts of disorder can be combined to give three hybrid types of disorder in crystalline compounds. The most important of these is Frenkel disorder, in which equal numbers of vacancies and interstitials of the same kind of atom are formed in a compound. The possibility of Schottky-antistructure disorder (in which a vacancy is formed by... 

There are three possible thermally generated intrinsic disorder reactions in ceria that do not involve exchange with the gas phase. These defects, which are of the Schottky (Eq. 2.2) and Frenkel (Eqs. 2.3 and 2.4) types, can be represented using the Kroger and Vink defect notation, which will be used throughout this chapter ... 

There are different types of formation reactions and equilibria, depending on the type of lattice and the type of defect. The types of disorders are known as Schottky, Frenkel, and anti-Frenkel,... 

Anti-Frenkel disorder similar to Frenkel disorder except that the interstitials are anions and vacancies are therefore in the anion sublattice. In Zr02 the reaction is 0 kS + 0[ and the anti-Frenkel equilibrium constant is K p = [ko ][On- This type of thermal defect is found in lattices that have a fluorite structure (CaF2, Zr02), which means that there are many large interstitial sites where the anions can be accommodated, but not the cations because their charge is larger, and they are less well screened from each other. 

The defects which disrupt the regular patterns of crystals, can be classified into point defects (zero-dimensional), line defects (1-dimensional), planar (2-dimensional) and bulk defects (3-dimensional). Point defects are imperfections of the crystal lattice having dimensions of the order of the atomic size. The formation of point defects in solids was predicted by Frenkel [40], At high temperatures, the thermal motion of atoms becomes more intensive and some of atoms obtain energies sufficient to leave their lattice sites and occupy interstitial positions. In this case, a vacancy and an interstitial atom, the so-called Frenkel pair, appear simultaneously. A way to create only vacancies has been shown later by Wagner and Schottky [41] atoms leave their lattice sites and occupy free positions on the surface or at internal imperfections of the crystal (voids, grain boundaries, dislocations). Such vacancies are often called Schottky defects (Fig. 6.3). This mechanism dominates in solids with close-packed lattices where the formation of vacancies requires considerably smaller energies than that of interstitials. In ionic compounds also there are defects of two types, Frenkel and Schottky disorder. In the first case there are equal numbers of cation vacancies... 

Fig. 3-1. Schematic representation of the disorder types in AgBr and in KCl. AgBr exhibits Frenkel disorder. KCl exhibits Schottky disorder. In both cases thermal disorder predominates.

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