Frenkel defect equilibrium

The estimation of the number of Frenkel defects in a crystal can proceed along lines parallel to those for Schottky defects by estimating the configurational entropy (Supplementary Material S4). This approach confirms that Frenkel defects are thermodynamically stable intrinsic defects that cannot be removed by thermal treatment. Because of this, the defect population can be treated as a chemical equilibrium. For a crystal of composition MX, the appropriate chemical equilibrium for Frenkel defects on the cation sublattice is... 

At all temperatures above 0°K Schottky, Frenkel, and antisite point defects are present in thermodynamic equilibrium, and it will not be possible to remove them by annealing or other thermal treatments. Unfortunately, it is not possible to predict, from knowledge of crystal structure alone, which defect type will be present in any crystal. However, it is possible to say that rather close-packed compounds, such as those with the NaCl structure, tend to contain Schottky defects. The important exceptions are the silver halides. More open structures, on the other hand, will be more receptive to the presence of Frenkel defects. Semiconductor crystals are more amenable to antisite defects. 

It is important that the complete diagram displays prominently information about the assumptions made. Thus, the assumption that Schottky defect formation was preferred to the formation of electronic defects is explicitly stated in the form Ks > Ke (Fig. 7.9e). As Frenkel defect formation has been ignored altogether, it is also possible to write Ks > Ke > > Kt , where A p represents the equilibrium constant for the formation of Frenkel defects in MX. 

S4.3 Equilibrium Concentration of Frenkel Defects Derived from Configurational Entropy... 

The equilibrium thus established is a Frenkel defect. In both the Schottky and Frenkel equilibria, the stoichiometry of the crystal is unaltered (figure 4.2). Assuming that the thermodynamic activity of the various species obeys Raoult s law, thus corresponding to their molar concentrations (denoted hereafter by square brackets), the constant of the Schottky process is reduced to... 

Defect Reaction Equilibrium Constants. Recall that a Frenkel disorder is a self interstitial-vacancy pair. In terms of defect concentrations, there should be equal concentrations of vacancies and interstitials. Frenkel defects can occur with metal... 

Self-diffusion of Ag cations in the silver halides involves Frenkel defects (equal numbers of vacancies and interstitials as seen in Fig. 8.116). In a manner similar to the Schottky defects, their equilibrium population density appears in the diffusivity. Both types of sites in the Frenkel complex—vacancy and interstitial— may contribute to the diffusion. However, for AgBr, experimental data indicate that cation diffusion by the interstitialcy mechanism is dominant [4]. The cation Frenkel pair formation reaction is... 

A Frenkel defect is formed when an ion leaves its position in the lattice, leaving behind a vacancy, and moves to a nearby interstitial site (see Figure 5.21). The Frenkel [41] equilibrium is expressed as follows... 

Figure 34. Boundary equilibrium in the level diagram for Frenkel defects (i, v e.g. Agj, v Ag, z = 1) or anti-Frenkel defects (i,v e.g. Oj", Vo, z...

A population of vacancies on one subset of atoms created by displacing some atoms into normally unoccupied interstitial sites constitute a second arrangement of paired point defects, termed Frenkel defects (Figure 2(b), (c)). Because one species of atom or ion is simply being redistributed in the crystal, charge balance is not an issue. A Frenkel defect in a crystal of formula MX consists of one interstitial cation plus one cation vacancy, or one interstitial anion plus one anion vacancy. Equally, a Frenkel defect in a crystal of formula MX2 can consist of one interstitial cation plus one cation vacancy, or one interstitial anion plus one anion vacancy. As with the other point defects, it is found that the free energy of a crystal is lowered by the presence of Frenkel defects and so a popnlation of these intrinsic defects is to be expected at temperatures above 0 K. The calculation of the number of Frenkel defects in a crystal can proceed along lines parallel to those for Schottky defects. The appropriate chemical equilibrium for cation defects is ... 

According to the Frenkel mechanism, an atom moves from its regular site into the nearest interstitial position hence, two types of defect are formed in the crystal, namely the vacancy and the interstitial atom. If the Frenkel defects form in the A-sublattice of AB2 crystal, this process and the corresponding equilibrium constant can be written as ... 

As long as the crystal is in equilibrium, this expression is always valid i.e., the left-hand side of the equation will always be equal to 2.7 x 10 . Under certain conditions, discussed below, the Frenkel defects can dominate, in which case... 

The presence of a small number of Frenkel defects reduces the Gibbs energy of a crystal and so Frenkel defects are intrinsic defects. The formula for the equilibrium concentration of Frenkel defects in a crystal is similar to that for Schottky defects. There is one small difference compared with the Schottky defect equations the number of interstitial positions that are available to a displaced ion, N, need not be the same as the number of normally occupied positions, N, from which the ion moves. The number of Frenkel defects, np. present in a crystal of formula MX at equilibrium is given by ... 

The equilibrium population of point defects is another example of a pseudochemical equUibrium. For the creation of a Frenkel defect on a cation array ... 

At this point let us briefly consider the formation of associates. The formation of associates between cation vacancies and divalent impurities in alkali halides has already been given as an example. Such reactions are homogeneous solid state reactions, and so the relaxation time for the formation of associates can be calculated in a completely analogous manner to the calculation of the relaxation time for the equilibration of Frenkel defects. The result of such calculations is precisely the same as the result given in eq. (6-5). It is only necessary, in the case of association, to replace the concentration c (eq) = in the denominator by the nearly constant concentration of the corresponding majority defect. In general, in the case of the formation of defect associates, we can conclude that the equilibrium concentration is attained rapidly compared to the time required by defect reactions which occur at sites of repeatable growth. 

In stoichiometric oxides, the equilibrium concentration of the defects depends only on temperature. For Frenkel defects at equilibrium ... 

Intrinsic defects such as lattice vacancies or interstitials are present in the pure crystal at thermodynamic equilibrium. The simplest of these crystalline defects involve single or pairs of atoms or ions and are therefore known as point defects. Two main types of point defect have been identified Schottky defects,in which an atom or ion pair are missing from the lattice (Figure 3.35a), and Frenkel defects, in which an atom or ion is displaced from its ideal lattice position into an interstitial site (Figure 3.35b). 

In a similar way the number of Frenkel defects at equilibrium Up, in a binary ionic crystal... 

Sehottky and Frenkel defects are continuously created by fluctuations in the molecular energy (thermal fluctuations) and destroyed by recombination. An equilibrium concentration of defects is established, the value of the concentration being a rapidly increasing function of temperature. 

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