Frenkel defects stoichiometry

Thus, if Frenkel Defects predominate in a given solid, other defects are usually not present. Likewise, for the Schottky Defect. Note that this applies for associated defects. If these are not present, there will still be 2 types of defects present, each having an opposite effect upon stoichiometry. 

Intrinsic disorder is observed in conditions of perfect stoichiometry of the crystal. It is related to two main defect equilibria Schottky defects and Frenkel defects. 

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... 

Small deviation from stoichiometry. II. Imbalanced Frenkel defects... 

Schottky and Frenkel defects do not alter the stoichiometry of the material as they are intrinsic. In non-stoichiometric materials, both types of point defect occur, but ... 

In ionic lattices in which there is a significant difference in size between the cation and anion (e.g. AgBr), the smaller ion may occupy a site that is vacant in the ideal lattice. This is a Frenkel defect Figure 5.27) and does not affect the stoichiometry or electrical neutrality of the compound. 

Eigure 4.7 shows Mrowec and Przybylski s results plotted accordingly. Erom this figure it can be deduced that n varies from 3.4 at 950 °C to 3.96 at 1300 °C whereas Q, the apparent activation energy, increases from 159.6 kJ mol at P02 = 0.658 atm to 174.7 kJ mol at 6.58x 10 atm. They further showed that these variations could be explained satisfactorily only by assuming that CoO contains defects arising intrinsically, i.e., Frenkel defects, as well as defects arising extrinsically as a result of deviations from stoichiometry. 

Figure 13.4. Effect of non-stoichiometry on effective diffusion coefficient in an MO oxide where both the Schottky and Frenkel defects are considerable.

In ionic lattices in which there is a significant difference in size between the cation and anion, the smaller ion may occupy a site that is vacant in the ideal lattice. This is a Frenkel defect (a point defect) and does not affect the stoichiometry or electrical neutrality of the compound. Figure 6.28 illustrates a Frenkel defect in AgBr which adopts an NaCl structure. The radii of the Ag" " and Br i(Mis are 115 and 196 pm. The central Ag" " ion in Fig. 6.28a is in an octahedral hole with respect to the fee arrangement of Br ions. Migration of the Ag ion to a vacant tetrahedral hole (Fig. 6.28b) creates a Frenkel defect in the lattice. 

On the other hand, Frenkel defects are combinations of interstitial cations and cation vacancies. Electrical neutrality and stoichiometry are also maintained. Thus, combinations of the type of defects provide the ionic diffusion mechanism for oxide growth, but stoichiometry may not be maintained due to the electrical nature of oxides having either metal-excess or metal-deficit conditions. 

To illustrate these, let us consider two isostmctural solids, NaCl and AgCl. Both these solids adopt the fee rock salt structure (Section V), with cep Cl and Na or Ag+ in the octahedral sites. In NaCl, Schottky defects are observed, with pairs of Na and Cl ions missing from their ideal lattice sites. As equal numbers of vacancies occur in the anion and cation sublattices, overall electroneutrality and stoichiometry are preserved. In AgCl a Frenkel defect is preferred with some of the silver ions displaced from their normal octahedral sites into interstitial tetrahedral sites. This leaves the anion sublattice intact, as for every cation vacancy introduced a cation interstitial is formed. The defects in AgCl and NaCl are illirstrated schematically in Figure 3.36. 

Let us consider a crystal similar to that discussed in Sections 1,3.3 and 1.3.4, which, in this case, shows a larger deviation from stoichiometry. It is appropriate to assume that there are no interstitial atoms in this case, because the Frenkel type defect has a tendency to decrease deviation. Consider a crystal in which M occupies sites in N lattice points and X occupies sites in N lattice points. It is necessary to take the vacancy-vacancy interaction energy into consideration, because the concentration of vacancies is higher. The method of calculation of free energy (enthalpy) related to is shown in Fig. 1.12. The total free energy of the crystal may be written... 

A/iAg as a function of time with a single and spatially fixed sensor at , or one can determine D with several sensors as a function of the coordinate if at a given time [K.D. Becker, et al. (1983)]. An interesting result of such a determination of D is its dependence on non-stoichiometry. Since >Ag = DAg d (pAg/R T)/d In 3, and >Ag is constant in structurally or heavily Frenkel disordered material (<5 1), DAg(S) directly reflects the (normalized) thermodynamic factor, d(pAg/R T)/ In 3, as a function of composition, that is, the non-stoichiometry 3. From Section 2.3 we know that the thermodynamic factor of compounds is given as the derivative of a point defect titration curve in which nAg is plotted as a function of In 3. At S = 0, the thermodynamic factor has a maximum. For 0-Ag2S at T = 176 °C, one sees from the quoted diffusion measurements that at stoichiometric composition (3 = 0), the thermodynamic factor may be as large as to 102-103. 

The composition of these oxides normally departs from the precise stoichiometry, expressed in their chemical formulae. For example, in the case of a stoichiometric oxide, such as A05, where 8 = 0, we will have only thermal disorder, where the concentration of vacancies, and interstitials will be determined by the Schottky, Frenkel, and anti-Frenkel mechanisms [40-42] (these defects are explained in more detail in Chapter 5). In the case of the Schotky mechanism, the following equilibrium, described with the help of the Kroger-Vink notation, [43] develops [40]... 

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. 

In Anderson s treatment, no account is taken of changes in the electronic disorder of the compound arising from changes in the stoichiometry. In the sense of the notation used previously this is equivalent to considering the presence of only neutral defects. For a binary compound exhibiting only Frenkel disorder in the metal lattice, the defects are therefore Vm° and Mf, with no defects in chalcogenide lattice. The presentation given here is equivalent to that of Anderson, since we can write ... 

Cobalt forms two oxides, CoO and C03O4, of NaCl and spinel structures, respectively CoO is a p-type cation-deficit semiconductor through which cations and electrons migrate over cation vacancies and electron holes. In addition to the usual extrinsic defects, due to deviations from stoichiometry above 1050 °C, intrinsic Frenkel-type defects are also present. The variations of oxidation-rate constant with oxygen partial pressure and with temperature are, therefore, expected to be relatively complex. Consequently, it is important to ensure that very accurate data are obtained for the oxidation reactions, over a wide range of oxygen pressure and temperature. 

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. 

The ratio of cations to anions is not altered by the formation of either a Frenkel or a Schottky defect. If no other defects are present, the material is said to be stoichiometric, stoichiometry Stoichiometry may be defined as a state for ionic compounds wherein there is the exact... 

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