Anti-Frenkel

In the case of the Frenkel defect, the "square" represents where the cation was supposed to reside in the lattice before it moved to its interstitial position in the cation sub-lattice. Additionally, "Anti-Frenkel" defects can exist in the anion sub-lattice. The substitutional defects axe shown as the same size as the cation or anion it displaced. Note that if they were not, the lattice structure would be disrupted from regularity at the points of ins tlon of the foreign ion. 

In this case, we use 6 as a small fraction since the actual number of defects is small in relation to the overall number of ions actually present. For the F-Center, the brackets enclose the complex consisting of an electron captured at an anion vacancy. Note that these equations encompass all of the mechanisms that we have postulated for each of the individual reactions. That is, we show the presence of vacancies in the Schottlqr case and interstitial cations for the Frenkel case involving either the cation or anion. The latter, involving an interstitlcd anion is called, by convention, the "Anti-Frenkel" case. The defect reaction involving the "F-Center" is also given. 

Anti-Frenkel Anti-Structure Vacancy- AntiStructure Xi Vx + Mx Vm -h Mx... 

CeHot. From thermodsmamic measurements, it was found that the intrinsic defects were Anti-Frenkel in nature, i.e.- (H + Vh). An equilibrium constant was calculated as ... 

Schottky Frenkel Anti-Frenkel Anti-Structure Vacancy-Structure. 

Given ksh = 3 x 10-3 for CaQ2, calculate the number of intrinsic defects present in this crystal. If CaCl2 is face-centered cubic, use the same equilibrium constant to calculate the intrinsic Frenkel, Anti-Frenkel and Interstial defects expected in this crystal. 

We now proceed as we did for the stoichiometric case, namely to develop defect- concentration equations for the non-stoichiometric case. Consider the effect of Anti-Frenkel defect production. From Table 2-1, we get Kaf with its associated equation, kAF In Table 2-2, we use Kxi for X-interstitial sites. Combining these, we get ... 

Frenkel type in which metal atoms on regular sites move to interstitial sites, leaving metal vacancies, i.e. (M = MJ (Fig. 1.9(b)). Anti-Frenkel type defects, in which anion atoms on regular sites move to interstitial sites, are also possible, but are rarely observed because the ionic radii of anions are usually larger than those of the metals under consideration. Frenkel type is stoichiometric. 

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]... 

In Section 2.3, the structure of oxides were studied. For a stoichiometric oxide, such as AOs, where 5 = 0, we will have thermal disorder, and the concentration of vacancies and interstitials will be determined by the Frenkel, anti-Frenkel [40,41], and Schottky [40,42] mechanisms (see Figures 5.21 through 5.23). 

In particular if the anions are as small as F", e.g. in CaF2) we can have Frenkel disorder in the anion sublattice, which is also referred to as anti-Frenkel-disorder ( F ),... 

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...

Let us consider an anti-Frenkel disordered material, taking into account both O and v , but initially neglecting the occurrence of variable valence states. (In addition, we will assume a quasi one-dimensional situation.) Doing this we reduce the problem to a relatively trivial case. For it is immediately evident that the source terms disappear on consideration of the total ion flux densityj (or current density i)... 

For the purpose of simplicity, the vacancies are taken to be neutral and we do not consider the Frenkel or anti-Frenkel processes. In both cases, extending the analysis does not alter the important parts of the results. From Eqs. 5-7, it can be shown that ... 

Anti- Frenkel, Anti-Structure, Vacancy-Structure. 

Of the nine defect-pairs possible, only 5 have actually been experimentally observed in solids. These are Schottky, Frenkel, Anti- Frenkel, Anti-Structure, Vacancy-Structure. 

Anti-Frenkel Disorder Equal concentrations of anion vacancies and anion interstitials... 

Anti-Frenkel defects Vacancies in the anion lattice are compensated by anions on interstitials (Figure 1.16b). 

Figure 1.16 (a) Frenkel defect, vacancies in the cation lattice are compensated by cations on interstitials (b) anti-Frenkel defect, vacancies in the anion lattice are compensated by anions on interstitials (c) Schottky defect, vacancies in the cation lattice are compensated by vacancies in the anion lattice and (d) anti-Schottky defect, cations on interstitials are compensated by anions on interstitials. 

Fig. 4.2. (a) Schematic representation of electrolytic domain, i.e. relative electronic (for instance n and p type for ZrOj Ca) and ionic conductivity as a function of partial pressure pXj of the more volatile element (e.g. Oj or Ij). Dotted zone corresponds to a mixed conduction domain where the ionic transport number (tj) goes from 0 to 1 (with permission). The Agl area is limited by the a-p transition and by melting on the low and high temperature sides, respectively, (b) Schematic defect structure of an oxide M2O3 as a function of the water pressure. The oxide is dominated by anti-Frenkel defects and protons ([H ]) and doped with cations, concentration of which is assumed to be constant ([MI ]). Metal vacancies are shown as examples of minority defects. 

Defect pairs Schottky pairs of anion and cation vacancies Frenkel pairs of a cation vacancy and the same cation as an interstitial Anti-Frenkel the same as Frenkel but for anions 0... 

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. 

Oxides that have a crystal structure that prefers interstitial formation have Frenkel and anti-Frenkel disorder and obey the following defect formation equations. 

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. 

Let us next calculate the defect strucmre of a more general case, AaO-doped MO. Here we assume that A substitutes M and MO has the anti-Frenkel disorder as the majority type of ionic disorder. Then we may list the defects of the most concern as... 

Besides the Frenkel and the Schottky disorders, also the anti-Frenkel and anti-Schottky disorders exist. But more important are the Frenkel and Schottky types. In the case of sodium sulfate, sodium ions on the normal lattice position (the notation of Krbger-Vink is used see entry Kroger-Vinks Notation of Point Defects ) go into free space of ions (interstitials) and sodium vacancies remain (Frenkel defects) ... 

The equations written in Kroger-Vink symbols with building elements can be used in mass action law. As an example the incorporation of oxide ions on interstitials (Anti-Frenkel disorder) can be expressed as follows ... 

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