Anti-Frenkel disorder

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 ),... 

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)... 

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

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 ... 

Frenkel disorder usually occurs in the cation sublattice. It is less common to observe Frenkel disorder in the anion snblattice (anti-Frenkel disorder), and this is because anions are conunonly larger than cations. An important exception to this generalization lies in the occurrence of anti-Frenkel disorder in fluorite-stractured compotmds, like alkaline earth halides (CaFj, SrFj, SrClj, BaFj), lead fluoride (PbFj), and thorium, uranium, and zirconium oxides (ThOj, UOj, ZrOj). One reason for this is that the anions have a lower electrical charge than the cations, while the other reason lies in the nature of the open stmcture of the fluorite lattice. 

The Kroger-Vink diagrams of compounds ABX4, in which A, B, and X are ions with charge +1, +3, and -1 are presented in Figures 5.4 and 5.5. It is assumed that the model material exhibits anti-Frenkel disorder. 

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. 

Here, anti-Frenkel disorder is assnmed to occur. The corresponding defect equiUbrium... 

For a doped oxide M2O3 exhibiting anti-Frenkel disorder, Colomban and Novak present a sehematie Kroger-Vink diagram of the extrinsic and intrinsic point defects as a function of the partial water pressure. With regard to electrical properties, the proton conductivity in the binary metal oxides is usually much lower than in the perovskite-type oxides. ... 

For YBajCujOy.x tentative Kroger-Vink diagrams have been constructed under the assumption of excess Y, and in the case of anti-Frenkel disorder.In the first case the electroneutrality conditions are with increasing Po,J... 

For Schottky-disorder in alkali halides [114], Frenkel-disorder in silver halides and anti-Frenkel-disorder in alkaline earth halides it is found that const 1.5...2 [115]. Anti-Schottky disorder, i.e. the interstitial incorporation of both cation md anion, is extremely rare, primarily on account of the high energy of large anions in the interstices and is only possible in loose structures. It has, for instance, been postulated for orthorhombic PbO [116,117]. A detailed description of these disorder types is given in Section 5.5. Table 5.1 lists formation data for a series of halides. 

Fig. 5.37 Coupling of the ionic and the electronic energy level pictures for an anti-Frenkel disordered material M+X" through MX = 5MX2 = MX- - Me-- According to the position of the electrochemical potentials the materied is in the I-regime on the r.h.s. ([h l > [e ]j [X(] [Vx]) of the intrinsic point (cf. Fig. 5.38). Note that jiie- =Jle = —Mh-and Mx- = Mx = Mvx> see Sections 5.2, 5.3). Instead of fix[ (or Mi) and (or Mv) it should more precisely read Jlx. -Wi and Mvx-Xx- (Cf- also Fig. 5.9 on page 127.) From Ref. [168].

As already mentioned analogous phenomena also occur in anti-Frenkel disordered anionic conductors (e.g. CaF2 or PbF2). In contrast to the above conductors, the more acidic Si02 is more effective here than AI2O3. This qualitative aspect and, even more, the precise analysis [267] lead to the conclusion that the adsorption of F is the dominant mechanism. 

In an asymmetrical disorder, the two defects that make it up pertain to the same sub-lattice of A or of B. In practice, we find only two families of anti-S5mmetrical disorders Frenkel disorder and so-called AS disorder. 

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