Frenkel defect disorder

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 equilibrium concentration of defects is obtained by applying the law of mass action to Eq. (7) or (8). This leads in the case of Frenkel disorder to... 

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... 

Results of the ideal solution approach were found to be identical with those arrived at on the basis of a simple quasichemical method. Each defect and the various species occupying normal lattice positions may be considered as a separate species to which is assigned a chemical potential , p, and at equilibrium these are related through a set of stoichiometric equations corresponding to the chemical reactions which form the defects. For example, for Frenkel disorder the equation will be... 

In this section we are concerned with the properties of intrinsic Schottky and Frenkel disorder in pure ionic conducting crystals and with the same systems doped with aliovalent cations. As already remarked in Section I, the properties of uni-univalent crystals, e.g. sodium choride and silver bromide which contain Schottky and cationic Frenkel disorder respectively, doped with divalent cation impurities are of particular interest. At low concentrations the impurity is incorporated substitutionally together with an additional cation vacancy to preserve electrical neutrality. At sufficiently low temperatures the concentration of intrinsic defects in a doped crystal is negligible compared with the concentration of added defects. We shall first mention briefly the theoretical methods used for such systems and then review the use of the cluster formalism. 

We now introduce a Fourier transform procedure analogous to that employed in the solution theory, s 62 For the purposes of the present section a more detailed specification of defect positions than that so far employed must be introduced. Thus, defects i and j are in unit cells l and m respectively, the origins of the unit cells being specified by vectors R and Rm relative to the origin of the space lattice. The vectors from the origin of the unit cell to the defects i and j, which occupy positions number x and y within the cell, will be denoted X 0 and X for example, the sodium chloride lattice is built from a unit cell containing one cation site (0, 0, 0) and one anion site (a/2, 0, 0), and the translation group is that of the face-centred-cubic lattice. However, if we wish to specify the interstitial sites of the lattice, e.g. for a discussion of Frenkel disorder, then we must add two interstitial sites to the basis at (a/4, a/4, a]4) and (3a/4, a/4, a/4). (Note that there are twice as many interstitial sites as anion-cation pairs but that all interstitial sites have an identical environment.) In our present notation the distance between defects i and j is... 

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... 

Let us finally estimate the relaxation times of homogeneous defect reactions. To this end, we analyze the equilibration course of a silver halide crystal, AX, with predominantly intrinsic cation Frenkel disorder. The Frenkel reaction is... 

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. 

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 ... 

A major difference between crystals and fluids refers to the necessity of distinguishing between different sites. So the autoprotolysis in water could, just from a mass balance point of view, also be considered e.g. as a formation of a OH vacancy and a IT vacancy. In solids such a disorder is called Schottky disorder (S) and has to be well discerned from the Frenkel disorder (F). In the densely packed alkali metal halides in which the cations are not as polarizable as the Ag+, the formation of interstitial defects requires an unrealistically high energy and the dominating disorder is thus the Schottky reaction... 

Figure 20. Sketch of the real structure of a Frenkel disordered crystal (only the affected sublattice is shown). The increase in temperature corresponds to an increase in defect concentration. Interaction leads to a narrowing of the spacing of the energy levels (electrochemical potentials minus configurational term) and eventually to a transition into the superionic state.12 (Reprinted from J. Maier and W. Munch, Z. Anorg. Allg. Chem., 626,264-269, Copyright 2000 with permission from WILEY-VCH Verlag GmbH.)...

Stoichiometric reaction is one in which no mass is transferred across the crystal boundaries. The three most common stoichiometric defects are Schottky defects, Frenkel defects, and antistructure disorder or misplaced atoms. 

Schottky and Frenkel disorder and to relate the lattice distortion and attendant expansion to particular defects. 

Frenkel disorder a pair of defects involving only cations, with the pair... 

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. 

As an example, the space charge properties of Ti02 were analyzed by Ikeda and Chiang [16]. Assuming cation Frenkel disorder - that is, cation vacancies and interstitials are the predominant defects - the free energy change due to the introduction of such defects into a perfect crystal is given by... 

For higher concentrations of defects the equation must be written in terms of activities instead of concentrations. Frenkel disorder occurs in the silver halides, for example, in AgCl. 

At this point of the discussion it is worthwhile to distinguish between two different kinds of disorder. If the concentrations of the majority defect centers, which constitute the disorder type, are independent of the component activities and are only determined by P and 7, then we speak of thermal disorder or intrinsic disorder (e. g. Frenkel disorder in silver bromide). However, the concentrations of minority defect centers do depend upon the component activities even in the case of a crystal with thermal disorder. This will be discussed more explicitly later. On the other hand, if the concentrations of the majority defects are dependent upon the component activities, then we speak of activity-dependent disorder or extrinsic disorder (e. g. cation vacancies and electron holes in transition metal oxides). 

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. 

From electrical conductivity measurements, from transference measurements, and by combining lattice constant and density measurements, it has been shown that Frenkel disorder predominates in pure undoped AgBr [6]. Estimates of disorder energies as in section 3.2.3 can be made. They confirm the experiments. For purposes of solving the system of equations (4-13) to (4-21), however, this means that only (V g) and (Agj) need to be considered in the balance equations. These are the concentrations of the majority defect centers which constitute the disorder type. All other concentrations (of the minority defect centers) can be neglected. Since 5 is not chemically measurable (i.e. 6 1), it follows from eqs. (4-14) and (4-16) that (V g) (Ag i). The relative partial pressure of bromine PsTjPBrzi 0) can be calculated... 

Fig. 4-3. The concentrations of point defects (/) in the ternary crystal AB2O4 at constant as a function of the activity of B2O3. The intrinsic disorder is assumed to be Frenkel disorder in the B-ion sublattice (By ) ai (Vb").

K is the equilibrium constant for Frenkel disorder. This can be seen if we set the time derivative in eq. (6-2) equal to zero, since this is the condition for thermodynamic equilibrium. On the basis of the simple theory of homogeneous reactions, two limiting cases arise. 1. The electrostatic interaction between electrically charged defects at a separation 2 is negligible compared... 

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