Disorder Frenkel

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... 

Intrinsic Frenkel disorder, in which some of the oxygens are displaced into normally unoccupied sites, is responsible for the oxide ion conduction in, for example, Zr2Gd207, Fig. 2.11. The interstitial oxygen concentration is rather low, however, and is responsible for the low value of the preexponential factor and for the rather low (by -Bi203 standards ) conductivity. 

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... 

A second example will now be discussed in order to illustrate the application of the internal equilibrium condition in combination with structural constraints. Let us regard a crystal AX, such as AgBr, having Frenkel disorder in the cation sublattice (see Fig. 1-2), Structure elements which must be considered here are Aa, Xx, Va, Vj, Aj. The structural constraint reads... 

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. 

Similar to a Frenkel disorder, or to the dissociation equilibrium in water, a mass action law according to... 

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... 

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 16. Concentrations of vacancies and interstitial particles in a positively doped Frenkel-disordered material taking account of association between cationic vacancies and dopant ions. The...

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.)...

Figure 30. Ionic space charge effects at grain boundaries in Frenkel disordered materials, (a) Theoretical profiles if u, > u .120 (b) The enhanced grain boundary conductivity can be verified by point electrode impedance spectroscopy.121 The number given are in units of nS / cm and refer to room temperature.

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)... 

Figure 1.2 Representation of internal (Frenkel-) disorder in the (free) energy level diagram and its coupling with the fundamental electronic excitation in the bulk (a) and at boundaries (b) [3]. The illustrations correspond to particular cases when Li (a) or generally monovalent M + cations (b) are excited in the lattice.

It should be noted that neither Schottky nor Frenkel disordering processes affect the stoichiometry of crystals. 

At room temperature the ionic properties of the pure silver halides are primarily determined by Frenkel disorder on the cation sublattice [18,19]. The equilibrium volume concentrations of silver ion interstitials ( ] and vacancies (nv) have the temperature dependence... 

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