Frenkel reaction

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... 

In view of the appreciable number of SE s involved in reaction (9.25), distinct serial reaction steps can be anticipated. Steps in which the electron hole is involved are assumed to be fast compared to steps involving ionic SE s (in line with the fact that >h > Dj). Thus, the (locally homogeneous) Frenkel reaction becomes rate determining for the overall internal process described by Eqn. (9.25). The Frenkel reaction is bimolecular. The rate equation for the formation of Frenkel defects is, according to standard kinetics,... 

The ionic charge carriers in ionic crystals are the point defects.1 2 23,24 They represent the ionic excitations in the same way as H30+ and OH-ions are the ionic excitations in water (see Fig. 1). They represent the chemical excitation upon the perfect crystallographic structure in the same way as conduction electrons and holes represent electronic excitations upon the perfect valence situation. The fact that the perfect structure, i.e., ground structure, of ionic solids is composed of charged ions, does not mean that it is ionically conductive. In AgCl regular silver and chloride ions sit in deep Coulomb wells and are hence immobile. The occurrence of ionic conductivity requires ions in interstitial sites, which are mobile, or vacant sites in which neighbors can hop. Hence a superionic dissociation is necessary, as, e.g. established by the Frenkel reaction ... 

Analogously [v Ag] = [Ag ] = V V >[e ] pci /2 > [h ] Pa2/2 (where now the decisive disorder reaction is the Frenkel reaction of the Ag sublattice the exponent 1/2 results since the effective charges are 1). Please note that in these examples with overwhelming ion disorder, e.g. 

Note that the Frenkel reaction as written does not violate rule 3 since interstitial sites do not constitute regular lattice sites. FeO. NiO, CoO, and CU2O are examples of oxides that exhibit Frenkel defects. 

The old-fashioned charge designations which are exclusively used for the defects, indicate particles and indicate the relative charge with respect to the perfect lattice. According to the arguments detailed in Section 5.2 we know that there is, at equilibrium, a mass action law of the form (F indicates Frenkel reaction)... 

It corresponds to the coupling of reaction (5.98a) with the anti-Frenkel reaction in the form (cf. Eq. (5.90))... 

Let us consider the Frenkel reaction in an ideally pure crystal. If we gradually reduce the temperature, the interstitial ion will eventually fall back into the vacancy (cf. Kp —> 0 for T- 0). In the same way, the semiconductor s conduction electrons will, via annihilation of holes, become valence electrons (Kb —> 0). In a similar sense, in strongly Cd-doped AgCl, there is a Coulomb attraction between Cd g and the counterdefect V g leading to a mutual trapping and thus to a deviation from random distribution at low temperatures. This can be described approximately by an exothermal production of associates [203,204] of the form... 

By analogy with similar materials in which free elecU ons and electron holes are formed, NiO is called a p-type compound having vacant site Schottky defects, and ZnO is an n-type compound having interstitial Frenkel defects. The concentrations of these defects and their relation to the oxygen pressure in the suiTounding atmosphere can be calculated, for a dilute solution of defects by the application of a mass action equation. The two reactions shown above are represented by the equations... 

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. 

It is apparent that the Frenkel process coupled with the electronic process are the predominating mechanisms in forming defects in AgBr through the agency of external reaction with Br2 gas. 

Johnson KJ, Cygan RT, Fein JB (2006) Molecular simulations of metal adsorption to bacteria surfaces. Geochim Cosmochim Acta 70 5075-5088 Jokic A, Frenkel AI, Vairavamurthy MA, Huang PM (2001) Bimessite catalysis of the Maillard reaction and its significance in natural humification. Geophys Res Lett 28 3899-3902... 

In many instances, the above reaction will then take place in the reverse direction, and the silver atom will revert to the normal stable state as a Frenkel... 

The addition of either donors or acceptors will, however, upset the charge balance, and these must be included in the electroneutrality equation. Consider donor doping by a trivalent ion D3+ due to reaction with D2X3 to introduce D defects, once again assuming that Frenkel defects are not important. The original electroneutrality Eq. (7.12) ... 

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... 

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... 

In 1937, dost presented in his book on diffusion and chemical reactions in solids [W. lost (1937)] the first overview and quantitative discussion of solid state reaction kinetics based on the Frenkel-Wagner-Sehottky point defect thermodynamics and linear transport theory. Although metallic systems were included in the discussion, the main body of this monograph was concerned with ionic crystals. There was good reason for this preferential elaboration on kinetic concepts with ionic crystals. Firstly, one can exert, forces on the structure elements of ionic crystals by the application of an electrical field. Secondly, a current of 1 mA over a duration of 1 s (= 1 mC, easy to measure, at that time) corresponds to only 1(K8 moles of transported matter in the form of ions. Seen in retrospect, it is amazing how fast the understanding of diffusion and of chemical reactions in the solid state took place after the fundamental and appropriate concepts were established at about 1930, especially in metallurgy, ceramics, and related areas. 

The first reaction is a site exchange reaction and so does not alter the number of lattice sites. The second reaction describes the formation of a complete lattice molecule M. An example of the first type of reaction (exchange reaction, Eqn. (2.59)) is the so-called Frenkel defect formation reaction in AX (e.g., in silver halides, see Fig. 1-2)... 

Let us apply Eqn. (4.143) to the homogeneous Frenkel defect formation reaction Aa+V = A +VA, which describes the formation of intrinsic majority defects in... 

Subsequent to the collision, the most important event concerning kinetics is the displacement of regular SE s and the formation of Frenkel-type point defects. The corresponding formation reaction is... 

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