External defect equilibria

However, every realistic treatment of defect chemistry of pure materials must take account of the fact that there are always sUght deviations from the exact stoidiio-metric composition, i.e. that the defects also interact with the external world. This finds expression in the phase rule, whereby, at fixed temperature smd constant partial pressure, all degrees of freedom are only utilized when there are as many phases present as components. Fixing the properties thermodynamically, thus, demands equilibration with a particular M or X activity and thus with a particular M or X2 partial pressure (cf. also Section 4.3). 

This can be generally achieved by bringing the sample into contact with the M mother material (or if existent, with the appropriate metal-rich thermod5Tiamically compatible compound), or by contact with pure X2 gas (or with the correspondingly phase-compatible X-rich compound). Then the extreme stoichiometries corresponding to the highest possible and lowest possible M-content (lowest possible and highest possible X-content), respectively, are set up. When there is low disorder in the compoimd MX, M activity and Pxa partial pressure are coupled by  

As demonstrated copiously in Section 4.3.5, Eq. (5.96) only applies if mx is approximately constant, i.e. in the case of slight deviations from stoichiometry which 

The interaction of the oxide PbO with oxygen can, for example, be formulated as the incorporation of oxygen interstitially. The oxygen incorporated sits in the lattice as, i.e. in our example as 0 and, therefore, conduction electrons e are annihilated (formally low valence lead is oxydized to Pb +)  

Note that this equation (in contrast to the pure redox or acid-base reactions discussed above) represents both a redox reaction and an acid base reaction. It hence couples electronic and ionic defect concentrations via the chemical potential of the 

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Let us analyze these results one step further and ask about a quantitative measure of the Kirkendall effect. This effect had been detected by placing inert markers in the interdiffusion zone. Thus, the lattice shift was believed to be observable for an external observer. If we assume that Vm does not depend on concentration and local defect equilibrium is established, the lattice site number density remains constant during interdiffusion. Let us designate rv as the production (annihilation) rate of the vacancies. We can derive from cA+cB+cv = l/Vm and jA +/ B +./v = 0 that... 

Several points are to be noted. Firstly, pores and changes of sample dimension have been observed at and near interdiffusion zones [R. Busch, V. Ruth (1991)]. Pore formation is witness to a certain point defect supersaturation and indicates that sinks and sources for point defects are not sufficiently effective to maintain local defect equilibrium. Secondly, it is not necessary to assume a vacancy mechanism for atomic motion in order to invoke a Kirkendall effect. Finally, external observers would still see a marker movement (markers connected by lattice planes) in spite of bA = bB (no Kirkendall effect) if Vm depends on composition. The consequences of a variable molar volume for the determination of diffusion coefficients in binary systems have been thoroughly discussed (F. Sauer, V. Freise (1962) C. Wagner (1969) H. Schmalzried (1981)]. 

The phenomenon of metastability is closely related to the defect equilibrium properties. Some external excitation - illumination, charge, current flow, energetic particles, etc. - induces defects (or dopants) which are subsequently removed by annealing. The metastable... 

The lanthanum fluoride crystal is a conductor for fluoride ions which being small can move through the crystal from one lattice defect to another, and equilibrium is established between the crystal face inside the electrode and the internal solution. Likewise, when the electrode is placed in a solution containing fluoride ions, equilibrium is established at the external surface of the crystal. In general, the fluoride ion activities at the two faces of the crystal are different and so a potential is established, and since the conditions at the internal face are constant, the resultant potential is proportional to the fluoride ion activity of the test solution. 

More generally, co is independent of the external gas pressure k is the Boltzmann constant (1.38 x 10 erg deg ) and T is the temperature in Kelvin. Furthermore, the equilibrium between co and a collapsed CS plane fault is maintained by exchange at dislocations bounding the CS planes. Clearly, this equilibrium cannot be maintained except by the nucleation of a dislocation loop and such a process requires a supersaturation of vacancies and CS planes eliminate supersaturation of anion vacancies (Gai 1981, Gai et al 1982). Thus we introduce the concept of supersaturation of oxygen point defects in the reacting catalytic oxides, which contributes to the driving force for the nucleation of CS planes. From thermodynamics. 

The importance of interactions amongst point defects, at even fairly low defect concentrations, was recognized several years ago. Although one has to take into account the actual defect structure and modifications of short-range order to be able to describe the properties of solids fully, it has been found useful to represent all the processes involved in the intrinsic defect equilibria in a crystal (with a low concentration of defects), as well as its equilibrium with its external environment, by a set of coupled quasichemical reactions. These equilibrium reactions are then handled by the law of mass action. The free energy and equilibrium constants for each process can be obtained if we know the enthalpies and entropies of the reactions from theory or... 

Since the state of a crystal in equilibrium is uniquely defined, the kind and number of its SE s are fully determined. It is therefore the aim of crystal thermodynamics, and particularly of point defect thermodynamics, to calculate the kind and number of all SE s as a function of the chosen independent thermodynamic variables. Several questions arise. Since SE s are not equivalent to the chemical components of a crystalline system, is it expedient to introduce virtual chemical potentials, and how are they related to the component potentials If immobile SE s exist (e.g., the oxygen ions in dense packed oxides), can their virtual chemical potentials be defined only on the basis of local equilibration of the other mobile SE s Since mobile SE s can move in a crystal, what are the internal forces that act upon them to make them drift if thermodynamic potential differences are applied externally Can one use the gradients of the virtual chemical potentials of the SE s for this purpose ... 

By a change of temperature or pressure, it is often possible to cross the phase limits of a homogeneous crystal. It supersaturates with respect to one or several of its components, and the supersaturated components eventually precipitate. This is an additive reaction. It occurs either externally at the surfaces, or in the crystal bulk by nucleation and growth. Reactions of this kind from initially homogeneous and supersaturated solid solutions will be discussed in Chapter 12 on phase transformations. Internal reactions in the sense of the present chapter occur after crystal A has been brought into contact with reactant B, and the product AB forms isothermally in the interior of A or B. Point defect fluxes are responsible for the matter transport during internal reactions, and local equilibrium is often established throughout. 

Defects or dopants are created because the external excitation drives the defect reaction away from the initial equilibrium. The formation energy of charged defects, and therefore their equilibrium density, depends on the position of the Fermi energy according to Eqs. (6.26) and (6.27). Thus, the equilibrium defect density is... 

The light-induced and thermal equilibrium defect reactions are aspects of the same general process. Indeed, the structural models proposed are virtually identical (compare Figs. 6.13 and 6.30). In the two-well description of Fig. 6.1, excitation over the barrier in either direction can, in principle, be thermal or by an external excitation. The... 

Several other sources of external excitation result in metastable defect or dopant creation in a-Si H. Most have the characteristic property that a shift in the Fermi (or quasi-Fermi) energy causes a slow increase in the density of states and that annealing to 150-200 °C reverses the effect. The phenomena are therefore similar in origin to the optically-induced states and fall within the same general description of departures from the thermal equilibrium state induced by excess carriers. 

In a crystal, perturbations can be classified as internal and external. The internal perturbations are disturbances from an equilibrium condition, taken as an ideal uniform distribution of impurities or defects which do not modify the crystal lattice and the average electronic density. Mechanical perturbations can be microscopic, like those introduced by impurities or defects producing large local volume changes, which reflect on crystal lattice spacings when their concentration is large, or macroscopic due to residual or accidental stresses. Permanent perturbations can also be produced by unrelaxed stresses... 

One of the aims of this chapter is to relate the concentration of defects to temperature and other externally imposed thermodynamic parameters such as oxygen partial pressure, a goal that is now almost at hand. This is accomplished by considering defects to be structural elements which possess a chemical potential and hence activity and expressing their equilibrium concentrations by a mass action expression similar to Eq. (5.30) ... 

To relate the concentrations of point and electronic defects to temperature and externally imposed thermodynamic conditions such as oxygen partial pressures, the defects are treated as chemical species and their equilibrium concentrations are calculated from mass action expressions. If the free-energy changes associated with all defect reactions were known, then in principle diagrams, known as Kroger-Vink diagrams, relating the defect concentrations to the externally imposed thermodynamic parameters, impurity levels, etc., can be constructed. 

We can express relationships between defect formation, the influence of various external factors, and the equilibrium constant thereby related. We do this in terms of 6, the degree of non-stoichiometry, as given in Table 2-... 

Consider this factor carefully by again examining Table 2-2. Also given is the reaction producing the defect, with its corresponding equilibrium constant. In most cases, the deviation, 6, is presented in terms of the equilibrium constant and the partial pressure of the external gaseous reactant. 

In general it can be shown that exactly enough internal defect equilibria exist to permit us to express all defect concentrations as functions of the independent variables, as long as the material, site, and charge balances are observed. This explains the free choice which we have in formulating eq. (4-5) (i.e. in formulating the external equilibrium conditions). 

As external equilibrium condition (i.e. as the relationship between the chosen independent variable and the dependent defect concentrations at fixed P and T), the following reaction equation between components and structural elements may be written ... 

As yet we have neglected the concentrations of foreign atoms relative to the concentrations of the inherent defects. If small additions of foreign substances are made, within their solubility limits, then the equilibrium constants of the internal and external equilibria as formulated above do not change. However, site balances and electroneutrality conditions must be modified. For example, if CdBr2 is added to AgBr, and minority defects are neglected, then the electroneutrality condition is as follows ... 

In this discussion, we are considering a quasi-binary system AO-B2O3, in which electronic point defects are not the majority defect centers (see page 39). Electronic defects are connected with an excess (or a deficiency) of oxygen, and therefore the composition of the compound cannot be located on the quasi-binary line AO B2O3 in case that one kind of electronic defects predominates along with one other majority center. In the quasi-binary system, the second external equilibrium condition can therefore be suitably formulated by means of the following reaction equation ... 

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