Defects, ionic

Finally, it is apparent that the principal defect type may change as the temperature increases, so that, for example, electronic defects may become more important than ionic defects. In such cases the diagram will change appreciably. 

These mle-of-thumb generalizations must be treated cautiously. For precise work it is necessary to recalculate values of the equilibrium constants at the new temperature and then replot the diagram. The equilibrium constants have a general form (Chapter 2)  

The approximations inherent in Brouwer diagrams can be bypassed by writing the appropriate electroneutrality equation as a polynomial equation and then solving this numerically using a computer. (This is not always a computationally trivial task.) To illustrate this method, the examples given in Sections 7.5 and 7.6, the MX system, will be rewritten in this form. 

Equilibrium partial pressure = 1 atm of X2 log px2 = 0. Note that the values for K0 and K, are obtained by using  

It is necessary to write the electroneutrality equation in terms of just two variables, a defect type and the partial pressure, to obtain a polynomial capable of solution. For example, the equation for the concentration of holes, [h ], is obtained thus Electroneutrality  

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In insulating oxides, ionic defects arise from the presence of impurities of different valence from the host cation. An aluminum ion impurity substituting in a magnesium oxide [1309-48-4] MgO, hostlattice creates Mg vacancies. 

The most important driving forces for the motion of ionic defects and electrons in solids are the migration in an electric field and the diffusion under the influence of a chemical potential gradient. Other forces, such as magnetic fields and temperature gradients, are commonly much less important in battery-type applications. It is assumed that the fluxes under the influence of an electric field and a concentration gradient are linearly superimposed, which... 

In most cases of practically useful ionic conductors one may assume a very large concentration of mobile ionic defects. As a result, the chemical potential of the mobile ions may be regarded as being essentially constant within the material. Thus, any ionic transport in such a material must be predominantly due to the influence of an internal electrostatic potential gradient,... 

The type of disorder may be determined by conductivity measurements of electronic and ionic defects as a function of the activity of the neutral mobile component [3]. The data are commonly plotted as Brouwer diagrams of the logarithm of the concentration of all species as a function of the logarithm of the activity of the neutral mobile component. The slope is fitted to the assumption of a specific defect-type model. 

Parkhutik and Shershulskii249 have modeled the distribution of the space charge of ionic defects inside oxides (sufficiently far from the interfaces that the charge distribution near them can be neglected) based on the following assumptions ... 

The effective charges on an ionic defect can be considered to be linked to the defect by an imaginary bond. If the bond is weak, the effective charge can be liberated, say by thermal energy, so that it becomes free to move in an applied electric field and so contribute to the electronic conductivity of the material. Whether the effective charge on a defect is considered to be strongly associated with the defect or free depends upon the results obtained when the physical properties of the solid are measured. 

Reactions involving the creation, destruction, and elimination of defects can appear mysterious. In such cases it is useful to break the reaction down into hypothetical steps that can be represented by partial equations, rather akin to the half-reactions used to simplify redox reactions in chemistry. The complete defect formation equation is found by adding the partial equations together. The mles described above can be interpreted more flexibly in these partial equations but must be rigorously obeyed in the final equation. Finally, it is necessary to mention that a defect formation equation can often be written in terms of just structural (i.e., ionic) defects such as interstitials and vacancies or in terms of just electronic defects, electrons, and holes. Which of these alternatives is preferred will depend upon the physical properties of the solid. An insulator such as MgO is likely to utilize structural defects to compensate for the changes taking place, whereas a semiconducting transition-metal oxide with several easily accessible valence states is likely to prefer electronic compensation. 

The only ionic defects of importance are vacancies on metal sites, V, and... 

This equation is substituted into Eqs. (7.7)-(7.10) to derive the defect concentrations relevant to the low partial pressure region. Once again these will be similar to those derived for ionic defects. 

Although Fig. 7.10e is similar to that for ionic defects (Fig. 7.9e), there are a number of significant differences. In particular, the stoichiometric range is now far... 

As in the case of ionic defects, the form of this diagram can easily be modified to smooth out the abmpt changes between the three regions by including intermediate electroneutrality equations. 

In the oxygen-deficient region, the predominant ionic defect is the oxygen vacancy, V". The charge neutrality in the solid is maintained by reduction of transition element in B-site to the lower valence state. This can be represented as [13] ... 

For ionic defects the individual terms in the formal virial expansions diverge just as they do in ionic solution theory. The essence of the Mayer theory is a formal diagram classification followed by summation to yield new expansions in which individual terms are finite. The recent book by Friedman25 contains excellent discussions of the solution theory. We give here only an outline emphasizing the points at which defect and solution theories diverge. Fuller treatment can be found in Ref. 4. 

We define an "i-th nearest neighbour complex to be a pair of oppositely charged defects on lattice sites which are i-th nearest neighbours, such that neither of the defects has another defect of opposite charge at the i-th nearest neighbour distance, Rit or closer. This corresponds to what is called the unlike partners only definition. A different definition is that the defects be Rt apart and that neither of them has another defect of either charge at a distance less than or equal to R. This is the like and unlike partners definition. For ionic defects the difference is small at the lowest concentrations the definition to be used depends to some extent on the problem at hand. We shall consider only the first definition. It is required to find the concentration of such complexes in terms of the defect distribution functions. It should be clear that what is required is merely a particular case of the specialized distribution functions of Section IV-D and that the answer involves pair, triplet, and higher correlation functions. In fact this is not the procedure usually employed, as we shall now see. 

Ionic crystals are compounds by necessity. Let us regard a binary compound (A[ X) and derive the electronic conductivity (transference) as a function of its component activity. From Eqn. (4.84) and the necessarily prevailing ionic defects, we can conclude that the ionic conductivity is independent of the component activities which, however, does not mean that the total conductivity is also constant. Let us first formulate the equilibrium between crystal A, X and component X2... 

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