Pairs of defects

According to our nomenclature, as used in the table, Vm is a vacancy at an M cation site, etc. The first five pairs of defects given above have been observed experimentally in solids, whereas the last four have not. This answers the question posed above, namely that defects in solids occur in pairs. 

Table 3-3, given on the next page, siunmarizes the various pairs of defects possible for binary compounds. Equilibria are given along with the appropriate equilibriiun constant. Note that these equations are rather simple and can be used to specify the equilibrium constants for these defects present in the lattice. These types of defects have been observed and studied in the compounds given under "Example in this Table. These are the major types of defects to be expected in most inorganic compounds, where the number of sites in the lattice is consteuit. 

To see how these equations might be used, consider the following. First, suppose we want to be able to determine the number of intrinsic defects in a given solid. Since pairs of defects predominate in a given solid, the number of each type of intrinsic defect, Ni (M) or Ni (X), will equal each... 

Draw one or more "plane-nets" for the "P" eation eombined with a "U" anion. Indicate all of the possible defects that can appear. Write the symbol of each as you proceed. Include pairs of defects as needed. 

Although we win not treat the other types of pairs of defects, it is well to note that similar equations can also be derived for the other intrinsic defects. What we have really shown is that external reactants can cause further changes in the non-stoichiometry of the soUd. Let us now consider ionized defects. It should be clear that an external gaseous factor has a major effect upon defect formation. The equations given above are very complicated and represent more closely what actually happens in the real world of defect formation in crystals. 

In this diagram, we have shown both charged ions involved in the migrational diffusion process as well as charged vacancies which also add to the overall diffusion process. We must have electroneutrality in diffusion, i.e.- pairs of defects, and using the above example, we can write nine equations, of which the following is just one case ... 

The defect population in the doped solid under moist conditions then consists of V%, OHo, h, and Y /x. The domains over which each species is dominant for conductivity can be represented diagrammatically when data concerning the conductivity of the solid has been measured (Fig. 8.20). In this representation, the conductivity fields are bounded by lines tracing the locus where the transport number for a pair of defects is equal to 0.5. (The diagram could equally well be drawn in terms of domains delineating the defect species that predominate.)... 

As it was discussed in Chapter 3, neutral point defects in all solids interact with each other by the elastic forces caused by overlap of deformation fields surrounding a pair of defects. These forces are effectively attractive for both similar and dissimilar defects (interstitial-interstitial, vacancy-vacancy and interstitial-vacancy, respectively) and decay with the distance between defects as... 

Here p is a creation rate (dose rate) of stable pairs of defects, while the quantity 5 is a fraction of the reaction volume overlapped by recombination spheres, i.e., the ratio of the effective recombination volume to the entire volume of the crystal. The problem of constructing the kinetic equation of... 

Equation (1) is the most general and rigorous analytical result. However, it does not take account of the aggregation of similar defects and hence it is applicable only at not very large irradiation doses (up to a concentration of defects (l/2)no, where no is concentration at saturation). In fact, here the existence of clusters of similar defects is allowed, but it is actually assumed that these clusters are statistical fluctuations of the Poisson distribution of similar defects, which does not reflect a real pattern of cluster formation with a substantially non-Poisson spectrum of fluctuations. It is assumed implicitly in equation (1) that, after each event of creating a new pair of defects, the entire system of defects is stirred to attain the Poisson distribution. In the case of the absence of the defect correlation in genetic pairs we arrive at equation (2). 

The same principle (from bottom to top) is used for numeration of the lengths of bonds in pair of defective 5- and 7-members cycles which are disposed in the boundary between (n+1,0) CNT and (n,0) CNT (see the picture on the left). 

Examination of the equilibrium conditions shows that the equilibrium constant for intrinsic disorder (Equation 1) involves only the (endothermic) energy needed to create a complementary pair of defects in the ordered, stoichiometric crystal lattice. Writing Nh, Nl as the number of cation vacancies and of interstitial cations, respectively, in a stoichiometric crystal with N lattice sites in all, at an expenditure of energy denoted by Eh, respectively, then the intrinsic disorder, f, is given by Equation 5. 

In this case of large stoichiometric deviations and interaction between like pairs of defects, a(Vm°) is not [Vm°]/ZVB, but is given by ... 

If one repeats the above statistical derivation for the formation of a pair of defects or for more complicated reactions of the type... 

Annealing of geminate pairs of defects with the boundary conditions... 

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