Brouwer’s approximation

To obtain the theoretical rate of the atomic transport, it is possible to assume that one of the six elementary steps is rate-determining. Then, the calculation needs the determination of the concentration of the intermediates, that can easily be achieved by using the Brouwer s approximation. 

We can see that the exact calculation quickly becomes complex. In order to simplify it, we apply Brouwer s approximation, known as the majority state approximation, to relation [3.8] the expression of electrical neutrality. Thus, relation [3.8] will assume one of the following four extreme forms, each of which defines a temperature region ... 

Equations (76) introduce sum relations in addition to the electroneutrality condition. Thus, in order to use Brouwer s approximation method, assumptions must be made for each sum relation, thereby making the treatment of partial equilibrium more complex than that of complete equilibrium. The situation becomes even more complicated if incomplete quenching occurs. More detailed discussions of incomplete equilibrium are given by Kroger and Van Gool. ... 

Impurity reactions such as Eqs. (104), (105), or (106) may be combined with the previously given reactions for the formation and ionization of native defects to compute concentrations of all defects present under various conditions. Brouwer s approximation method is used when necessary. A discussion of all the possibilities which may occur is far beyond the scope of this chapter. However, one simple case will be considered for illustrative purposes. 

From Eqs. (112)-(114) and Eq. (7), the authors were able to compute the concentration of each defect structon, as well as n and p, as a function of oxygen pressure using Brouwer s approximation method. 

In fact, the system thus obtained is complex to solve, particularly when we do not have the numerical values of the equilibrium constants. We can simplify the calculation by using Brouwer s majority defect approximation, which, remember, involves considering only the two largest terms (one with each sign) in the expression of electrical neutrality [3.36]. Thus, in our example, we would have four Brouwer cases, defined by ... 

At this point, all the necessary information needed to relate the concentrations of the various defects to the oxygen potential or partial pressure surrounding the crystal is available. In Eqs. (6.23) to (6.27), there are four unknowns [n, p, Vq, Vm] and five equations. Thus in principle, these equations can be solved simultaneously, provided, of course, that all the Ag s for the various reactions are known. Whereas this is not necessarily a trivial exercise, fortunately the problem can be greatly simplified by appreciating that under various oxygen partial pressure regimes, one defect pair will dominate at the expense of all other pairs and only two terms remain in the neutrality condition. How this Brouwer approximation is used to solve the problem is illustrated now ... 

The graphs in Figs. 5.47 and 5.48 have been constructed on the basis of Eqs. (5.154, 5.155). Notice that the transition regions are curved compared with the discontinuous change in the slope as predicted by the Brouwer approximations (curved sections). Regarding AgCl, for which UAg > Uy , Eq. (5.155) exhibits a minimum at s... 

---