Brouwer approximation

The results, plotted as log (defect concentration) versus log pXl, give the complete diagram (Fig. 7.11 b). It is seen that the Brouwer approximation bears a good correspondence to this diagram and indicates that the simplification procedure may be adequate in most cases. 

The variation of the defect species present as a function of dopant concentration, oxygen partial pressure, or temperature can then be determined by solution of the polynomial equations that connect the various defect populations. The Brouwer approximation may be satisfactory in many cases. As always, comparison with experimental data is essential. 

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 ... 

However, this is usually prohibitively messy, the problem being that the algebraic structure of the charge neutrality condition, Eq. (7), is different from the remainder [Eqs. (3)-(6)]. Thus, the normal practice is to approximate the charge neutrality condition by a limiting condition or in terms of an oppositely charged pair of disorders in the majority (Brouwer approximation [4]), depending on the thermodynamic conditions. 

By solving these three equations, one can get each defect concentration as a function of T and x independently of the component activities. The present situation can be solved even analytically, but for the consistency sake, we will employ the Brouwer approximations as [e ] 2[VJ j] [m -] [V (i]. 

General equilibrium of an oxide with oxygen in the Brouwer approximation of majority defects... 

Table 3.4. Concentrations ofdefect points in cerium oxide in both cases of the Brouwer approximation...

To cany out calculation, we eonsider the Brouwer approximation, which consists of keeping only a single term of eaeh sign in the expression of the electric neutrality. 

The predictions by the Brouwer approximation could be proven experimentally with aluminum and silicon doped LiFeP04. in case of aluminum doped LiFeP04 a further effect was observed besides the changes of the conductivities due to the doping. Here the electronic conductivity flattens at low temperatures, while the decrease of the ionic conductivity becomes steeper at low temperatures (the starting temperatures being the same]. The quantitative analysis succeeds by taking account of defect association ... 

In order to make the logarithmic form of Eq. (5.103b) linear it is necessary that one of the two additive terms on each side of the equation is negligible. This corresponds to the Brouwer approximation [169], which we will now discuss for a somewhat simplified disorder model and specific2dly for an oxide. 

Fig. 5.38 Defect concentrations of our model oxide as a function of the partial pressure of oxygen (Kroger-Vink or Brouwer diagram). The Brouwer approximation is naturally no longer applicable at the crossing points, and the transitions are smoothed. The partial pressure of the intrinsic point (index i) is indicated. Note that the charge number z is included in the definition of the ordinate.

When mass action laws are valid and Brouwer approximation can be used then power laws of the type... 

The nuances obtained in the transition regions in comparison to Figs. 5.46a,c come from a precise calculation avoiding the Brouwer approximation (see, for example, Eq. (5.142) instead of Eq. (5.144)). 

If we apply the Brouwer approximation we also directly obtain the results for the minoriiy charge carriers. Prom Eqs. (5.135, 5.136) it follows (lower temperatures) ... 

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... 

The formal treatment and the calculation of the dependence of the charge carrier concentration, in particular of [Hj] as a function of T, Pqj, PhjO and the concentration of the negative dopant, is carried out as described above using the Brouwer approximation. Under reversible conditions the result is formally given by... 

The individual concentrations can be calculated using Eqs. (5.188, 5.189) and the electroneutrality condition. We have added one more variable by introducing associates, but there is also an additional equation. Although the precise calculation is simple (see results in Fig. 5.60), let us restrict to the Brouwer approximation and... 

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