Brouwer diagram construction

The four Eqs. [(8.3)—(8.6)] are simplified using chemical and physical intuition and appropriate approximations to the electroneutrality Eqs. (8.7) and (8.10). Brouwer diagrams similar to those given in the previous chapter can then be constructed. However, by far the simplest way to describe these equilibria is by way of polynomials. This is because the polynomial appropriate for the doped system is simply the polynomial equation for the undoped system, together with one extra term, to account for the donors or acceptors present. For example, following the procedure described in Section 7.9, and using the electroneutrality equation for donors, Eq. (8.9), the polynomial appropriate to donor doping is ... 

A number of factors must be taken into account when the diagrammatic representation of mixed proton conductivity is attempted. The behavior of the solid depends upon the temperature, the dopant concentration, the partial pressure of oxygen, and the partial pressure of hydrogen or water vapor. Schematic representation of defect concentrations in mixed proton conductors on a Brouwer diagram therefore requires a four-dimensional depiction. A three-dimensional plot can be constructed if two variables, often temperature and dopant concentration, are fixed (Fig. 8.18a). It is often clearer to use two-dimensional sections of such a plot, constructed with three variables fixed (Fig. 8.18h-8.18<7). 

These equations then allow the Brouwer diagrams to be constructed (Fig. 8.19). 

Considering different pairs of majority defects, all relationships between defect concentrations and partial pressure can be constructed from simplified situations, and this leads to so-called Brouwer diagrams. Figs. 2a and 3a show such Brouwer diagrams for a pure oxide MO with Schottky disorder, and for a Schottky-disordered oxide with a negative dopant. (Please notice that the exact curves calculated from the complete electroneutrality equation (Eq. (11)) exhibit smooth transitions rather than sharp bends.)... 

In addition, an equation for the electtoneuttality condition for the bulk crystal is always necessary. Problems requiring the construction of Brouwer diagrams are given at the end of the chapter. 

On the basis of such an exercise one may construct schematic diagrams of how defect pairs dominate defect stmctures and how minority defects behave and eventually take over. Transition zones are not solved explicitly in this way and are thus often drawn sharp and schematically. Logarithmic depictions are common, and such plots are then called Brouwer diagrams. 

We will explore the defect structure of the oxide by considering limiting conditions and follow the procedure listed before to construct Brouwer diagrams. 

To construct a Brouwer diagram to depict the relative concentration of these defects with P02, it is necessary to define a series of approximations to the full neutrality condition. The first of these corresponds to a region (R I) where 6 > 0 and where the material is hyperstoichiometric. 

Describe the process of doping by a set of chemical reactions of the appropriate defects. Construct a Brouwer diagram to illustrate the effect of gas partial pressure on the concentration of defects in a solid. 

Construct a Brouwer diagram, including regions of low pg, intermediate pg, and high pg,. In the region of intermediate pg, assume that the concentration of electronic defects is greater than the vacancy concentration. When is this material intrinsic When is it n-type When is itp-type ... 

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