Thus far, we have considered both structural and electronic defects. In addition, we have derived the relationship between oxygen vacancies and the oxy­gen partial pressure, Po2 , which gives rise to nonstoi­chiometry. It should therefore not come as any surprise that we now consider the equilibrium between isolated structural defects, electronic de­fects, and Po2. Of course, we have also considered the equilibrium that exists between isolated struc­tural defects and defect clusters, but defect clusters will not be considered in the present context. Nevertheless, defect clustering does play an impor­tant role in the equilibrium between electronic and structural defects and cannot, in a research context, be ignored.

In solving defect equilibria in previous sections, we have generally ignored the role that minority defects might have. For example, when considering Schottky disorder in MgO, which we know from experiments is the dominant defect formation pro­cess, the effect that oxygen interstitials might have was not taken into account.2 This is certainly reason­able within the context of determining the oxygen vacancy concentration of MgO. The oxygen vacancy concentration is the important parameter to know when predictions of the oxygen diffusivity in MgO are required. However, minority defects may well play an important role in other physical processes. For example, the electrical conductivity or resistivity will depend on the hole or electron concentration; these may be minority defects compared to oxygen vacancies, but understanding them is nevertheless crucial. Thus, we must be concerned with four different defect processes2 simultaneously:

1. The dominant intrinsic structural disorder process (e. g., Schottky or Frenkel).

2. The intrinsic electronic disorder reaction.

3. The REDOX reaction.

4. Dopant and impurity effects.

Again we begin by considering MgO.6 If we ignore impurity effects, the three reactions are2

MgMg+og ! v(Mg+VO*+MgO Ks = [vmJ [vo*]

Null! e’ + h* Kele = [e’] [h*]

Og! І02 + VO* + 2e’ Kredox = pO=2 [e’]2 [vo*]

These equations contain six unknown quantities: four are defect concentrations, the other two vari­ables are the Po2 and the temperature, which are experimental variables and are thus given. Of course, we must know the enthalpies of the defect reactions. Nevertheless, to solve these equations simultaneously, we need a further relationship. This is provided by the electroneutrality condition, which, for MgO states that2

2[vm J +[e’]= 2[VO*] + [h*]

To make the problem more tractable, we now intro­duce the Brouwer approximations, which simplify the form of the electroneutrality condition. These effectively concern the availability of defects via the partial pressure of oxygen. For example, if the Po2 is very low, the REDOX reaction equilibrium will require that the [VO*] and [e’] concentrations are relatively high so that these are the dominant positive and negative defect concentrations. Therefore, for low Po2,2

[e’] = 2 [VO*]

Conversely at high Po2, both oxygen vacancies and their charge-compensating electrons must have rela­tively low concentrations and therefore, the electro­neutrality condition becomes dominated by the [VmJ and [h*] defects so that2

[h*]= [VMg]

Between these two regimes, the Brouwer approxima­tion depends on whether structural or electronic defects dominate. In the case of MgO, we know that Schottky disorder dominates over electronic disorder (as it is a good insulator) and therefore, at intermedi­ate values of Po2, the appropriate electroneutrality condition is

[VO*] = [VMg]

If the electronic disorder was dominant, this last reaction would be replaced by

[e’] = [h*]

We are now in a position to be able to construct a Brouwer diagram, which is usually in the form of ln (defect concentration) versus lnPO2 for various defect components at a constant temperature. In the case of MgO, as indicated above, the diagram will clearly

Neutrality condition

2[V0] = [e’j [V0′] = [V M] [H]=2[VM]

have three regimes corresponding to the three Brouwer conditions (refer to Figure 18 and Chiang

et at2).