Brouwers Approximation Method

Let us consider an MXs crystal with Schottky defects (this is one of the more easily defined types, mathematically). We define S=l. The types of intrinsic defects was given in 2.6.1., the equilibrium constants of intrinsic defects was given in Table 2-1, numbers of intrinsic defects was given in 

ionization of vacancies in 2.6.22, 2.6.23 2.6.24, and 2.6.21. gave the effect of an external factor (X2 gas) on the production of nonstoichiometry. We can rewrite these equilibrium constant equations for the MX compound in the form given in 2.7.1. on the next page. 

This set of equations completely defines the defect concentrations in an MX crystal containing Schottky defects. 

It is important to realize that e is the sum of the electrons, not a single negative charge, i.e.- 

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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. 

The approximations inherent in Brouwer diagrams can be bypassed by writing the appropriate electroneutrality equation as a polynomial equation and then solving this numerically using a computer. (This is not always a computationally trivial task.) To illustrate this method, the examples given in Sections 7.5 and 7.6, the MX system, will be rewritten in this form. 

De Brouwer et al. (2010) identified 5 folate vitamers in their study 5-methyltet-rahydrofolate (the predominant vitamer), 5,10-methenyltetrahydrofolate and tetrahy-drofolate (the most labile vitamers), folic acid, and 10-formyl folic acid. This study reported that folate levels in wild-type rice were approximately 20 pg/lOO g lower than in bio-fortified rice [41], While separation of folate vitamers was obtained in 8 min run time, a previous LC-MS/MS method required 20 min [49]. Also, UPLC improved the sensitivity by reducing the limit of quantitation. Another study documented that endogenous 5-methyltetrahydrofolate in bread ranged between 1.3 and 3.4 pg/100 g [14]. Measurements of folic acid and 5-methyltetrahydrofolate in flour and infant milk formula have been successfully made using this method, described in the latter study. 

We realize that the more the number of defects taken into account in the solid, the more the number of quasi-chemical equilibriums and the more conqtlex the system becomes. Brouwer recommended a method of usually very sufficient approximation. It consists of considering the expression of electric neutrahty in oidy the two majority enhhes (one on each side of the equality). 

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