Calculate the ratio of the concentration of Schottky to Frenkel defect as a funtion of the enthalpy difference  
The equations for the concentrations of the point defects in the "mixed" case are  


Note that c_{V}(C) or c_{i}(C) is, by definition, identical to the concentration c_{S }or c_{FP }of Schottky or Frenkel defects, respectively. If you have problems with this, refer to the link.  
We abbreviated the root of the expression in square brackets by K for writing efficiency.  
The ratio c_{S }/c_{FP} is easy to obtain. The K's cancel, we are left with  


That is  of course  what we should have expected. The concentrations of Schottky and Frenkel defects are independent of each other and their relation could have been derived straight from the basic equations defining their equilibrium concentrations.  
Show in in particular, how large the difference must be if 90% or 99% of the defects are to be of one kind.  
We want to evaluate the equation for c_{S}/c_{FP} = 0,011 or = 0,001 (prevalence of Frenkel defects) and c_{S}/c_{FP} = 90 or = 99 (prevalence of Schottky defects).  
For the difference DH of the formation enthalpies as defined above we obtain  


We have to define a value for N/N' ; we simply take this relation to be 1 or 0,1 as limiting cases.  
Values are easily obtained, we arrange them in a little table  


Discuss the result.  
We have two interesting results:  
If the formation enthalpies of the two defect kinds differ by just about 1/10 of an eV, we are fully justified to consider that only one defect kind is present.  
The preexponential factor N'/ N, which describes the differences in the basic geometry for interstitials relative to vacancies, accounts at most for about 1/20 of an eV if expressed in enthalpy differences.  
Formation Enthalpies and Entropies for Vacnacies an SelfInterstitials
Exercise 2.16: Enthalpy difference for the limiting cases of Schottky or Frenkel Defects
Exercise 2.18 Quick Questions
Point Defects in Ionic Crystals
Formation Enthalpies and Entropies for Frenkel and Schottky Defects
© H. Föll (Defects  Script)