Here are some quick questions:  
The answers are sometimes (and possibly only indirectly) contained in the links.  
Let's look at combined defects  double vacancies, impurity atom  vacancy complex (and so on):  
Derive from the mass action law and
write down the essential equations for the
concentrations of


Now let's do something important. You really should do this, it will teach you a lot!  
Make sketches
of various concentrations in an Arrhenius plot. Try to produce intellegent and
neat sketches with parameters as follows:


First, produce one Arrhenius diagram showing single and double vacancies.  
Second, produce an Arrhenius diagram for single vacancies, impurities, and impurity atom  vacancy complex  
Third, produce one Arrhenius diagram showing single and double vacancies but assume that the single vacancy concentration cannot decline anymore at some lower temperature.  
Discuss you curves (in particular the 2nd and 3rd), take into account how the temperature changes in "theory" and in "real life"  
Now a few really quick ones:  
If all vacancies present at thernal equilibrium near the melting point at a concentration of c_{V} » 10^{–4} end up in vacancy clusters with an average of 100 vacancies, what is the concentration of these clusters? What is their average cluster distance compared to the average vacancy distance (assume a typical lattice constant around 0.3. nm)?  
Given a equilbrium vacancy concentration of c_{V}, an (substitutional) impurity concentration c_{F}, and some binding enthalpy and entropy H_{C} and S_{C}, the concentration c_{C} of vacancy  (substitutional) impurity complexes should be proportional to:........?  
What would you expect for the case of no binding enthalpy and entropy?  
2.2.1 Extrinsic Point Defects and Agglomerates
© H. Föll (Defects  Script)