 

What have we gained by this? We now can describe all kinds of structure elements
 atoms, molecules and defects  and their reactions in a clear and unambiguous way relative to
the empty space. Lets look at some examples 


Formation of Frenkel defects in, e.g., AgCl: 
 
Ag_{Ag} + V_{i} = V^{/}_{Ag} + Ag
^{·}_{i} 




We see why we need the slightly strange construction of a vacancy on an interstitial site. 


Formation of Schottky defects for an AB crystal

 
^{ }A_{A} + B_{B} 
= 
V^{/}_{A} + V ^{·}_{B}
+ A_{A} + B_{B} 
^{ }A_{A} + B_{B} 
=  V^{/}_{A} + V_{B}
^{·} + AB  




The second equation simply considers the two dislodged atoms as a molecule that must be put somewhere. 

This looks good. The question is, if we now
can use the mass action law to determine equilibrium concentrations.
If the Frenkel defect example could be seen as analogous to the chemical reaction A + B = AB, we could write a mass
action law as follows: 
 
[Ag_{Ag}] · [V_{i}] [V^{/}_{Ag}] · [Ag_{i}
^{·}]  = const 




with [A] meaning "concentration of A". The reaction
constant is a more or less involved function of pressure p and temperature T, and especially
the chemical potentials of the particles involved. 
 
Unfortunately, this is wrong! 



Why? Well, the notion of chemical equilibrium and thus the mass action law, at the normal conditions of constant temperature
T and pressure p, stems from finding the minimum of the free
enthalpy
G (also called Gibbs energy) which in our case implies the equality of
all chemical potentials. You may want to read up a bit on the concept of chemical
potentials, this can be done in the link. 


In other words, we are searching for the equilibrium concentration of the particles n_{i}
involved in the reaction, which, at a given temperature and pressure, lead to dG = 0. 


The equation dG = 0 can always be written as a total
differential with respect to the variables dn_{i}: 
 
dG  =  
¶G
¶n_{1} 
· dn_{1} + 
¶G
¶n_{2} 
· dn_{2} + ... 




The partial derivatives are defined as the chemical
potentials of the particles in question and we always have to keep in mind that the long version
of the above equation has a subscript at every partial derivative, which we, like many others, conveniently "forgot".
If written correctly the partial derivative for the particle n_{i} reads (in HTML somewhat awkwardly), 
 
¶G
¶n_{i}  ÷
÷ 
p, T, n_{j ¹ i} = const 




Meaning that T, p, and all other
particle concentrations must be kept constant. 

Only if that condition is fulfilled, a mass action equation
can be formulated that involves all particles present in the reaction equation! And fulfilling the condition means that
you can  at least in principle  change the concentration of any kind of particle (e.g.
the vacancy concentration) without
changing the concentration of all the other particles. 


This "independence condition"
is automatically not fulfilled if we have additional constraints which link some of
our particles. And such constraints we do have in the KrögerVink notation, as
alluded to before! 


There is no way within the system to produce a vacancy, e.g. V_{A} without
removing an Aparticle, e.g. generating an A_{i} or adding another Bparticle, B_{B}. 

S... ! We now have a very useful way of describing chemical reactions, including
all kinds of charged defects, but we cannot use simple thermodynamics! That is the point where other notations come in. 

You now may ask:
Why not introduce a notation that has it all and be done with it? 


The answer is: It could be done, but only by losing simplicity in describing
reactions. And simplicity is what you need in real (research) life, when, in sharp contrast to text books, you do not know what is going on, and you try to get an answer by mulling over various possibility in
your mind, or on a sheet of paper. 

So "defectsinceramics" people live with several kinds of notation,
all having pro and cons, and, after finding a good formulation in one notation, translate it to some other notation to get
the answers required. We will provide a glimpse of this in the next subchapter. 