
Now you may be tempted to write this down in a first reaction equation
as follows 
 



While this is not necessarily wrong, it is at least strange: You create, in a kind of chemical
reaction, something from nothing  what keeps you from applying this equation to vacuum, which is surely not sensible? Maybe
you should somehow get the crystal involved as the reference system within which things happen? 

So let's device a more elaborate system by looking at our crystal before
and after a Frenkel pair was formed 


Before a Frenkel pair is formed, the site occupied by
the vacancy after the formation process is a Na site, we denote it by Na_{Na}.
This simply means that a Na atom occupies a Na site before a vacancy is formed there. 


At the interstitial site, where the Na interstitial after
the formation process is going to be, you have nothing before the process. However,
all those possible interstitial sites also form a lattice (e.g. the lattice of the octahedral
sites); in a perfect crystal all those sites are occupied by vacancies, we consequently
denote an empty interstitial site by V_{i} = vacancy on an interstitial site. 


A Na ion on an interstitial site then is Na_{i}, and a Na vacancy
becomes V_{Na} . Now we can write down a reaction equation that reads 
 
Na_{Na} + V_{i} Û Na_{i}
+ V_{Na} 



This
looks like a cool reaction equation, we now create a Frenkel pair within a
crystal and not out of thin air. 


Indeed, the reaction equation does look
so much better this way! Small wonder, we just invented part of the socalled KrögerVink notation, in use since the fifties of the 20th century  not all that long ago, actually.



This notation is also called notation by structure elements
and it is very useful for formulating all kinds of reactions involving point defects. However, the first
law of economics applies ("There is no such thing as a free lunch"): 
 
Don't use the mass
action law uncritically with these kinds of reaction equations!. The reason for this is simple,
but usually never mentioned in the context of chemical reaction formulation: 
 
A proper reaction equation contains only
reaction partners that are independent 



This means that you can, in principle, change the concentration of every
reaction partner without changing the others. 

Consider for example the following purely chemical simple reaction equation: 
 
2H_{2} + O_{2} Û_{ }
2 H_{2} O 




You can put arbitrary amounts of all three reaction partners in a container and change any
individual amount at will without changing the others. 

In our reaction equation for point defects, however, you cannot
do this. If you consider, e.g., to change the Na_{Na} concentration a little, you automatically
change V_{i}, too  those quantities are not independent! 


This was the bad news about using KrögerVink notation. The good news are: In most practical
cases it doesn't matter! Chapter 2.4  often alluded to  will contain details about all of this.


It is not easy to grasp the reaction equation concept for point defects in all
its complexity, but it is worthwhile if you want to dig deeper into point defects. For the purpose of this paragraph let's
just postulate that the two sums left and right of the reaction equation would constitute
the proper reactants (those sums, by the way, are called building elements in the
Schottky notation). 


Be that as it may, we now apply the mass action law, keeping in mind that
the reaction equation from above in full splendor actually contains a reaction enthalpy G_{Reaction},
i.e.: Na_{Na} + V_{i} + G_{Reaction} Û Na_{i}
+ V_{Na}) 
 
[Na_{Na}] · [V_{i}] [Na_{i}] · [V_{Na}] 
= const = exp 
G_{Reaction} kT 
[Na_{i}] · [V_{Na}]  = 
[Na_{Na}] · [V_{i}] 
· exp – 
G_{Reaction} kT 




G_{Reaction}, of course, is the free enthalpy change of the crystal
upon the formation of one mol of Frenkel pairs. If we relate it to 1
Frenkel pair, it becomes H_{FP}. 


The [...] are the molar concentrations of the respective
quantities if we use molar reaction enthalpies. 


OK, now let's spell it out. If we have a crystal with N
mols of NaCl, we have the molar concentration
of [Na_{Na}] = N for really obvious reasons. 


[V_{i}] = N, most likely, will hold, too, but here we may have to dig
deeper. How many different places for interstitials do we have in the given unit cell? We can figure it out, but for the
sake of generality their molar concentration could be larger or N'  so it can be different in principle from
N as we have seen before. 

Since we usually go for atomic concentrations, we note that c_{V}(C)
= atomic concentration of the cationvacancy = [V_{Na}] /N and
c_{i}(C) =
atomic concentration of the cationinterstitial = [Na_{i}] /N
we now obtain one equation for the two unknowns
c_{V}(C) · and c_{i}(C), which will be the first
of the equations we will need for what follows. 


c_{V}(C) · c_{i}(C) 
=  N' N 
· exp – 
H_{FP} kT_{ } 
 (1) 


Note that this is
not our old result, because it does not imply
that c_{V} = c_{i}. All the mass action law can do is to supply one
equation for whatever number of unknowns. 

We need a second independent equation. This is  of course (?)  always electroneutrality.
Looking just at Frenkel pairs, we have directly 


c_{V}(C)  = 
c_{i}(C)   
 for Frenkel Pairs only 




Now we have two equations for two unknown concentrations that we could easily solve. 

However, we are interested in mixed defects
here, so we must also consider Schottky defects and then mix them with Frenkel defects,
always maintaining electroneutrality. 

We might now go through the same procedure as before by using
a similar reaction equation for Schottky defects  with a few more complications in finding the proper reaction equation.
We will not do this here (do it yourself or use the link), just note the rather
simple result: 


With c_{V}(A) and c_{V}(C)
denoting the vacancies on the anion or cation sublattice, resp., and with H_{S} = formation enthalpy
of a Schottky pair, we obtain for a second equation 
 
c_{V}(A) · c_{V}(C) 
= exp – 
H_{S} kT_{ } 
 (2) 


Again, this is not the old equation
for Schottky defects  the concentrations are not necessarily equal once more 


Note that the vacancies on the anion or cation sublattice are positively
or negatively charged  opposite to the charge of the (negatively
charged) anion or (positively charged) cation that was removed! A cation
vacancy thus carries a negative charge and so on, whereas a cation interstitial carries a positive charge. Look at the
illustrations if you are not clear about this! 

Knowing that electroneutrality has to be maintained (look at the direct calculation for Schottky defects), we introduce electroneutrality now for the more general
case of our three charged defects: The sum of all charges on the point defects must
be zero; we obtain the third equation 


c_{V}(C)  = 
c_{V}(A) + c_{i}(C) 
 (3) 



Or:
Sum over all negative charges = Sum over all positive charges. 

Now we have 3 equations for 3 unknown concentrations,
which can be solved with ease (haha). We obtain for the general situation of mixed defects 
 
 c_{V}(C) 
= exp – 
H_{S} 2kT 
· 
æ ç è 
1 +  N' N 
· exp 
H_{S} – H_{FP}
kT  ö ÷ ø
 1/2 
 c_{V}(A) 
= exp – 
H_{S} 2kT 
· 
æ ç è 
1 +  N' N 
· exp 
H_{S} – H_{FP}
kT  ö ÷ ø
 – 1/2 
 c_{i}(C) 
=  N' N 
· exp 
H_{s} 2kT 
· exp – 
H_{FP} kT  ·
 æ ç è 
1 +  N' N 
· exp 
H_{S} – H_{FP}
kT  ö ÷ ø
 – 1/2 




These equations contain the "pure" Frenkel and Schottky case as limiting cases. 

Was that worth the effort? Probably not  as long as you just look at simple ionic
crystals (where one defect type will prevail anyway). being in simple equilibrium without considering surfaces and the environment. 


However! In real life, where point defects in ionic (and
oxide) crystal are used for sensor applications, this kind of approach
is the only way to go! It will be far more complicated, there will be approximations
and "shortcuts", but the basic kind of reasoning will be the same. 

Now it is time for an exercise 
 
