
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 


