


This calls for a little exercise 




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. 