Equilibrium Concentration of the Vacancy-Impurity Atom Complex

We consider a complex of one vacancy and one foreign atom (a Johnson complex) in thermodynamic equilibrium.
We start with a number nF of foreign atoms that is given by external circumstances. Some, but not necessarily all of these atoms will form a complex with a vacancy. The number of these complexes we call nC.
We can calculate the equilibrium number of Johnson complexes exactly analogous to the equilibrium number of vacancies by simply defining a formation enthalpy GC and doing the counting of arrangements and minimization of the free enthalpy procedure.
This obviously will give us
 
nC  = (NCnC) · exp – GC
kT 
 
With NC = number of sites in the crystal where a vacancy could sit in order to form a Johnson complex.
We take NCnC in full generality because the places already occupied ( = nC) are no longer available, and we do not assume at this point that nC << NC applies as in the case of vacancies.
NC, of course, is not the number of atoms of the crystal as in the case of vacancies, but roughly the number of foreign atoms - after all, only where we have a foreign atom, can we form a complex.
If we don't look at the situation roughly but in detail, we need to consider that there are as many possibilities to form a foreign atom - vacancy pair as there are nearest neighbors. We thus find
 
NC  = nF · z  –  nC · z
 
z is the coordination number of the lattice considered, i.e. the number of nearest neighbors. Again we do not neglect the places already taken, i.e. we subtract nC · z from the total number of places.
If we look at concentrations, we refer the numbers to the number of lattice atoms N which gives us for the concentration cC of impurity atom - vacancy complexes
 
cC  =  nC
N

cC
cFcC
 = z · exp – GC
kT
 
We are essentially done. We have the concentration of Johnson complexes as a function of the concentration of the foreign atoms, the lattice type (defining z) and their formation enthalpy.
However, we would feel happier, if we could base the equation on material parameters which we already know - in particular on the equilibrium concentration of vacancies in the given material.
This needs a closer view on the formation enthalpy of the complex.
As in the case of double vacancies, we may simply assume that there is a binding enthalpy between a vacancy and a foreign atom (otherwise there would be no driving force to form a complex in the first place).
We thus can write for GC
 
GC  =  GVF  –  (HCT · DSC)
 
GVF is the free enthalpy of vacancy formation, HC is the binding enthalpy of a Johnson complex, and T · DSC is the "association entropy" of the complex, accounting for the entropy change of the crystal upon the formation of a complex.
Inserting this in the equation above gives for the concentration of Johnson complexes in terms of vacancy parameters and binding energies:
 
cC
cFcC
 =  z · exp – GVF
kT 
 ·  exp HC
kT
 ·  exp DSC
k

cC  =  (cFcC) · cV · z · exp  HC
kT 
 ·  exp  DSC
k 
             
    = c'F · cV · z · exp  HC
kT 
 ·  exp  DSC
k 
 
We used the familiar equation cV = exp – (GVF/ kT) to get this result.
We abbreviated the difference of the total concentration of foreign atoms and the concentration of Johnson complexes by c'F; i.e. c'F = (cFcC) because this allows a simple interpretation of the equation.
The point now is to recognize that c'F is nothing but the concentration of foreign atoms which are still available for a reaction with a vacancy, and that the last equation therefore is nothing but the mass action law written out for the reaction
 
1F + 1V  Û  1C
 
With F = (available) foreign atom; V = vacany, and C = Johnson complex.
Looking closely (= thinking hard) you will notice that we now have a certain inconsistency in our book keeping:
We always took into account that Johnson complexes already formed can not be neglected in counting possibilites, and we always corrected for that by using cFcC and so on - but we did not correct for the now more limited possibilities for positioning a single vacancy. We must ask ourselves if the presence of foreign atoms will change the equilibrium concentration of free vacancies.
In other words, while we took the number of available positions for a vacancy in a complex to be nF · znC · z, we implicitly took the number of available positions for a free vacancy in the crystal to be simply N = number of lattice atoms.
Being more precise, we have to subtract nF · z from N because nF · z positions are, after all, not available for free vacancies. We thus have to replace N by N' = NnF · z when we consider the number of free vacancies.
The concentration of the free vacancies thus becomes cV = (1 – z · cF) · exp – (GVF/ kT), or exp – (GVF/ kT) = cV / (1 – z · cF)
Using this in the equation for the concentration yields
 
cC  =  (cFcC) · cV · z
(1 – z · cF)
· exp  HC
kT 
 ·  exp  DSC
k 
             
   »  cF · cV · z
(1 – z · cF)
· exp  HC
kT 
 ·  exp  DSC
k 
 
The last approximation is, of course, attainable if cC << cF, and that is the equation given in the backbone text.
 

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