
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
G_{C} 


G_{C} = G^{V}_{F}
– (H_{C} – T · DS_{C}) 




G^{V}_{F} is the
free enthalpy of vacancy formation, H_{C} is the
binding
enthalpy of a Johnson complex, and T · DS_{C} 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: 


c_{C}
c_{F} – c_{C} 
= z · exp – 
G^{V}_{F}
kT_{ } 
· exp 
H_{C}
kT 
· exp 
DS_{C}
k 
c_{C} 
= 
(c_{F} – c_{C}) ·
c_{V} · z · 
exp_{ } 
H_{C}
kT_{ } 
· exp_{ } 
DS_{C}
k_{ } 








= 
c'_{F} · c_{V} · z
· 
exp_{ } 
H_{C}
kT_{ } 
· exp_{ } 
DS_{C}
k_{ } 




We used the familiar equation
c_{V} = exp – (G^{V}_{F}/
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} =
(c_{F} – c_{C}) 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 





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
c_{F} – c_{C} 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 n_{F}
· z – n_{C} · 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
n_{F} · z from N because
n_{F} · z positions are, after all, not available for free vacancies. We thus have to replace
N by N' = N – n_{F} ·
z when we consider the number of free vacancies. 


The concentration of the free vacancies thus
becomes c_{V} = (1 – z ·
c_{F}) · exp – (G^{V}_{F}/
kT), or exp – (G^{V}_{F}/ kT)
= c_{V} / (1 – z · c_{F})



Using this in the equation for the concentration yields 


c_{C} 
= 
(c_{F} – c_{C}) ·
c_{V} · z
(1 – z · c_{F}) 
· exp_{ } 
H_{C}
kT_{ } 
· exp_{ } 
DS_{C}
k_{ } 








» 
c_{F} · c_{V} · z
(1 – z · c_{F}) 
· exp_{ } 
H_{C}
kT_{ } 
· exp_{ } 
DS_{C}
k_{ } 




The last approximation is, of
course, attainable if c_{C} <<
c_{F}, and that is the equation given in the
backbone text. 

