
In global equilibrium all crystals
contain point defects with a concentration c_{PD} given
by an Arrhenius expression of the form: 

c_{PD}^{ } 
= A · exp – 
G_{F}
kT 
= A · exp 
S_{F}
k 
· exp – 
H_{F}
kT 




A is a constant around
(1 ....10), reflecting the geometric possibilities to introduce 1
PD in the crystal (A = 1 for a simple vacancy). 



G_{F},
H_{F}, S_{F} are the free energy of
formation, enthalpy (or colloquial "energy") of formation, and
entropy of formation, respectively, of 1 PD 



The entropy of formation reflects the disorder
introduced by 1 PD; it is tied to the change in lattice vibrations
(circle frequency w) around a PD and
is a measure of the extension of the PD. It must not be confused with
the entropy of mixing for many PDs! 

S_{F}^{ } 
= k · 
å
i 
ln 
w_{i}
w'_{i} 




Formation enthalpies are roughly around 1
eV for common crystals ("normal" metals"); formation
entropies around 1 k. 







Small PD clusters (e.g.
divacancies) are still seen as PDs, their concentration follows from
the same considerations as for single PDs to: 

c_{2V} 
= 
z
2 
· exp 
S_{2V}
k 
· 
exp – 
H_{F(2V)}
kT 
c_{2V} 
= 
z
2 
· c_{1V}^{2} ·
exp 
DS_{2V}
k 
· 
exp – 
E_{2V}
kT 




The constant A for divacancies is
half the coordination number z (= number of possibilities to
arrange the axis of a divacancy
dumbbell) 



The formation enthalpy and entropy of a PD
cluster can always be expressed as the sum of these parameters for single
PDs minus a binding enthalpy E and a binding entropy
DS 



The term c_{1V}^{2}
or c_{1V}^{n} for a cluster of
n vacancies makes sure that the concentration of clusters is
always far smaller than the concentration of single PDs. 







The same relations can be obtained by
"making" divacancies (or any cluster) by a "chemical"
reaction between the PDs and employing the mass action law: 

1V + 1V Û V_{2V}
+ E_{2V} 
(c_{1V})^{2}
c_{2V} 
= 
K(T) = 
const · exp – 
DE
kT 




There are, however, some pitfalls in using the
mass action law; we also loose any information about the factor
A 



Most important in doing "defect
chemistry" with mass action, is a proper definition of the
"ingredients" to chemical reaction equations. A vacancy, after all,
is not an entity like an atom that can exist on its own. More to that in
chapter 2.4. 







Note: All of the above is generally
valid for all independent PDs: "A" and
"B" vacancies, interstitials, antisite defects,. ... . 

c_{V}^{ } 
= A_{V} · 
exp 
S_{F}^{V}
k _{ }^{ } 
· exp – 
H_{F}^{V}
kT _{ }^{ } 
c_{i}^{ } 
= A_{i} · 
exp 
S_{F}^{i}
k_{ }^{ } 
· exp – 
H_{F}^{i}
kT_{ }^{ } 




However: If
there are additional restraints (like
charge neutrality), we may have to consider pairs of (atomic) PDs as one
point defect; e.g. Frenkel or Schottky defects. 



First principle" calculation show that
charge neutrality can only be locally violated on length scales given by the
Debye length of the crystal. 






Frenkel and Schottky defects are
vacancy interstitial or vacancy^{–} vacancy^{+} pairs in
ionic crystals. 

Frenkel defect: 

V^{–} +
i^{+} 
AntiFrenkel defect: 

V^{+} +
i^{} 
Schottky defect: 

V^{–} +
V^{+} 

Frenkel disorder in: 

AgCl, AgBr, CaF_{2}, BaF_{2},
PbF_{2}, ZrO_{2}, UO_{2}, ... 
Schottky disorder in: 

LiF, LiCl, LiBr, NaCl, KCL, KBr, CsI, MgO,
CaO, ... 




They are extreme cases of the general "mixed
defect case" containing all possible PDs (e.g.
V^{–}, V^{–}, i^{+}, i^{+})
while maintaining charge neutrality. 



Usually, one finds either Frenkel defects or
Schottky defects  if the respective formation enthalpies
H_{Fre} or H_{Scho} differ by some
0.1 eV, one defect type will dominate. 



It is, however, hard to predict the dominating
defect type from "scratch". 
