Environmental Engineering Reference
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where O sublattice remains undisturbed. Now, according to the law of mass action
one can write the following expression:
G 2
kT
S 2
k
H 2
kT
[V x M ][M i ]
K F
K 2 exp
K 2 exp
exp
(5.11)
Some of the various point defects may also be ionized according to the following
reaction:
V x M
V
M
h
(5.12)
That is, metal vacancies are acceptors that take up electrons from the valence
band and inject positive holes to the valence band. Applying law of mass action
in Eq. 5.12, one can write the following expression:
[V
M ]
p
K
2
(5.13)
[V x M ]
where p represents the concentration of positive holes in the valence band. Simi-
larly,
M i
M i
e
(5.14)
That is, interstitial metal atoms are donors that donate electrons to the conduction
band. Hence one can write:
[M i ]
n
K
3
(5.15)
[M i ]
where n represents the concentration of excess electrons in the conduction band.
For a metal oxide (MO) exhibiting Frenkel disorder, the defect equilibrium can
also be written as:
1
2 O 2 (g)
M i
MO
(5.16)
for which
K R
[M i ] P 1/2
(5.17)
O 2
Moreover, the concentrations of excess electrons in the conduction band and
positive holes in the valence band are interrelated through intrinsic ionization in
the manner:
Null
e
h
(5.18)
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