 |
Calculate the ratio of the
concentration of Schottky to Frenkel defect as a funtion of the enthalpy
difference |
|
|
|
|
|
|
|
The equations for the concentrations
of the point defects in the "mixed" case
are |
|
|
|
|
|
| |
cV(C) |
= cS = exp – |
HS
2kT |
· |
( |
1 + |
N'
N |
· exp |
HS – HFP
kT |
) |
1/2 |
= exp – |
HS
2kT |
· K |
| |
cV(A) |
= exp – |
HS
2kT |
· |
( |
1 + |
N'
N |
· exp |
HS – HFP
kT |
) |
– 1/2 |
= exp – |
HS
2kT |
· K–1 |
| |
ci(C) |
= cFP = |
N'
N |
· exp |
Hs
2kT |
· exp – |
HFP
kT |
· |
( |
1 + |
N'
N |
· exp |
HS – HFP
kT |
) |
– 1/2 |
= |
N'
N |
· exp |
Hs
2kT |
· exp – |
HFP
kT |
· K–1 |
|
|
|
|
|
|
 |
Note that cV(C) or
ci(C) is, by definition, identical to the
concentration cS or cFP of
Schottky or Frenkel defects, respectively. If you have problems with this,
refer to the link. |
|
|
We abbreviate the root of the expression in
square brackets by K for writing efficiency. |
|
The ratio cS
/cFP is easy to obtain. The K's cancel, we
are left with |
|
|
|
|
|
cS
cFP |
= |
N
N' |
· exp – |
(HS – HFP)
kT |
= |
N
N' |
· exp – |
ΔH
kT |
|
|
|
|
|
|
|
That is - of course - what we should have
expected. The concentrations of Schottky and Frenkel defects are independent of
each other and their relation could have been derived straight from the basic
equations defining their equilibrium concentrations. |
|
|
|
|
Discuss the result. Show in in
particular, how large the difference must be if 90% or 99%
of the defects are to be of one kind. |
|
|
|
|
|
|
|
We want to evaluate the equation for
cS/cFP = 0,011 or = 0,001
(prevalence of Frenkel defects) and
cS/cFP = 90 or = 99
(prevalence of Schottky defects). |
|
|
For the difference ΔH of the formation enthalpies as defined above
we obtain |
|
|
|
|
|
| ΔH |
= – |
kT |
· |
( |
ln |
N'
N |
+ ln |
cS
cFP |
) |
|
|
|
|
|
|
|
We have to define a value for
N/N' ; we simply take this relation to be 1 or
0,1 as limiting cases. |
|
Values are easily obtained, we
arrange them in a little table |
|
|
|
|
|
|
|
99 |
90 |
10 |
0,1 |
0,011 |
0,010 |
| ΔH [eV] |
|
–0,115 |
–0,112 |
–0,058 |
0,058 |
0,112 |
0,115 |
|
|
–0,172 |
–0,169 |
–0,115 |
–0,0004 |
0,054 |
0,057 |
|
|
|
|
|
We have an interesting result: If the
formation enthalpies of the two defect kinds differ by just about 1/10
of an eV, we are fully justified to consider one defect kind only. |
|
|
The pre-exponential factor N'/
N which describes the differences in the basic geometry for
interstitials relative to vacancies accounts at most for about 1/20 of
an eV if expressed in enthalpy differences. |
|
|
|
© H. Föll
Point Defects in Ionic Crystals 
To index