 |
The basic equation for the concentration of Johnson complexes is |
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| cC | = |
z · cF · cV
1 – z · cF |
· exp |
DSC
k |
· exp |
HC
kT |
|
|
|
 |
We first need to chose a coordination number, we take z = 12 for fcc
and hcp crystals. All other coordination numbers are smaller; we thus have the maximal effect of z. |
|
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The given concentration of impurity atoms of 1 % and 1 ppm correspond to cF
= 10–2 and cF = 10–6, respectively. |
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First we note that the factor 1 – z · cF
equals 0,88 or 0,999..; i.e. we can forget it - at least for the low concentration. |
|
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Next we calculate the ratios cC / cF and cC
/ cV in order to get a feeling how the Johnson complex concentration relates to the (fixed) concentration
of impurity atoms and the (temperature dependent) equilibrium concentration of vacancies. We have |
| |
cC
cF |
= | (12 ... 13,6) · cV |
· exp |
DSC
k |
· exp |
HC
kT | = |
(12 ... 13,6) · exp – |
(HFV – GB)
kT
|
cC
cV |
= | (12 ... 13,6) · cF |
· exp |
DSC
k |
· exp |
HC
kT | = |
(12 ... 13,6) · cF | · exp |
GB
kT |
|
|
|
|
The numbers in the bracket span the range of the given cF concentrations. |
|
Our first result thus is simple: The ratios asked for are directly proportional
to the concentration of vacancies or foreign atoms, respectively. The proportionality factor is about 2 times the
Boltzmann factor of the free enthalpy of complex formation. So let's look at the role of the binding energy. |
|
|
Let's look at binding energies (more precisely: binding free enthalpies GB)
of – ¥ eV (i.e. extreme repulsion between a vacancy and the foreign atom),
0 eV (no interaction), ½ HFV (strong interaction), and HFV
(extreme interaction). This gives us |
|
|
| GB | |
– ¥ | |
0 | |
½ HFV | |
HFV | |
| | |
| | |
| |
cC
cF |
| 0 | |
» 12cV |
|
» 12 · (cV)½ |
|
» 12 |
| | | |
| | |
| |
| cC
cV | | 0 |
|
» 12cF |
|
» 12 · cF · (cV)–
½ | |
» |
12 · cF
cV |
|
|
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What does it mean? |
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First, for extreme repulsion, we simply do not
form Johnson complexes as we would expect. |
|
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Second, for zero interaction, we form
Johnson complexes just at random - a vacancy just does not care if it sits next to an
impurity atom or not. The concentration thus is directly given by the product of the concentrations of the partners (the
factor 12 just accounts for the 12 different ways to form a Johnson complex with one vacancy). |
|
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Third, for appreciable but not extreme binding energies
the quotient cC / cF is always < 1, because (cv)
½ << 1; it decreases rapidly with temperature. This means that in equilibrium
only a small part of the foreign atoms will form Johnson complexes. |
|
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Fourth, for appreciable but not extreme binding energies
the quotient cC / cV can be >1 or <1, depending on 12cF
being larger or smaller than (cV)½. Below some temperature the vacancy concentration
will always be so low that the ratio is >1, we then have more Johnson complexes
than free vacancies. But that does not mean we have many - just more then the extremely
few vacancies. |
|
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Fifth, for extreme binding energies we have a problem.
The relations given just must be wrong - we cannot for example, have 12 times as many Johnson complexes as we have
foreign atoms. What went wrong? |
|
Well, our starting formula is only valid under
the assumption that cC << cF. This assumption is obviously violated
for binding energies too large; we then must not use the simple formula. |
|
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If we take the correct formula, we simply find that cV times the
exponential vanishes (i.e. cC /cV does not make sense anymore), and cC
/ cF »
z /(1 + z) » 1 under all conditions, as we would expect. |
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© H. Föll (Defects - Script)
Equilibrium Concentration of theVacancy-Impurity Atom Complex 
With frame