|
The basic equation for the
concentration of Johnson complexes is |
|
|
|
|
|
cC |
= |
z · cF · cV
1 z · cF |
· exp |
ΔSC
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. |
|
|
The given concentration of impurity atoms of 1
% and 1 ppm correspond to cF =
102 and cF =
106, respectively. |
|
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. |
|
|
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 |
ΔSC
k |
· exp |
HC
kT |
= |
(12 ... 13,6) · exp |
(HFV GB)
kT |
cC
cV |
= |
(12 ... 13,6) · cF |
· exp |
ΔSC
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 |
|
|
|
|
|
|
What does it mean? |
|
|
First, for
extreme repulsion, we simply do not form
Johnson complexes as we would expect. |
|
|
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). |
|
|
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. |
|
|
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. |
|
|
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. |
|
|
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. |
|
|
|
© H. Föll