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We consider a complex of one vacancy
and one foreign atom (a Johnson complex) in thermodynamic equilibrium. |
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We start with a number nF of foreign atoms
that is given by external circumstances. Some, but not necessarily all of these
atoms will form a complex with a vacancy. The number of these complexes we call
nC. |
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We can calculate the equilibrium
number of Johnson complexes exactly analogous to the equilibrium number of
vacancies by simply defining a formation enthalpy GC
and doing the counting of arrangements and minimization of the free enthalpy
procedure. |
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This obviously will give us |
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nC |
= (NC nC) · exp
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GC
kT |
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With NC = number of
sites in the crystal where a vacancy could sit in order to form a Johnson
complex. |
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We take NC
nC in full generality because the places already occupied
( = nC) are no longer available, and we do not assume at this point that
nC << NC applies as in the
case of vacancies. |
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NC, of
course, is not the number of atoms of the
crystal as in the case of vacancies, but roughly the number of foreign atoms - after all,
only where we have a foreign atom, can we form a complex. |
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If we don't look at the situation roughly but in detail, we need to consider that
there are as many possibilities to form a foreign atom - vacancy pair as there
are nearest neighbors. We thus find |
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z is the coordination number of the
lattice considered, i.e. the number of nearest neighbors. Again we do not
neglect the places already taken, i.e. we subtract nC
· z from the total number of places. |
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If we look at concentrations, we refer the numbers to the number
of lattice atoms N which gives us for the concentration
cC of impurity atom - vacancy complexes |
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cC
cF cC |
= z · exp |
GC
kT |
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We are essentially done. We have the
concentration of Johnson complexes as a function of the concentration of the
foreign atoms, the lattice type (defining z) and their formation
enthalpy. |
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However, we would feel happier, if we could base
the equation on material parameters which we already know - in particular on
the equilibrium concentration of vacancies in the given material. |
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This needs a closer view on the formation
enthalpy of the complex. |
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As in the case of double vacancies,
we may simply assume that there is a binding enthalpy between a vacancy and a
foreign atom (otherwise there would be no driving force to form a complex in
the first place). |
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We thus can write for
GC |
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GC = GVF
(HC T · ΔSC) |
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GVF is the
free enthalpy of vacancy formation, HC is the
binding
enthalpy of a Johnson complex, and T · ΔSC is the "association
entropy" of the complex, accounting for the entropy change of the
crystal upon the formation of a complex. |
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Inserting this in
the equation above gives for the concentration of Johnson complexes in terms of
vacancy parameters and binding energies: |
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cC
cF cC |
= z · exp |
GVF
kT |
· exp |
HC
kT |
· exp |
ΔSC
k |
cC |
= |
(cF cC) ·
cV · z · |
exp |
HC
kT |
· exp |
ΔSC
k |
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= |
c'F · cV · z
· |
exp |
HC
kT |
· exp |
ΔSC
k |
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We used the familiar equation
cV = exp (GVF/
kT) to get this result. |
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We abbreviated the difference of the
total concentration of foreign atoms and the concentration of Johnson complexes
by c'F; i.e. c'F =
(cF cC) because this allows a
simple interpretation of the equation. |
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The point now is to recognize that
c'F is nothing but the concentration of foreign atoms
which are still available for a reaction with a vacancy, and that the last
equation therefore is nothing but the mass action law written out for the
reaction |
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With F = (available) foreign atom;
V = vacany, and C = Johnson complex. |
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Looking closely (= thinking hard) you
will notice that we now have a certain inconsistency in our book keeping: |
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We always took into account that Johnson
complexes already formed can not be
neglected in counting possibilites, and we always corrected for that by using
cF cC and so on - but we did
not correct for the now more limited
possibilities for positioning a single vacancy. We must ask ourselves if the
presence of foreign atoms will change the equilibrium concentration of free
vacancies. |
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In other words, while we took the number of
available positions for a vacancy in a complex to be nF
· z nC · z, we implicitly
took the number of available positions for a free vacancy in the crystal to be
simply N = number of lattice atoms. |
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Being more precise, we have to subtract
nF · z from N because
nF · z positions are, after all, not available for free vacancies. We thus have to replace
N by N' = N nF ·
z when we consider the number of free vacancies. |
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The concentration of the free vacancies thus
becomes cV = (1 z ·
cF) · exp (GVF/
kT), or exp (GVF/ kT)
= cV / (1 z · cF)
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Using this in the equation for the concentration yields |
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cC |
= |
(cF cC) ·
cV · z
(1 z · cF) |
· exp |
HC
kT |
· exp |
ΔSC
k |
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» |
cF · cV · z
(1 z · cF) |
· exp |
HC
kT |
· exp |
ΔSC
k |
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The last approximation is, of
course, attainable if cC <<
cF, and that is the equation given in the
backbone text. |
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© H. Föll