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The Kröger-Vink notation defined
structure elements - atoms, molecules,
point defects and even electrons and holes relative to empty space. Despite the problem with the
inapplicability of the mass action law, this notation is in use throughout the
scientific community dealing with point defects. |
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The other important notation - the
"Schottky
notation" or
"building element
notation" is defined as follows: |
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Defects are defined relative to the perfect crystal. |
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Charges are notated as in the Kröger-Vink way, i.e.
relative to the perfect crystal. We again
use the "·" for positive
(relative) charge and the "/" for (relative) negative
charge. |
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To make things a bit more
complicated, there are two ways of writing
the required symbols, the "old" and the "new" Schottky notation. |
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The "old" Schottky notation used special graphical
symbols, like black circles or squares which are not available in HTML
anyway. |
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So we only give the new Schottky notation in direct comparison with the
Kröger-Vink notation, again for the example NaCl with Ca
impurities, i.e.
A = Na+, B = Cl, C = Ca++. |
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A on B site |
A-vacancy |
A-interstitial |
Schottky
(new) |
Na|Cl| ·· |
|Na|/ |
Na · |
Kröger-Vink |
NaCl ·· |
VNa/ |
Nai · |
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So far, the difference between the
Schottky notation and the Kröger-Vink notation seems superficial. The
important difference, however, becomes clear upon writing down defect
reactions. Lets look at the formation of Frenkel and Schottky defects in the
two notations. |
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Frenkel defects |
Schottky defects |
Schottky
(new) |
|Ag|/ + Ag · = 0 |
|Ag|/ + |Cl|· + AB = 0 |
Kröger-Vink |
AgAg + Vi = V/Ag + Ag
·i |
AgAg + ClCl = V/Ag + V
·Cl + AB |
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In words, the Schottky notation
says: |
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For Frenkel
defects: A negatively charged Ag vacancy plus a positively
charged Ag interstitial gives zero. |
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For Schottky
defects: A neatively charged Ag vacancy plus a positively
charged Cl vacancy plus a AgCl "lattice molecule" gives
zero. |
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This is clear enough for these simple
cases, but not as clear and easy as the Kröger-Vink notation. |
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But, and that is
the big advantage, we can apply the mass action law directly to the reactions
in the Schottky notation. |
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This is not directly obvious. After all, Frenkel
defects, e.g., do not only appear to be
linked (where there is an interstitial, there is also a vacancy), but actually
are linked if the defects are charged
(otherwise there would be neither net charge in the crystal, or we would have
to invoke electrons or holes to compensate the ionic charge - but then we would
have to include those into the reaction equation). |
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Theoretically, however, you can introduce one more
vacancy or one more interstitial into a crystal with a given concentration of
each and look at the change of the free enthalpy, i.e. the chemical potential
of the species under consideration. The independence condition does not require that it is
easy to change individual concentrations,
only that it is possible! |
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If you do neglect the energy associated with
charge (i.e. you look at the chemical and not the electrochemical potential),
the answers you get will not contain the coupling between the defects and you
have to consider that separately. We will see how this works later. |
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Now, why don't we use just the
Schottky notation and forget about Kröger-Vink? We
asked that question before; the
answer hasn't changed: If we look at more complicated reactions, e.g. between
point defects in an ionic crystal, a gas on its outside, and with electrons and
holes for compensating charges, it is much
easier to formulate possible reaction in the Kröger-Vink
notation. The trick now is, to convert your reaction equations from the
Kröger-Vink structure elements to the Schottky building elements. There is
a simple recipe for doing this. |
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All we have to do, is to combine the
two structure elements of Kröger-Vink
that refer to the same place in the lattice
and view the combination as a building
element. |
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Lets first look at an example and
then generalize. Consider the Frenkel disorder in AgCl. Using structure
elements, we write |
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Combining the terms
referring to the same place in the lattice (with the actual defects always as
the first term in the combination) yields |
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(V/Ag AgAg) +
(Ag i Vi) |
= |
0 |
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Now all we have to do is to write down the
corresponding Schottky notation and identify the terms in brackets with the
Schottky structure elements. We see that |
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We can generalize
this into a "translation
table": |
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A on B site |
A-vacancy |
A-interstitial |
AB molecule |
C on
B site |
Free
electron |
hole |
All defects
neutral |
Always charged
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Schottky
(new)
Building elements |
A|B| |
|A| |
A |
AB |
C|B| |
e/ |
h · |
Kröger-Vink
Structure elements |
AB |
VA |
Ai |
AB |
CB |
e/ |
h · |
Combined structure elements =
Building elements |
AB - BB |
VA - AA |
Ai - Vi |
AB |
CB - BB |
e/ |
h · |
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We take the Schottky defect as a
fitting elementary example and go through the movements: |
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1. Kröger-Vink structure element equation |
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AgAg + ClCl |
= |
V/Ag + V
·Cl + AB |
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After rearranging (remember, the defect comes first!) so we can use the translation
table, we have |
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(V/Ag AgAg) + (V
· Cl ClCl) +
AB |
= |
0 |
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2. Switch to building elements using the the expressions in
brackets; we have |
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3. Charge
neutrality demands |
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Σ(pos. charge) |
= |
Σ(neg. charge) |
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[|Ag|/] |
= |
[|Cl| ·] |
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4.
The
mass action law now gives
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[|Ag|/] · [|Cl| ·]
[AB] |
= exp |
Σν
i μ i0
kT |
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And this leads to |
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[|Ag|/] |
= |
[|Cl| ·] |
= [AB] · exp |
Σ ν i
μ i0
kT |
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With μ
i = standard chemical potentials of the two vacancies and a lattice
molecule, resp., and ν i =
stoichiometric coefficients of the reaction (1,1, and
1 in our case). |
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This sure looks strange compared to
the formula derived in the
"physical" way. But it is the
same. Lets see why. |
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First, the activity (or concentration) of the
lattice molecule AB is simply [AB] = 1 since it is nothing but
the number of mols of AgCl molecules in one mol of AgCl; i.e. it
is = 1. This gives us |
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[|Ag|/] · [|Cl| ·] |
= exp |
Σ ν
iμ i0
kT |
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Now lets look at the energies in
the exponent. As always, the energy scale is relative. From whatever zero point
you measure your energy to make an Ag vacancy or a Cl vacancy,
you must subtract the energy of the AB molecule as measured in that
system. If you take it to be zero - which then defines the energy origin of
your standard system - the standard chemical potentials of the two vacancies
are just the usual formation energies. |
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Note that
as in treatment given before, the
mass action law alone does not specify the
vacancy concentration, only their product.
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Only invoking the electroneutrality condition,
which demands [|Ag|/] = [|Cl| ·], allows to compute the individual
concentrations. Writing H+ and H for
the formation enthalpies of the positively or negatively charged vacancy,
resp., we obtain the familiar
result |
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[|Ag|/] |
= [|Cl| ·] = exp |
H+ + H
2kT |
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First we
write down the reaction with structure
elements (= Kröger-Vink notation). |
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After rearranging (remember, the defect comes first!) so we can use the translation
table, we have |
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(Ai· Vi) +
(VA/ AA) |
= |
0 |
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2. The expressions in brackets are the
building elements, we have |
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3. Charge
neutrality demands |
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4.
The
mass action law now gives |
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A problem?
What are we going to do with the "0"? Well, there really is no
problem with the zero - just take the
mass action law as it
is |
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P(ai)ni |
= |
exp |
G0
kT |
= K = |
Reaction
Constant |
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Nowhere was it required that in the product there
must be terms with negative stoichiometry coefficients νi. This gives us |
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[A·] · [|A|/] |
= exp |
GF
kT |
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And we identify G0 with the
formation energy GF of a Frenkel pair
as before. |
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Together with the charge neutrality condition we
have |
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[A·] |
= [|A|/] = exp |
GF
2kT |
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almost the
familiar result - except that we do not have the factor
(N/N')1/2. |
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OK - we do have a problem, but
not with the zero. Where did we lose the
factor (N/N')1/2 ? |
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Lets look at equilibrium another way.
We do not involve the mass action law but go one step back to the
equilibrium condition for
the chemical potentials: μA· = μ|A|/
. We write the chemical potentials in the standard form and obtain |
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μA· |
= μ0A· + kT · ln |
nA·
N' |
μ|A|/ |
= μ0|A|/ + kT
· ln |
n|A|/
N |
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For equilibrium we now obtain (if you wonder at
the n/N and n/N',
consult the link) |
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nA· ·
n|A|/ |
= N · N' · exp |
μ0A· + μ0|A|/
kT |
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Charge neutrality tells us that |
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For the concentration of Frenkel pairs
cFP = nFP/N we now
obtain the correct old formula |
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cFP |
= |
( |
N
N' |
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1/2 |
· exp |
μ0A· + μ0
2kT |
= |
( |
N
N' |
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1/2 |
· exp |
GFP
2kT |
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Aha! Applying the mass action law uncritically
causes a problem: The standard chemical potentials of vacancies and
interstitials were for different standard
conditions: |
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In one case (the vacancies) the standard condition was for adding
N vacancies to the system, in the other case (the interstitials) it was for adding
N' = N ·
l interstitials (and l being some factor
taking into account that there are more positions for interstitials than for
vacancies in a crystal). |
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If that appears to be incredibly complicated and
prone to errors - that's because it is! But
take comfort: You get used to it, and working with it is not all that difficult
after overcoming an intial "energy barrier". |
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Many books and other texts to not
dwell extensively on the fine differences between notations, problems with the
standard condition in the chemical potentials, meaning of reaction equations
and so on - they write down a reaction equation, in the worst case a mix of
Kröger-Vink and Schottky notations, throw in electrons or holes right away
to achieve charge neutrality, and write down the mass action law in the
form |
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P
i |
(ai)ni |
= exp |
G
kT |
= K(T) = const · exp
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G
kT |
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And not much attention is given to the constant
K(T) in front of the exponential. |
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Even though it's faulty
thermodynamics, let's see what happens if we do that for Frenkel
defects in the Kröger-Vink notation: |
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The reaction equation
was |
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AgAg + Vi
V/Ag Ag · |
= |
0 |
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The mass action law uncritically applied gives
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[AgAg] · [Vi]
[V/] · [Ag ·] |
= const · exp |
G'
kT |
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As long as the defect concentration is small
compared to the concentrations of atoms and lattice sites, we may simply equate
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Which leaves us with |
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[V/] · [Ag ·] |
= const · exp |
G
kT |
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With G = G', but
that is irrelevant - we simply know that the exponential always has a minus
sign for the reactions we are interested in and that G must be
the formation enthalpy of a Frenkel pair. |
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That is the
correct result, expressed in Kröger-Vink terms. What that means
is that you don't have to worry all that much about the finer details as long
as you are not terribly interested in the exact value of the constant in front
of the exponential - you will mostly get your reaction equation right! |
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Luckily, there are only a few fundamental
reaction equations involving point defects - everything else can be expressed
as linear combinations of the fundamental reactions (like Frenkel and Schottky
defect equilibrium) - after some initiation, you will feel quite comfortable
with defect reactions. |
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As shown above for Frenkel defects,
it is often advisable not to use the mass action law directly, but to go one
step back and use the equilibrium condition for the chemical potentials. This
gives not only a clearer view of what constitutes the standard conditions, but
also circumvents a number of other problems associated with the law of mass
action (if you really want to know, consult the
advanced module accessible
by the link). |
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© H. Föll