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How do we treat point defects in
perfect analogy to atoms and molecules in chemical reaction equations? A very
clear way was suggested by Kröger and Vink, it is therefore called "Kröger-Vink notation" or
notation by "structure
elements" - we already had a
glimpse of this. |
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We define vacancies and interstitials as particles which
occupy a defined site in a crystal and which may have a charge. |
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Sites in a crystal are the points where the atoms, the
interstitials, or the vacancies can be. For a crystal composed of two kinds of
atoms we have, e.g., the "A-sites" and the
"B-sites". An A-atom on an A-site we denote by
AA, a vacancy on a B-site is a
VB |
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This leaves the interstitials out of the picture. We therefore
simply name all possible interstitials sites with their own place symbol and
write Ai or Bi for an A-atom or a
B-atom, resp., on its appropriate interstitial site. |
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An interstitial site not
occupied by an interstitial atom then, by definition, is occupied by a vacancy
and symbolized by Vi. A perfect crystal in the
Kröger-Vink notation thus is full of vacancies on interstitial sites! |
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In order to facilitate book keeping
with respect to the electrical charge, we only note the excess charge relative to the neutral lattice.
Positive excess charge is marked by a point (e.g. A·), negative charge by a hyphen or dash or
whatever you like to call it (e.g. A/)
to distinguish this relative charge from the absolute charge. If we consider a
positively charged Na+ ion in the NaCl lattice, we
write NaNa as long as it is sitting on its regular lattice
position, i.e. without a charge symbol. If we now consider a vacancy on the
Na-site, the Na-ion as interstitial, or a Ca++
ion on the Na-site, we write |
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V /Na,
Na·i, and Ca·Na because this defines the charge
relative to the neutral lattice. |
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Running through all the possible combinations for our
NaCl crystal with some Ca, we obtain the matrix |
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A atom (Na+) |
B atom (Cl) |
Vacancy |
C atom (Ca++) |
A-site |
NaNa |
Cl / /Na |
V /Na |
Ca ·Na |
B-site |
Na ··Cl |
ClCl |
V · Cl |
Ca ···Cl |
i-site |
Na ·i |
Cl /i |
Vi |
Ca ··i |
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This calls for a little exercise |
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What have we gained by this? We now
can describe all kinds of structure elements - atoms, molecules and defects -
and their reactions in a clear and unambiguous way relative to the empty space. Lets look at some
examples |
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Formation of Frenkel defects in, e.g., AgCl: |
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AgAg + Vi = V
/Ag + Ag ·i |
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We see why we need the slightly strange construction of a
vacancy on an interstitial site. |
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Formation of Schottky defects for an AB crystal
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AA +
BB |
= |
V /A + V ·B + AA +
BB |
AA +
BB |
= |
V /A + VB · + AB |
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The second equation simply considers the two dislodged atoms
as a molecule that must be put somewhere. |
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This looks good. The question is, if we now can use the
mass action law to
determine equilibrium concentrations. If the Frenkel defect example could be
seen as analogous to the chemical reaction A + B = AB, we could write a
mass action law as follows: |
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[AgAg] · [Vi]
[V /Ag] · [Agi ·] |
= const |
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with [A] meaning
"concentration of A". The reaction constant is a more or less
involved function of pressure p and temperature T,
and especially the chemical potentials of
the particles involved. |
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Unfortunately, this is
wrong! |
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Why? Well, the notion of chemical equilibrium and thus the mass
action law, at the normal conditions of constant temperature T
and pressure p, stems from finding the minimum of the
free enthalpy G
(also called Gibbs energy) which in our
case implies the equality of all chemical potentials. You may want to read up a
bit on the concept of chemical
potentials, this can be done in the link. |
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In other words, we are searching for the
equilibrium concentration of the particles ni involved
in the reaction, which, at a given temperature and pressure, lead to
dG = 0. |
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The equation dG = 0 can always be
written as a total
differential with respect to the variables
dni: |
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dG |
= |
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∂G
∂n1 |
· dn1 + |
∂G
∂n2 |
· dn2 + ... |
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The partial derivatives are defined as the
chemical
potentials of the particles in question and we always have to keep in mind
that the long version of the above equation
has a subscript at every partial derivative, which we, like many others,
conveniently "forgot". If written correctly the partial derivative
for the particle ni reads (in
HTML somewhat awkwardly), |
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∂G
∂ni |
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p, T, nj ¹ i =
const |
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Meaning that T, p,
and all other particle concentrations must
be kept constant. |
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Only if that
condition is fulfilled, a mass action equation can be formulated that involves
all particles present in the reaction equation! And fulfilling the condition
means that you can - at least in principle - change the concentration of
any kind of particle (e.g. the vacancy
concentration) without changing the
concentration of all the other particles. |
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This "independence condition" is automatically
not fulfilled if we have additional
constraints which link some of our particles. And such constraints we do have in the Kröger-Vink notation, as
alluded to before! |
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There is no way within the system to produce a
vacancy, e.g. VA without removing an A-particle, e.g.
generating an Ai or adding another B-particle,
BB. |
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S... ! We now have a very useful way
of describing chemical reactions, including all kinds of charged defects, but
we cannot use simple thermodynamics! That is the point where other notations
come in. |
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You now may ask: Why not
introduce a notation that has it all and be done with it? |
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The answer is: It could be done,
but only by losing simplicity in describing reactions. And simplicity is what
you need in real (research) life, when, in sharp contrast to text books, you do
not know what is going on, and you try to
get an answer by mulling over various possibility in your mind, or on a sheet
of paper. |
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So "defects-in-ceramics"
people live with several kinds of notation, all having pro and cons, and, after
finding a good formulation in one notation, translate it to some other notation
to get the answers required. We will provide a glimpse of this in the next
subchapter. |
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© H. Föll