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Frenkel defects are, like Schottky defects, a speciality of ionic crystals.
Consult this illustration modul for pictures and more
details. |
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In fact, the discussion of this defect in AgCl in 1926 by Frenkel more or less introduced the concepts of point
defects in crystals to science. |
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In ionic crystals, charge neutrality
requires (as we will see) that defects come in pairs with opposite charge, or
at least the sum over the net charge of all charged point defects must be zero. |
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"Designer defects" (defects carrying name
tags) are special cases of the general point defect situation in non-elemental crystals.
Since any ionic crystal consists of at least two different kinds of atoms, at least two kinds of vacancies and interstitials
are possible in principle. |
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Thermodynamic equilibrium always allows all possible kinds
of point defects simultaneously (including charged defects) with arbitrary concentrations, but always requiring a minimal
free enthalpy including the electrostatic energy components in this case. |
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However, if there is a charge inbalance, electrostatic energy will quickly override everything
else, as we will see. As a consequence we need charge neutrality in total and in any
small volume element of the crystal - we have a kind of independent boundary condition for equilibrium. |
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Charge neutrality calls for at least two kinds of differently
charged point defects. We could have more than just two kinds, of course, but again as we will see, in real crystals usually
two kinds will suffice. |
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One of two simple ways of maintaining charge neutrality with
two different point defects is to always have a vacancy - interstitial pair, a combination we will call a Frenkel pair. |
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The generation of a Frenkel defect is easy to visualize: A lattice ion moves to an interstitial site, leaving a vacancy behind. The ion will always be the
positively charged one, i.e. a cation interstitial, because it is pretty much always smaller than the negatively charged
one and thus fits better into the interstitial sites. In other words; its formation enthalpy will be smaller than that of
a negatively charged interstitial ion. Look at the pictures
to see this very clearly. |
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It may appear that electrostatic forces keep the interstitial and the vacancy in close proximity.
While there is an attractive interaction, and close Frenkel pairs do exist (in analogy to excitons,
i.e. close electron-hole pairs in semiconductors), they will not be stable at high temperatures. If the defects can diffuse,
the interstitial and the vacancy of a Frenkel pair will go on independent random walks and thus can be anywhere, they do
not have to be close to each other after their generation. |
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Having vacancies and interstitials is called Frenkel
disorder, it consists of Frenkel pairs or the Frenkel defects. |
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Frenkel disorder is an extreme case of general disorder; it is prevalent
in e.g. Ag - halogen crystals like AgCl. We thus have |
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This implies, of course, that vacancies carry a
charge; and that is a bit of a conceptual problems. For ions as interstitials, however, their charge is obvious.
How can we understand a charge "nothing"? |
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Well, vacancies can be seen as charge carriers in analogy to holes
in semiconductors. There a missing electron - a hole - is carrying the opposite charge of the electron.
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For a vacancy, the same reasoning applies. If a Na+ lattice ion is missing,
a positive charge is missing in the volume element
that contains the corresponding vacancy. Since "missing" charges are non-entities, we have to assign a negative charge to the vacancy in the volume element to get
the charge balance right. |
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Of course, any monoatomic crystal could (and will) have arbitrary numbers of vacancies and interstitials at the same time as intrinsic point defects; but only if charge consideration
are important ni = nv holds exactly; otherwise the two concentrations are uncorrelated
and simply given by the formula for the equilibrium concentrations. |
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Indeed, since the equilibrium concentrations are never exactly zero, all
crystals will have vacancies and interstitials present at the same time, but since the
formation energy of interstitials is usually much larger than that of vacancies, they may be safely neglected for most considerations
(with the big exception of Silicon!). |
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Of course, in biatomic ionic crystals, there could (and will) be two
kinds of Frenkel defects: cation vacancy and cation interstitial; anion vacancy and anion interstitial; but in any given
crystal one kind will always be prevalent. |
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We will take up all these finer points in modules to come, but now let's just look at the
simple limiting case of pure Frenkel disorder. |
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With
the equilibrium condition ¶G/¶n = 0
we obtain for the concentration cFP of Frenkel pairs |
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| cFP | = |
nFP
N |
= |
æ
ç
è |
N'
N |
ö
÷
ø |
1/2 | · exp |
SFP
2k |
· exp – |
HFP
2kT |
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The factor 1/2 in the exponent comes from equating the formation energy HFP
or entropy, resp., with a pair of point defects and not with an individual defect.
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What is the reality, i.e. what kind of formation enthalpies are encountered? Surprisingly,
it is not particularly easy to find measured values; the link, however,
will give some numbers. |
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That was rather straight forward, and we will not discuss Frenkel defects much
more at this point. We will, however, show in the next subchapter from first principles that, indeed, charge neutrality
has to be maintained. |
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© H. Föll (Defects - Script)
