 |
It is important to be clear about the
possibilities of producing defects in ionic crystals. |
|
|
|
|
|
|
|
 |
Even for the most simple ionic crystals of the type
A+B– like LiCl or NaCl, we can,
in principle, produce arbitrary
concentrations of two kinds of vacancies and two kinds of interstitials as
shown on the left. |
|
|
|
|
However, as we already learned in dealing with
Schottky defects, global
charge neutrality must be
maintained. Arbitrary concentrations are
thus not really allowed, we must demand
that the the sum of the positively charged defects equals the sum of the
negatively charged defects. |
|
|
|
|
If we also keep the number of atoms constant, we must add an
A or B atom to the surface of the crystal for every pure vacancy
we produce. |
|
|
|
|
The picture on the left would not have needed the AB
molecule, because we have two interstitials, too. But since it is supposed to
illustrate the general case, with arbitrary
numbers of defects, it needs to include A and B atoms on the
surface. |
|
|
|
|
|
|
|
Looking closely at the picture above, we realize
that in this case it must be considerably
more difficult to stuff an anion on an
interstitial place than a cation. Without a considerable relaxation of the
atoms surrounding the defect as shown it is simply impossible. |
|
|
As always, you must bear in mind that
pictures as shown here are schematic - in
more realistic pictures the ions would touch! However, in more realistic
pictures it also would be harder to show what is intended. |
|
|
This will always be true if the anion is larger as
the cation which is the case for many, but not all ionic crystals. We thus can
safely assume that the concentration of one
kind of interstitial, here the anion interstitial, is
always far smaller than that of the other three defect kinds and we will simply
neglect it from now on. |
|
|
However, for crystals with a big and heavy cation
(e.g. Ca+) and a light anion (e.g.
F–), the cation might just be as big as the anion, and
occur as interstitial (e.g. in the so-called "Anti-Frenkel
defects"). |
|
|
|
|
If we forget about the
anion interstitial, we are left with three possible point defects.
|
|
|
|
|
|
|
|
The three now possible defect types are shown on the
left. |
|
|
|
|
This is the general case of the mixed defects treated in the
backbone |
|
|
|
|
Note that charge equilibrium demands that you always have more cation vacancies than anion
vacancies or cation interstitials: |
| |
|
|
|
|
| |
|
|
|
|
| |
|
|
|
|
|
|
|
|
This necessitates that some AB molecules must be added
to the surface of the crystal if we keep the atom count in order, too (same
concentration as the anion vacancy, to be precise). |
|
|
|
|
|
|
|
The realistic mixed case thus contains
Schottky and
Frenkel defects in
parallel. |
|
|
Note that the picture above does not show the equilibrium case, because we do not
have charge neutrality - for that it would need another cation vacancy. |
|
|
Every cation vacancy finds an anion vacancy as a
fictive partner, forming a formal Schottky defect, and every cation
interstitial finds a cation vacancy, too, for a formal Frenkel pair, and the
concentration of the anion vacancies is just so that it meets both demands for
partners. |
|
|
We thus can identify
cV(A) with the concentration of Schottky defects and
ci(C) with the concentration of Frenkel defects. |
|
|
|
Schottky and Frenkel disorder may now simply be
seen as extreme cases of the mixed
disorder. |
|
|
|
|
|
|
|
|
Frenkel disorder is
predominant if the formation enthalpy of Frenkel pairs is smaller than that of
Schottky pairs, i.e.
HFP < HS |
|
|
|
|
We then only - or at least predominantly - have Frenkel pairs,
i.e. equal numbers of cation vacancies and cation interstitials |
|
|
|
|
We do not have an AB molecule for Frenkel
disorder. |
|
|
|
|
|
|
|
|
|
|
|
Schottky disorder is
predominant if the formation enthalpy of Schottky pairs is smaller than that of
Frenkel pairs, i.e.
HS < HFP. |
|
|
|
|
We only - or at least predominantly - have vacancy pairs, i.e.
equal numbers of cation vacancies and anion vacancies |
| |
|
|
|
And - in contrast to Frenkel disorder - we always need to form
a lattice molecule, our AB molecule, to preserve atom numbers. |
|
|
|
|
|
|
|
Just how much smaller the relevant formation
enthalpy has to be for factual predominance of one defect type remains to be
seen - in an exercise. |
|
|
© H. Föll
Exercise 2.1-6: Enthalpy difference for the limiting cases of Schottky
or Frenkel Defects 
To index