 |
In global equilibrium all crystals
contain point defects with a concentration cPD given
by an Arrhenius expression of the form: |
|
| cPD |
= A · exp – |
GF
kT |
= A · exp |
SF
k |
· exp – |
HF
kT |
|
|
|
 |
A is a constant around
(1 ....10), reflecting the geometric possibilities to introduce 1
PD in the crystal (A = 1 for a simple vacancy). |
|
|
|
GF,
HF, SF are the free energy of
formation, enthalpy (or colloquial "energy") of formation, and
entropy of formation, respectively, of 1 PD |
|
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The entropy of formation reflects the disorder
introduced by 1 PD; it is tied to the change in lattice vibrations
(circle frequency w) around a PD and
is a measure of the extension of the PD. It must not be confused with
the entropy of mixing for many PDs! |
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Formation enthalpies are roughly around 1
eV for common crystals ("normal" metals"); formation
entropies around 1 k. |
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Small PD clusters (e.g.
di-vacancies) are still seen as PDs, their concentration follows from
the same considerations as for single PDs to: |
|
| c2V |
= |
z
2 |
· exp |
S2V
k |
· |
exp – |
HF(2V)
kT |
| c2V |
= |
z
2 |
· c1V2 ·
exp |
DS2V
k |
· |
exp – |
E2V
kT |
|
|
|
|
The constant A for di-vacancies is
half the coordination number z (= number of possibilities to
arrange the axis of a di-vacancy
dumbbell) |
|
|
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The formation enthalpy and entropy of a PD
cluster can always be expressed as the sum of these parameters for single
PDs minus a binding enthalpy E and a binding entropy
DS |
|
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The term c1V2
or c1Vn for a cluster of
n vacancies makes sure that the concentration of clusters is
always far smaller than the concentration of single PDs. |
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The same relations can be obtained by
"making" di-vacancies (or any cluster) by a "chemical"
reaction between the PDs and employing the mass action law: |
|
| 1V + 1V Û V2V
+ E2V |
(c1V)2
c2V |
= |
K(T) = |
const · exp – |
DE
kT |
|
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There are, however, some pitfalls in using the
mass action law; we also loose any information about the factor
A |
|
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Most important in doing "defect
chemistry" with mass action, is a proper definition of the
"ingredients" to chemical reaction equations. A vacancy, after all,
is not an entity like an atom that can exist on its own. More to that in
chapter 2.4. |
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Note: All of the above is generally
valid for all independent PDs: "A" and
"B" vacancies, interstitials, antisite defects,. ... . |
|
| cV |
= AV · |
exp |
SFV
k |
· exp – |
HFV
kT |
| ci |
= Ai · |
exp |
SFi
k |
· exp – |
HFi
kT |
|
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|
|
However: If
there are additional restraints (like
charge neutrality), we may have to consider pairs of (atomic) PDs as one
point defect; e.g. Frenkel or Schottky defects. |
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First principle" calculation show that
charge neutrality can only be locally violated on length scales given by the
Debye length of the crystal. |
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Frenkel and Schottky defects are
vacancy- interstitial or vacancy–- vacancy+ pairs in
ionic crystals. |
|
| Frenkel defect: |
|
V– +
i+ |
| Anti-Frenkel defect: |
|
V+ +
i- |
| Schottky defect: |
|
V– +
V+ |
|
| Frenkel disorder in: |
|
AgCl, AgBr, CaF2, BaF2,
PbF2, ZrO2, UO2, ... |
| Schottky disorder in: |
|
LiF, LiCl, LiBr, NaCl, KCL, KBr, CsI, MgO,
CaO, ... |
|
|
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|
They are extreme cases of the general "mixed
defect case" containing all possible PDs (e.g.
V–, V–, i+, i+)
while maintaining charge neutrality. |
|
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Usually, one finds either Frenkel defects or
Schottky defects - if the respective formation enthalpies
HFre or HScho differ by some
0.1 eV, one defect type will dominate. |
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It is, however, hard to predict the dominating
defect type from "scratch". |
|
© H. Föll (Defects - Script)
