 |
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 | |
|
|
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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|
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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) | |
|
|
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. | |
| | |
| |
|
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. | |
|
|
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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| |
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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)
