### 2.1.5 Essentials to Chapter 2.1: Point Defect Equilibrium

| 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! |  |
SF | = k · | å i | ln | wi w'i
--- | --- | --- | --- | ---
|  | Formation enthalpies are roughly around 1 eV for common crystals ("normal" metals"); formation entropies around 1 k. |
| 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 | D S2V 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 D S |
|  | 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 – | D E kT
--- | --- | --- | --- | ---
|  | There are, however, some pitfalls in using the mass action law; we also loose any information about the factor A |
|  | 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. |

| 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
--- | --- | --- | --- | --- | ---
|  | 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. |
| 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, ...
|  | They are extreme cases of the general "mixed defect case" containing all possible PDs (e.g. V–, V–, i+, i+) while maintaining charge neutrality. |
|  | 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. |
|  | It is, however, hard to predict the dominating defect type from "scratch". |

© H. Föll (Defects - Script)