# Equilibrium Concentration of the Vacancy-Impurity Atom Complex

| We consider a complex of one vacancy and one foreign atom (a Johnson complex) in thermodynamic equilibrium.
|  | We start with a number nF of foreign atoms that is given by external circumstances. Some, but not necessarily all of these atoms will form a complex with a vacancy. The number of these complexes we call nC.
| We can calculate the equilibrium number of Johnson complexes exactly analogous to the equilibrium number of vacancies by simply defining a formation enthalpy GC and doing the counting of arrangements and minimization of the free enthalpy procedure.
|  | This obviously will give us
nC | = (NC – nC) · exp – | GC kT
--- | --- | ---
|  | With NC = number of sites in the crystal where a vacancy could sit in order to form a Johnson complex.
|  | We take NC – nC in full generality because the places already occupied ( = nC) are no longer available, and we do not assume at this point that nC << NC applies as in the case of vacancies.
| NC, of course, is not the number of atoms of the crystal as in the case of vacancies, but roughly the number of foreign atoms - after all, only where we have a foreign atom, can we form a complex.
|  | If we don't look at the situation roughly but in detail, we need to consider that there are as many possibilities to form a foreign atom - vacancy pair as there are nearest neighbors. We thus find
NC | = nF · z – nC · z
--- | ---
|  | z is the coordination number of the lattice considered, i.e. the number of nearest neighbors. Again we do not neglect the places already taken, i.e. we subtract nC · z from the total number of places.
| If we look at concentrations, we refer the numbers to the number of lattice atoms N which gives us for the concentration cC of impurity atom - vacancy complexes
cC | = | nC N
--- | --- | ---
cC cF – cC | = z · exp – | GC kT
--- | --- | ---
| We are essentially done. We have the concentration of Johnson complexes as a function of the concentration of the foreign atoms, the lattice type (defining z) and their formation enthalpy.
|  | However, we would feel happier, if we could base the equation on material parameters which we already know - in particular on the equilibrium concentration of vacancies in the given material.
|  | This needs a closer view on the formation enthalpy of the complex.

| As in the case of double vacancies, we may simply assume that there is a binding enthalpy between a vacancy and a foreign atom (otherwise there would be no driving force to form a complex in the first place).
|  | We thus can write for GC
GC = GVF – (HC – T · DSC)
|  | GVF is the free enthalpy of vacancy formation, HC is the binding enthalpy of a Johnson complex, and T · DSC is the "association entropy" of the complex, accounting for the entropy change of the crystal upon the formation of a complex.
| Inserting this in the equation above gives for the concentration of Johnson complexes in terms of vacancy parameters and binding energies:
cC cF – cC | = z · exp – | GVF kT | · exp | HC kT | · exp | DSC k
--- | --- | --- | --- | --- | --- | ---
cC | = | (cF – cC) · cV · z · | exp | HC kT | · exp | DSC k
--- | --- | --- | --- | --- | --- | ---
| = | c'F · cV · z · | exp | HC kT | · exp | DSC k
|  | We used the familiar equation cV = exp – (GVF/ kT) to get this result.
| We abbreviated the difference of the total concentration of foreign atoms and the concentration of Johnson complexes by c'F; i.e. c'F = (cF – cC) because this allows a simple interpretation of the equation.
|  | The point now is to recognize that c'F is nothing but the concentration of foreign atoms which are still available for a reaction with a vacancy, and that the last equation therefore is nothing but the mass action law written out for the reaction
1F + 1V | Û | 1C
--- | --- | ---
|  | With F = (available) foreign atom; V = vacany, and C = Johnson complex.
| Looking closely (= thinking hard) you will notice that we now have a certain inconsistency in our book keeping:
|  | We always took into account that Johnson complexes already formed can not be neglected in counting possibilites, and we always corrected for that by using cF – cC and so on - but we did not correct for the now more limited possibilities for positioning a single vacancy. We must ask ourselves if the presence of foreign atoms will change the equilibrium concentration of free vacancies.
|  | In other words, while we took the number of available positions for a vacancy in a complex to be nF · z – nC · z, we implicitly took the number of available positions for a free vacancy in the crystal to be simply N = number of lattice atoms.
|  | Being more precise, we have to subtract nF · z from N because nF · z positions are, after all, not available for free vacancies. We thus have to replace N by N' = N – nF · z when we consider the number of free vacancies.
|  | The concentration of the free vacancies thus becomes cV = (1 – z · cF) · exp – (GVF/ kT), or exp – (GVF/ kT) = cV / (1 – z · cF)
|  | Using this in the equation for the concentration yields
cC | = | (cF – cC) · cV · z (1 – z · cF) | · exp | HC kT | · exp | DSC k
--- | --- | --- | --- | --- | --- | ---
| » | cF · cV · z (1 – z · cF) | · exp | HC kT | · exp | DSC k
|  | The last approximation is, of course, attainable if cC << cF, and that is the equation given in the backbone text.

With frame

2.2.1 Extrinsic Point Defects and Agglomerates

Solution to Exercise 2.2-1

© H. Föll (Defects - Script)