Basic Equilibrium Considerations
We start with the most simple point defects imaginable and consider an uncharged vacancy in a simple crystal with a base consisting of only one atomic species  that means mostly metals and semiconductors.  
Some call this kind of defect "Schottky Defect, although the original Schottky defects were introduced for ionic crystals containing at least two different atoms in the base.  
We call vacancies and their "opposites", the selfintersitals, intrinsic point defects for starters. Intrinsic simple means that these point defects can be generated in the ideal world of the ideal crystal. No external or extrinsic help or stuff is needed.  
To form one vacancy at constant pressure (the usual situation), we have to add some free enthalpy G_{F} to the crystal, or, to use the name commonly employed by the chemical community, Gibbs energy.  
G_{F}, the free enthalpy of vacancy formation, is defined as  
 
The index F always means "formation"; H_{F} thus is the formation enthalpy of one vacancy, S_{F} the formation entropy of one vacancy, and T is always the absolute temperature.  
The formation enthalpy H_{F} in solids is practically indistinguishable from the formation energy E_{F} (sometimes written U_{F}) which has to be used if the volume and not the pressure is kept constant.  
The formation entropy, which in elementary considerations of point defects usually is omitted, must not be confused with the entropy of mixing or configurational entropy; the entropy originating from the many possibilities of arranging many vacancies, but is a property of a single vacancy resulting from the disorder introduced into the crystal by changing the vibrational properties of the neighboring atoms (see ahead).  
The next step consists of minimizing the free enthalpy G of the complete crystal with respect to the number n_{V} of the vacancies, or the concentration c_{V} = n_{V} /N, if the number of vacancies is referred to the number of atoms N comprising the crystal. We will drop the index "V" from now now on because this consideration is valid for all kinds of point defects, not just vacancies.  
The number or concentration of vacancies in thermal equilibrium (which is not necessarily identical to chemical equilibrium!) then follows from finding the minimum of G with respect to n (or c), i.e.  
 
with G_{0} = Gibbs energy of the perfect crystal, G_{1} = Work (or energy) needed to generate n vacancies = n · G_{F}, and G_{2} = – T · S_{conf} with S_{conf} = configurational entropy of n vacancies, or, to use another expression for the same quantity, the entropy of mixing n vacancies.  
We note that the partial derivative of G with respect to n, which should be written as [¶G/¶n]_{everything else = const.} is, by definition, the chemical potential µ of the defects under consideration. This will become important if we consider chemical equilibrium of defects in, e.g., ionic crystals.  
The partial derivatives are easily done, we obtain  
 
which finally leads to  
 
We now need to calculate the entropy of mixing or configurational entropy S_{conf} by using Boltzmann's famous formula  

With k_{B} = k = Boltzmanns constant and P = number of different configurations (= microstates) for the same macrostate.  
The exact meaning of P is sometimes a bit confusing; activate the link to see why.  
A macrostate for our case is any possible combination of the number n of vacancies and the number N of atoms of the crystal. We obtain P(n) thus by looking at the number of possibilities to arrange n vacancies on N sites.  
This is a standard situation in combinatorics; the number we need is given by the binomial coefficient; we have  
 
If you have problems with that, look at exercise 2.11 below.  
The calculation of ¶S/¶n now is straight forward in principle, but analytically only possible with two approximations:  
1. Mathematical Approximation: Use the Stirling formula in its simplest version for the factorials, i.e.  
 
2. Physical Approximation: There are always far fewer vacancies than atoms; this means  
 
As a first result we obtain "approximately"  
 
If you have any doubts about this point, you should do the following exercise.  
 
With n/N = c_{V} = concentration of vacancies as defined before, we obtain the familiar formula  
 
or, using G_{F} = H_{F} – T S_{F}  
 
For selfinterstitials, exactly the same formula applies if we take the formation energy to be now the formation energy of a selfinterstitial.  
However, the formation enthalpy of selfinterstitials is usually (but not necessarily) considerably larger than that of a vacancy. This means that their equilibrium concentration is usually substantially smaller than that of vacancies and is mostly simply neglected.  
Some numbers are given in this link; far more details are found here. The one number to remember is:  
 
It goes without saying (I hope) that the way you look at equations like this is via an Arrhenius plot. In the link you can play with that and refresh your memory  
Instead of plotting c_{V}(T) vs. T directly as in the left part of the illustration below, you plot the logarithm lg[c_{V}(T)] vs. 1/T as shown on the right.  
In the resulting "Arrhenius plot" or "Arrhenius diagram" you will get a straight line. The (negative) slope of this straight line is then "activation" energy of the process you are looking at (in our case the formation energy of the vacancy), the yaxis intercept gives directly the preexponential factor.  
 
Compared to simple formulas in elementary courses, the factor exp(S_{F}/k) might be new. It will be justified below.  
Obtaining this formula by shuffling all the factorials and so on is is not quite as easy as it looks  lets do a little fun exercise  
 
Like always, one can secondguess the assumptions and approximations: Are they really justified? When do they break down?  
The reference enthalpy G_{0} of the perfect crystal may not be constant, but dependent on the chemical environment of the crystal since it is in fact a sum over chemical potentials including all constituents that may undergo reactions (including defects) of the system under consideration. The concentration of oxygen vacancies in oxide crystals may, e.g., depend on the partial pressure of O_{2} in the atmosphere the crystal experiences. This is one of the working principles of Ionics as used for sensors. Chapter 2.4 has more to say to that.  
The simple equilibrium consideration does not concern itself with the kinetics of the generation and annihilation of vacancies and thus makes no statement about the time required to reach equilibrium. We also must keep in mind that the addition of the surplus atoms to external or internal surfaces, dislocations, or other defects while generating vacancies, may introduce additional energy terms.  
There may be more than one possibility for a vacancy to occupy a lattice site (for interstitials this is more obvious). This can be seen as a degeneracy of the energy state, or as additional degrees of freedom for the combinatorics needed to calculate the entropy. In general, an additional entropy term has to be introduced. Most generally we obtain  
 
with Z_{d} or Z_{0} = partition functions of the system with and without defects, respectively. The link (in German) gets you to a short review of statistical thermodynamics including the partition function.  
Lets look at two examples where this may be important:  
The energy state of a vacancy might be "degenerate", because it is charged and has trapped an electron that has a spin which could be either up or down  we have two, energetically identical "versions" of the vacancy and Z_{d}/Z_{0} = 2 in this case.  
A double vacancy in a bcc crystals has more than one way of sitting at one lattice position. There is a preferred orientation along <111>, and Z_{d}/Z_{0} = 4 in this case.  
Calculation and Physical Meaning of the Formation Entropy
The formation entropy is associated with a single defect, it must not be mixed up with the entropy of mixing resulting from many defects.  
It can be seen as the additional entropy or disorder added to the crystal with every additional vacancy. There is disorder associated with every single vacancy because the vibration modes of the atoms are disturbed by defects.  
Atoms with a vacancy as a neighbour tend to vibrate with lower frequencies because some bonds, acting as "springs", are missing. These atoms are therefore less well localized than the others and thus more "unorderly" than regular atoms.  
Entropy residing in lattice vibrations is nothing new, but quite important outside of defect considerations, too:  
Several bcc element crystals are stable only because of the entropy inherent in their lattice vibrations. The – TS term in the free enthalpy then tends to overcompensate the higher enthalpy associated with non closepacked lattice structures. At high temperatures we therefore find a tendency for a phase change converting fcc lattices to bcc lattices which have "softer springs", lower vibration frequencies and higher entropies. For details compare Chapter 6 of Haasens book.  
The calculation of the formation entropy, however, is a bit complicated. But the result of this calculation is quite simple. Here we give only the essential steps and approximations.  
First we describe the crystal as a sum of harmonic oscillators  i.e. we use the wellknown harmonic approximation. From quantum mechanics we know the energy E of an harmonic oscillator; for an oscillator number i and the necessary quantum number n we have  
 
We are going to derive the entropy from the allencompassing partition function of the system and thus have to find the correct expression.  
The partition function Z_{i} of one harmonic oscillator as defined in statistical mechanics is given by  
 
The partition function of the crystal then is given by the product of all individual partition function of the p = 3N oscillators forming a crystal with N atoms, each of which has three degrees of freedom for oscillations. We have  
 
From statistical thermodynamics we know that the free energy F (or, for solids, in a very good approximation also the free enthalpy G) of our oscillator ensemble which we take for the crystal is given by  
 
Likewise, the entropy of the ensemble (for const. volume) is  
 
Differentiating with respect to T yields for the entropy of our  so far  ideal crystal without defects:  
 
Now we consider a crystal with just one vacancy. All eigenfrequencies of all oscillators change from w_{i} to a new as yet undefined value w'_{i}. The entropy of vibration now is S'.  
The formation entropy S_{F} of our single vacancy now can be defined, it is  
 
i.e. the difference in entropy between the perfect crystal and a crystal with one vacancy.  
It is now time to get more precise about the w_{i}, the frequencies of vibrations. Fortunately, we know some good approximaitons:  
At temperatures higher then the Debye temperature, which is the interesting temperature region if one wants to consider vacancies in reasonable concentrations, we have  
 
which means that we can expand hw_{i}/2p into a series of which we (as usual) consider only the first term.  
Running through the arithmetic, we obtain as final result, summing over all eigenfrequencies of the crystal  
 
This now calls for a little exercise:  
 
For analytical calculations we only consider next neighbors of a vacancy as contributors to the sum; i.e. we assume w = w' everywhere else. In a linear approximation, we consider bonds as linear springs; missing bonds change the frequency in an easily calculated way. As a result we obtain (for all cases where our approximations are sound):  
S_{F} (single vacancy) » 0.5 k (Cu) to 1.3 k (Au).  
S_{F} (double vacancy) » 1.8 k (Cu) to 2.2 k (Au).  
These values, obtained by assuming that only nearest neighbors of a vacancy contribute to the formation entropy, are quite close to the measured ones. (How formation entropies are measured, will be covered in chapter 4). Reversing the argumentation, we come to a major conclusion:  
The formation entropy measures the spatial extension of a vacancy, or, more generally, of a zerodimensional defect. The larger S_{F}, the more extended the defect will be because than more atoms must have changed their vibrations frequencies.  
As a rule of thumb (that we justify with a little exercise below) we have:  
S_{F} » 1k corresponds to a truly atomic defect, S_{F} » 10k correponds to extended defects disturbing a volume of about 5  10 atoms.  
This is more easily visualized for interstitials than for vacancies. An "atomic" interstitials can be "constructed" by taking out one atom and filling in two atoms without changing all the other atoms appreciably. An interstitial extended over the volume of e.g. 10 atoms is formed by taking out 10 atoms and filling in 11 atoms without giving preference in any way to one of the 11 atoms  you cannot identify a given atom with the interstitial.  
Vacancies or interstitials in elemental crystal mostly have formation entropies around 1k, i.e. they are "point like". There is a big exception, however: Si does not fit this picture.  
While the precise values of formation enthalpies and entropies of vacancies and interstitials in Si are still not known with any precision, the formation entropies are definitely large and probably temperature dependent; values around 6k  15k at high temperatures are considered. Historically, this led Seeger and Chik in 1968 to propose that in Si the selfinterstitial is the dominating point defect and not the vacancy as in all other (known) elemental crystals. This proposal kicked of a major scientific storm; the dust has not yet settled.  
 
Multi Vacancies (and Multi  Interstitials by Analogy)
So far, we assumed that there is no interaction between point defects, or that their density is so low that they "never" meet. But interactions are the rule, for vacancies they are usually attractive. This is relatively easy to see from basic considerations.  
Let's first look at metals:  
A vacancy introduces a disturbance in the otherwise perfectly periodic potential which will be screened by the free electrons, i.e. by a rearrangement of the electron density around a vacancy. The formation enthalpy of a vacancy is mostly the energy needed for this rearrangement; the elastic energy contained in the somewhat changed atom positions is comparatively small.  
If you now introduce a second vacancy next to to the first one, part of the screening is already in place; the free enthalpy needed to remove the second atom is smaller.  
In other word: There is a certain binding enthalpy (but from now on we will call it energy, like everybody else) between vacancies in metals (order of magnitude: (0,1  0,2) eV).  
Covalently bonded crystals  
The formation energy of a vacancy is mostly determined by the energy needed to "break" the bonds. Taking away a second atom means that fewer bonds need to be broken  again there is a positive binding energy.  
Ionic crystals  
Vacancies are charged, this leads to Coulomb attraction between vacancies in the cation or anion sublattice, resp., and to repulsion between vacancies of the same nature. We may have positive and negative binding energies, and in contrast to the other cases the interaction can be longrange.  
The decisive new parameter is the binding energy E_{2V} between two vacancies. It can be defined as above, but we also can write down a kind of "chemical" reaction equation involving the binding energy E_{2V} (the sign is positive for attraction):  
 
V in this case is more than an abbreviation, it is the "chemical symbol" for a vacancy.  
If you have some doubts about writing down chemical reaction equation for "things" that are not atoms, you are quite right  this needs some special considerations. But rest assured, the above equation is correct, and you can work with it exactly as with any reaction equation, i.e. apply reaction kinetics, the mass action law, etc.  
Now we can do a calculation of the equilibrium concentration of Divacancies. We will do this in two ways.  
First Approach: Minimize the total free enthalpy (as before):  
First we define a few convenient quantities  
 
With D
S_{2V} = entropy of association (it is in the order of 1k
 2k in metals), and E_{2V} = binding energy between two vacancies. We obtain in complete analogy to single vacancies  
 
The factor z/2 (z = coordination number = number of (symmetrically identical) next neighbors) takes into account the different ways of aligning a divacancy on one point in the lattice as already noticed above. We have z = 12 for fcc, 8 for bcc and 4 for diamond lattices.  
The formula tells us that the concentration of divacancies in thermal equilibrium is always much smaller than the concentration of single vacancies since c_{V} << 1. "Thermal equilibrium" has been emphasized, because in nonequilibrium things are totally different!  
Some typical values for metals close to their melting point are  
 
In the second approach, we use the mass action law.  
With the reversible reaction  
 
With DE = energy of the forward reaction (you have to be extremely careful with sign conventions whenever invoking mass action laws!). This leads to  
 
In other words: Besides the "const.^{–1}" we get the same result, but in an "easier" way.  
The only (small) problem is: You have to know something additional for the determination of reaction constants if you just use the mass action law. And that it is not necessarily easy  it involves the concept of the chemical potential and does not easily account for factors coming from additional freedoms of orientation. e.g. the factor z/2 in the equation above.  
The important point in this context is that the reaction equation formalism also holds for nonequilibrium, e.g. during the cooling of a crystal when there are too many vacancies compared to equilibrium conditions. In this case we must consider local instead of global equilibrium, see chapter 2.2.3.  
There would be much more to discuss for single vacancies in simple monoatomic crystals, e.g. how one could calulate the formation enthalpy, but we will now progress to the more complicated case of point defects in crystals with two different kinds of atoms in the base.  
That is not only in keeping with the historical context (where this case came first), but will provide much food for thought.  

 
© H. Föll (Defects  Script)