Making Vacancies 

Calculating the Equilibrium Vacancy Concentration from Scratch  
How many
vacancies does a
real crystal need for achieving 

First we need to determine the free
N = number of atoms forming the crystal. n = number of vacancies in the crystal. T = (absolute, of course) temperature of the crystal. I will give you the full function G(N, n, T) first and then discuss it. What we have is 



The first
term, G(N, T) , simply describes the total free
enthalpy of a perfect crystal with
N atoms at the temperature T and no vacancies. It
contains the energy E_{cryst} of the crystal, essentially
contained in the vibrations of the atoms, and the entropy
S_{vib} resulting from that. If you like you can write it out as G(N, T) = E_{cryst}(N, T) – TS_{vib}(N, T) but that will just make the equation longer. For any number of atoms N and temperature T, the function G(N, T) would produce some number if we put our mind to it. We won't do that, however, since it will turn out that we don't really need G(N, T) 

The second
term acknowledges that we expect to have n vacancies in our
crystal. Making a vacancy requires that we sever the bonds between the atoms an and rip out the cutoff atom somehow. That will take some effort or energy. The energy needed to make one vacancy in a given crystal is always the same, no matter how "we" make the vacancy. It is aptly named formation energy of a vacancy and abbreviated with E_{F}. We want to make n vacancies so we need to add n · E_{F} to the energy contained in the otherwise perfect crystal. 

The third
term acknowledges that having vacancies increases the disorder and thus the
entropy S of the crystal. This is called the "entropy of mixing" since it comes
exclusively from mixing up "things" and not from the
"things" itself. We thus must subtract TS from the total energy of the situation. The entropy depends only on the number of atoms N and the number of vacancies n so we can write it S(N, n). 

The third term with the entropy
S(N, n) is the decisive term.
Boltzmann's famous formula for the
entropy, 



P(N, n) is the number of possibilities to arrange n vacancies in a crystal consisting of N atoms; k = Boltzmann's constant. And of course, you know what "ln", the symbol for a natural logarithm means  
How large is P(N, n)? You might also ask: In how many ways can I distribute 7 socks in my bedroom, assuming I have 42 places (in the actual sock drawer, in any of the 11 other drawers, under the bed, on the bed, ....).  
Question like that are answered by combinatorics, a muchhated branch of stochastic math. Another combinatorial question (but not mathematically equivalent) could be "How many different car license plates can you make by using a combination of two letters and three numbers" or "How many ways to dress differently with two pairs of pants, three shirts, five pairs of socks", or "how many different hands are there at poker", or... You get the idea.  
The answer to our question is so elementary (and simple) that the combinatoric folks came up with a special mathematical symbol, the binomial coefficient, to abbreviate equation writing. We have  


In case you forgot: An exclamation mark behind a
mathematical symbol or number means the factorial of that number. A factorial is an extremely simple thing: n! = 1 · 2 · 3 · 4 · 5 · ..... · n. Take 5! as example. We have 

Now we are done. The full expression for the free enthalpy G(N, n, T) of a crystal consisting of N atoms that contains n vacancies at the temperature T is  


We are seeking nirvana for a crystal
of given size (i.e. given and constant N) at some given and
constant temperature T and that means we are seeking the minimum
value of that function. The only variable left is n, the number of vacancies. We now need to determine the minimum of the function G(N, n, T) with respect to the number of vacancies n. 

Obviously we have to differentiate G(N, n, T) with respect to n and set it to zero: dG(N, n, T)/dn = 0. The first constant term disappears if we do that (that's why we didn't need to worry about it) and we get  


Oh Sh...! If you had an extremely good high school education in the US, or an average one in Germany, you know how to differentiate a natural logarithm: But only with an extremely good high school education in Germany you know how to differentiate a factorial function: y = x!; dy/dx = can't be done! It can't be done! It's not a smooth function. 

Thank God for Stirling's formula that makes an approximate smooth
function out of factorials 



Since we have a hell of a lot of atoms (if you
can see a crystal it has more than about 10^{15} atoms) and
therefore also a lot of vacancies, Stirling's formula is extremely precise for
our undertaking. We now replace all the factorials by Stirling's expression, do the differentiation (lengthy but simple), use the fact that 



To be honest, it is a bit too simple. In full generality our basic equation for the equilibrium or nirvana concentration c_{PD} of all point defects we have:  


The only difference is that we now have a
preexponential
factor c_{0} that takes into account that there
is a bit more to the entropy than just the
entropy of mixing atoms and
vacancies. For example, a single vacancy, or just about any point defect, also
carries some direct entropy around, since
it disturbs the arrangement and in particular the vibration spectra of its
neighbors and thus increases the local disorder. The preexponential factor c_{0} carries a lot of information about what is going on in detail. However, its numerical value is typically around 1  5, which means it doesn't matter much as long as we just look at the basics of what is going on. 

Puhhh! Quite a bit of work. And I didn't even run through the details.  
The result, however, is of supreme beauty. We
have derived the most important mathematical expression in sword making. No, it's not the vacancy concentration, it is the exp–(E/kT) term known as Boltzmann factor. The link takes you there. Here we look at bit more closely at what we can learn from our vacancy concentration formula. 

Discussion and Generalization  
The first thing to realize is that we calculated far more than just the vacancy equilibrium concentration. We can use this basic equation also to calculate, for example:  
The selfinterstitial concentration. Everything is pretty much the same, except that the quantity E_{F} now would be the formation energy of one selfinterstitial. In most cases this energy is somewhat larger than the formation energy of vacancies, meaning that we have a lot less interstitials than vacancies in equilibrium. That's what exponential functions will do.  
The divacancy concentration, i.e. the concentration of vacancy pairs. E_{F} is now the formation energy of a divacancy, which we can safely assume to be twice the formation energy of a single vacancy minus some binding energy.  
The equilibrium concentration of
foreign atoms 
interstitial or substitutional. Same procedure once more (eschewing some
details that really don't matter much). The energy in the exponent then is
called "solution energy" and
tells you how much a crystal likes or hates to have a foreign atom sitting
around. If the solution energy would be zero, the crystal doesn't care  the
foreign atom is just as good as one of the own kind. Positive values mean the
crystal does not like the foreign atom, negative values (possible in this case)
that it actually prefers a bit of change. You realize, of course, that the equilibrium concentration value of foreign atoms depends exponentially on the temperature, while the real value is more or less constant. You can't easily change a given concentration of foreign atoms, after all. 

In essence the major difference between point defects is only how much energy E  formation energy, solution energy, whatever  they carry around with them.  
The basic equation for the equilibrium concentration of all those point defects always contains the "Boltzmann factor" exp–(E/kt). The link will show that the vacancy concentration equation derived here is just a special case of a much more general principle around the Boltzmann factor.  
We now need a number for E_{F} so we can evaluate our basic equations for the concentration of point defects. So are we now going to calculate formation or solution energies for all kinds of point defects?  
No, we won't. It simply can't be done with pencil
and paper. Just about now, it might be possible to get reliable numbers with
the biggest computers around. It's not that we don't know how to do it; it's
just too much number crunching. Since E_{F} comes up in
an exponent, we need a rather precise number in order to avoid big
uncertainties. So far we actually do experiments and measure those numbers, as far as possible (which is not very far). It's not easy and we don't have as many reliable numbers as we would like. Here are some examples: 



As a rule of thumb, "normal" metals
have vacancy formation energies H^{F}(V) roughly around
1 eV, and selfinterstitial formation energies
H^{F}(i) a bit larger, say 1,5 eV. No reliable
numbers are known for selfinterstitials. However, if their formation energies
are just a few tenth of an eV larger than those of vacancies, the interstitial
concentrations will be far smaller (factor 100 or more) than the vacancy
concentration. In this case some interstitials will be around but are neither
of any importance for diffusion nor experimentally accessible. Another rule of thumb asserts that the maximum vacancy concentration close to the melting point is around 10^{–4} 

Now let's look at some graphs, representing the point defect concentration.  


A direct plot is not very helpful. Except for
temperatures close to the melting point, the curve is indistinguishable from
the Taxis. That's why we mostly plot equations with
exponentials in a
socalled Arrhenius plot. We simply take as new parameter the
logarithm of the concentration,
lg(c_{V}), and the reciprocal value of the absolute temperature,
1/T. If we call them "y" and
"x", respectively, the equation reads 

Now we interpret the equation as giving the equilibrium concentration for some foreign atom in some crystal, e.g. carbon in bcc iron. Then we typically plot it directly, but with the axes' switched as shown below:  


The dotted line indicates some fixed
concentration. Only concentrations on the
blue line are the proper nirvana or equilibrium concentrations for the
extrinsic point
defect "carbon in iron" at the temperature T. Since
the concentration of foreign atoms typically can't change but has some fixed
value, any point in the yellow region signifies that the crystal is undersaturated, i.e. has less foreign atoms
dissolved than necessary for nirvana. There is absolutely nothing the crystal can do about undersaturations of extrinsic point defects, in contrast to the intrinsic point defects that it can make (or remove) "at will". In the blue region, the extrinsic point defect is supersaturated; there are too many. The crystal can and will do something about that: it can imprison the surplus atoms in a precipitate and thus remove them. What exactly a crystal does about supersaturations of extrinsic point defects is at the very core of steel technology (and about any other materialbased technology). 

Hey, I'm just about to invent phase diagrams and thus get ahead of myself. So let's stop here. You might come back after you went through chapter 6.  
Steel Revolution. 2. The Thomas  Gilchrist Process
Experimental Techniques for Measuring Diffusion Parameters
Phenomenological Modelling of Diffusion
The Story of SelfInterstitials in Silicon
Atomic Mechanisms of Diffusion
Segregation at Room Temperature
Global and Local Equilibrium for Point Defects
© H. Föll (Iron, Steel and Swords script)