## Making Vacancies | |||||||||||||||||

Calculating the Equilibrium Vacancy Concentration from
Scratch | |||||||||||||||||

How many vacancies
does a real crystal need for achieving thermodynamic equilibrium. It is about the most simple thing one can calculate
"from scratch" by using the second law. | |||||||||||||||||

First we need to determine the free free enthalpy but that is just a different word for the
same thing. What matters is the mathematical definition: G(N, n, T)
= number of atoms forming the crystal.N = number of vacancies in the crystal.n = (absolute, of course) temperature of the crystal.TI will give you the full function first and then discuss it. What we have is G(N, n, T) | |||||||||||||||||

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The first term, , simply describes the total free enthalpy of a G(N, T)
perfect crystal with
atoms at the temperature N and no vacancies. It contains the energy T of the crystal, essentially contained in the vibrations of the atoms, and the entropy E_{cryst}
resulting from that. S_{vib}If you like you can write it out as but that will just make the equation longer.G(N, T) = E_{cryst}(N,
T) – TS_{vib} (N, T)For any number of atoms and temperature N, the function 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)G(N, T) | |||||||||||||||||

The second
term acknowledges that we expect to have vacancies in our crystal. nMaking a vacancy requires that we sever the bonds between the atoms an and rip out the cut-off 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 vacancies
so we need to nadd
n · to the energy contained in the otherwise perfect crystal.E_{F} | |||||||||||||||||

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

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

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is the number of possibilities to arrange P(N, n)
vacancies in a crystal consisting of n atoms; Nk = Boltzmann's constant. And of course, you know what
"ln", the symbol for a natural logarithm means
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How large is ? You might also ask: In how many
ways can I distribute P(N, n)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 much-hated 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 | |||||||||||||||||

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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: . Take n!
= 1 · 2 · 3 · 4 · 5 · ..... · n5! as example. We have 5!
= 1 · 2 · 3 · 4 · 5 = 96 | |||||||||||||||||

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

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

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

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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:y = ln; xdy/dx = 1/xBut only with an extremely good high school education in Germany you know how to differentiate a factorial function: y = ; x!dy/d! x = can't be doneIt 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 x | |||||||||||||||||

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

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To be honest, it is a bit too simple. In full generality our basic equation for the equilibrium or nirvana concentration
of all c_{PD}point defects we have: |
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The only difference is that we now have a pre-exponential
factor
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 c_{0}direct entropy around, since it disturbs the arrangement and
in particular the vibration spectra of its neighbors and thus increases the local disorder. The pre-exponential factor carries a lot of information about what is going on in detail. However, its numerical value
is typically around c_{0}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–(
term known as E/kT)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 |
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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 self-interstitial concentration.
Everything is pretty much the same, except that the quantity now would be the formation energy
of one self-interstitial. 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.E_{F} | ||||||||||||||||||||||||||||||||

The di-vacancy concentration, i.e. the concentration of vacancy
pairs. is now the formation energy of a di-vacancy, which we can safely
assume to be twice the formation energy of a single vacancy E_{F}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
- formation energy, solution energy, whatever - they carry around with them.E | ||||||||||||||||||||||||||||||||

The basic equation for the equilibrium concentration of all those point defects
always contains the "Boltzmann factor" exp–(.
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. E/kt) | ||||||||||||||||||||||||||||||||

We now need a number for 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?E_{F} | ||||||||||||||||||||||||||||||||

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 comes up in an exponent, we need a rather precise number in order to
avoid big uncertainties.E_{F}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: | ||||||||||||||||||||||||||||||||

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As a rule of thumb, "normal" metals have vacancy formation energies
roughly around H^{F}(V)1 eV, and self-interstitial formation energies a bit larger, say H^{F}(i)1,5
eV. No reliable numbers are known for self-interstitials. 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. | ||||||||||||||||||||||||||||||||

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A direct plot is not very helpful. Except for temperatures close to the melting point, the
curve is indistinguishable from the -axis. That's why we mostly plot equations with exponentials
in a so-called TArrhenius
plot
. We simply take as new parameter the logarithm of the concentration, lg(c,
and the reciprocal value of the absolute temperature, _{V})1/. If we call them "T" and
"y", respectively, the equation reads xy = –(E_{F}/k) · x–(.E_{F}/k) | ||||||||||||||||||||||||||||||||

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: | ||||||||||||||||||||||||||||||||

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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 . 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 Tundersaturated
, 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 material-based
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 Self-Interstitials in Silicon

Atomic Mechanisms of Diffusion

Segregation at Room Temperature

Global and Local Equilibrium for Point Defects

© H. Föll (Iron, Steel and Swords script)