We have already derived the formula for calculating the density of vacancies in equilibrium, it reads:
G(n)  =  E._{0} + n · E.^{F.}  k_{B.}T · Ln  N! n! · (N – n)! 


The calculation of the equilibrium number of vacancies is thus over
dG(n) / dn = 0 now become a task.
The hard part is
dS.(n) / dn, so
dS._{n} dn  = k_{B.} ·  d dn 
 ln N!  [ln n! + ln (N  n)!] 


The math problem is reduced to calculating
d [lnn!] dn  +  d [ln (N – n)!] dn 


Since one cannot really differentiate functions with factorials (they are not continuous at all), it is now necessary to make some approximations:
Using the simplest version of the
formula for faculties:
This simple formula not only generates a very wellfitting numerical value for not too small
xe.g.
x = 17, but also produces a function, i.e. it also supplies values for e.g.
x = 17,31. What
17,31! may mean, let's leave it open  but in any case we can now differentiate with this approximation.
The following always applies to a real crystal:
This is justified because the number of voids will always be much smaller than the number of atoms.
Furthermore, instead of the or
n at spaces also their (relative frequency)
c_{V} about the now familiar relationship
n N  = c_{V} =   Concentration of vacancies (index V) 


Again: Frequencies as defined here have unit of measure; a
c_{V}Value of
0,01 corresponds to
1 % Spaces related to the. Instead of using the following abbreviations:
That the
English or American trillion the
German billion (
= 10^{9}) and the (
= 10^{12}) is a source of constant errors in all German newspapers, but of course not in
ET&ITEngineers!
We leave the math, which is now quite simple, to an exercise.
We can now generalize this detailed consideration immediately, because it also applies analogously to atomic defects:
Let's take that
Formation energy of selfinterstitial atomsE.^{F.}(i), we have the equation for that.
Let us take a specific energy to describe the incorporation of a foreign atom,
E.^{L.}(FA), we use it to describe the
solubility a foreign atom, d. H. the optimal concentration at a given temperature.
E.^{L.}(FA) describes the energy that has to be applied to build a foreign atom into the lattice.
You have to be careful here, however. While one is working on, i.e. installing an apprenticeship or a
ZGA, Energy must, can
E.^{L.}(FA) sometimes being, i.e. you get energy through the incorporation of a foreign atom (simply because crystal atoms sometimes prefer to have a foreign atom as a neighbor rather than one of their own kind). Also can
E.^{L.}(FA) be very small (i.e. the crystal does not really care who is sitting on the lattice places).
As long as the concentrations are small, i.e. as long as the various types of atomic defects "do not see" each other, all concentrations are simply additive 
GG calls for the correct concentration of feasible atomic defects in the system.
In any case, this requires that a certain concentration of atomic defects be present. For high formation or solubility energies or low temperatures, this concentration can be as small as desired, but it never becomes zero!
A concentration is zero at the latest when there is less than an atomic defect on all atoms of the crystal in question. In macroscopic (visible to the naked eye) crystals this is round and crude at concentrations of
c_{V} 10^{–21} the case.
In terms of measurement technology, however, concentrations of
c_{V} 10^{–10} usually no longer directly detectable. However, this rules out that atomic defects in such small concentrations can still influence the properties of a material.
Another of the very sad consequences of minimizing the free energy via atomic defects is that crystals tend to get dirty at high temperatures.
If the installation energy
E.^{L.}(FA) of a foreign atom is not too high, the crystal "would like" to "have" some at a high temperature. If these atoms are available  they are in the atmosphere, on the surface  the crystal incorporates them by diffusion until it has the correct equilibrium concentration.
The only problem now is that it doesn't get rid of the dirt when it cools down  unlike intrinsic ones
AFthat can also disappear again. This is a theoretical problem, but a "plague" that can only be fought with a lot of money in any semiconductor technology.
Here are the quick questions:
© H. Föll (MaWi for ET&IT  Script)