A (big) crystal cools down from its melting point T_{m} to room temperature T_{r} ( about 0^{o} C) with T = T_{m} · exp – (l · t). The point defects present have a diffusion coefficient given by D = D_{0} · exp – (E_{m}/kT).  
How large is the average distance L that they cover during cooling down from some temperature T to T_{r}?  
This is not an easy question. What you should do is:  
Use the Einstein relation for the diffusion length (and forget about lattice factors), but consider that the diffusion coefficient is a function of time, i.e.  


Proceed by first finding the values of l for initial cooling rates at the melting point of 1 ^{o}C/s, 10 ^{o}C/s, 50 ^{o}C/s and, for fun, 10^{4} ^{o}C/s.  
Using the following substitution will help with the integration  


The integral now runs from u_{0} corresponding to t'_{0} to whatever value of u corresponds to t' = ¥.  
You will obtain the following integral:  


This integral cannot be solved analytically. In order to get a simple and good approximation, you may use the linear Taylor expansion for 1/u around u_{0}.  
Show that for realistic u_{0} values you can replace 1/u by 1/u_{0} in a decent approximation and that you now can do the integral.  
Now use typical values for melting temperatures, migration activation energies E_{m}, and D_{0}; e.g. from the backbone, two tables or diagrams given here. For missing values (e.g. D_{0}), make some reasonable assumptions.  
Plot L as a function of T for activation energies E = 1.0 eV, E = 2.0 eV, and E = 5 eV with the four cooling rates given above as parameter.  
Play around a bit and draw some conclusions, e.g. with respect to


Link to the Solution 

4.2.1 Point Defects in NonEquilibrium
Numbers for Point Defect Diffusion
SelfDiffusion and some Related Quantities in Si
Impurity Diffusion in Si  Arrhenius Plot
© H. Föll (Defects  Script)