# Exercise 4.2-1

# Diffusion During Cooling

| A (big) crystal cools down from its melting point Tm to room temperature Tr ( about 0o C) with T = Tm · exp – (l · t). The point defects present have a diffusion coefficient given by D = D0 · exp – (Em/kT).
|  | How large is the average distance L that they cover during cooling down from some temperature T to Tr?
| 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.
L2 | = | 6D · t | = | t' = ¥ ó õ t ' = t0 | D(t') · dt'
|  | Proceed by first finding the values of l for initial cooling rates at the melting point of 1 oC/s, 10 oC/s, 50 oC/s and, for fun, 104 oC/s.
|  | Using the following substitution will help with the integration
u(t) | = | Em · exp l · t kTm
--- | --- | ---
|  | The integral now runs from u0 corresponding to t'0 to whatever value of u corresponds to t' = ¥.
|  | You will obtain the following integral:
L2 | = | 2D0 | ? ó õ ? | 1 u | · exp – u · du
|  | 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 u0.
|  | Show that for realistic u0 values you can replace 1/u by 1/u0 in a decent approximation and that you now can do the integral.
| Now use typical values for melting temperatures, migration activation energies Em, and D0; e.g. from the backbone, two tables or diagrams given here. For missing values (e.g. D0), 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 Average density of precipitates of point defects obtained in big crystals with few internal sinks. Average size of these precipitates for some equilibrium concentration c0 at Tm. Possible errors made in quenching experiments. Influence of sinks for point defects as a function of the average distance between sinks
| Link to the Solution

With frame

History of Steel

4.2.1 Point Defects in Non-Equilibrium

4.2.2 Essentials to Chapter 4.2: Experimental Techniques for Studying Point Defects in Non-Equilibrium

Numbers for Point Defect Diffusion

Self-Diffusion and some Related Quantities in Si

Impurity Diffusion in Si - Arrhenius Plot

Solution to Exercise 4.2-1

© H. Föll (Defects - Script)