1. D for a random walk – Simple Picture
A simple look-
Consider 2 planes of atoms with different concentrations of solute: n1 n2
Plane (1)
Plane (2) Δx
n1 - # at/ unit area on plane (1)
n2 - # at/ unit area on plane (2)
# of atoms leaving (1) = Γ n1, where Γ is the average # of jumps an atom makes between planes per unit time

2. The flux from (1) → (2)
(2) → (1)
The concentrations ( # at / unit volume) in the planes
and
let

3. so
In 3 – D since
; isotropy →

4. Atomic jumps and the diffusion coefficient
the overall displacement is composed of a number of elementary jumps (all equal in a cubic crystal).
If is the overall displacement per unit time,
where is the elementary jump vector and there are jumps per unit time.
Squaring both sides,

5. The sum can be written by separating terms for which i = j from those for which i ≠ j.
Rewriting the second term in terms of partial sums;
R2 can now be written as,

6. In a cubic crystal all the ri are equal. Also
where θi, i+j is the angle between i th and (i + j)th jump. R is the magnitude of the displacement of a single atom in unit time.
To obtain the mean square displacement we average the expression for R2 over a large number of atoms i.e.,
where
and is called the correlation factor. f is usually hard to calculate and for simple diffusion mechanisms f is a constant near unity

7. Recall so
For interstitial and vacancy diffusion f =1 and
where rI or v is the distance between nearest – neighbor points and I or v is the jump frequency.

8. Self - Diffusion
For self-diffusion by a vacancy mechanism the jump frequency Γv must be multiplied by xv the probability that an atom is next to a vacancy.
(Here xv is the mole fraction of vacancies.) This is so because Γv is the actual number of jumps an atom makes per unit time and an atom can not move unless a vacancy is next to it.
Recall the general form
Thus for self-diffusion, and,

9. So that
Assuming that the vacancies are in thermodynamic equilibrium:
And combination with the previous equation yields

10. Where we have taken f ≈ 1. Using
The relation for Ds is often written more concisely as
where
and

11. Note that a similar form can be written for an interstitial except the xv term equivalent is of course absent . For most close-packed metals:
and
A consequence of these correlations is that the diffusion coefficient of all
materials with a given crystal structure and bond type will be approximately the same at the same fraction of their melting temperature, i.e.,
where ( T / Tm ) is known as the homologous temperature.