Diffusion Solids Fundamentals Diffusion Controlled Solid State Episode 1 Part 4 potx

4.1 Random Walk and Diffusion 61



R2

= n

i=1



r 2

i



+ 2

n

−1

i=1

n

j=i+1



rirj



,



X2

= n

i=1



x2

i



+ 2

n

−1

i=1

n

j=i+1



xixj



. (4.20)

The first term contains squares of the individual jump lengths only. The

double sum contains averages between jump i and all subsequent jumps j.

Uncorrelated random walk: Let us consider for the moment a random

walker that executes a sequence of jumps in which each individual jump is

independent of all prior jumps. Thereby, we deny the ‘walker’ any memory.

Such a jump sequence is sometimes denoted as a Markov sequence (memory

free walk) or as an uncorrelated random walk. The double sum in Eq. (4.20)

contains n(n − 1)/2 average values of the products

xixj  or

rirj . These

terms contain memory effects also denoted as correlation effects. For a Markov

sequence these average values are zero, as for every pair xixj one can find

for another particle of the ensemble a pair xixj equal and opposite in sign.

Thus, we get from Eq. (4.20) for a random walk without correlation

R2

random = n

i=1

r 2

i ,

X2

random = n

i=1

x2

i . (4.21)

The index ‘random’ is used to indicate that a true random walk is considered

with no correlation between jumps.

In a crystal lattice the jump vectors can only take a few definite values. For

example, in a coordination lattice (coordination number Z), in which nearest￾neighbour jumps occur (jump length d with x-projection dx), Eq. (4.21) re￾duces to



R2

random

= 

n



d2 , 

X2

random

= 

n



d2

x . (4.22)

Here

n denotes the average number of jumps of a particle. It is useful to

introduce the jump rate Γ of an atom into one of its Z neighbouring sites via

Γ ≡

n

Zt . (4.23)

We then get

D = 1

6

d2ZΓ = d2

6¯τ . (4.24)

62 4 Random Walk Theory and Atomic Jump Process

Table 4.1. Geometrical properties of cubic Bravais lattices with lattice parameter a

Lattice Coordination number Z Jump length d

Primitive cubic 6 a

Body-centered cubic (bcc) 8 a

√3/2

Face-centered cubic (fcc) 12 a

√2/2

This equation describes diffusion of interstitial atoms in a dilute interstitial

solid solution1. The quantity

τ¯ = 1

ZΓ (4.25)

is the mean residence time of an atom on a certain site. For cubic Bravais

lattices, the jump length d and the lattice parameter a are related to each

other as indicated in Table 4.1. Using these parameters we get from Eq. (4.24)

D = a2Γ . (4.26)

4.1.4 Correlation Factor

Random walk theory, to this point, involved a series of independent jumps,

each occurring without any memory of the previous jumps. However, several

atomic mechanisms of diffusion in crystals entail diffusive motions of atoms

which are not free of memory effects. Let us for example consider the vacancy

mechanism (see also Chap. 6). If vacancies exchange sites with atoms a mem￾ory effect is necessarily involved. Upon exchange, vacancy and ‘tagged’ atom

(tracer) move in opposite directions. Immediately after the exchange the va￾cancy is for a while available next to the tracer atom, thus increasing the

probability for a reverse jump of the tracer. Consequently, the tracer atom

does not diffuse as far as expected for a completely random series of jumps.

This reduces the efficiency of a tracer walk in the presence of positional mem￾ory effects with respect to an uncorrelated random walk.

Bardeen and Herring in 1951 [7, 8] recognised that this can be ac￾counted for by introducing the correlation factor

f = limn→∞

R2

R2

random = 1 + 2 limn→∞

n−1

i=1

n

j=i+1

rirj 

n

i=1

r 2

i 

,

1 In a non-dilute interstitial solution correlation effects can occur, because some

of the neighbouring sites are not available for a jump.