Diffusion Solids Fundamentals Diffusion Controlled Solid State Episode 1 Part 6 potx

7.3 Vacancy Mechanism of Self-diffusion 113

from giving a comprehensive review of all methods. Instead we rather strive

for a physical understanding of the underlying ideas: we consider explicitely

low vacancy concentrations and cubic coordination lattices. Then, the aver￾ages in Eq. (7.7) refer to one complete encounter. Since for a given value of

n there are (n − j) pairs of jump vectors separated by j jumps, and since all

vacancy-tracer pairs immediately after their exchange are physically equiva￾lent, we introduce the abbreviation

cos θj ≡

cos θi,j  and get:

f = 1 + limn→∞

2

n

n

−1

j=1

(n − j)

cos θj . (7.21)

Here

cos θj  is the average of the cosines of the angles between all pairs of

vectors separated by j jumps in the same encounter. With increasing j the

averages

cos θj  converge rapidly versus zero. Executing the limit n → ∞,

Eq. (7.21) can be written as:

f =1+2∞

j=1

cos θj 

= 1+2(

cos θ1 +

cos θ2 + ...) . (7.22)

To get further insight, we consider – for simplicity reasons – the x-dis￾placements of a series of vacancy-tracer exchanges. For a suitable choice of

the x-axis only two x-components of the jump vector need to be considered2,

which are equal in length and opposite in sign. Since then cos θj = ±1, we

get from Eq. (7.22)

f =1+2∞

j=1

"

p+

j − p−

j

#

, (7.23)

where p+

j (p−

j ) denote the probabilities that tracer jump j occurs in the same

(opposite) direction as the first jump. If we consider two consecutive tracer

jumps, say jumps 1 and 2, the probabilities fulfill the following equations:

p+

2 = p+

1 p+

1 + p−

1 p−

1 ,

p−

2 = p+

1 p−

1 + p−

1 p+

1 . (7.24)

Introducing the abbreviations tj ≡ p+

j − p−

j and t1 ≡ t, we get

t2 = p+

1

"

p+

1 − p−

1

#

   t

−p−

1

"

p+

1 − p−

1

#

   t

= t

2 . (7.25)

From this we obtain by induction the recursion formula

tj = t

j . (7.26)

2 Jumps with vanishing x-components can be omitted.

114 7 Correlation in Solid-State Diffusion

The three-dimensional analogue of Eq. (7.26) was derived by Compaan

and Haven [20] and can be written as

cos θj  = (

cos θ)

j , (7.27)

where θ is the angle between two consecutive tracer jumps. With this recur￾sion expression we get from Eq. (7.22)

f =1+2

cos θ

$

cos θ + (

cos θ)

2 + ... %

. (7.28)

The expression in square brackets is a converging geometrical series with the

sum 1/(1 −

cos θ). As result for the correlation factor of vacancy-mediated

diffusion in a cubic coordination lattice, we get

f = 1 +

cos θ

1 −

cos θ

. (7.29)

We note that Eq. (7.29) reduces correlation between non-consecutive pairs

of tracer jumps within the same encounter to the correlation between two

consecutive jumps. Equation (7.29) is valid for self- and solute diffusion via

a vacancy mechanism.

The remaining task is to calculate the average value

cos θ. At this point,

it may suffice to make a few remarks: starting from Eq. (7.29), we consider

the situation immediately after a first vacancy-tracer exchange (Fig. 7.3).

The next jump of the tracer atom will lead to one of its Z neighbouring sites

l in the lattice. Therefore, we have

cos θ = 

Z

l=1

Pl cos δl . (7.30)

In Eq. (7.30) δl denotes the angle between the first and the second tracer

jump, which displaces the tracer to site l. Pl is the corresponding probability.

A computation of Pl must take into account all vacancy trajectories in the

lattice which start at site 1 and promote the tracer in its next jump to

site l. An infinite number of such vacancy trajectories exist in the lattice.

One example for a vacancy trajectory, which starts at site 1 and ends at site

4, is illustrated in Fig. 7.3. Some trajectories are short and consist of a small

number of vacancy jumps, others comprise many jumps.

A crude estimate for the correlation factor can be obtained as follows: we

consider the shortest vacancy trajectory, which consists of only one further

vacancy jump after the first displacement of the tracer, i.e. we disregard the

infinite number of all longer vacancy trajectories. Then, nothing else than the

immediate back-jump of the tracer to site 1 in Fig. 7.3 can occur. Any other

tracer jump requires vacancy trajectories with several vacancy jumps. For

example, for a tracer jump to site 4 the vacancy needs at least 4 jumps (e.g.,