Geology Reference
In-Depth Information
Anderson (
1981
) for extension to ternary systems. The practical difficulties
increase greatly with more components and attention has therefore been given to
methods of estimating diffusion coefficients using ionic conductances or trace
diffusion coefficients (Anderson
1981
; Lasaga
1979
).
3.5.4 Atomic Theory of Diffusion
We turn now from the macroscopic or phenomenological approach to consider the
atom movements involved in diffusion, which are the basis of the atomic or kinetic
theory of diffusion. In any material, diffusion arises essentially from the tendency
for an atom to move randomly relative to its neighbors. In crystals, this movement
involves displacements at a certain jump frequency between structural or inter-
stitial sites. Many mechanisms have been proposed, taking into account any
necessary rearrangements to accommodate the atom in its new position. Basically,
the various mechanisms involve (1) more or less direct exchange of structural
sites, (2) exchange with a defect (especially a vacancy), or (3) movement via
interstitial sites (Howard and Lidiard
1964
; Manning
1968
,
1974
).
If the probability that a given atom will jump to a neighboring site is inde-
pendent of the direction of that site and of the previous jump history, then it can be
shown (Flynn
1972
, Chap. 6; Manning
1974
) that the resultant random walk motion
can be expressed through a diffusion coefficient that is the sum of terms of the form
ad
2
C
;
where d is the jump distance, C the jump frequency, and a a numerical factor
of order unity (a
¼
1
=
6 for a simple cubic crystal), the sum being taken over all
combinations of the different types of sites between which jumps can occur,
weighted according to probability of occupation. However, a given jump may be
influenced by the previous jump; for example, in the case of a vacancy mechanism
there is a bias in favor of a reverse exchange with a vacancy with which an
exchange has just occurred. This effect can be allowed for by multiplying the
previous terms by a correlation factor f. The factor f will in general be a tensor,
depending on diffusion mechanism and on temperature, but for simple crystal
structures with one type of jump it is a scalar constant, the calculation of which is
given in most treatments of the kinetic theory of diffusion; for example, f
¼
0
:
78
for diffusion by a simple vacancy mechanism in f.c.c. crystals and f
¼
1 for
interstitial mechanisms (Manning
1968
). Thus, under the assumption that other
factors (drift forces) affecting the relative probabilities of forward and backward
jumps are absent, we obtain the theoretical result that the tracer diffusion coefficient
D
for a single mobile component having a single type of site should be given by
D
¼
ad
2
f C
ð
3
:
32
Þ
In other cases, the diffusing atom may be subject to a drift force F affecting the
relative probabilities of forward and backward jumps and giving rise to a drift
velocity v
F
proportional to F. The flux can then be written as
