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We are only interested in the flux of
vacancies in the x-direction, the diffusion current
j of the vacancies. The flux or
diffusion current of atoms that move via a
vacancy mechanism, would have the same magnitude in the opposite direction.
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We do not assume equilibrium, but a
space-dependent vacancy concentration cV(x,
y, z). Being one-dimensional, we only assume a concentration
gradient in the x-direction, cV(x,
y, z) = cV(x). |
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On any lattice plane perpendicular to
x we have a certain number of vacancies per unit area (the area
density in cm2), which is computable by
c(x). We distinguish this particular concentration with
the index of the plane; i.e. P1 is the number of
vacancies on 1 cm2 area on plane No. 1, etc. |
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We then have |
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P1 |
= |
a · cV (x) |
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P2 |
= |
a · cV (x + dx)
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With dx = a = lattice constant, because smaller increments make no
physical sense, we obtain
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Next we consider the jump rates in
x-direction, i.e. that part of all vacancy jumps out of the plane
that are in +x-direction. We define |
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r12 |
= |
jump rate in x direction
from P1 to P2 |
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r21 |
= |
jump rate in x
direction
from P2 to P1 |
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We obtain for our geometry: |
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r12(T) |
= r21(T) =
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1
6 |
· r (T) |
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This means that 1/6 of the total number of
possible jumps of a vacancy is in the +x or
x direction, the other possibilities are in the y- or
z-direction. |
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The jump rate itself is given by the
usual Boltzmann formula |
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With ν0
= vibration frequency of the particle, HM =
enthalpy of migration. |
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We obtain for the number of vacancies
per cm2 and second, which jump from
P1 to P2, i.e. for the
component of the diffusion current j12 flowing
to the right (and this is not yet the
diffusion current from Ficks law!): |
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This is the current of vacancies flowing
out in x-direction from P1. This
current will be compensated to some extent by the current component
j21 which flows into
P1. This current component is given by |
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With the equation from above we obtain for the
two components of the current |
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j12 |
= |
r
6 |
· a · c(x) |
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j21 |
= |
r
6 |
· a · c(x +
dx) |
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The net jx
current in x-direction, which is the current in Ficks laws, is exactly the
difference between the two partial currents, we obtain |
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jx |
= |
j12
j21 |
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= |
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a · r
6 |
· {c(x + dx)
c(x)} |
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If we now multiply by dx/dx =
a/dx we obtain directly
Ficks first
law for one dimension: |
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jx |
= |
a2 · r
6 |
· |
c(x + dx) c(x)
dx |
= |
a2 · r
6 |
· |
dc(x)
dx |
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All we have to do is to indentify
(a2 · r)/6 with the diffusion coefficient
D of Fick's first law; we then have it in full splendor: |
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Ficks first law thus can be deduced
in an unambiguous and physically sensible way for primitive cubic crystals in
one dimension. (Mathematicians may have problems with the equality dx
= a; but never mind). |
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We also obtain an equation for the phenomenological diffusion coefficient D in terms of the
atomic parameters lattice constants and jump
rate (for the simple cubic lattice). |
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Considering arbitrary crystals now is
easy. |
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The only parameters different in different
crystal systems are the factor 1/6 and the jump distance, which does not
have to be only a, but , in general, for jump type i will
be Δxi. With i we enumerate
all geometrically different variants of jumps and take into account that the
x- component may depend on i. |
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The diffusion coefficient then is given by |
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And g is a constant
which is specific for the lattice under consideration, it is the so-called geometry factor of the lattice
for diffusion. |
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If we reconsider how
we obtained the factor 1/6 for the cubic primitive lattice
used above, it is clear that in a general case
the geometry factor is defined by the equation |
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The factor 1/2 takes into account that
only 1/2 of all possible jumps must be counted, because the other half
would be the jumps back. Δxi/a simply expresses the component
of the jump in x-direction in units of a. |
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For simple lattices g is easily
calculated; for the fcc and bcc lattice we have g =
1. |
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Taking into account three dimension
is easy, too: |
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In isotropic lattices (which, besides the cubic
lattices, covers all poly-crystals) no direction is special, the above
equations are equally valid for the y- and
z-direction. We obtain then a vector equation for Ficks first
law |
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j(r) |
= D0 · exp |
EM
kt |
·
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c(x,y,z) |
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In anisotropic crystals things are
messy. Every direction has to be considered separately, the so far scalar quantity D evolves into a
second-rank tensor. Fortunately, we do not
have to consider this here. |
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© H. Föll