
We are only interested in the flux of
vacancies in the xdirection, 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.



We do not assume equilibrium, but a
spacedependent vacancy concentration c_{V}(x,
y, z). Being onedimensional, we only assume a concentration
gradient in the xdirection, c_{V}(x,
y, z) = c_{V}(x). 


On any lattice plane perpendicular to
x we have a certain number of vacancies per unit area (the area
density in cm^{–2}), which is computable by
c(x). We distinguish this particular concentration with
the index of the plane; i.e. P_{1} is the number of
vacancies on 1 cm^{2} area on plane No. 1, etc. 


We then have 


P_{1} 
= 
a · c_{V} (x) 



P_{2} 
= 
a · c_{V} (x + dx)





With dx = a = lattice constant, because smaller increments make no
physical sense, we obtain



P_{2} 
= 
a · c_{V} (x + a) 



Next we consider the jump rates in
xdirection, i.e. that part of all vacancy jumps out of the plane
that are in +xdirection. We define 


r_{1–2} 
= 
jump rate in x – direction
from P_{1} to P_{2} 



r_{2–1} 
= 
jump rate in – x
– direction
from P_{2} to P_{1} 




We obtain for our geometry: 


r_{1–2}(T) 
= r_{2–1}(T) =

1
6 
· r (T) 




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
zdirection. 

The jump rate itself is given by the
usual Boltzmann formula 


r = n_{0} ·
exp – 
H^{M}
kT^{ } 




With n_{0}
= vibration frequency of the particle, H^{M} =
enthalpy of migration. 

We obtain for the number of vacancies
per cm^{2} and second, which jump from
P_{1} to P_{2}, i.e. for the
component of the diffusion current j_{1–2} flowing
to the right (and this is not yet the
diffusion current from Ficks law!): 


j_{1–2} 
= 
P_{1} · r_{1–2} 




This is the current of vacancies flowing
out in xdirection from P_{1}. This
current will be compensated to some extent by the current component
j_{2–1} which flows into
P_{1}. This current component is given by 


j_{2–1} 
= 
P_{2} · r_{2–1} 




With the equation from above we obtain for the
two components of the current 


j_{1–2} 
= 
r
6 
· a · c(x) 




j_{2–1} 
= 
r
6 
· a · c(x +
dx) 



The net j_{x}
current in xdirection, which is the current in Ficks laws, is exactly the
difference between the two partial currents, we obtain 


j_{x} 
= 
j_{1–2} –
j_{2–1} 






= 
– 
a · r
6 
· {c(x + dx) –
c(x)} 




If we now multiply by dx/dx =
a/dx we obtain directly
Ficks first
law for one dimension: 


j_{x} 
= – 
a^{2} · r
6 
· 
c(x + dx) – c(x)
dx 
= – 
a^{2} · r
6 
· 
dc(x)
dx 




All we have to do is to indentify
(a^{2} · r)/6 with the diffusion coefficient
D of Fick's first law; we then have it in full splendor: 


