 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 x
direction, 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: 
  