
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: 
 
