 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: 
 
