Contents
The nucleus
Radioactivity
Radiation Measurements
Black body radiation
Statistical Mechanics
Radiation and scattering
Related topics
Related links
© The scientific sentence. 2010
|
Fermi-Dirac distribution
The case of fermions is a little bit similar, except that each sublevel within a degeneracy can take just ONE or ZERO particle.
Here the number of ways to place ni in gi degeneracies is:
C(ni, gi) = gi!/ni!(gi - ni)!
That is the number of combinations of ni among gi
The total number of macrostates is then:
Ω = Π gi!/ni!(gi - ni)!
Using Stirling's approximation, we get:
ln(Ω) = Σ giln gi - gi - niln i + ni - (gi - ni)ln(gi - ni) + (gi - ni)
= Σ giln gi - niln ni - (gi - ni)ln(gi - ni)
Differentiating to maximaze, we get:
dln(Ω) =
Σ - dniln ni
- dni + dniln(gi - ni) + d ni =
Σ dniln[(gi - ni)/ ni]
It follows that by adding the same above constraints:
Σ dni {ln[(gi - ni)/ ni] + α - βEi]} = 0
Which gives:
(gi - ni)/ ni = exp[- α] exp[βEi]
Therefore:
The Fermi-Dirac population numbers is :
ni = gi/(exp[- α] exp[βEi] + 1)
|
