Statistical Mechanics - Physics at Oregon State University

4.5. FERMI GAS. 81
fF D(ɛ) =
1
e ɛ−µ
k B T + 1
(4.51)
The chemical potential µ depends on the temperature and the value of µ at
T = 0K is called the Fermi energy µ(T = 0) = ɛF . It is easy to show that in
the limit T → 0 the Fermi-Dirac function is 1 for ɛ < µ and 0 for ɛ > µ. The
chemical potential follows from
N =
fF D(ɛo) = g
orb
nxnynz
fF D(ɛ(nx, ny, nz)) (4.52)
where g = 2S+1 is the spin degeneracy; g = 2 for electrons. The spin factors out
because the energy levels do not depend on spin, only on the spatial quantum
numbers. Note that we use the fact that the particles are identical, we only
specify the number of particles in each orbital, not which particle is in which
orbital! The description is slightly more complicated when magnetic fields are
included, adding a dependency of the energy on spin.
Convergence of series.
When we have an infinite sum, we always need to ask the question if this
series converges. This means the following:
∞
xn = S (4.53)
n=1
if we can show that for SN = N
n=1 xn the following is true:
lim
N→∞ SN = S (4.54)
This means that for any value of ɛ > 0 we can find a value Nɛ such that
N > Nɛ ⇒ |SN − S| < ɛ (4.55)
If the terms in the series are dependent on a variable like the temperature, we
need to ask even more. Does the series converge uniformly? In general we want
∞
xn(T ) = S(T ) (4.56)
n=1
and this is true if for every ɛ > 0 we can find a value Nɛ(T ) such that
N > Nɛ(T ) ⇒ |SN (T ) − S(T )| < ɛ (4.57)
The problem is that the values of Nɛ depend on T . What if, for example,
limT →0 Nɛ(T ) does not exist? Than the sum of the series is not continuous at
T = 0. That is bad. In order to be able to interchange limits and summations,