Statistical Mechanics - Physics at Oregon State University

5.1. FERMIONS IN A BOX. 103
This relates the Fermi energy to the density and this relation is very important
in the theory for the conduction electrons in a metal. In that case, of course,
S = 1
2 and the spin factor 2S + 1 is equal to two.
The correction for low temperatures is obtained by using this T = 0 result
in the expression for N together with the second term in the series expansion:
n
4
= (2S + 1)
nQ(T ) 3 √ 3
(βµ) 2 (1 +
π π2
8 (βµ)−2 ) (5.75)
and by comparing with the T = 0 result we obtain:
or
(2S + 1) 4
3 √ π (βEF ) 3
2 = (2S + 1) 4
3 √ 3
(βµ) 2 (1 +
π π2
8 (βµ)−2 ) (5.76)
( µ
)
EF
− 3
2 = 1 + π2
8 (βµ)−2
(5.77)
For low temperatures we write µ = EF + ∆µ, with ∆µ small, and we can
expand the left hand side up to first order:
− 3 ∆µ
≈
2 EF
π2
8 (βµ)−2
(5.78)
On the right hand side we only need the leading order term, and we can replace
µ by EF . This gives
or
∆µ
EF
µ(T, N, V ) ≈ EF
What are low temperatures?
≈ − π2
12 (kBT ) 2 E −2
F
1 − π2
12
2
kBT
EF
(5.79)
(5.80)
The previous equation also gives us a direct measure for the magnitude of
the temperature. For a given density n = N
V we can find the Fermi energy from
equation (5.74 ). The corrections depend on the ratio of the thermal energy and
the Fermi energy, and hence low temperature means kBT ≪ EF .
Helmholtz free energy al low temperature.
The Helmholtz free energy is obtained by integrating µ from 0 to N. Since
EF ∝ N 2
3 we can use