Carsten Timm: Theory of superconductivity

where O(λ 3 /V ) is a correction of order λ 3 /V ≪ 1. Thus, by Taylor expansion,
N ϵ = V λ 3 g 3/2(1) − O(1) = V ζ(3/2) − O(1). (3.25)
λ3 The intensive term O(1) can be neglected compared to the extensive one so that
N ϵ
∼
V = ζ(3/2). (3.26)
λ3 This is the maximum possible value at temperature T . Furthermore, N 0 = y/(1 − y) is solved by
For y to be very close to unity, N 0 must be N 0 ≫ 1. Since
y = N 0
N 0 − 1 = 1
1 + 1/N 0
. (3.27)
must be positive, we require
N 0 = N − N ϵ
∼
V = N − ζ(3/2) (3.28)
λ3 We conclude that the fraction of particles in excited states is
The fraction of particles in the ground state is then
N
V > ζ(3/2)
λ 3 (3.29)
⇒ T < T c . (3.30)
N ϵ
N ∼ =
V
( ) 3/2
Nλ 3 ζ(3/2) = λ3 (T c ) T
λ 3 (T ) = . (3.31)
T c
In summery, we find in the thermodynamic limit
( ) 3/2
N 0
N ∼ T
= 1 − . (3.32)
T c
(a) for T > T c :
(b) for T < T c :
N ϵ
N ∼ = 1,
( ) 3/2
N ϵ
N ∼ T
= ,
T c
N 0
N
≪ 1, (3.33)
( ) 3/2
N 0
N ∼ T
= 1 − . (3.34)
T c
1
N / N
0
N / N
ε
0
0
0.5
1
T / T c
16