438 Quantum ideal gases
from which it can be seen that the average occupation of each energy level is given by
〈fnm〉=
ζe−βεn
1 −ζe−βεn=
1
eβ(εn−μ)− 1. (11.6.30)
Eqn. (11.6.30) is known as theBose–Einstein distribution function. For the ground
state (n= (0, 0 ,0)), the occupation number expression is
〈f 0 m〉=ζ
1 −ζ. (11.6.31)
Substituting the ansatz in eqn. (11.6.23) forζinto eqn. (11.6.31) gives
〈f 0 m〉≈V
a=
V
λ^3(
ρλ^3
g−R(3/2)
)
(11.6.32)
forρλ^3 /g > R(3/2). Atρλ^3 /g=R(3/2),ζ→0, and the occupation of the ground
state becomes 0. The temperature at which then= (0, 0 ,0) level starts to become
occupied can be computed by solving
ρλ^3
g=R(3/2)
ρ
g(
2 π ̄h^2
mkT 0) 3 / 2
=R(3/2)
kT 0 =(
ρ
gR(3/2)) 2 / 3
2 π ̄h^2
m. (11.6.33)
For temperatures less thanT 0 , the occupation of the ground state becomes
〈f 0 m〉=ρV
g[
1 −
g
ρλ^3R(3/2)
]
=
〈N〉
g[
1 −
g
ρλ^3R(3/2)
]
=
〈N〉
g[
1 −
gR(3/2)
ρ(
mkT
2 π ̄h^2) 3 / 2 (
kT 0
kT 0) 3 / 2 ]
=
〈N〉
g[
1 −
(
T
T 0
) 3 / 2 ]
〈f 0 m〉
〈N〉=
1
g[
1 −
(
T
T 0
) 3 / 2 ]
. (11.6.34)
AtT= 0,