436 Quantum ideal gases
0 0.2 0.4 0.6 0.8 1
z0
0.5
1
1.5
2
2.5
3
g3/2(z)Fig. 11.2 The functiong 3 / 2 (ζ).prefactor. It is, therefore, useful to ask precisely how closeζ must be to 1 for the
divergent term to be important. Because of theλ^3 /Vprefactor,ζcan only be different
from 1 by an amount on the order of 1/V. In order to see this, let us assume thatζ
can be written in the form
ζ= 1−
a
V, (11.6.23)
whereais a positive constant to be determined. The magnitude ofais a measure of
the amount by whichζdeviates from 1 at a given volume. Substituting this ansatz
into eqn. (11.6.19) gives
ρλ^3
g=g 3 / 2 (1−a/V) +λ^3
V1 −a/V
a/V. (11.6.24)
Sinceg 3 / 2 (ζ) does not change its value much ifζis displaced slightly from 1, we can
replace the first term to a very good approximation byR(3/2), which yields
ρλ^3
g≈g 3 / 2 (1) +
λ^3
V1 −a/V
a/V. (11.6.25)
Eqn. (11.6.25) can be solved for the unknown parameterato give
a=λ^3
ρλ^3
g −R(3/2), (11.6.26)
where we have neglected a term proportional toλ^3 /V, which vanishes in the thermody-
namic limit. Sinceamust be positive, this solution is only valid forρλ^3 /g > R(3/2).