Statistical Mechanics - Physics at Oregon State University

5.3. BOSE-EINSTEIN CONDENSATION. 113
limit, however, the contribution of these terms will go to zero! For finite volumes
these terms can give some contributions, leading to rounding off of sharp
curves near the phase transition.
By splitting the range in these two parts, we can show that on each part the
convergence is uniform. Hence we have
V > Vɛ ⇒
error(T, µ, V )
Ω
V
< ɛ (5.131)
with Vɛ independent of T and µ. Therefore, in all cases the error term is small
compared with the value of Ω, assuming a large volume. A value of 1 m 3 for
the volume certainly satisfies the criteria of being large. Hence for all practical
applications we can ignore the error term. Keep in mind, though, that if you
do experiments on a small container of helium gas with only 1000 atoms at
normal pressure and temperature, you will find deviations from this formula.
To analyze such experiments you really have to calculate the sum of the series.
5.3 Bose-Einstein condensation.
The expression for the density N
V is
n =
2S + 1
fBE(ɛ111) +
V
2S + 1
λ 3 T
g 3 (λ) (5.132)
2
We now take the thermodynamic limit of this expression. For large values of
the volume we may replace ɛ111 by zero. This gives us
n =
2S + 1
V
It is easy to show that g 3
2
the derivative is proportional to g 1
2
λ
1 − λ + (2S + 1)nQ(T )g 3
2
(λ) (5.133)
(λ) is a monotonically increasing function of λ, since
which is always positive for 0 λ 1. As
a result, the particle density is an monotonically increasing function of λ. We
define G = g 3 (1) = 2.612 · · ·.
2
It is possible to distinguish two regimes. If
n
< (2S + 1)G (5.134)
nQ(T )
it is possible to find a value of λ < 1 satisfying n = (2S +1)nQ(T )g 3 (λ) . In the
2
expression for n we therefore are able to take the thermodynamic limit V → ∞,
and the first term in equation (5.133) is zero. This means that in the density
regime n < nQ(T ) the ground state orbital does not play an important role,
and the gas behaves like a normal gas.
On the other hand, if
n
> (2S + 1)G (5.135)
nQ(T )