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