564
10 Quantum Statistics
and
λ =
3 ρ −
6
2
3
2
ρ
2
+
1
4
−
1
3
3
2
9 ρ
3
+ . . . .
(10.6.8)
Substitution of Eq. (10.6.8) into Eq. (10.6.6), followed by equating the powers of
ρ BE on the two sides of the resulting equation, gives the pressure for a dilute (ideal)
Bose–Einstein gas as [see also Table 10.3.]
P
k B T
BE
= ρ −
3
2
5
2
ρ
2
+
6
1
8
−
2
3
5
2
ρ
3
+ . . . ,
(10.6.9)
with
B 2 (T ) = −
3
2
5
2
,
(10.6.10)
B 3 (T ) =
1
8
−
2
3
5
2
6 .
(10.6.11)
The effective interaction in this case is attractive, and is just the opposite to what
would be obtained for a gas of spinless fermions (could such a thing exist!).
Table 10.3 Contributions to
the second virial coefficient
of 4 He at low temperatures a
Contributions to B
Total
T /K B ideal
B interaction B bound B 2 (T )
0.5 −200.08 −1172.00 −5.20 −1377.30
1.0 −70.74
−402.61
−0.92 −474.26
2.0 −25.01
−167.36
−0.16 −192.53
3.0 −13.61
−105.12
−0.12 −118.79
4.0 −8.84
−75.09
−0.06 −83.96
5.0 −6.33
−57.07
−0.03 −63.42
6.0 −4.81
−44.98
−0.02 −49.80
7.0 −3.82
−36.28
−40.11
8.0 −3.13
−29.72
−32.85
9.0 −2.62
−24.59
−27.21
10.0 −2.24
−20.46
−22.70
a Contributions to the second virial coefficient of 4 He.
All values of B 2 are given in cm 3 mol
−1 . Data from
Ref. [2], with permission of Taylor and Francis
10 Quantum Statistics
and
λ =
3 ρ −
6
2
3
2
ρ
2
+
1
4
−
1
3
3
2
9 ρ
3
+ . . . .
(10.6.8)
Substitution of Eq. (10.6.8) into Eq. (10.6.6), followed by equating the powers of
ρ BE on the two sides of the resulting equation, gives the pressure for a dilute (ideal)
Bose–Einstein gas as [see also Table 10.3.]
P
k B T
BE
= ρ −
3
2
5
2
ρ
2
+
6
1
8
−
2
3
5
2
ρ
3
+ . . . ,
(10.6.9)
with
B 2 (T ) = −
3
2
5
2
,
(10.6.10)
B 3 (T ) =
1
8
−
2
3
5
2
6 .
(10.6.11)
The effective interaction in this case is attractive, and is just the opposite to what
would be obtained for a gas of spinless fermions (could such a thing exist!).
Table 10.3 Contributions to
the second virial coefficient
of 4 He at low temperatures a
Contributions to B
Total
T /K B ideal
B interaction B bound B 2 (T )
0.5 −200.08 −1172.00 −5.20 −1377.30
1.0 −70.74
−402.61
−0.92 −474.26
2.0 −25.01
−167.36
−0.16 −192.53
3.0 −13.61
−105.12
−0.12 −118.79
4.0 −8.84
−75.09
−0.06 −83.96
5.0 −6.33
−57.07
−0.03 −63.42
6.0 −4.81
−44.98
−0.02 −49.80
7.0 −3.82
−36.28
−40.11
8.0 −3.13
−29.72
−32.85
9.0 −2.62
−24.59
−27.21
10.0 −2.24
−20.46
−22.70
a Contributions to the second virial coefficient of 4 He.
All values of B 2 are given in cm 3 mol
−1 . Data from
Ref. [2], with permission of Taylor and Francis