10.4 Fermi–Dirac Statistics
529
10.4 Fermi–Dirac Statistics
10.4.1 The Ideal Fermi Gas
As has been pointed out in the introductory section of this chapter, the classical
Gibbs treatment of indistinguishable particles is valid for NN 3 (T )/V 1. The
3 He isotope of helium is a fermion (due to its nuclear spin I a =
1
2 ) and is the
lightest element for which the atom is the naturally-occurring form on our planet.
If we consider 3 He at temperature T = 300 K, for example, it has a thermal
de Broglie wavelength, (T ), of approximately 0.45 nm. For a sample of 3 He at
a pressure P ◦ = 1 bar and molar volume V
◦ = 24.943 L, we obtain V
◦ /N 0
(6.44 nm) 3 , so that N 0 3 (T = 300 K)/V
◦ (300 K) 3.42 × 10 −4 , which is indeed
quite small. This means that even for 3 He, we do not really require the use of Fermi–
Dirac statistics to describe properties of an ideal gas with this mass at a temperature
of 300 K. However, if we carry out the same estimate for a sample of 3 He at a
temperature of 5 K, for example, we obtain N 0 3 (5 K)/V
◦ (5 K) 61.4: even for
a temperature T = 50 K, this ratio has a value of approximately 0.108, so that it
becomes clear that a quantum-statistical description of 3 He will be necessary for
3 He at any temperature significantly below room temperature.
In the following we shall focus upon a gas of spin1
2 fermions, so that
FD (() = 4π
2m
h 2
3
2
V V
1
2 ,
(10.4.1)
since the number of spin1
2 fermion states is twice the number of classical
translational states [note that for arbitrary spin I a , we would require a factor 2I a + 1
rather than simply 2, see Eq. (10.2.13b)].
From Eq. (10.2.12), together with the relation = −P V , we obtain the equation
of state for the Fermi–Dirac gas as
P V
k B T
FD
=
k
ln
1 + λe
−ββ k
.
We begin by replacing this discrete variable expression by the continuum approximation
P V
k B T
FD
∞
0
FD (() ln
1 + λe
−ββ
d
= 4π
2m
h 2
3
2
V
∞
0
1
2 ln
1 + λe
−ββ
d .
(10.4.2)
529
10.4 Fermi–Dirac Statistics
10.4.1 The Ideal Fermi Gas
As has been pointed out in the introductory section of this chapter, the classical
Gibbs treatment of indistinguishable particles is valid for NN 3 (T )/V 1. The
3 He isotope of helium is a fermion (due to its nuclear spin I a =
1
2 ) and is the
lightest element for which the atom is the naturally-occurring form on our planet.
If we consider 3 He at temperature T = 300 K, for example, it has a thermal
de Broglie wavelength, (T ), of approximately 0.45 nm. For a sample of 3 He at
a pressure P ◦ = 1 bar and molar volume V
◦ = 24.943 L, we obtain V
◦ /N 0
(6.44 nm) 3 , so that N 0 3 (T = 300 K)/V
◦ (300 K) 3.42 × 10 −4 , which is indeed
quite small. This means that even for 3 He, we do not really require the use of Fermi–
Dirac statistics to describe properties of an ideal gas with this mass at a temperature
of 300 K. However, if we carry out the same estimate for a sample of 3 He at a
temperature of 5 K, for example, we obtain N 0 3 (5 K)/V
◦ (5 K) 61.4: even for
a temperature T = 50 K, this ratio has a value of approximately 0.108, so that it
becomes clear that a quantum-statistical description of 3 He will be necessary for
3 He at any temperature significantly below room temperature.
In the following we shall focus upon a gas of spin1
2 fermions, so that
FD (() = 4π
2m
h 2
3
2
V V
1
2 ,
(10.4.1)
since the number of spin1
2 fermion states is twice the number of classical
translational states [note that for arbitrary spin I a , we would require a factor 2I a + 1
rather than simply 2, see Eq. (10.2.13b)].
From Eq. (10.2.12), together with the relation = −P V , we obtain the equation
of state for the Fermi–Dirac gas as
P V
k B T
FD
=
k
ln
1 + λe
−ββ k
.
We begin by replacing this discrete variable expression by the continuum approximation
P V
k B T
FD
∞
0
FD (() ln
1 + λe
−ββ
d
= 4π
2m
h 2
3
2
V
∞
0
1
2 ln
1 + λe
−ββ
d .
(10.4.2)