536
10 Quantum Statistics
in Eqs. (10.2.12), (10.2.21) and (10.2.23) can be replaced by integrations over
energy. This approximation will remain accurate in the thermodynamic limit N →
∞, V → ∞, N/V = n, irrespective of temperature.
10.4.4 Strongly Degenerate Fermi–Dirac Gas: Electrons in
Metals
A Fermi–Dirac gas at low temperatures and/or high densities represents the strongly
degenerate case. We shall examine the free electron model of metals, in which the
valence electrons of the metal atoms are represented by an ideal gas of electrons.
For the simple monovalent metals the valence electrons interact with one another
and with the atomic cores through a Coulombic potential. The R −1 dependence is
so long-range that each valence electron is subjected to a total electronic potential
that is almost constant from location to location in the metallic crystal. The resulting,
and essentially free, electrons are referred to as conduction electrons. In fact, as we
shall see, many of the physically observable properties of metals can be associated
more readily with quantum-statistical effects than with the details of the electronelectron and electron-ionic core interactions.
The electron occupation number n k for the state with energy k is given by
Eq. (10.3.13), with the + sign, namely
n k =
λe −ββ k
1 + λe −ββ k
=
1
1 + e β(( k −μ) .
(10.4.16)
Now, if we assume, as in the previous section, a continuous distribution of energy
states, we can write this in the form
f (() =
1
1 + e β((−μ) ,
(10.4.17)
in which f (() represents the probability that a given state is occupied (see Fig. 10.4).
For T = 0 the distribution is unity for < μ and zero for > μ. All states having
energy less than F ≡ μ(0) are occupied, while those having energy greater than F
are unoccupied: in this sense, F serves as a cut-off energy. As the number of states
lying between and + d is given by Eq. (10.4.1), we are able to determine F
directly from the fact that all states lying below the energy F are occupied, while
all states lying above F remain unoccupied.
We shall obtain the equation of state for the conduction electrons by obtaining
firstly an expression for their internal energy, U 0 ≡ U FD (T = 0), then utilizing the
thermodynamic relation
P = −
∂U
∂V
S,N
10 Quantum Statistics
in Eqs. (10.2.12), (10.2.21) and (10.2.23) can be replaced by integrations over
energy. This approximation will remain accurate in the thermodynamic limit N →
∞, V → ∞, N/V = n, irrespective of temperature.
10.4.4 Strongly Degenerate Fermi–Dirac Gas: Electrons in
Metals
A Fermi–Dirac gas at low temperatures and/or high densities represents the strongly
degenerate case. We shall examine the free electron model of metals, in which the
valence electrons of the metal atoms are represented by an ideal gas of electrons.
For the simple monovalent metals the valence electrons interact with one another
and with the atomic cores through a Coulombic potential. The R −1 dependence is
so long-range that each valence electron is subjected to a total electronic potential
that is almost constant from location to location in the metallic crystal. The resulting,
and essentially free, electrons are referred to as conduction electrons. In fact, as we
shall see, many of the physically observable properties of metals can be associated
more readily with quantum-statistical effects than with the details of the electronelectron and electron-ionic core interactions.
The electron occupation number n k for the state with energy k is given by
Eq. (10.3.13), with the + sign, namely
n k =
λe −ββ k
1 + λe −ββ k
=
1
1 + e β(( k −μ) .
(10.4.16)
Now, if we assume, as in the previous section, a continuous distribution of energy
states, we can write this in the form
f (() =
1
1 + e β((−μ) ,
(10.4.17)
in which f (() represents the probability that a given state is occupied (see Fig. 10.4).
For T = 0 the distribution is unity for < μ and zero for > μ. All states having
energy less than F ≡ μ(0) are occupied, while those having energy greater than F
are unoccupied: in this sense, F serves as a cut-off energy. As the number of states
lying between and + d is given by Eq. (10.4.1), we are able to determine F
directly from the fact that all states lying below the energy F are occupied, while
all states lying above F remain unoccupied.
We shall obtain the equation of state for the conduction electrons by obtaining
firstly an expression for their internal energy, U 0 ≡ U FD (T = 0), then utilizing the
thermodynamic relation
P = −
∂U
∂V
S,N