The conduction electrons in a metal are nonlocalized (i.e., they are
not tied to any particular atoms). In conventional metals, each atom contributes
a fixed number of such electrons (corresponding to its valency). To a first approximation, it is possible
to neglect the mutual interaction of the conduction electrons, because this
interaction
is largely shielded out by the stationary ions. The conduction electrons
can, therefore, be treated as an ideal gas. However, the number density of
such electrons in a metal far exceeds the number density of
molecules in a conventional gas.
Electrons are subject to the Pauli exclusion principle, according to which a given electron state can either be unoccupied, or singly occupied. (See Section 4.4.3.) The socalled Fermi energy is the energy at which electrons are available without doing work.
Thus, an electron state of energy has an available free energy 0 when it is unoccupied, and an available free energy
when it is occupied. According to the Boltzmann distribution (see Section 5.4.7), the relative probabilities of unoccupied and occupied
states are thus and
, respectively, where is the temperature. (See Section 5.4.7.) Thus, the mean occupancy number of the state is

(5.511) 
which reduces to

(5.512) 
Here,
is termed the Fermi function.
Let us investigate the behavior of the Fermi function as varies. Here, the energy is measured from
its lowest possible value
.
The Fermi energy for conduction electrons in a metal is
such that
.
In this limit, if
then
,
so that
. On the other hand, if
then
, so that
falls off exponentially with increasing
. Note that
when
. The transition region in which goes from
a value close to unity to a value close to zero corresponds to an
energy interval of order , centered on
. In fact, when
, and when
. The behavior of the Fermi function is
illustrated in Figure 5.10.
Figure 5.10:
The Fermi function.

In the limit as
, the transition region becomes
infinitesimally narrow. In this case, for
, and
for
, as illustrated in Figure 5.10.
This is an obvious result, because when the conduction
electrons attain their lowest energy, or groundstate, configuration.
Because the Pauli exclusion principle requires that there be no
more than one electron per singleparticle quantum state, the lowest
energy configuration is obtained by piling
electrons into the lowest available unoccupied states, until all of
the electrons are used up. Thus, the last electron added to the
pile has a quite considerable energy,
, because all of the
lower energy states are already occupied. Clearly, the exclusion principle
implies that free electrons in a metal possess a large mean energy, even at a temperature of absolute
zero.
We can calculate the Fermi energy by equating the number of occupied electron states to the
total number of electrons in the metal, . In other words,

(5.513) 
where
is the density of electron states specified in Equation (5.466). Assuming that
, we can make the approximation

(5.514) 
Thus, making use of Equation (5.466), we obtain

(5.515) 
which can be rearranged to give

(5.516) 
where is the number density of conduction electrons. The mean electron energy is

(5.517) 
(See Section 4.4.3.)
Copper at room temperature has a number density of conduction electrons of
. According to Equation (5.516), the corresponding Fermi energy is

(5.518) 
The associated Fermi temperature is

(5.519) 
Thus, at room temperature, K, we obtain

(5.520) 
which confirms that
for conduction electrons in a metal.
Figure 5.11:
Approximate Fermi function.

Let us crudely approximate the Fermi function at finite temperature in the fashion shown in Figure 5.11.
As can be seen from the figure, the proportion of thermally excited electrons is the ratio of the area of a triangle of height and
base to that of a rectangle of height 1 and base
. In other words,

(5.521) 
Now, the centroid of a rightangled triangle is rd of the distance along its base from the rightangle. Thus, the
mean energy of the excited electrons increases by

(5.522) 
Hence, the thermal energy per conduction electron is

(5.523) 
which implies that the internal energy (i.e., the difference between the energy at a finite temperature and the energy at absolute zero) of the conduction electrons is

(5.524) 
where is the number of moles of electrons.
Finally, the molar specific heat of the electrons at constant volume is

(5.525) 
which can also be written

(5.526) 
The exact result is

(5.527) 
Thus, we conclude that the contribution of the conduction electrons to the molar specific heat capacity of a
metal is proportional to the temperature. However, this contribution is much less that the classical contribution, ,
predicted by the equipartition theorem (see Section 5.5.5), given that each conduction electron possesses three translational
degrees of freedom. This is the case because the conduction electrons in a metal are highly degenerate. (See Section 4.4.3.)
In fact, Equations (5.520) and (5.527) imply that the contribution of the conduction electrons to the molar specific heat of copper at room temperature is a factor 85 times smaller than the classical
contribution.
Using the superscript to denote the electronic
specific heat due to conduction electrons, the molar specific heat of such electrons can be written

(5.528) 
where is a (positive) constant of proportionality.
At room temperature, is completely masked by the much
larger specific heat, , due to lattice vibrations. However, at
very low temperatures,
, where is a (positive)
constant of proportionality. (See Section 5.6.5.) Clearly,
at low temperatures,
approaches zero far more rapidly than
the electronic specific heat, as is reduced. Hence, it should be possible
to measure the electronic contribution to the molar specific
heat at low temperatures.
Figure: 5.12
The lowtemperature heat capacity of potassium, plotted as versus
. The straightline shows the fit
.

The total molar specific heat of a metal at low temperatures takes the
form

(5.529) 
Hence,

(5.530) 
It follows that a plot of versus should yield a
straightline whose intercept on the vertical axis gives the
coefficient . Figure 5.12 shows such a plot.
The fact that a good straightline, with a nonzero intercept, is obtained verifies that the temperature
dependence of the heat capacity predicted by Equation (5.529) is indeed
correct.