Recall, from Sect. 8.5, that the mean number of particles occupying
state (energy ) is given by

(662) |

(663) |

(664) |

Let us investigate the behaviour of the *Fermi function*

(665) |

(666) |

In the limit as
, the transition region becomes
infinitesimally narrow. In this case, for and
for , as illustrated in Fig. 10.
This is an obvious result, since when the conduction
electrons attain their lowest energy, or *ground-state*, configuration.
Since the Pauli exclusion principle requires that there be no
more than one electron per single-particle 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 quite a considerable energy, , since all of the
lower energy states are already occupied. Clearly, the exclusion principle
implies that a Fermi-Dirac gas possesses a large mean energy, even at absolute
zero.

Let us calculate the Fermi energy of a Fermi-Dirac
gas at . The energy of each particle is related to its
momentum
via

(667) |

Thus, at all quantum states with are filled, and all those with are empty.

Now, we know, by analogy with Eq. (514), that there are
allowable translational states per unit volume of -space. The volume of
the sphere of radius in -space is
. It
follows that the *Fermi sphere* of radius contains
translational states. The number of
quantum states inside the sphere is *twice* this, because electrons
possess two possible spin states for every possible translational state. Since the
total number of occupied states (*i.e.*, the total number of quantum
states inside the Fermi sphere) must equal the total number of particles
in the gas, it follows that

(669) |

The above expression can be rearranged to
give

(670) |

(671) |

According to Eq. (668), the Fermi energy at takes the form

(672) |

The majority of the conduction electrons in a metal occupy a
band of completely filled states with energies far below the Fermi
energy. In many cases, such electrons have very little effect
on the macroscopic properties of the metal. Consider, for example, the
contribution of the conduction electrons to the specific heat of the metal.
The heat capacity at constant volume of these electrons can
be calculated from a knowledge of their mean energy as a
function of : *i.e.*,

(673) |

However, the actual situation, in which has the form shown in Fig. 10, is very different. A small change in does not affect the mean energies of the majority of the electrons, with , since these electrons lie in states which are completely filled, and remain so when the temperature is changed. It follows that these electrons contribute nothing whatsoever to the heat capacity. On the other hand, the relatively small number of electrons in the energy range of order , centred on the Fermi energy, in which is significantly different from 0 and 1, do contribute to the specific heat. In the tail end of this region , so the distribution reverts to a Maxwell-Boltzmann distribution. Hence, from Eq. (675), we expect each electron in this region to contribute roughly an amount to the heat capacity. Hence, the heat capacity can be written

(676) |

(677) |

Since in conventional metals, the molar specific heat of
the conduction electrons is clearly very much less than the classical value
. This accounts for the fact that
the molar specific heat capacities of metals at room temperature are
about the same as those of insulators. Before the advent of
quantum mechanics, the classical theory predicted incorrectly that the
presence of conduction electrons should raise the heat capacities of
metals by 50 percent [*i.e.*, ] compared to those of insulators.

Note that the specific heat (678) is not temperature independent.
In fact, using the superscript to denote the *electronic*
specific heat, the molar specific heat can be written

(679) |

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

(681) |