Figure 5.2 shows the fermion population density
(see equ. 5.4 with the positive sign in the denominator.)
It is evidently quite different from its classical counterpart,
the Boltzmann factor. In particular, when the temperature is low then
it approaches a step function: in the limit
states having an energy below a certain value are inhabited,
while the states with higher energies remain empty.
The threshold energy is called ``Fermi energy''.
Mean population numbers of the states in a Fermi-Dirac system
with . For comparison we have also drawn the classical
for a temperature of .
It is only for high temperatures and small values of that the Fermi-Dirac
density approaches the classical Boltzmann density; and as it does so it strictly
obeys the inequality
. For the purpose of comparison we
have included a graph of
in Figure 5.2.
ELECTRONS IN METALS
Conduction electrons in metallic solids may be considered as an ideal
fermion gas - with an additional feature: since electrons have two
possible spin states the maximum number of particles in a state
is instead of .
At low temperatures all states with
are populated. The number of such states is, as we can see from
Assuming a value of
typical for conduction electrons we find
Applying the general formulae 4.51 and 4.53
we may easily derive the pressure and internal energy of the
electron gas. We may also recapture the relation ,
consistent with an earlier result for the classical ideal gas.