Recall that the density equation could be expressed as an integral
which lead to an expansion in powers of 
. It is also possible
to develop an expansion in powers of 
.
This is accomplished by letting
and developing an expansion in powers of 
. In order to see how this
is done, consider making a change of variables in the integral
, 
, 
. Then
Integrate by parts using
so that
If we now expand 
about 
:
substitute this expansion into the integral and perform the resulting integrals over y, we find
where the fact that 
has been used owing to the
low temperature.
Since we are in the high density limit, 
is expected to
be large as well so that the series, whose error goes as powers
of 
will converge. As 
, 
and only one term in the above expansion survives:
Solving for 
gives
which is independent of T.
The special value of the chemical potential 
is known as the Fermi energy. To see what its physical
meaning is, consider the expression for the
average number of particles:
However, recall that
for a specific number of particles. Averaging both sides gives
Comparing these two expressions, we see that the
average occupation number of a given state with
quantum number 
and m is
As , 
, and
if
, and
if
. Thus, at T=0, we have the
result
A plot of the occupation average occupation number vs. 
at T=0 is shown in the plot below:
Thus, at T=0, the particles will exactly fill up all of the
energy levels up to an energy value 
above which no
energy levels will be occupied.
As T is increased, the probability of an excitation above the
Fermi energy becomes nonzero, and the average occupation
(shown for several different values of 
) appears as follows:
Thus, there is a finite probability that some of the levels just above the Fermi energy will become occupied as T is raised slightly above T=0. At T=0, the highest occupied energy eigenvalue must satisfy
This defines a spherical surface in space, which is
known as the Fermi Surface. Note that the Fermi surface
is only a sphere for the ideal gas. For systems in which
interactions are included, the Fermi surface can be a much
more complicated surface, and studying the properties of this
surface is a task that occupies the time of many a
solid-state physicist.
