The high density, low temperature limit
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# The high density, low temperature limit

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:

Figure 1:

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:

Figure 2:

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.

Next: Zero-temperature thermodynamics Up: No Title Previous: The high temperaturelow

Mark Tuckerman
Sat Jan 4 22:04:30 EST 2003