Statistical Mechanics - Physics at Oregon State University

80 CHAPTER 4. STATISTICS OF INDEPENDENT PARTICLES.
valid anymore, because the condition n ≪ nQ(T ) ∝ T 1.5 will not be true for
small temperatures. Every gas will show deviations from ideal gas behavior
at low temperatures. A gas is called a quantum gas or degenerate gas when
n ≈ nQ(T ). In a degenerate gas the differences between fermions and bosons
become important.
At these higher densities the interactions between the particles also play a
role, but the independent particle description is still a very good first approximation
for most quantities. Inter-particle interaction can be included using
perturbation theory. Of course, when n ≫ nQ(T ) that approach does not work
anymore, and one really has to start with a model including inter-particle interactions.
An estimate of the temperature T0 where these effects start to play a role
comes from solving the equation n = nQ(T0). We will calculate some typical
examples of T0. A gas of helium atoms has a density of about 2.5 × 10 25 m −3 .
With M ≈ 4 × 10 −27 kg, ≈ 10 −34 Js, and kB ≈ 10 −23 JK −1 we get T0 ≈ 1K.
Although helium is a liquid at these temperatures, special effects are seen in 4 He
and also there are big differences between 4 He and 3 He at these temperatures.
Electrons in a metal have much higher densities that normal gases. A typical
number is n ≈ 10 29 m −3 . With M ≈ 10 −30 kg we find T0 ≈ 10 5 K. Such an
electron gas is always degenerate, and the Pauli principle has to be taken into
account. This is clearly seen in the physics of metals. The famous exchange
energy for the electrons in a solid has to be taken into account. The conduction
electrons in a semiconductor, on the other hand, have much lower densities.
A typical range is 10 23 · · · 10 26 m −3 , and hence T0 ranges from 10K to 1000K.
At room temperature the electron gas in a semiconductor can be classical or
degenerate, depending on the doping. This leads to interesting effects in these
semiconductors.
4.5 Fermi gas.
Quantum gases come in two varieties, named after Fermi and Bose. There are
some other cases of theoretical interest, but they are probably not important
from a practical point of view. If the particles in a gas have integral spin we
need Bose-Einstein statistics, if the spin is half-integral Fermi-Dirac statistics.
We will first study a gas of identical, independent particles obeying Fermi-
Dirac statistics. Kittel’s book has a number of simple examples and is a good
source of information for simple techniques. The same techniques are used in
Sommerfeld’s theory of the conduction electrons in a metal, and hence any text
book on solid state physics will also be useful.
Fermi energy.
The average number of particles in an orbital with energy ɛ is given by the
Fermi-Dirac distribution function