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