Biomedical Engineering Reference
In-Depth Information
12.5.2 The Pauli Treatment
It was W. Pauli who devised a quantummechanical treatment of the electron gas; in Pauli's
model, the conduction electrons are still treated as an ideal gas of noninteracting particles,
but the analysis is done according to the rules of quantum mechanics. My starting point is
the energy level diagram for a cubic three-dimensional infinite potential well (Figure 12.9).
20
15
10
5
0
Figure 12.9
Electrons in a metal
Pauli assumed that the metal could be modelled as an infinite potential well, and that the
electrons did not interact with each other. He also assumed that each quantum state could
hold no more than two electrons. At 0 K, the
N
conduction electrons fill up the
N
/2 lowest
energy quantum states. I have shown 8 conduction electrons in Figure 12.9, for illustration.
The highest occupied quantum state corresponds to the Fermi level, and the Fermi factors
are therefore
1
for ε
≤
ε
F
p
(ε)
=
0
for ε>ε
F
At higher temperatures, the electrons will be spread amongst the available quantum states,
but subject still to the restriction that each quantum state can hold a maximum of two
electrons. For electron energies much above the Fermi level so that ε
−
ε
F
k
B
T
, the
Fermi factor can be approximated by ignoring the 1 in the nominator to give
exp
ε
ε
F
k
B
T
−
p
(ε)
≈
−
