Biomedical Engineering Reference
In-Depth Information
12.5.1 The Drude Model
The first attempt to model metals was the
free electron
model developed independently
by P. Drude and by Lorentz. According to this model, each metallic atom in the lattice
donates its valence electrons to form a sea of electrons. So a sodium atom would lose one
electron and an aluminium atom three; these conduction electrons move freely about the
cation lattice points, rather like the atoms in an ideal gas.
The model had some early successes; electrical conduction can be explained because the
application of an electric field to the metal will cause the conduction electrons to move from
positions of high to low electrical potential and so an electric current can be established. In
fact, Ohm's law can be recovered from the Drude model with a little analysis.
Eventually, the Drude model gradually fell into disrepute. An obvious failing was its
inability to explain the heat capacity of metals. If the model of solid aluminium is correct
and eachAl atom loses three valence electrons, we might expect that these valence electrons
wouldmake a contribution to the internal energy of 3
(3/2)
RT
, with a correspondingmolar
heat capacity of (9/2)
R
(i.e. 32 J K
−
1
mol
−
1
). In fact, Al obeys the Dulong-Petit rule with a
molar heat capacity of about 25 J K
−
1
mol
−
1
, so the equipartition of energy principle seems
to overestimate the contribution from the conduction electrons.
The electrons are of course in motion, and we should ask about their energy distribution
just as we did in earlier sections. There is no experimental technique that can measure the
spread with the same accuracy, as can be done with photons in the black body experiment, or
with particles in the kinetic theory experiments.Avariation on the theme of the photoelectric
effect (called high-energy photoemission) can however be used to probe electron kinetic
energies in a metal. If I write the familiar expression for the distribution function
g
(ε)
×
=
×
g
(ε)
(density of states,
D
(ε))
(average occupancy of a state,
p
(ε))
then it turns out that the density of states is proportional to ε
1/2
, just as it was for the spread
of kinetic energies in an ideal gas. The average occupancy factor is quite different from
anything we have met so far:
1
p
(ε)
=
exp
(ε
ε
F
)
k
B
T
−
+
1
Parameter ε
F
is called the
Fermi energy
, and I have illustrated the factor for the case of a
Fermi energy of 2.5 eV at 600 and 6000 K in Figure 12.7. The lower temperature shows a
factor of 1 for all energies until we reach the Fermi level, when the factor becomes zero.
The lower the temperature, the sharper is the cut-off. The corresponding Boltzmann factors
are shown in Figure 12.8, from which it is seen that the two sets of results are radically
different.
The experimental distribution function is therefore
1
B
ε
1/2
exp
(ε
g
(ε)
=
(12.16)
ε
F
)
k
B
T
−
+
1
and the average occupancy of states factor cannot be explained by the Drude model.
