Set of supplementary notes.

1.2. DRUDE MODEL 13
Im(ε)
Permittivity ε
0
Re(ε)
0 ω p
*
Frequency
Figure 1.5: Real and imaginary part of the dielectric permittivity in the Drude model. The
peak at ω T1 illustrates the possibility of additional resonances due to the bound, core electrons.
1.2 Drude model
The Lorentz, or dipole oscillator model extends naturally to metals, if we imagine that some
of the electrons are no longer bound to the ions. The remaining inner, or core electrons are
still closely bound and will continue to contribute to the permittivity according to the Lorentz
oscillator model. The outer, or conduction electrons, however, have been cut loose from the
ions and are now free to roam around the entire piece of metal. This corresponds to the
spring constant in their linearised force law going to zero, and hence the natural frequency ω T
vanishes. To model their contribution to ɛ ω we can then simply use our earlier expressions for
the frequency dependent permittivity, but drop ω T : the resonance peak now occurs at zero
frequency. This picture, in which some of the electrons are cut free from the ionic cores, is
called the Drude model.
ω T1
1.2.1 Optical properties of metals in the Drude model
Setting ω T in the dipole oscillator response (1.2) to zero and inserting a background permittivity
ɛ ∞ to take account of the polarisability of the bound core electrons, leads us to the Drude
response
ɛ ω = ɛ ∞ − N V
q 2
mɛ 0 (ω 2 + iωγ) = ɛ ∞ −
ω 2 p
ω 2 + iωγ
, (1.5)
where ωp 2 = ne2
mɛ 0
(defining n = N/V , the number density of mobile electrons). ω p is called the
plasma frequency.
We can draw three immediate conclusions from the above form of the permittivitiy (Fig. 1.2):
1. |ɛ ω | diverges for ω → 0 =⇒ metals are highly reflecting at low frequency.
2. The imaginary part of ɛ ω peaks at ω = 0, giving rise to enhanced absorption at low
frequency, the ’Drude peak’.
3. ɛ ω crosses through zero at high frequency ωP ∗ =⇒ metals become transparent in the
ultraviolet.