Set of supplementary notes.

1.2. DRUDE MODEL 17
1. Rather than starting with an equation of motion for the electronic displacement, we
consider the rate of change of the velocity v = ˙u.
2. Instead of considering the velocity of an individual electron, we average over a large
number electrons, and look for a differential equation for the average velocity of the
electrons. Even if the individual electrons have wildly different velocities, the average, or
drift velocity will be seen to follow a simple equation of motion, and it is the drift velocity
which determines the optical and transport properties of the metal.
Writing ρ = qN/V for the charge density, we can express the current density j in terms of
the average velocity v of the mobile, or conduction electrons, and we can link j to E via the
conductivity σ and Ohm’s law:
j = ρv = σE (1.7)
Because the polarisation P = (N/V )uq, we can link the current density to the timederivative
of the polarisation due to the conduction electrons, P c :
j = Ṗc (1.8)
Note that while there can be some ambiguity about P due to the ambiguity about the
displacement u for travelling electrons, this drops out of the time derivative, so the expression
for j is correct. 1
Adding in the polarisation of the core electrons, which is given by the background polarisability
χ ∞ , we obtain
Ṗ = j + ɛ 0 χ ∞ Ė (1.9)
Considering, as before, the oscillatory response v ω e −iωt , j ω e −iωt , P ω e −iωt to an oscillating
electric field E ω e −iωt , we find for the Fourier components
from which we obtain the key expressions
j ω = σ ω E ω = −iωɛ 0 (χ ω − χ ∞ )E ω , (1.10)
σ ω = −iωɛ 0 (ɛ ω − ɛ ∞ )
ɛ ω = i σ ω
ɛ 0 ω + ɛ ∞ (1.11)
These expressions relate the imaginary part of the permittivity (which determines optical
absorption) to the real part of the frequency dependent conductivity. That means that we can
indirectly determine the conductivity of a metal at high frequencies by optical measurements.
If the atomic (or background) polarisability is zero, then ɛ ∞ = 1 and σ ω = −iωɛ 0 (ɛ ω − 1).
1 An alternative approach, which avoids introducing the field u(r, t) altogether, could consider ∇P c = −ρ c
and ˙ρ c = −∇j to find j = Ṗc + j 0 , where ∇j 0 = 0 so that the boundary conditions on the sample fix j 0 = 0.