We take a step back and ignore the band structure, we also ignore the fact that electrons are Fermions and obey the Pauli Exclusion Principle.
9.2 The Drude Model of Conductivity in Free Electron Gases
The model dates back to the 1890's before Quantum Mechanics and it was good up to Ohm's Law, the relations between thermal and electrical materials. However it also predicts that everything is a metal.
9.3 Charge Density Oscillation Plasmon
If we take a cylindrical volume of a solid and pull the electrons a small distance, D x , away from the ions.
Figure 9.1 - Charge Density Oscillation Plasmon
By doing this a dipole moment is set up due to the seperation of the charges
p=-ne.AD x .l
where ne is the charge per unit volume, AD x is the volume of charged material and l is the seperation of the charges.
This gives a dipole moment per unit volume
P=-neD x
If the medium is a dielectric with constant er then this polarisation gives a field
E=
-P
e0er
=
neD x
e0er
This sets up a restoring force F which acts on the charge
F=m
d2D x
dt2
=
ne2D x
e0er
where m is the mass of a material in the charged region. Looking at the equation we quickly identify that the procession of this dipole moment is Simple Harmonic Motion.
This can be modeled as a Simple Harmonic Oscillator with Plasma Frequency
wp2=
ne2
e0erm0
where m0 is for the moment.
9.4 How Does The Prediction Compare With Reality
Electricity and Magnetism says that for w>wp (where w is the AC frequency of the electromagnetic wave) then K is reall and the electromagnetic wave propagates through the solid. However if w<wp then the wave cannot propagate and so it reflects. Typical numbers for n in a metal is generally 6× 1028m-3 . The Plasma Frequency is 2.2× 1015Hz , corresponds to light of wavelength of about 160nm .
