Dielectric Function of The Electron Gas
On a scale much larger than the inter atomic distance a solid can be viewed as an aggregate of a negatively charged plasma of the free electron gas and a positively charged background of atomic cores. The background is the rather stiff and massive background of atomic nuclei and core electrons which we will consider to be infinitely massive and fixed in space. The negatively charged plasma is formed by the valence electrons of the free electron model that are uniformly distributed over the interior of the solid. If an oscillating electric field is applied to the solid, the negatively charged plasma tends to move a distance x apart from the positively charged background. As a result the sample is polarized and there will be an excess charge at the opposite surfaces of the sample. The surface charge density is
which produces a restoring electric field in the sample
The dielectric function of the sample is expressed as
where is the electric displacement and is the polarization density.
The electric field and polarization densities are
and the polarization per atom with n electrons is
The force F of the oscillating electric field causes the electrons with charge e and mass m to accelerate with an acceleration a
which, after substitution of E, P and x, yields an harmonic oscillator equation.
After a little algebra the relation between polarization density and electric field can be expressed as
The frequency dependent dielectric function of the solid is
At a resonance frequency, called the plasma frequency, the dielectric function changes sign from negative to positive and real part of the dielectric function drops to zero.
This is a plasma oscillation resonance or plasmon. The plasma frequency is a direct measure of the square root of the density of valence electrons in a solid. Observed values are in reasonable agreement with this theoretical prediction for a large number of materials. Below the plasma frequency, the dielectric function is negative and the field cannot penetrate the sample. Light with angular frequency below the plasma frequency will be totally reflected. Above the plasma frequency the light waves can penetrate the sample.
Read more about this topic: Free Electron Model
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