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High Frequency Limit; Plasma Frequency

Way above the highest resonant frequency the dielectric constant takes on a simple form (factoring out $ \omega >> \omega_i$ and doing the sum to the lowest surviving order in $ \omega_p/\omega$ . As before, we start out with: &epsi#epsilon;(&omega#omega;) & = & &epsi#epsilon;_0 (1 + N e^2m &sum#sum;_i f_i(&omega#omega;_i^2 - &omega#omega;^2 - i &omega#omega;&gamma#gamma;_i) )
& = & &epsi#epsilon;_0 ( 1 - N e^2&omega#omega;^2 m &sum#sum;_i f_i(1 + i&gamma#gamma;_i&omega#omega; - &omega#omega;_i^2&omega#omega;^2) )
& &ap#approx;& &epsi#epsilon;_0 ( 1 - NZ e^2&omega#omega;^2 m )
& &ap#approx;& &epsi#epsilon;_0 (1 - &omega#omega;_p^2&omega#omega;^2) where

$\displaystyle \omega_p^2 = \frac{n e^2}{m} .$ (11.68)

This is called the plasma frequency, and it depends only on $ n =
NZ$ , the total number of electrons per unit volume.

The wave number in this limit is given by:

$\displaystyle ck = \sqrt{\omega^2 - \omega_p^2}$ (11.69)

(or $ \omega^2 = \omega_p^2 + c^2 k^2$ ). This is called a dispersion relation $ \omega(k)$ . A large portion of contemporary and famous physics involves calculating dispersion relations (or equivalently susceptibilities, right?) from first principles.

Figure 11.5: The dispersion relation for a plasma. Features to note: Gap at $ k = 0$ , asymptotically linear behavior.
\begin{figure}
\centerline{\epsfbox{figures/plasma_dispersion.eps}}\end{figure}

In certain physical situations (such as a plasma or the ionosphere) all the electrons are essentially ``free'' (in a degenerate ``gas'' surrounding the positive charges) and resonant damping is neglible. In that case this relation can hold for frequencies well below $ \omega_p$ (but well above the static limit, since plasmas are low frequency ``conductors''). Waves incident on a plasma are reflected and the fields inside fall off exponentially away from the surface. Note that

$\displaystyle \alpha_p \approx \frac{2 \omega_p}{c}$ (11.70)

shows how electric flux is expelled by the ``screening'' electrons.

The reflectivity of metals is caused by essentially the same mechanism. At high frequencies, the dielectric constant of a metal has the form

$\displaystyle \epsilon(\omega) \approx \epsilon_0(\omega) - \frac{\omega_p^2}{\omega^2}$ (11.71)

where $ \omega_p^2 = n e^2/m^\ast$ is the ``plasma frequency'' of the conduction electrons. $ m^\ast$ is the ``effective mass'' of the electrons, introduced to describe the effects of binding phenomenologically.

Metals reflect according to this rule (with a very small field penetration length of ``skin depth'') as long as the dielectric constant is negative; in the ultraviolet it becomes positive and metals can become transparent. Just one of many problems involved in making high ultraviolet, x-ray and gamma ray lasers -- it is so hard to make a mirror!


next up previous contents
Next: Penetration of Waves Into Up: Dispersion Previous: Low Frequency Behavior   Contents
Robert G. Brown 2017-07-11