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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:

$\displaystyle \epsilon(\omega)$ $\textstyle =$ $\displaystyle \epsilon_0\left (1 + \frac{N e^2}{m}
\sum_i \frac{f_i}{(\omega_i^2 - \omega^2 - i \omega \gamma_i)}\right)$  
  $\textstyle =$ $\displaystyle \epsilon_0 \left (
1 - \frac{N e^2}{\omega^2 m}
\sum_i \frac{f_i}{(1 + i\frac{\gamma_i}{\omega} -
\frac{\omega_i^2}{\omega^2})} \right)$  
  $\textstyle \approx$ $\displaystyle \epsilon_0 \left (
1 - \frac{NZ e^2}{\omega^2 m} \right)$  
  $\textstyle \approx$ $\displaystyle \epsilon_0 \left(1 -
\frac{\omega_p^2}{\omega^2}\right)$ (9.131)

where
\begin{displaymath}
\omega_p^2 = \frac{n e^2}{m} .
\end{displaymath} (9.132)

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:

\begin{displaymath}
ck = \sqrt{\omega^2 - \omega_p^2}
\end{displaymath} (9.133)

(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 9.5: 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

\begin{displaymath}
\alpha_p \approx \frac{2 \omega_p}{c}
\end{displaymath} (9.134)

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

\begin{displaymath}
\epsilon(\omega) \approx \epsilon_0(\omega) - \frac{\omega_p^2}{\omega^2}
\end{displaymath} (9.135)

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 2014-08-19