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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 and doing the sum to the lowest surviving order in . 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 (11.68)

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

The wave number in this limit is given by: (11.69)

(or ). This is called a dispersion relation . A large portion of contemporary and famous physics involves calculating dispersion relations (or equivalently susceptibilities, right?) from first principles. 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 (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 (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 (11.71)

where is the plasma frequency'' of the conduction electrons. 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: Penetration of Waves Into Up: Dispersion Previous: Low Frequency Behavior   Contents
Robert G. Brown 2017-07-11