plasma lect.2

Plasma Frequency
The plasma frequency arises as a basic consequence of the restoring
Coulomb interaction between oppositely charged particles. It is the plasma
“ringing” response to a charge perturbation. We now consider the dynamic
response of the plasma to an imposed charge separation. To study this, we
displace electrons and ions and calculate the transient response. Since the
electrons (-e) are much lighter than the ions, we assume the ions are
immobile. Then
F ? qE
me d2x
dt2 ? ?e ?x
?0
me d2x
dt2 ? ??pe
2 x
which is the equation of a simple harmonic oscillator with electron plasma
frequency:
?pe ? ? e?
me?0
?1/2 ? ? ne2
me?0
?1/2
Note that the solution contains no wavenumber dependence (kx ? 0), that
is, x ? x0 exp (i?cet) and the oscillation is not a propagating wave.
The ions oscillate much more slowly (by factor me/mi ) about the centre
of mass. At these frequencies, one can regard the ions as an irregular lattice
of charge.
As a quick estimate, one can use
fpe ? 9 ne Hz
where ne is in the unit of m?3.
For example, for ne ? 1 ? 1018m?3(H-1NF), weh have fpe ? 9 GHz, well
into the microwave region of the spectrum. For ne ? 1 ? 1014m?3 (a flame)
we have fpe ? 90 MHz (FM radio)
It is also useful to not that the Debye length is a typical distance
traversed by an electron with speed uTe ? ?kTe/me?1/2, in a time ?p?e1,
?D ? ? ?0kTe
nee2 ?1/2 ? uTe
?pe