Environmental Engineering Reference
In-Depth Information
frequency of plasma oscillations to be given by (1.12), we find that condition (5.84)
is equivalent to the criterion
N
1/2
e
N
10
12
cm
3/2
.
This shows that in some gas discharge plasmas the conditions for the existence of
plasma oscillations are not fulfilled.
We now consider the mixture of plasma oscillations with electromagnetic waves
that is given by dispersion relation (5.69). We now insert damping in (5.68) that
follows from the Maxwell equations with inclusion of electron currents. Intro-
ducing the dependence of the electric field strength on time in the form
E
exp(
ikx
i
ω
t
), we obtain in the limit
ωτ
1 the following dispersion equation:
p
1
,
i
ωτ
2
c
2
k
2
2
ω
D
C
ω
which in the limit
ω
ω
p
when a wave is damped in a plasma gives
i
k
2
c
2
ω
ω
D
.
(5.85)
p
τ
This dispersion relation allows us to analyze the character of penetration into a
plasma. Defining the dependence of the signal amplitude on the distance
x
from
the plasma boundary in the form exp(
x
/
Δ
), we have the depth of penetration of
an electromagnetic wave in a plasma:
1
Imk
D
c
Δ
D
p
p
2
.
(5.86)
ω
ωτ
As is seen, the penetration depth becomes infinite in the limit of low frequencies,
until this depth attains the Debye-Hückel radius. This phenomenon is the normal
skin effect.
5.3.2
Interaction Between Plasma Oscillations and Electrons
The above damping mechanism for plasma oscillations is due to electron-atom
collisions, since these collisions cause a phase shift of the oscillations of the collid-
ing electron, leading to the damping of the oscillations. Now we consider another
mechanism for interaction of electrons with waves. Electrons can be captured by
waves (see Figure 5.10), and then a strong interaction takes place between these
electrons and the waves. To analyze this process in detail, we note that in the frame
of reference where the wave is at rest, a captured electron travels between the po-
tential “walls” of the wave. Reflecting from one wall, an electron exchanges energy
with the wave. If the electron velocity in this frame of reference is
u
, the energy is
1
2
m
e
(
v
p
1
2
m
e
(
v
p
u
)
2
u
)
2
Δ
ε
D
C
D
2
m
e
v
p
,
