Environmental Engineering Reference
In-Depth Information
1.1.6
Oscillations of Plasma Electrons
The Debye-Hückel radius is the parameter that characterizes an ideal quasineutral
plasma. We can estimate a typical time for the response of a plasma to an external
field. For this purpose we study the behavior of a uniform infinite plasma if all the
plasma electrons are shifted at the initial time by a distance
x
0
to the right starting
from a plane
x
0. This creates an electric field whose strength corresponds to
thePoissonequation(1.4):
D
dE
dx
D
4
π
e
(
N
i
N
e
).
Assuming the electric field strength at
x
<
0 is zero, the Poisson equation gives an
electric field strength for
x
eN
0
x
0
,where
N
0
is the average number
density of charged particles in the plasma. The movement of all the electrons under
the influence of the electric field leads to a change in the position of the boundary.
The equation of motion for each of the electrons can be written as
>
x
0
of
E
D
4
π
m
e
d
2
(
x
C
x
0
)
D
eE
,
dt
2
where
m
e
is the electron mass and
x
is the distance of an electron from the bound-
ary. Because
x
is a random value, not dependent on the phenomenon being con-
sidered, one can assume this value to be independent of time. Thus, the equation
of motion of an electron is
d
2
x
0
dt
2
2
p
x
0
,
D
ω
where [2, 20-23]
4
1/2
e
2
m
e
ω
D
π
N
0
(1.13)
p
is called the plasma frequency, or Langmuir frequency.
The solution of the equation obtained predicts an oscillatory character for the
electron motion. Accordingly, 1/
p
is a typica
l time f
or a plasma to respond to an
external signal. Note that the value
r
D
ω
D
p
2
T
/
m
e
is the thermal electron veloc-
ity. From this it follows that a typical time for a plasma to respond to an external
signal is the time during which the electrons experience a displacement of the or-
der of the Debye-Hückel radius. Thus, we have two fundamental parameters of an
ideal quasineutral plasma: the Debye-Hückel radius
r
D
, which is a shielding dis-
tance for fields in a plasma, and the plasma frequency
ω
p
ω
1
p
ω
p
,so
is a typical time
for the plasma to respond to external signals.
Thus, the Debye-Hückel radius and the plasma frequency are two fundamental
parameters which reflect a long-range character of interaction of charged particles
in an ionized gas. These parameters do not depend on a short-range interaction
