Environmental Engineering Reference
In-Depth Information
in the beam as
N
b
C
N
b
exp[
i
(
kx
ω
t
)] and the velocity of the electrons in the
beam as
u
t
)], where the
x
-axisisparalleltothevelocityofthe
beam and
N
b
and
u
are, respectively, the electron number density and the electron
velocity in the unperturbed beam. Then we have
C
w
b
exp[
i
(
kx
ω
ikE
0
N
b
m
e
(
N
b
D
.
(5.95)
ω
ku
)
2
As was done in arriving at the last equation of system (5.56), the Poisson equa-
tion (3.2) yields
ikE
0
D
N
b
) (5.96)
for the amplitude of the system's parameters. Eliminating the amplitudes
N
0
,
N
b
,
and
E
0
from the set of equations (5.94), (5.95), and (5.96), we obtain the dispersion
relation
ω
e
(
N
e
C
4
π
2
p
2
p
ω
N
b
N
0
C
D
1 .
(5.97)
ω
2
(
ω
ku
)
2
N
0
e
2
/
m
e
)
1/2
is the frequency of plasma oscillations in accordance
with (1.12). When the number density of the beam electrons is zero (
N
b
Here
ω
D
(4
π
p
0),
dispersion relation (5.97) reduces to (5.57), where the electron temperature is taken
to be zero. The strongest interaction between the beam and plasma occurs when
the phase velocity of the plasma waves
D
/
k
is equal to the velocity of the electron
beam. Let us analyze this case. Since the number density of the beam electrons
N
b
is small compared with the number density
N
0
of the plasma electrons, the
frequency of the plasma oscillations is close to the plasma frequency
ω
ω
p
.Hence,
we shall consider waves with wave number
k
/
u
, which have the most effective
interaction with the electron beam. We write the frequency of these oscillations as
ω
D
ω
D
ω
and insert it into dispersion relation (5.97). Expanding the result in a
series in terms of the small parameter
C
δ
p
δ
/
ω
p
,weobtain
p
N
b
2
N
0
1/3
exp
2
,
π
in
3
δ
D
ω
(
N
b
/
N
0
)
1/3
where
n
is an integer. One can see that
δ
/
ω
1. That is, the
p
expansion over the small parameter
p
is valid.
If the imaginary component of the frequency (which is equal to the imaginary
component of
δ
/
ω
) is negative, the wave is attenuated; if it is positive, the wave is
amplified. The maximum value of the amplification factor is given by (
n
δ
D
1)
p
3
2
N
b
N
0
1/3
p
N
b
N
0
1/3
γ
D
ω
D
0.69
ω
.
(5.98)
p
The amplitude
N
b
varies with time as exp(
t
); this result is valid if the plasma
oscillations do not affect the properties of the plasma. This type of instability is
known as beam-plasma instability. As a result of this instability, the distribution
function for the beam electrons expands (see Figure 5.11), and the energy surplus is
transferred to plasma oscillations. In reality, various factors influence the character
and rate of a beam of charged particles propagating in a plasma [54].
γ
