Environmental Engineering Reference
In-Depth Information
5.3.5
Buneman Instability
We now consider instability of another type that develops if the mean velocity of
the electrons differs from the mean velocity of the ions. We formulate the prob-
lem by taking all the plasma ions to be at rest and all the electrons to be traveling
with velocity
u
with respect to the ions. The plasma is quasineutral, that is, the
number densities of the electrons and ions are equal. Our goal is to determine
the maximum amplification factor of the plasma oscillations. The electron beam is
decelerated owing to the transfer of energy from the beam to the plasma oscilla-
tions [55].
With this formulation, the problem is equivalent to the previous case of inter-
action of an electron beam and a plasma. In both problems an electron beam is
penetrating a plasma, so the dispersion relation can be derived in a similar way.
Denoting the ion mass as
M
and taking into account the equality of ion and elec-
tron number densities, we obtain the dispersion relation
M
ω
2
p
ω
2
p
m
e
C
D
1
(5.99)
ω
2
(
ω
ku
)
2
instead of (5.97). If we let the ratio
m
e
/
M
go to zero, we obtain the dispersion rela-
tion
ω
D
ω
C
ku
. Hence, one can write the frequency of the plasma oscillations
p
as
ω
D
ω
C
ku
C
δ
.
p
Substituting this frequency into dispersion relation (5.99) and expanding the result
in a power series in terms of the small parameter
δ
/
ω
p
,weobtain
2
p
ω
2
δ
ω
m
e
M
p
D
)
2
.
(
ω
C
ku
C
δ
p
The electron beam has the strongest interaction with the wave whose wave number
is
k
D
ω
p
/
u
. For this wave we have
p
exp
2
,
m
e
M
π
in
3
1/3
δ
D
ω
where
n
is an integer. The highest amplification factor corresponds to
n
D
1andis
given by
p
3
2
m
e
2
M
1/3
p
m
e
M
1/3
γ
D
Im
δ
D
ω
D
0.69
ω
.
(5.100)
p
Note that the frequency of oscillations is the order of the amplification factor. This
type of instability of the electron beam due to interactionwith plasma ions is known
as Buneman instability [55]. This instability can exist in an ionospheric plasma [56]
and in a hot or nonequilibrium plasma [57].
