Environmental Engineering Reference
In-Depth Information
Figure 5.11
The velocity distribution func-
tion for plasma and beam electrons when an
electron beam is injected in a plasma. The
solid curve corresponds to the initial electron
distribution in the plasma and beam. Ranges
with a positive derivation are unstable, and
development of the instability transforms the
distribution function into that indicated by the
dotted line.
velocity
v
p
of the wave. When condition (5.88) is not satisfied, the wave takes energy
from the electrons and its amplitude increases.
In deriving condition (5.88) for wave damping, we stated that condition (5.87) is
satisfied when the field of the wave does not affect the electron distribution func-
tion. However, criterion (5.88) holds also when condition (5.87) is not satisfied and
the electron distribution function is altered owing to the interaction with the wave.
Interaction with the wave tends to equalize the number densities of electrons with
velocities
v
p
D
v
p
,
albeit without a change in its sign. That is, criterion (5.88) remains valid even when
the interaction with the wave alters the electron distribution function.
Condition (5.88) is satisfied for the Maxwell electron energy distribution function
or if it is simply a monotonically decreasing function, and oscillations in such a
plasma decrease with time owing to the interaction with electrons. However, when
an electron beam is injected into the plasma producing the electron distribution
function in Figure 5.11, plasma oscillations in this system will be amplified, get-
ting the necessary energy from the electrons. The interaction between plasma os-
cillations and electrons will reduce the velocity of the electrons until the electron
distribution function again becomes monotonically decreasing.
Amplification of the oscillations can cause their amplitude to increase rapidly
with time. If the oscillation amplitude is small and the oscillations do not alter the
electron distribution function, that is, if condition (5.87) is satisfied, then the oscil-
lation amplitude will increase exponentially. When particles of the plasma transfer
their energy to the plasma wave, thus amplifying it, such a plasma state is termed
an unstable state.
C
u
and
v
p
u
and, hence, it decreases the derivative
@
f
/
@
v
x
at
v
x
