Environmental Engineering Reference
In-Depth Information
the thermal velocity of the ions, and the attenuation factor is of the order of the
wave frequency. Therefore, ion sound can propagate only in plasmas in which the
electron temperature
T
e
is considerably higher than the ion temperature
T
i
,thatis,
when we have
T
e
T
i
.
(5.93)
5.3.4
Beam-Plasma Instability
Assume that an electron beam penetrates into a plasma, where the velocity of the
electrons in the beam is much higher than the thermal velocity of the plasma elec-
trons and the number density
N
b
of the electrons in the beam is considerably lower
than the number density
N
0
of plasma electrons. Deceleration of the electron beam
can occur owing to the scattering of electrons of the beam by electrons and ions of
the plasma. There is, however, another mechanism for deceleration of the electron
beam: beam instability. In the early days of plasma physics this phenomenon was
known as the Langmuir paradox [26-28]. It was revealed in these investigations
that when electrons are ejected from a cathode surface in the form of a beam, the
average energy of ejected electrons becomes equal to the thermal energy of the
electrons of the gas discharge plasma into which this electron beam penetrates, at
small distances from the cathode as compared with the mean free path of the elec-
trons. Because it was assumed in that era that energy exchange between beam and
plasma electrons results only from collisions because oscillations of the plasma
were weak, the observed effect was considered to be a paradox. In actuality, interac-
tion of the beam electrons with plasma electrons proceeds more effectively through
collective degrees of freedom of the beam-plasma system than through collisions
due to the resonant character of this interaction, and this explains the effect un-
der consideration [29, 30, 52, 53]. This interaction can be succinctly described as
follows. Suppose plasma oscillations are generated in a plasma. Interacting with
electrons of the beam and taking energy from them, these oscillations are ampli-
fied. Thus, some of the energy of the electron beam is transformed into the energy
of plasma oscillations and remains in the plasma. This energy may subsequently
be transferred to other degrees of freedom of the plasma.
One can estimate the amplification of the plasma oscillations in the above sce-
nario assuming the oscillation amplitude to be small, and taking the temperatures
of the electrons in the plasma and in the beam to be zero. Hence, the pressure of
the electrons in the plasma and beam is zero. Applying the continuity equation (4.1)
and the Euler equation (4.6) to the plasma electrons, we derive equations for the
amplitudes of the plasma parameters that follow from the two first equations of
set (5.91) with
p
0
0. Elimination of the electron velocity
w
0
in the wave in these
D
equations yields
kE
0
m
e
N
e
D
i
2
N
0
.
(5.94)
ω
One can obtain the expression for the amplitude of oscillations of the electron num-
ber density
N
b
in the beam in a similar way by writing the electron number density
