Environmental Engineering Reference
In-Depth Information
5.2.3
Ion Sound
Since an ionized gas contains two types of charged particles, electrons and ions,
there are two types of plasma oscillations due to electrons and ions [28]. Above
we considered fast oscillations, in which ions as a slow plasma component do not
partake, and now we analyze the oscillations due to the motion of ions in a homo-
geneous plasma. The special character of these oscillations is due to the large mass
of ions. This stands in contrast to the small mass of electrons that enables them to
follow the plasma field, so the plasma remains quasineutral on average:
N
e
D
N
i
.
Moreover, the electrons have time to redistribute themselves in response to the
electric field in the plasma. Then the Boltzmann equilibrium is established, and
the electron number density is given by the Boltzmann formula (1.43):
N
0
exp
e
N
0
1
,
T
e
e
T
e
N
e
D
C
where
is the electric potential due to the oscillations and
T
e
is the electron tem-
perature. These properties of the electron oscillations allows us to express the am-
plitude of oscillations of the ion number density as
'
N
0
e
T
e
N
i
D
.
(5.59)
We can now introduce the equation of motion for ions. The continuity equa-
tion (4.1) has the form
@
N
i
/
@
t
C @
(
N
i
w
i
)/
@
x
D
0andgives
N
i
ω
D
kN
0
w
i
,
(5.60)
where
is the frequency,
k
isthewavenumber,and
w
i
is the mean ion velocity
duetotheoscillations.Hereweassumetheusualharmonicdependence(5.48)for
oscillation parameters. The equation of motion for ions due to the electric field of
the wave has the form
m
i
d
w
i
dt
ω
D
e
E
D
e
r
'
.
Taking into account the harmonic dependence (5.48) on the spatial coordinates and
time, we have
m
i
ω
w
i
D
ek
'
.
(5.61)
,and
w
i
in the set of equa-
tions (5.59), (5.60), and (5.61), we obtain the dispersion relation
Eliminating the oscillation amplitudes of
N
i
,
'
k
s
T
e
m
i
ω
D
(5.62)
