Environmental Engineering Reference
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and the electric field strength
E
are zero. Hence, we obtain
kp
0
m
e
N
0
eE
0
m
e
N
e
C
ikN
0
w
0
D
w
0
C
i
ω
0,
i
ω
i
C
D
0,
p
0
p
0
N
e
N
0
ikE
0
D
D
γ
eN
.
,
4
π
(5.56)
Here
k
and
are the wave number and the frequency of the plasma oscillations,
N
0
is the mean number density of charged particles,
p
0
ω
x
˛
is the electron gas
pressure in the absence of oscillations,
m
e
is the electron mass,
v
x
is the electron
velocity component in the direction of oscillations, and the angle brackets denote
averaging over electron velocities.
N
e
,
w
0
,
p
0
,and
E
0
in (5.56) are the oscillation
amplitudes for the electron number density, mean velocity, pressure, and electric
field strength, respectively.
Eliminating the oscillation amplitudes from the system of equations in (5.56),
we obtain the dispersion relation for plasma oscillations;
N
0
m
e
˝
v
D
C
γ
˝
v
x
˛
k
2
,
2
2
p
2
ω
D
ω
(5.57)
D
p
4
where
N
0
e
2
/
m
e
is the plasma frequency defined by (1.13). Plasma os-
cillations or Langmuir oscillations are longitudinal, in contrast to electromagnetic
oscillations. Hence, the electric field due to plasma waves is directed along the wave
vector. This fact was used in deducing the set of equations in (5.56).
The dispersion relation (5.57) is valid for adiabatic propagation of plasma oscil-
lations. If heat transport is due to thermal conductivity of electrons, the adiabatic
condition takes the form
ω
π
p
k
2
)
ωτ
ω
/(
1 (compare this with criterion (5.55)),
k
2
)
1
where
ω
is the frequency of oscillations,
τ
(
is a typical time for heat
transport in the wave,
is the electron thermal diffusion coefficient, and
k
is the
wave number of the wave. Since
v
e
λ
,
ω
ω
v
e
/
r
D
,where
v
e
is the mean
p
electron velocity,
is the electron mean free path, and
r
D
is the Debye-Hückel
radius given by (1.9). The adiabatic condition yields
λ
r
D
k
2
λ
1.
If this inequality is reversed, then isothermal conditions in the wave are fulfilled.
In this case the adiabatic parameter
in the dispersion relation (5.57) must be
replaced by the coefficient 3/2 since for the pressure we use the equation of the
gaseous state,
p
γ
N
/
T
.Asaresult,wehaveintheisothermalcasethefollowing
dispersion relation for plasma oscillations instead of (5.57) [29, 30]:
D
3
k
2
T
e
m
e
2
2
p
ω
D
ω
C
.
(5.58)
Note that because the frequency of plasma oscillations is much greater than the
inverse of a typical time interval between electron-atom collisions, we have
ω
p
N
a
v
e
σ
v
e
/
λ
. From this it follows that
λ
r
D
.
ea
