Environmental Engineering Reference
In-Depth Information
Typical values for the quantities indicated in Figure 4.21 for ions produced
in
air
at
room
temperature
are
N
1
10
17
cm
3
,
N
2
10
20
cm
3
,
10
6
cm
3
/s. Figure 4.22 shows measured
rate constants [106, 107] for recombination of positive and negative ions in air as a
function of pressure. The maximum of the recombination coefficient can be seen
to occur at atmospheric pressure.
(10
9
10
7
)cm
3
/s, and
α
α
1
max
4.5
Plasma in a Magnetic Field
4.5.1
Electron Hydrodynamics in a Gas in an External Field
We consider below the behavior of electrons in a gas in external fields. For simplic-
ity, this analysis will be based on hydrodynamics of electrons in a gas in external
fields, which corresponds to the tau approximation for the electron-atom collision
integral (3.5) in the regime of a low electron number density, when one can ignore
electron-electron collisions. In the tau approximation, the kinetic equation (3.4) for
the distribution function for electrons in a gas in an external field has the form
@
f
m
e
@
f
@
v
D
f
f
0
t
C
,
@
τ
where
F
is the force acting on the electron from external fields,
m
e
is the electron
mass, 1/
τ
D
N
a
v
σ
ea
is the rate of electron-atom collisions,
N
a
is the number den-
sity of atoms,
σ
ea
is the cross section for electron-atom collisions, and the collision
time
is assumed to be independent of the electron velocity. We take the force
F
acting on an electron in a general form:
τ
e
c
[
v
H
],
F
D
e
E
exp(
i
ω
t
)
that is, the external field consists of a harmonic electric field and a constant mag-
netic field. In the standard method, by multiplication of the kinetic equation by the
electron velocity and by subsequent integration over electron velocities, we obtain
the equation of electron motion in external fields:
m
e
d
w
e
m
e
w
e
e
c
[
w
e
dt
D
τ
e
E
exp(
i
ω
t
)
H
] ,
(4.130)
where
w
e
is the electron drift velocity. As is seen, electron-atom collisions give rise
to a frictional force
m
e
w
e
/
.
Let us write the motion equation in terms of its components. If the magnetic
field
H
is directed along the
z
-axis and the electric field direction is in the
xz
plane
(
E
D
i
E
x
τ
C
k
E
z
), (4.130) leads to the following set of equations:
