Environmental Engineering Reference
In-Depth Information
relation
L
2
D
Ir
0
JND
.
(5.137)
Being guided by the reduced diffusion coefficient of atoms in a plasma,
DN
10
19
cm
1
s
1
according to the data in Tables 4.2-4.4, we find that radiation pen-
etrates to a depth
L
10 cm, if the radiation power is
Ir
0
100W. This shows
that conditions for passage of photoresonant radiation through an ionized gas are
available in reality.
5.5.4
Ionization Instability of a Plasma in a Magnetic Field
We now examine the behavior of a dense plasma in crossed electric and magnetic
fields at low electric currents. In this problem, local ionization equilibrium exists
everywhere within the plasma, so the Saha relation for the electron number density
is valid, but heat transport processes are not essential. We can examine a time for
evolution of a perturbation of the electron number density. The electron energy
per unit volume is
W
3
N
e
T
e
/2, and the balance equation for this quantity has,
according to (3.35), the form
D
dW
e
dt
3
m
e
D
eEw
e
N
e
M
(
T
e
T
)
ν
N
e
.
(5.138)
Here we account for the negative direction of the electric field,
m
e
and
M
are the
electron and atom masses,
T
e
and
T
are the electron and atom temperatures, and
for simplicity we assume that the electron-atom collision rate
N
a
v
σ
ea
is inde-
ν
D
σ
ea
is the diffusion
cross section of electron-atom collisions, and
v
is the electron velocity.
Because of the local ionization equilibrium, we have
N
e
pendent of the electron velocity,
N
a
is the atom number density,
exp(
J
/2
T
e
), where
J
is the atom ionization potential. This gives the relation
N
e
N
e
T
e
T
e
J
2
T
e
between perturbation values of the electron number density
N
e
and temperature
T
e
, so we can conclude that
T
e
T
e
D
N
e
N
e
.
On the basis of this inequality, the term
T
e
dT
e
/
dt
can be ignored compared with
the term
N
e
dN
e
/
dt
on the left-hand side of (5.138). This yields the result
dN
e
dt
4
T
e
J
m
e
M
ν
N
e
,
D
γ
γ
D
.
(5.139)
Thus, the ionization perturbation under consideration has a timescale for damping
that is much longer than a typical time for electron-atom collisions.
