Environmental Engineering Reference
In-Depth Information
the electric field to electrons, and then from collisions between electrons and gas
atoms, which leads to energy transfer from electrons to atoms. These processes
establish the energy balance for electrons and determine the electron temperature
T
e
. To find this temperature, we use directly the stationary kinetic equation for
electrons in a gas in an electric field. This equation has the form
e
m
e
@
f
@
v
D
I
ea
(
f
)
C
I
ee
(
f
0
) .
(3.35)
Because collisions between electrons are stronger in this regime than collisions
between electrons and atoms, we have from this in the first approximation
I
ee
(
f
0
)
0. This gives the Maxwell distribution function for electrons, where
the electron temperature
T
e
is a parameter. The meaning of this is that the equi-
librium of the energy distribution function is established by electron-electron
collisions, whereas the drift velocity of the electrons is maintained by electron-
atom collisions. To determine the electron temperature, we multiply this equation
by
m
e
v
D
2
/2 and integrate over electron velocities. We have
R
(
m
e
v
2
/2)
I
ee
(
f
0
)
d
v
D
0,
because collisions between electrons do not change the total energy of the electron
subsystem. Then we obtain the following balance equation:
Z
1
˝
v
2
m
e
M
ν
˛
,
m
e
v
T
T
e
2
eEw
e
D
I
ea
(
f
0
)
d
v
D
2
where we use (3.18) for the collision integral of electrons and atoms. This is the
balance equation for electrons, so its right-hand side is the energy per electron that
is transferred from an external electric field, and the left-hand side is the specific
energy transferred from an electron to gas atoms. From this we have for the differ-
ence between the electron and gas temperatures
˝
v
ν
˛
2
/
Ma
2
3
T
e
T
D
,
(3.36)
h
v
2
ν
i
where
a
eE
/
m
e
and an average is made with the Maxwell distribution function
for electrons.
Note that the regime of high electron number density holds true under the crite-
rion opposite to (3.30), which has the form
N
e
N
a
D
2
σ
ea
m
e
M
T
T
e
ε
.
(3.37)
2
π
e
4
ln
Λ
Under this condition one can ignore the term
I
ee
on the right-hand side of (3.35).
Let us consider the limiting cases for the dependence
ν
(
v
). If
ν
D
const, we have
from (3.34) and (3.36)
eE
m
e
M
3
w
e
.
w
e
D
,
T
e
T
D
ν
σ
)
1
.
σ
(
If
)
D
const, we introduce the mean free path of electrons as
λ
D
(
N
a
v
Then (3.34) and (3.36) yield
4
eE
λ
3
32
Mw
e
.
w
e
D
p
2
,
T
e
T
D
π
T
e
m
e
