Environmental Engineering Reference
In-Depth Information
One can see that the drift velocity of electrons is small compared with a thermal en-
ergy, and the contribution of the drift velocity of electrons to the total kinetic energy
of electrons is small:
m
e
w
e
/
T
e
m
e
/
M
. Therefore, we used the Maxwell distri-
bution function for electrons with zero drift velocity. One can rewrite the above
fo
rmulas, expressing the electron drift velocity
w
e
and the average electron energy
ε
through the electric field strength. In the limit
T
e
T
,wehave
1/4
s
eE
p
eE
0.990
m
e
M
1/2
16
λ
(
m
e
M
)
1/4
D
λ
m
e
w
e
D
π
p
3
,
3
s
M
m
e
eE
0.433
s
M
p
3
4
3
2
T
e
ε
D
D
λ
D
m
e
eE
λ
.
(3.38)
These quantities have the same dependence on the problem parameters and are
characterized by numerical factors which are close to those in the regime of low
electron number densities according to (3.29).
Let us consider the case when the diffusion cross section of elastic electron-atom
collision is independent of the velocity, as occurs in the helium case. Let us fix the
electric field strength and let the electron density vary from small values to large
ones. Then the drift velocity increases as a result of transition from the regime
of low electric field strengths to the regime of high electric field strengths, but this
value jumps by 10%. It should be noted that because of a small variation of the elec-
tron energy in elastic collisions with atoms, the remarkable difference between the
electron energy and the thermal energy corresponds to low electric field strengths.
Let us demonstrate this with the helium example when the diffusion cross section
of elastic electron-atom collision varies weakly over a wide range of energies and is
approximately 6 Å
2
. Let us consider the regime of high electron number densities,
so that (3.36) has the form
M
12
m
e
(
eE
)
2
,
(
T
e
T
)
T
e
D
λ
σ
) is the mean free path for electrons in helium. The solution of
this equation for the electron temperature is
where
λ
D
1(
N
a
T
2
2
3
5
s
1
4
C
r
12
m
e
M
E
E
0
1
2
C
T
e
4
T
e
D
,
E
0
D
.
(3.39)
λ
E
0
characterizes the transition from low electric field strengths to high ones. In
particular, for helium (
σ
D
6Å
2
) at the gas temperature
T
D
300 K we have
E
0
/
N
a
0.06 Td.
We also derive the equation for relaxation of the electron temperature
T
e
.We
begin from a nonstationary kinetic equation (3.4). Multiplying this equation by the
electron energy and integrating the result over electron velocities, we obtain for the
D
