Environmental Engineering Reference
In-Depth Information
given by
M
2
m
e
eEw
e
(
ε
max
)
ε
D
.
max
ν
(
ε
max
)
Thus, within the framework of the scenario, when we ignore the chaotic character
of variation of the electron energy that corresponds to the first term of the collision
integral (3.18), the electron energy under the given conditions varies monotonically
until it reaches its maximum value
max
. Accounting for the diffusion term of the
collision integral (3.18) leads to broadening of the electron distribution function,
but violation of criterion (3.32) proceeds with long time.
We demonstrate these results for the helium example when the diffusion cross
section
ε
σ
of electron-atom collisions is independent of the collision velocity.
Then, introducing the mean free path for electrons
σ
), we have
λ
D
1/(
N
a
s
M
6
m
e
eE
ε
D
λ
D
0.96
ε
,
max
where the average electron energy
in the limit of long times is given by (3.30).
Next, one can represent the equilibrium electron energy in the form
ε
ε
D
ax
,
max
where
x
E
/
N
a
is the reduced electric field strength and
a
is independent of the
electric field strength. In particular, in the helium case, where
D
σ
D
6Å
2
,wehave
a
0.58 eV/Td, and formula (3.29) gives the drift velocity of electrons in the limit
of high electric field strengths. Next, we have from (3.26) in the limit of low electric
field strengths
D
s
1
v
2
eE
λ
3
m
e
2
3
eE
2
w
e
D
D
λ
m
e
T
,
π
σ
istheelectronmeanfreepathinthegas.Combiningthese
limiting cases, we obtain for the electron drift velocity
where
λ
D
1/
N
a
0.53
eE
λ
w
e
D
p
m
e
T
1
.
0.59
m
e
1/4
q
eE
λ
C
T
In the helium case, this formula takes the form
v
0
w
e
D
C
p
x
,
1
C
10
6
cm/s,
C
where
v
0
D
E
/
N
a
is expressed in Townsends. This formula is compared with experimental da-
ta [32] for the electron drift velocity in helium in Figure 3.3.
D
2.8
D
3.7, and the reduced electric field strength
x
