Environmental Engineering Reference
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where the electron temperature T e is in electronvolts and ln
Λ D
10. In particular,
if N i / N a
D
10 4 and the electron temperature T e
D
1 eV, the electron drift velocity
10 5 the decrease of the electron drift velocity is about
decreases by 6%, at N i / N a
D
10 6 , this drift velocity decrease is approximately
2%, and in the case N i / N a
D
0.3%.
3.2.5
Electrons in a Gas in an Alternating Electromagnetic Field
As follows from the above consideration, there are two typical times for electron-
atomcollisionswhenanelectronismovinginanatomicgasinanexternalfield.
The first one,
is a typical rate of electron-atom collisions), characterizes
an electron momentum variation, and the second time, approximately M /( m e
τ D
1/
ν
(
ν
ν
)
τ
M / m e , is a typical time for variation of the electron energy as a result of elastic
collisions with atoms. We now consider electron motion in a gas in a harmonic
electric field of strength E cos
m e / M ,sotheelec-
tron energy does not vary during the field period. This condition simplifies the
problem [37-39]. The above condition for the field frequency corresponds to the
following form of the distribution function similar to (3.10) [37]:
ω
t under the condition
ωτ
f ( v , t )
D
f 0 ( v )
C v x f 1 exp( i
ω
t )
C v x f 1 exp(
i
ω
t ),
where the x -axis is directed along the field. Substituting this expansion into the
kinetic equation and separating the corresponding harmonics by the standard
method, we obtain the following set of equations instead of (3.11):
a
2
df 0
d
a
2
df 0
d
v C
ν C
i
f 1
D
v C
ν
i
f 1
D
(
ω
) v
0,
(
ω
) v
0,
2 v
f 1 ) D
a
6 v
3 ( f 1
C
I ea ( f 0 ) .
(3.46)
From this we have for the electron drift velocity instead of (3.26)
Z
x [ f 1 exp( i
w e ( t )
D
t )
C
f 1 exp(
i
t )] d v
v
ω
ω
1
v
eE
3 m e
d
d
ν
cos
ω
t
C ω
sin
ω
t
3
D
.
(3.47)
v
2
ω
2
C ν
2
v
This expression corresponds to expansion over the small parameter m e /
( M
1, which allows us to ignore other terms of expansion over the spheri-
cal harmonics of the electron distribution function. Note that
ωτ
)
may be found
larger and smaller than one, and this ratio determines the phase shift for a drift-
ing electron with respect to an external field. The above expressions are valid for
criteria (3.30) and (3.37). We now write these formulas when criterion (3.37) holds
true, which allows us to introduce the electron temperature T e .Then,theelectron
drift velocity is
ω
/
ν
2 ν
eE
3 T e
cos
ω
t
C ω
sin
ω
t
w e ( t )
D
,
(3.48)
v
ω
2
C ν
2
 
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