Environmental Engineering Reference
In-Depth Information
where
E
is the electric field strength,
m
e
and
m
a
are the electron and atom masses,
and
σ
)
1
is the mean free path of electrons in a gas with number density
of atoms
N
a
. Substituting this distribution function into (4.64), we obtain for the
electron diffusion coefficient in this case
λ
D
(
N
a
s
eE
0.292
m
a
m
e
1/4
λ
m
e
D
?
D
λ
.
(4.67)
In the limit of hig
h
electric field strength
E
the diffusion coefficient of electrons is
proportional to
p
E
.Taking
σ
D
6Å
2
in the helium case, we obtain
D
0
p
x
,
D
?
D
1500 cm
2
/s, the diffusion coefficient, is reduced to the normal num-
ber density of atoms
N
a
where
D
0
D
10
19
cm
3
, and the reduced electric field strength
D
2.69
x
E
/
N
a
is expressed in Townsend. Note that the case of high electric field
strengths at room temperature of helium, if the average electron energy exceeds
remarkably the thermal energy of atoms, corresponds to the criterion
x
D
0.05 Td.
4.3.2
Diffusion of Electrons in a Gas in a Magnetic Field
We now determine the transverse diffusion coefficient for electrons in a strong
magnetic field if the directions of the electric and magnetic fields coincide. This
corresponds to the condition
eH
/(
m
e
c
)istheelectronLar-
mor frequency. The projection of the electron trajectory on a plane perpendicular to
the field consists of circles whose centers and radii change after each collision. The
diffusion coefficient, by its definition, is
D
?
Dh
ν
D
ω
,where
ω
H
H
is the square of
the displacement for time
t
in the direction
x
perpendicular to the field. We have
x
x
2
i
/
t
,where
h
x
2
i
H
t
,where
x
0
is the
x
-coordinate of the center of the electron's
rotational motion and
r
H
x
0
D
r
H
cos
ω
D
v
/
ω
H
is the Larmor radius, where
is the electron
v
velocity in the direction perpendicular to the field, and
ω
H
is the Larmor frequency
for an electron. From this it follows that
2
n
˝
(
x
x
0
)
2
˛
D
n
v
x
2
h
iD
,
2
H
2
ω
where
n
is the number of collisions. Since
t
D
n
/
ν
,where
ν
is the frequency of
electron-atom collisions, we obtain
*
v
+
v
,
2
ν
2
ν
D
?
D
D
ω
ν
,
H
2
H
2
H
2
ω
3
ω
where angle brackets mean averaging over electron velocities. Combining this re-
sult with (4.64), we find that the transverse diffusion coefficient for electrons in a
gas, moving perpendicular to electric and magnetic fields, is
.
2
1
3
ν
v
D
?
D
(4.68)
2
H
ω
C
ν
2
