Environmental Engineering Reference
In-Depth Information
where the angle brackets signify an average over atomic energies and
d
is the
electron-atom cross section corresponding to a given variation of the electron en-
ergy. We can make use of the constancy of the relative electron-atom velocity in the
collision process, or
σ
,where
v
and
v
0
are the electron velocities
before and after the collision and
v
a
is the velocity of the atom, unvarying in a colli-
sion with an electron. From this it follows that
jDj
v
0
v
a
j
(
v
v
a
)
j
2
v
0
)
2
2
v
a
(
v
v
0
), which yields
(
D
v
the result [30]
v
Z
m
e
2
2
a
3
T
m
e
M
m
e
v
2
(
v
v
0
)
2
N
a
v
N
a
v
σ
(
v
) .
B
(
ε
)
D
d
σ
D
(3.17)
2
In the determination of this expression we use
h
v
a
/3
2
iD
T
/
M
,where
T
is the gas
temperature,
M
is the atom mass, and
j
v
v
0
jD
2
v
sin(
#
/2), where
#
is the scat-
D
R
(1
σ
(
tering angle and
is the diffusion cross section of electron-
atom scattering. Thus, the collision integral from the spherical part of the electron
distribution function has the form [12, 30]
)
cos
#
)
d
σ
v
T
f
0
m
e
M
@
f
0
m
e
v
@
v
C
@
3
I
ea
(
f
0
)
D
ν
,
(3.18)
v
2
@
v
v
N
a
v
σ
(
v
) is the rate of electron-atom collisions.
where
ν
D
3.1.5
Collision Integral for Fast Electrons in an Electron Gas
When a fast electron is moving in an ionized gas, its scattering by electrons and
ions of this gas proceeds mostly into small angles because such scattering gives the
main contribution to the diffusion cross section for collision of a fast electron and
charged particles in a plasma. We determine below the spherical part of the elec-
tron distribution function
I
ee
(
f
0
) for electron scattering by electrons and ions. In
this way we account for the main part of the electron distribution function being al-
most spherically symmetric since because of the small parameter
m
e
/
M
(
m
e
is the
electron mass,
M
is the atom mass) change of the electron momentum proceeds
more effectively than change of the electron energy. Next, variation of the electron
energy as a result of one collision with charged particles is relatively small, so the
collision integral is given by (3.16), which has the form
)
@
m
e
v
f
0
m
e
v
@
v
C
@
f
0
T
e
I
ee
(
f
0
)
D
B
(
ε
,
v
2
@
v
where
B
(
) is determined by (3.13), and the Maxwell distribution function is used
for the electron gas with electron temperature
T
e
, and the energy of a fast electron
satisfies the criterion
ε
ε
T
e
.
Under this criterion, we can take the hydrodynamic flux into account in the Fokker-
Planck equation (3.16), that is,
@
m
e
v
I
ee
(
f
0
)
D
A
(
)
f
0
].
[
v
ε
2
@
v
