Environmental Engineering Reference
In-Depth Information
are
Z
(
x
1
τ
A
(
z
,
t
)
D
lim
τ
!
z
)
W
(
x
,
t
I
z
,
t
C
τ
)
dx
,
0
Z
1
2
z
)
2
W
(
x
,
t
B
(
z
,
t
)
D
lim
τ
!
(
x
I
z
,
t
C
τ
)
dx
.
(3.13)
τ
0
The corresponding equation for the probability is called the Fokker-Planck equa-
tion [28, 29], and has the form
2
(
BW
)
@
@
W
@
t
D
@
(
AW
)
@
C
@
.
(3.14)
z
z
2
This equation can be generalized for the case when the normalization condition
has the form
Z
(
z
)
W
(
z
0
,
t
0
I
z
,
t
)
dz
D
1,
that is, the state density is
(
z
). Then the quantity
W
in (3.13) must be replaced by
W
, and the Fokker-Planck equation then takes the form
2
(
@
W
@
t
D
@
(
AW
)
@
C
@
BW
)
@
.
(3.15)
z
z
2
The right-hand side of this equation can be used as the collision integral
of the spherical part of the electron distribution function because it describes
the incremental changes of the electron energy. To accomplish this, we replace
W
(
z
0
,
t
0
z
,
t
) in (3.15) with the distribution function
f
0
,andinplaceof
z
we sub-
stitute the electron energy
I
1/2
, and the collision
ε
.Then
(
ε
) behaves as
(
ε
)
ε
integral takes the form
f
0
)
1
@
@
ε
C
@
@
ε
I
ea
(
f
0
)
D
A
f
0
(
B
.
(
ε
)
The connection between
A
and
B
is found from the condition that if the distribu-
tion function coincides with the Maxwell distribution function, the collision inte-
gral will be zero. This condition yields [30]
)
@
1
@
@
ε
f
0
@
ε
C
f
0
T
I
ea
(
f
0
)
D
(
ε
)
B
(
ε
,
(3.16)
(
ε
)
where
T
is the gas temperature. This collision integral is zero for the Maxwell dis-
tribution function for electrons when the electron temperature is equal to the gas
temperature
T
.
In determining the quantity
B
(
ε
) for the electron-atom collision integral, we
account for its definition
Z
ε
!
ε
0
)
1
2
ε
ε
0
)
2
N
a
v
B
(
ε
)
D
(
d
σ
(
,
