Environmental Engineering Reference
In-Depth Information
repeat this operation after multiplication of this equation by cos
θ
.Asaresult,we
obtain the above kinetic equation in the form
v
f
1
D
x
df
0
v
a
d
d
a
v
3
v
D
I
ea
(
v
x
f
1
),
I
ea
(
f
0
) ,
(3.11)
d
3
2
v
v
where
a
eE
/
m
e
.
First let us determine the collision integral from a nonsymmetric part
I
ea
(
v
x
f
1
)
of the collision integral. Using (3.7) for the collision integral and accounting for
conservation of the atom velocity
v
a
in an electron-atom collision, since the mag-
nitude of
v
a
is small compared with that of the electron velocity
v
,wehave
D
Z
(
v
0
v
)
x
v
I
ea
(
v
x
f
1
)
D
d
σ
f
1
(
v
)
'
(
v
a
)
d
v
a
.
The electron velocity after collision has the form
v
0
D
v
cos
is
the scattering angle, and the unit vector
k
is directed perpendicular to the electron
velocity
v
. Accounting for random directions of vector
k
, we obtain after integration
#C
k
v
sin
#
,where
#
I
ea
(
v
x
f
1
)
D
ν
v
x
f
1
(
v
) ,
(3.12)
N
a
v
σ
(
σ
(
where
ν
D
) is the rate of electron-atom collisions and
)
D
(1
v
v
cos
is the diffusion cross section of electron-atom scattering.
For determination of
I
ea
(
f
0
) we take into account that the change of the electron
energy in any single collision is small compared with the total electron energy.
Therefore, the collision integral is a divergence of a flux in the electron energy space
and is the right-hand side of the Fokker-Plank equation [28, 29], which takes into
account a small variation of the electron energy in a single collision. Let us consider
a group of processes with a small variation of a variable
z
in each individual event,
that is, the system is diffusive in nature in the variable
z
. We define the probability
W
(
z
0
,
t
0
#
)
d
σ
z
,
t
) such that the value
z
occurs at time
t
if at time
t
0
it was
z
0
.The
normalization condition for this probability is
Z
W
(
z
0
,
t
0
I
I
z
,
t
)
dz
D
1.
Because of the continuous character of the evolution of the probability
W
,itsatis-
fies the continuity equation
@
W
@
t
C
@
j
z
D
0,
@
where the flux
j
can be represented in the form
B
@
W
@
j
D
AW
.
z
Here the first term is associated with the hydrodynamic flux, and the second one
corresponds to the diffusion flux. By definition, the coefficients for these processes
