Environmental Engineering Reference
In-Depth Information
can differ remarkably from the Maxwell one. On the other hand, the analysis of the
electron behavior is simplified because of a small mass compared with the mass of
other atomic particles in a gas. This allows one to represent the integral of electron-
atom collisions in the analytical form, as (3.12) and (3.18) for the collision integral
in the case of elastic electron-atom collisions. Formula (3.26) gives the electron
drift velocity in a gas in external fields with elastic collisions between electrons
and atoms. We now derive the expression for the transverse diffusion coefficient of
electrons in a gas under these conditions.
By definition, the diffusion coefficient of electrons
D
e
connects the electron num-
ber density gradient
r
N
e
and the electron flux
j
e
by the relation
j
e
D
D
e
r
N
e
.
On the other hand, the kinetic equation for the electron distribution function has
the following form if the electron number density varies in space:
r
f
D
I
ea
(
f
),
v
x
where
I
ea
is the integral of electron-atom collisions. We take the electron distribu-
tion function in the standard form (3.10)
f
D
f
0
(
v
)
C
v
x
f
1
(
v
),
where the
x
-axis is directed along the gradient of the electron number density.
Because the distribution function is normalized to the electron number density
f
N
e
/
N
e
, and the kinetic equation for the electron dis-
tribution function takes the form
N
e
,wehave
r
f
D
f
r
x
f
0
N
e
N
e
D
ν
v
x
f
1
r
v
if we use (3.12) for the collision integral taking into account elastic electron-atom
collisions. From this we obtain for the nonsymmetric part of the electron distribu-
tion function
f
0
r
N
e
f
1
D
.
ν
N
e
This leads to the following expression for the electron flux:
N
e
R
v
Z
Z
N
e
v
,
x
f
0
d
v
x
ν
x
f
1
d
v
D
r
2
j
e
D
v
fd
v
D
D
r
v
ν
N
e
where angle brackets mean averaging over the electron distribution function. Com-
paring this formula with the definition of the diffusion coefficient according to the
equation
j
e
D
D
e
r
N
e
, we find for the diffusion coefficient of electrons in a gas
v
.
2
D
e
D
3
ν
