Environmental Engineering Reference
In-Depth Information
3.2.2
Evolution of Electrons in an Atomic Gas in an Electric Field
When an electron is moving in an atomic gas in an electric field, its energy varies
weakly after a strong collisio
n w
ith gas atoms, and after
n
1 collisions this vari-
ation is proportional to
m
e
p
n
/
M
. But its drift velocity is established fast. Hence,
one can define the electron drift velocity as a function of the current electron en-
ergy [34]. Its value follows from (3.26), where we take the distribution function to
be
f
0
(
0
is the current
electron energy. If the corresponding electron velocity is
v
, according to (3.26) we
have
ε
)
D
A
δ
(
ε
ε
0
), where
A
is the normalization factor and
ε
,
3
eE
3
m
e
1
v
d
d
v
w
e
(
ε
)
D
(3.31)
2
ν
(
v
)
v
2
/2 is the current electron energy.
This consideration is valid when we consider evolution of an individual electron
in a gas in an external electric field and ignore a chaotic change of its energy, which
corresponds to the criterion
where now
ε
D
m
e
v
ε
Δ
ε
T
.
(3.32)
Here
is the uncertainty in the electron en-
ergy or the width of its distribution function. Hence, here we consider the electron
energy to be strictly determined in the course of its evolution. Let us determine the
character of the variation of the electron energy
ε
is the current electron energy and
Δ
ε
if at the beginning it is relatively
small. We multiply the kinetic equation (3.4) by the electron energy
ε
2
/2
ε
D
m
e
v
and integrate over electron velocities, which gives
Z
d
dt
D
I
ea
N
e
d
v
.
eEw
e
C
ε
This equation has a simple physical sense: the first term on the right-hand side of
this equation is the energy transferred from the field to the electron, and the second
term accounts for collision of this electron with atoms. We take into account elastic
electron-atom collisions only and use (3.18) for the collisi
on
integral. Because of
criterion (3.32), one can replace the average electron energy
ε
by its current value
ε
and ignore the first term in (3.18) for the collision integral. As a result, we obtain
d
dt
D
2
m
e
eEw
e
M
εν
.
(3.33)
This equation exhibits the character of variation of the electron energy when an
electron travels in an atomic gas in an external electric field and has low energy
at the beginning. We have that the electron energy increases in time, and in the
course of its increase the relative energy portion given to gas atoms from an elec-
tron increases until the maximum electron energy
ε
max
is reached. This energy is
