Environmental Engineering Reference
In-Depth Information
This means that a fast electron loses energy only.
Accordingtodefinition(3.13),wehavefortherateofelectronbraking
Z
Z
(
Δ
p
)
2
2
m
e
N
e
v
ε
ε
0
)
N
e
v
A
(
ε
)
D
(
d
σ
D
2
π
d
,
where
v
) is the momentum which is transferred to a slow electron
from the fast one for collision impact parameter
Δ
p
D
2
e
2
/(
and
N
e
is the number density of
slow electrons. From this we find for the rate of braking of a fast electron in a gas
of slow electrons
e
4
N
e
m
e
v
4
π
A
(
ε
)
D
ln
Λ
,
(3.19)
where ln
is the Coulomb logarithm. Adding formally the diffusion term to the
Fokker-Plank equation and requiring zero collision integral for the Maxwell distri-
bution function, we obtain the collision integral for a fast electron in an electron
gas:
Λ
T
e
f
0
e
4
N
e
ln
4
π
Λ
@
m
e
v
@
v
f
0
m
e
v
@
v
C
@
I
ee
(
f
0
)
D
,
ε
T
e
.
(3.20)
m
e
v
when a fast
electron is moving in an electron gas. Then we use the kinetic equation in the
form
We now derive the equation for variation of the electron energy
ε
@
f
0
T
D
I
ee
(
f
0
),
@
where the collision integral is given by (3.20) with
T
e
D
0. We take the distribution
function for fast electrons as
D
δ
(
u
v
)
f
0
(
u
)
,
4
π
u
2
where the normalization condition for fast electrons is taken in the form
Z
u
2
f
0
(
u
)
du
4
π
D
1.
Let us multiply the kinetic equation for the distribution function for fast electrons
by the electron energy
ε
and integrate over electron velocities. We obtain
d
dt
D
4
π
e
4
N
e
m
e
v
A
(
ε
)
ln
Λ
,
(3.21)
where
is the average electron energy, which exceeds significantly a thermal ener-
gy of slow electrons.
ε
