Environmental Engineering Reference
In-Depth Information
where
is
the relative velocity of the colliding electrons. From this we obtain for the diffusion
coefficient in electron-velocity space
α
is a component of the collision impact parameter and
g
Dj
v
v
0
j
Z
Z
Z
1
2
1
2
4
e
4
m
e
g
π
α
D
α
D
Δ
α
Δ
Wd
Δ
v
D
Δ
α
Δ
gd
σ
D
d
.
3
Let us take the frame of reference where the relative velocity
g
is directed along the
x
-axis and the impact parameter is directed along the
y
-axis, that is, the plane of
motion is the
xy
plane. Then only
y
is nonzero, so only tensor component
D
yy
is nonzero. For this component of the tensor we obtain
Δ
Z
2
e
4
m
e
g
1
4
e
4
m
e
g
π
D
yy
D
2
2
π
d
D
ln
Λ
,
where the integral over collision impact parameters is evaluated in a straightfor-
ward way, and ln
is the Coulomb logarithm.
Transferring to an arbitrary frame of reference, we take into account that tensor
D
α
is symmetric with respect to its indices and can be constructed on the basis of
symmetric tensors
Λ
δ
α
and
g
α
g
. Therefore, it has the form
e
4
m
e
g
3
(
g
α
g
2
π
g
2
D
α
D
δ
α
)ln
Λ
.
As a result, the Landau collision integral, which accounts for collisions between
electrons, has the form [31]
Z
d
v
2
f
1
@
f
2
D
α
D
@
j
@
v
1
@
v
2
@
f
2
f
1
@
v
1
I
ee
(
f
)
,
j
D
,
e
4
m
e
g
3
(
g
α
g
2
π
g
2
D
α
D
δ
α
)ln
Λ
.
(3.23)
The Landau collision integral is a version of the right-hand side of the Fokker-
Planck equation.
Let us consider the case when a beam of fast electrons propagates through an
electron gas, and braking of fast electrons results from collisions with slow elec-
trons of the electron gas. Then the collision integral is given by (3.20), where the
first term in parentheses is small and may be ignored, so the rate of variation of
the average energy of fast electrons is given by (3.21). We now analyze the kinetics
of the braking process using the Landau collision integral.
We divide electrons into two types, a beam of fast electrons with the distribution
function
f
(
v
)
f
1
(
v
) and an electron gas of slow electrons with the distribution
function
f
2
(
v
). Taking the
x
-axisasthebeamdirection,wefindonlytwocompo-
nents of the diffusion tensors are nonzero:
D
e
4
m
e
v
2
π
D
yy
D
D
zz
D
ln
Λ
,
