Environmental Engineering Reference
In-Depth Information
3.1.6
The Landau Collision Integral
Expression (3.20) gives the electron-electron collision integral in the limit when
the energy of a test electron is high compared with the energy of most of the
other electrons and is based on the assumption that the variation of the electron
energy resulting from a single electron-electron collision is relatively small. This
assumption is valid when the Coulomb logarithm is large, so small-angle scatter-
ing gives the main contribution to the electron-electron scattering cross section.
This fact allows us to formulate the problem in another way by ignoring the as-
sumption of high energy of a test electron. Because the electron momentum varies
weakly in small-angle scattering, one can represent the collision integral by the
three-dimensional Fokker-Planck equation. Such a form of the collision integral is
named the Landau collision integral. We develop its form below [31].
We start from the general expression (3.6) for the collision integral and account
for the symmetry of the rate
W
that follows from the principle of detailed balance
for elastic collisions of identical particles and which is given by (3.7). Let us in-
troduce the variation
Δ
v
of the electron velocity as a result of collision, that is,
v
1
D
v
1
C
Δ
v
. Because of conservation of the total momentum in the collision
we have
v
1
D
v
1
C
Δ
v
. Using the symmetry of the rate
W
due to the principle of
detailed balance, we represent it in the form
W
C
Δ
2
Δ
2
!
v
1
,
v
2
)
W
(
v
1
,
v
2
D
v
1
,
v
2
Δ
v
,
,
and from the principle of detailed balance it follows that
W
is an even function of
Δ
v
,thatis,
W
W
C
Δ
2
Δ
2
Δ
2
C
Δ
2
v
1
,
v
2
,
Δ
v
D
v
1
,
v
2
,
Δ
v
.
We use this symmetry in the expression for the electron-electron collision integral,
which has the form
Z
f
(
v
1
)
f
(
v
2
)
Δ
v
)
I
ee
(
f
)
D
f
(
v
1
C
Δ
v
)
f
(
v
2
W
C
Δ
2
Δ
2
v
1
,
v
2
,
Δ
v
,
(3.22)
and our goal is expansion of this expression over a small parameter
Δ
v
.
The first term in the expansion of the collision integral over the small parameter
Δ
v
is
Z
f
(
v
2
)
@
f
(
v
1
)
@
v
1
f
(
v
1
)
@
f
(
v
2
)
@
v
2
I
ee
(
f
)
D
Δ
v
Wd
Δ
v
d
v
2
.
Since
W
is an even function of
Δ
v
and because of the symmetry of this function
with respect to transformation
v
1
$
v
2
, this approximation gives zero.
