Environmental Engineering Reference
In-Depth Information
where
g
is the relative velocity of colliding particles. From this we have the connec-
tion between the scattering angle
#
and the collision impact parameter
in the
limit of small scattering angles,
# D
Δ
p
p
D
2
e
2
μ
,
g
2
an for the differential cross section of particles interacting through the Coulomb
interaction potential this gives [1]
D
π
e
4
d
Δ
ε
d
σ
D
2
π
d
.
(2.35)
ε
(
Δ
ε
)
2
This is the Rutherford formula [46] for scattering of two charged particles if their
charges are equal to the electron charge. The criterion for validity of (2.35) for scat-
tering into small angles has the form
e
2
ε
,
(2.36)
and the energy change for a reduced particle in the center-of-mass frame of refer-
ence is
p
2
2
m
D
2
e
4
e
4
Δ
ε
D
Δ
g
2
D
,
2
μ
2
ε
where
. Note that in the limit of
small scattering angles the differential cross section for collision of two charged
particles is identical for both the same and opposite signs of the particle charge.
We now determine the diffusion cross section of collision of two charged par-
ticles in an ideal plasma, which is a measure of the exchange of energy and mo-
mentum between charged particles in a plasma and is very useful in expressing
plasma properties. Assuming that small scattering angles give the main contribu-
tion to the diffusion cross section and using the Rutherford formula (2.35) for the
differential cross section of scattering into small angles, we obtain for the diffusion
cross section
Δ
ε
is small compared with the particle energy
ε
Z
1
Z
1
2
g
4
Z
d
e
4
4
π
σ
D
2
(1
cos
#
)
2
π
d
D
#
π
d
D
.
μ
0
0
As is seen, the integral diverges in both the small and the large impact parameter
limits. This means that in reality this integral must be cut short for small and large
impact parameters, and the diffusion cross section of scattering of two charged
particles can be represented in the form
e
4
4
π
ln
max
σ
D
,
μ
2
g
4
min
