dσ C ¼ 2πbdb Δθ
j j
2
ð2:4:14Þ
Here, the small-angle scattering angle is roughly evaluated as the ratio of impulse to
momentum:
Δθ %
F Á Δt
mv
¼
Ze
2
4πε 0 b
2
b
mv 2 ¼
b 0
b
ð2:4:15Þ
Here F is Coulomb force at r ¼ b and b 0 in (2.4.4) is used. The relation of (2.4.15) is
also obtained in the precise calculation with Rutherford scattering (Appendix A1).
Integrating (2.4.14) as follows, Coulomb collision cross section is obtained:
σ C ¼ 8πb
2
0
Z b max
b min
1
b
db ¼ 8πb
2
0 ln
b max
b min
ð2:4:16Þ
Within the classical plasma assumption, the following evaluation may be used:
b min ¼ b 0
b max ¼ λ D
ð2:4:17Þ
The cut of integral at the minimum of b 0 is usually called Landau cut. Then,
scattering cross section (2.4.16) is given:
σ C ¼ 8ln Λ
ð Þσ
R
90
ð2:4:18Þ
where (2.4.4) is used and
Λ ¼
b max
b min
(
)
¼ 4πn e λ
3
D
In (2.4.18), < > means taking average to thermal electrons and the following relation
was used:
1
2
mv
2
D
E
¼
3
2
T e
It is noted that the large-angle scattering cross section is roughly πb 0
2 and the
Coulomb logarithm
lnΛ
ð2:4:19Þ
which is much larger than unity in normal plasmas. Such plasmas are called ideal
plasma. In the ideal plasma, Debye sphere with radius λ D contains a huge number of
2.4 Electron Coulomb Collision by Ions in Plasma
59