molecular diffusion but convective and turbulent diffusion are more effective in
general. This physics, however, is very complicated and out of the scope of this
chapter. Readers may see the physics of turbulence in Vol. 2.
For the case like molecule gas, the mean free path is the same for any particles
with different velocity if the collision cross section σ is constant. In this case, it is
easy to imagine the collision process like hard ball collision, and transport
coefficients are easily calculated. However, plasmas are complicated because it is
made of ions and electrons with mass ratio of about 2000, and collision is due to
Coulomb force, and its effective cross section is inversely proportional to the square
of the thermal energy as seen below. The main idea of collisional process is the same
as molecule gas, and the mean free path is derived after calculating the Coulomb
collision cross section based on Rutherford scattering to be shown in Appendix-1.
It is instructive to evaluate the Coulomb cross section intuitively without
invoking to the detail calculation with Rutherford scattering. Assume that an electron
impacts an ion with the ionic charge Z and impact parameter r as shown in Fig. 2.11,
then strong Coulomb collision may work if the following relation is satisfied:
Ze
2
4πε 0 r
¼
1
2
mv
2
ð2:4:2Þ
Here v is the velocity of the impacting electron. The Coulomb collision cross section
obtained from this radius is:
πr
2
¼ π
2Ze
2
4πε 0 mv 2
2
¼ 4πb
2
0
ð2:4:3Þ
Here, b 0 is the impact parameter providing the 90 degree scattering in the calculation
by Rutherford [see (A1-9)], and the 90 degree scattering cross section is given as
follows:
b 0 ¼
Ze
2
4πε 0 mv 2
σ
R
90 ¼ π b 0
ð Þ
2
ð2:4:4Þ
The impact parameter constant b 0 is only a function of Z and electron kinetic energy
and
2
mv
2
0
Ze
4 r
r
Fig. 2.11 A schematic
trajectory of an electron
scattered due to the
Coulomb force of an ion at
the center
54
2 Laser Absorption by Coulomb Collision
general. This physics, however, is very complicated and out of the scope of this
chapter. Readers may see the physics of turbulence in Vol. 2.
For the case like molecule gas, the mean free path is the same for any particles
with different velocity if the collision cross section σ is constant. In this case, it is
easy to imagine the collision process like hard ball collision, and transport
coefficients are easily calculated. However, plasmas are complicated because it is
made of ions and electrons with mass ratio of about 2000, and collision is due to
Coulomb force, and its effective cross section is inversely proportional to the square
of the thermal energy as seen below. The main idea of collisional process is the same
as molecule gas, and the mean free path is derived after calculating the Coulomb
collision cross section based on Rutherford scattering to be shown in Appendix-1.
It is instructive to evaluate the Coulomb cross section intuitively without
invoking to the detail calculation with Rutherford scattering. Assume that an electron
impacts an ion with the ionic charge Z and impact parameter r as shown in Fig. 2.11,
then strong Coulomb collision may work if the following relation is satisfied:
Ze
2
4πε 0 r
¼
1
2
mv
2
ð2:4:2Þ
Here v is the velocity of the impacting electron. The Coulomb collision cross section
obtained from this radius is:
πr
2
¼ π
2Ze
2
4πε 0 mv 2
2
¼ 4πb
2
0
ð2:4:3Þ
Here, b 0 is the impact parameter providing the 90 degree scattering in the calculation
by Rutherford [see (A1-9)], and the 90 degree scattering cross section is given as
follows:
b 0 ¼
Ze
2
4πε 0 mv 2
σ
R
90 ¼ π b 0
ð Þ
2
ð2:4:4Þ
The impact parameter constant b 0 is only a function of Z and electron kinetic energy
and
2
mv
2
0
Ze
4 r
r
Fig. 2.11 A schematic
trajectory of an electron
scattered due to the
Coulomb force of an ion at
the center
54
2 Laser Absorption by Coulomb Collision