τ e ¼
1
32πn e
4πε 0 m e
e 2
2 v
3
lnΛ
τ i ¼
1
32πn i
4πε 0 m i
Z
2 e 2
2 v
3
lnΛ
ð2:4:25Þ
From (2.4.25), it is found that the relaxation times of electron gas and ion gas for
the fully ionized hydrogen plasma with Z ¼ 1 have the following relation:
τ e : τ i ¼ 1 :
m i
m e
1=2
ð2:4:26Þ
It is noted that this root mass ratio is very large. If the velocity distribution of initial
plasma is very far from Maxwellian, at first, electron gas becomes Maxwellian via
Coulomb collision among electrons, and then the ion gas tends to Maxwellian via
Coulomb collision among ions after the time more than 40 times.
Then, how long is the Coulomb collision relaxation time between electron gas
and ion gas in fully ionized plasma? As mentioned above, the energy exchange
between an electron and an ion in Coulomb collision is very inefficient. If the time
scale of the plasma phenomenon is fast, it is reasonable to assume both charged
gases have different temperatures, T e and T i . After the time scale of electron
relaxation time τ e and ion relaxation time τ i , two gases relax their energy and
momentum to finally become in thermodynamic equilibrium state with the same
temperature. The exact derivation of this time scale is not so easy, but it is known to
have the following simple relation including the electron-ion energy relaxation
time τ ei
e as briefly proofed soon below:
τ e : τ i : τ
e
ei ¼ 1 :
m i
m e
1=2
:
m i
m e
ð2:4:27Þ
As seen later in the case of laser heating of plasmas, most of the heating process or
production process of plasmas is due to the input energy from outside like laser to
deposit the energy to electrons. In the case where the main purpose is to produce
high-temperature ions for the purpose like driving hydrodynamic phenomena or
nuclear fusion, it takes time to transfer the electron energy to the ion energy.
The fact that energy relaxation between electron gas and ion gas takes long time
as shown in (2.4.27) can be understood by a simple consideration. Assume that the
time has passed and electron gas and ion gas are already in Maxwellian velocity
distribution. This means the velocity distribution is isotropic, spherically symmetric
in velocity space. Consider the following one-dimensional collision model to
evaluate the energy transfer fraction via one collision event:
64
2 Laser Absorption by Coulomb Collision
1
32πn e
4πε 0 m e
e 2
2 v
3
lnΛ
τ i ¼
1
32πn i
4πε 0 m i
Z
2 e 2
2 v
3
lnΛ
ð2:4:25Þ
From (2.4.25), it is found that the relaxation times of electron gas and ion gas for
the fully ionized hydrogen plasma with Z ¼ 1 have the following relation:
τ e : τ i ¼ 1 :
m i
m e
1=2
ð2:4:26Þ
It is noted that this root mass ratio is very large. If the velocity distribution of initial
plasma is very far from Maxwellian, at first, electron gas becomes Maxwellian via
Coulomb collision among electrons, and then the ion gas tends to Maxwellian via
Coulomb collision among ions after the time more than 40 times.
Then, how long is the Coulomb collision relaxation time between electron gas
and ion gas in fully ionized plasma? As mentioned above, the energy exchange
between an electron and an ion in Coulomb collision is very inefficient. If the time
scale of the plasma phenomenon is fast, it is reasonable to assume both charged
gases have different temperatures, T e and T i . After the time scale of electron
relaxation time τ e and ion relaxation time τ i , two gases relax their energy and
momentum to finally become in thermodynamic equilibrium state with the same
temperature. The exact derivation of this time scale is not so easy, but it is known to
have the following simple relation including the electron-ion energy relaxation
time τ ei
e as briefly proofed soon below:
τ e : τ i : τ
e
ei ¼ 1 :
m i
m e
1=2
:
m i
m e
ð2:4:27Þ
As seen later in the case of laser heating of plasmas, most of the heating process or
production process of plasmas is due to the input energy from outside like laser to
deposit the energy to electrons. In the case where the main purpose is to produce
high-temperature ions for the purpose like driving hydrodynamic phenomena or
nuclear fusion, it takes time to transfer the electron energy to the ion energy.
The fact that energy relaxation between electron gas and ion gas takes long time
as shown in (2.4.27) can be understood by a simple consideration. Assume that the
time has passed and electron gas and ion gas are already in Maxwellian velocity
distribution. This means the velocity distribution is isotropic, spherically symmetric
in velocity space. Consider the following one-dimensional collision model to
evaluate the energy transfer fraction via one collision event:
64
2 Laser Absorption by Coulomb Collision