2.4.1 Debye Shielding
Electrolyte that helps to make water electrically conducting is salt (NaCl) which is
decomposed Na
+ and Cl
À in water. This is a good example of shielding media.
Inserting electrodes in such an electrolyte, Debye found that when applying a
voltage, negative ions gather around the positive electrode, and positive ions gather
around the negative electrode; consequently, the electric field is shielded in the
electrolyte. This is Debye shielding.
Since plasma is a mixture of positive charge ions and negative charge electrons,
the same phenomenon is expected. In the case of electrons and ions, the masses and
thermal speeds are greatly different, so the shielding distance and time response of
shielding are not the same. The shielding distance is called Debye length and is one
of the most fundamental physical quantities in plasma physics. The picture of
electron cloud around the central ion is shown in Fig. 2.12b. Debye length is
obtained as follows mathematically.
Let’s see how the electrostatic potential due to an ion charge located at the center
spreads in the plasma. The equation to be solved is Poisson Eq. (2.2.3), and it is
enough to assume spherical symmetry. Since electrons and ions are in Boltzmann
distribution to the potential ϕ (r), the following equation is obtained for r > 0:
ε 0
1
r 2
d
dr
r
2 d
dr
ϕ
¼ À e Zn i r
ð Þ À n e r
ð Þ
f
g
¼ À en 0 exp ÀZ
eϕ
T i
À exp
eϕ
T e
!
ð2:4:7Þ
where n 0 is the average density of electrons. This is a nonlinear differential equation
for the potential ϕ (r). This is called the Debye-Huckel equation. Although this
cannot be solved analytically, it can be assumed in the high-temperature plasma that
the energy due to the Coulomb force is sufficiently smaller than the kinetic energy.
Then, the exponential function of (2.4.7) can be approximated by Taylor expansion:
exp ÀZ
eϕ
T i
À exp
eϕ
T e
% À
eϕ
T e
À Z
eϕ
T i
Note that RHS of (2.4.7) cancels for ϕ (r) ¼ 0 and change the variable as:
ϕ r
ð Þ ¼
Ze
r
y r
ð Þ
It is easy to find the following solution of the Debye shielded potential from (2.4.7):
56
2 Laser Absorption by Coulomb Collision
Electrolyte that helps to make water electrically conducting is salt (NaCl) which is
decomposed Na
+ and Cl
À in water. This is a good example of shielding media.
Inserting electrodes in such an electrolyte, Debye found that when applying a
voltage, negative ions gather around the positive electrode, and positive ions gather
around the negative electrode; consequently, the electric field is shielded in the
electrolyte. This is Debye shielding.
Since plasma is a mixture of positive charge ions and negative charge electrons,
the same phenomenon is expected. In the case of electrons and ions, the masses and
thermal speeds are greatly different, so the shielding distance and time response of
shielding are not the same. The shielding distance is called Debye length and is one
of the most fundamental physical quantities in plasma physics. The picture of
electron cloud around the central ion is shown in Fig. 2.12b. Debye length is
obtained as follows mathematically.
Let’s see how the electrostatic potential due to an ion charge located at the center
spreads in the plasma. The equation to be solved is Poisson Eq. (2.2.3), and it is
enough to assume spherical symmetry. Since electrons and ions are in Boltzmann
distribution to the potential ϕ (r), the following equation is obtained for r > 0:
ε 0
1
r 2
d
dr
r
2 d
dr
ϕ
¼ À e Zn i r
ð Þ À n e r
ð Þ
f
g
¼ À en 0 exp ÀZ
eϕ
T i
À exp
eϕ
T e
!
ð2:4:7Þ
where n 0 is the average density of electrons. This is a nonlinear differential equation
for the potential ϕ (r). This is called the Debye-Huckel equation. Although this
cannot be solved analytically, it can be assumed in the high-temperature plasma that
the energy due to the Coulomb force is sufficiently smaller than the kinetic energy.
Then, the exponential function of (2.4.7) can be approximated by Taylor expansion:
exp ÀZ
eϕ
T i
À exp
eϕ
T e
% À
eϕ
T e
À Z
eϕ
T i
Note that RHS of (2.4.7) cancels for ϕ (r) ¼ 0 and change the variable as:
ϕ r
ð Þ ¼
Ze
r
y r
ð Þ
It is easy to find the following solution of the Debye shielded potential from (2.4.7):
56
2 Laser Absorption by Coulomb Collision