Fundamentals of Plasma Simulation (I)
核融合基礎学（プラズマ・核融合基礎学）
李継全（准教授）/岸本泰明（教授）/今寺賢志（D1）
2007.4.9 — 2007.7.13
Lecture One (2007.4.)
Part one: Basic concepts & theories of plasma physics
➣ Plasma & plasma fluctuations
Definitions & basic properties of plasma
Basic parameters describing the plasma
Plasma oscillation & fluctuations
Reference books: S Ichimaru, Basic principles of plasma physics, Chpt.1
1
……
What is a plasma?
Plasma is quasi-neutral ionized gas containing enough free charges to make
collective electromagnetic effects important for its physical
Basic properties— quasi-neutrality & collective behaviour (motion of charged
particle may produce electric and magnetic fields, then influence other particles)
A plasma is regarded as the fourth state of matter in addition to the solid, liquid,
and gaseous states. It is remarked that 99% of universe consists of PLASMA.
State of matter: solid → fluid → gas → plasma
By heating a gas (to a temperature of 105– 106 K) one can make a plasma.
(collisions → ionization)
2
Where is a plasma?
In nature: Sun & solar corrona & solar wind in space; Aurora & lighting on earth
In nature:
Sun & solar corrona & solar wind
in space; Aurora & lighting on
earth
In man-made devices for
applications:
Fluorescent and neon lights;
Plasma TV;
Magnetic fusion (tokamak,
stellarator, Magnetic mirror ……)
Inertial fusion (laser plasma )
3
Where is a plasma?
Plasma in nature
corona
Magnetic reconnection
And solar physics
Vortex structures in
non-neutral plasma
Lightning and
discharge physics
4
Aurora physics
Where is a plasma?
Plasma in man-made devices
Plasma lamp
Fluorescent & neon lamps
Plasma TV
Magnetic
fusion
Inertial
fusion
5
Basic properties of plasma – Debye shielding
In plasma, binary Coulomb scattering CANNOT correctly describe the behavior of
charged particles interacting. Remarkable difference from neutral gas for plasmas is
COLLECTIVE behavior – Debye shielding
Binary Coulomb
interaction
+e
-e
+q
+q
Consider a positive point charge +q at the origin, Ti=Te, ni=ne.
Now think about what the positive charge does. It will attract the electrons and
repel the ions, making a cloud of net negative charge around itself, reducing
(shielding) the electric field the point charge makes.
6
Basic properties of plasma – Debye shielding (cont.)
In plasma
-e
+q
-e
+q
+q
But the electrons can’t just collapse onto the point charge to completely neutralize
it because they have too much thermal energy.
If we wait for inter-particle collisions to allow this competition between Coulomb
attraction and thermal motion to come to equilibrium, we have the situation first
studied by Peter Debye and called Debye shielding.
7
What is Debye Length?
Coulomb potential of a test charge +q at origin is:
 
q
4 0 r
In thermal equilibrium of a plasma with temperature T, the probability of a particle
being in a state with energy ε is proportional to the Boltzmann factor,
f e
  T
Since probability and number density are proportional to each other in a plasma
and since the energy of a particle is simply ε = qU by potential U, we may write the
electron and ion densities as
ni  Be  eU T
ne  Ae eU T
When U approaches to zero, no electric field to disturb the thermal equilibrium, so
A=B=n0. Therefore the potential equation can be determined by Poisson’s equation
 2U  
n0e
0
e
 eU T
 e eU T

8
What is Debye Length? (Cont.)
Consider the case that the potential energy of particle in the electric field is much
smaller than its kinetic energy, then, using the Taylor expansion to get
2


2
n
e
2
0
U
 U  
  0T 
We define a characteristic length (namely, Debye length)
D 
 0T
2n0e 2
the potential equation becomes
U
2
U

2
D
1   2 U 
U
r
U



r 2 r  r 
2D
We can solve this equation as
 qe  r D
U (r ) 
4 0 r
Debye potential
(Yukawa potential)
9
Physical meaning of Debye length
The electric field tends to 0 much faster, or in other
words, the electric field from the test charge is
effectively shielded at distances larger than the
Debye length.
U→0
λD
It is the “screening” distance, or the distance over which the usual Coulomb 1/r
field is killed off exponentially by the polarization of the plasma. This is the most
important length in plasma physics.
If you have a gas with equal numbers of charged particles in which this length is
larger than the size of the gas, you don’t have to do plasma physics. But if the
Debye length is smaller than the size of the gas, then you have to consider the fact
that electric fields applied to such plasmas don’t penetrate into them any deeper
than a few Debye lengths.
10
Debye length in different plasmas
(A Otto)
11
Plasma oscillation & plasma frequency
The Debye length governs plasma behavior in equilibrium, but dynamics depends on
another fundamental parameter called the plasma frequency.
Simple model to understand plasma oscillation
+
+
+
+
+
+
+
+
+
ions
electrons n0
-
d
Consider an infinite slab of electrons and ion with a
width of d (in x ) and particle density of n0. Assume
that the electrons are displaced by a small distance δx
in the x direction. This creates two regions of nonzero
charge density. The electric field is produced as
U ene
E 
x
d
0
Homework: problem 1
derive this expression of electric field.
12
Plasma oscillation & plasma frequency (cont.)
From the equation of particle motion
ne e 2x
d 2x
eE
2







pex
2
dt
me
 0 me
+
+
+
+
+
+
+
+
+
ions
electrons n0
-
plasma oscillation frequency:
2
n
e
e
 2pe 
 0 me
Electron oscillation equation:
d
You may also find this relation:
x  x0 cos( pe t   )
 D pe 
 0T
2n0e
2
ne e 2

 0 me
T
2 me
  th
13
Physical meaning of plasma frequency
Assuming the plasma is perturbed in some local place, how long time will the
plasma respond to it? In other words, if the plasma may locally deviate from the
quasi-neutrality due to some reason, how long time can it recover the charge
neutrality? This is about the response time.
From the oscillation equation of electron, the energy of oscillating electron is
2
about me ( x0 pe ) . If this energy is coming from the thermal energy,
2
me ( x0 pe )2
2

T
2
The amplitude of electron oscillation is approximately about the Debye length x0 ~  D .
If the response time of plasma to the perturbation is defined as the time that a
thermal electron needs to travel the distance of Debye length,
D
1
tD 

 th  pe
Then, inverse plasma frequency corresponds to the plasma response time to local
perturbation.
14
Parameters describing a plasma
Two important parameters: λD, and ωp, describe medium-like properties of plasma
due to static and dynamic consequences of long-range Coulomb force. On the other
hand, plasma consists of a large number of discrete particles. Hence, the interplay
between medium-like character and individual particle-like behavior is one of the
most interesting aspects of plasma physics.
D 
 0T
2n0e 2
 pe
ne e 2

 0 me
1
( D ; p )  f ( m; e; ;T )
n
Discreteness parameters for per particle:
mass(m); electric charge(e); average volume occupied(1/n); average kinetic
energy(κT)
Fluid-like parameters:
mass density (nm); charge density (ne); kinetic energy density (nT)
15
How to understand the fluid limit of plasma?
Imaging a process to cut each particle into finer and finer pieces, the discreteness
parameters all approach zero, but keep the fluidlike parameters ( nm; ne; nT) finite,
regarding the discreteness parameters as infinitesimal quantities of the same order.
This procedure is called fluid limit.
16
Plasma parameter
Discreteness parameters (m,e,1/n,T) are very useful in plasma kinetic theory. They
have finite physical dimensions. However, it is practical to conveniently use
dimensionless parameter to treat with plasma.
To construct a dimensionless parameter by using four discreteness parameters,
write an equation
[m x (1 / n) y T z e]  1
Notice: T here actually means κT, κ is Boltzmann constant.
Dimensional analysis:
The dimensions of a physical quantity are associated with mass; length and time,
represented by symbols m, l, and t , each raised to rational powers.
17
Plasma parameter (cont.)
Since the electric charge e has dimensions of [mass]1/2[length]3/2[time]-1, and κT has
dimensions [mass]1[length]2[time]-2 , i.e.,
T  m 1 l 2 t 2
e  m 1 / 2 l 3 / 2 t 1
1
1
x  0; y   ; z  
6
2
For the defined dimensionless parameter, we have
Defining a parameter with the same order of the discreteness parameters,
x
y
z
1/ 6
3
[n T
[m (1 / n) T e]
Plasma parameter
g
1 / 2
1
e] 
8 3 / 2 n3D
3
1
n3D
This parameter is also defined as the ratio of average (Coulomb) potential energy
and the kinetic energy of particle,
e 2 / 2 D
1
 3
3T / 2
n D
18
Physical meaning of plasma parameter
g
1
n3D
Density n
λD
Particle number in a Debye sphere
N  n3D
It implies that the number of particles in a Debye sphere N=4πnλD3/3 is much
larger than unity. This is consistent with the shielding. A considerable shielding of
individual charges can occur only on the Debye length if there are sufficient
charges in the Debye sphere of each individual particle.
For a plasma
g
1
n
3
D
 1
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Collision frequency
-- Role of binary Coulomb collision in plasma
Coulomb collision frequency for momentum exchange
4ne 4
 m ( )  nQm  2 3 ln 
m
See: Ichimaru textbook
Qm is Coulomb collision section
For the particles with Maxwellian distribution, the average value of υ3 is,
1
  3  8 
 
Hence
 
m   
2
3/ 2
1/ 2
T 
 
m
3/ 2
ne 4
ln 
1/ 2 3 / 2
m T
20
Collision frequency
-- Role of binary Coulomb collision in plasma ( cont.)
For the discrete parameters (m; 1/n; T; e) with the same order, νm is a quantity of
the first order. Plasma frequency is of the zeroth order of the discreteness
parameters. Hence, their ratio is
m
1

ln   1
 p 32 2 n3D
In a plasma, binary collisions are less important than collective plasma effects!!
Homework: problem 2
Calculate the average value lm of the Coulomb free path
(nQm)-1 and show that λD/lm is of the same order in the plasma
parameter as νm/ωp.
21
Collective vs particle behavior of density fluctuations
(S Ichimaru, Basic principles of plasma physics, Chpt.1.4)
In order to understand the essential features of the collective and individualparticle behaviors in a plasma, it is instructive to investigate the equation of motion
for the density fluctuations of an electron gas.
Assuming the point particles are treated, the density field of electrons is expressed
n
as
 (r )   (r  r )

i
i 1
The Fourier transformation of the density fluctuation is
  
 
 k    (r ) exp( ik  r )dr  exp( ik  ri )
i
Differentiate this equation above twice with respect to time,
 2  
 
 2 k

t 2
  [( k  v )  ik  vi )]exp( ik  ri )
i
The potential produced by all charged particles is
  
4e
1
 (r )  
   4e  2 exp[ ik  ( r  rj )]
j
r  rj
k ( 0)
j
k
Here the summation does not include the component k=0 since it is cancelled by
the contribution of background positive charge, i.e.
 
 (r )  e  i  k exp( ik  r )  i  0 
k
 
k exp(ik  r )  i
k (  0)
22
So the electric field is

  
k
E ( r )     i  4e  2 exp[ ik  ( r  rj )]
k
j k
The acceleration of the ith electron is calculated from the force acting on it from all
other particles(electrons and ions)

4e 2
v  i
m
It can derive

  
k
4e 2
exp[ ik  ( ri  r j )]   i


2
k
m
j (  i )k (  0 )

 
k
 k exp( ik  ri )

2
k
k ( 0)
 
  2
 
 2k
4e 2
k q
   ( k  v ) exp( ik  ri ) 
k  q q

2
t 2
m
q
i
q(  0)
The first term represents the influence of the translational motion of individual
electrons; the second term comes from the mutual interaction, which can be
separated as two terms with q=k and q≠k. the term for q=k is expressed by plasma
frequency, i.e.
  2
 
 2k
4e 2
2
  p    ( k  v ) exp( ik  ri ) 
t 2
m
i
 
k q
 k q  q

2
q
q (  0, k )
From here, it can be seen that the fluctuations in electron density oscillate at a
plasma frequency if the last two terms can be ignored.
23
Now it will be analyzed that in what condition the last two terms may be less
important.
Assuming the velocity distribution is a Maxwellian, average the first term to get
  2
 
 
  2
2 T
(
k

v
)
exp(

i
k

r
)

exp(

i
k

r
)
(
k

v
)
f
(

)
d


k
k
i
i
i
i 
M
m
The second term is nonlinear term which involves a product of two density
fluctuations. It is expected this term may be negligible in the first approximation for
small perturbation.
It can be seen that if the first term is small, the plasma (charged gas) is
characterized by collective oscillation, i.e.
k 2  k D2
with
k D2  4ne 2 T
Debye wave number
Therefore, whether a plasma behaves collectively or like an assembly of individual
particles depends on the wavelengths of the fluctuations.
L   D
24