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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 ……

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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)

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In nature: Sun & solar corrona & solar wind in space; Aurora & lighting on earth

Where is a plasma?

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 )

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Lightning and discharge physics

Magnetic reconnectionAnd solar physics

Vortex structures in non-neutral plasma

Aurora physics

Where is a plasma?

Plasma in naturecorona

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Where is a plasma?

Plasma in man-made devices

Plasma TV

Fluorescent & neon lampsPlasma lamp

Magnetic fusion

Inertial fusion

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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

Consider a positive point charge +q at the origin, T i=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.

+q

+e+q-e

Binary Coulomb interaction

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Basic properties of plasma – Debye shielding　 (cont.)

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.

+q+q

+q

-e

-e

In plasma

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Coulomb potential of a test charge +q at origin is:

r

q

04

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,

Tef 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

What is Debye Length?

TeUe Aen TeU

i Ben 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

TeUTeU eeen

U

0

02

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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

UT

enU

0

202 2

20

0

2 en

TD

We define a characteristic length (namely, Debye length)

the potential equation becomes

22

D

UU

22

2

1

D

UU

r

Ur

rr

r

qerU

Dr

04)(

We can solve this equation as

Debye potential(Yukawa potential)

What is Debye Length? (Cont.)

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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.

Physical meaning of Debye length

λD

U→0The 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.

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(A Otto)

Debye length in different plasmas

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The Debye length governs plasma behavior in equilibrium, but dynamics depends on another fundamental parameter called the plasma frequency.

Plasma oscillation & plasma frequency

Simple model to understand plasma oscillation

xen

d

UE e

0

+++++++++

---------

ions

Consider an infinite slab of electrons and ion with a width of d (in x ) and particle density of n0. Assumethat 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

d

electrons

n0

Homework: problem 1 derive this expression of electric field.

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Plasma oscillation & plasma frequency (cont.)

e

epe

m

en

0

22

plasma oscillation frequency:

xm

xen

m

eE

dt

xdpe

e

e

e

2

0

2

2

2

Electron oscillation equation:

)cos(0 txx pe

You may also find this relation: thee

epeD m

T

m

en

en

T

22 0

2

20

0

+++++++++

---------

ions

d

electrons

n0

From the equation of particle motion

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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 about . If this energy is coming from the thermal energy, 2

)( 20 pee xm

22

)( 20 Txm pee

Dx ~0The amplitude of electron oscillation is approximately about the Debye length . 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,

peth

DDt

1

Then, inverse plasma frequency corresponds to the plasma response time to local perturbation.

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Parameters describing a plasma

Two important parameters: λD, and ω p, 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.

20

0

2 en

TD

e

epe m

en

0

2

);

1;;();( Tn

emfpD

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)

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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.

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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

1])/1([ eTnm zyx

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.

Notice: T here actually means κ T, κ is Boltzmann constant.

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Plasma parameter (cont.)

3

1

Dng

32/332/16/1

8

1][

DneTn

3

2 1

2/3

2/

D

D

nT

e

Since the electric charge e has dimensions of [mass]1/2[length]3/2[time]-1, and κ T has dimensions [mass]1[length]2[time]-2 , i.e.,

Defining a parameter with the same order of the discreteness parameters,

This parameter is also defined as the ratio of average (Coulomb) potential energy and the kinetic energy of particle,

2

1;

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1;0 zyx

12/32/1 tlme

For the defined dimensionless parameter, we have

3])/1([ eTnm zyx

Plasma parameter

221 tlmT

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Physical meaning of plasma parameter3

1

Dng

3DnN

It implies that the number of particles in a Debye sphere N=4π nλD3/3 is much larger t

han 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.

λD

Density n

Particle number in a Debye sphere

For a plasma 11

3 Dn

g

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Collision frequency -- Role of binary Coulomb collision in plasma

ln4

)( 32

4

m

nenQmm

2/32/1

3 18

m

T

ln

2 2/32/1

42/3

Tm

nem

Qm is Coulomb collision section

Coulomb collision frequency for momentum exchange

For the particles with Maxwellian distribution, the average value of υ3 is,

Hence

See: Ichimaru textbook

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Collision frequency -- Role of binary Coulomb collision in plasma (cont.)

For the discrete parameters (m; 1/n; T; e) with the same order, ν m is a quantity of the first order. Plasma frequency is of the zeroth order of the discreteness parameters. Hence, their ratio is

1ln232

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Dp

m

n

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 ν m/ω p.

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n

iirrr

1

)()(

i

ik rkirdrkir )exp()exp()(

iii

k rkivkivkt

)exp()])[( 22

2

)0(

2 )](exp[1

44

)(k j

jj j

rrkik

err

er

ik

kik

kie rkirkir )0(

0 )exp()exp()(

Collective vs particle behavior of density fluctuations(S Ichimaru, Basic principles of plasma physics, Chpt.1.4)

The Fourier transformation of the density fluctuation is

Differentiate this equation above twice with respect to time,

The potential produced by all charged particles is

Here the summation does not include the component k=0 since it is cancelled by the contribution of background positive charge, i.e.

In order to understand the essential features of the collective and individual-particle 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 as

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k j

jrrkik

keirE )](exp[4)( 2

)0(

2

2

)( )0(2

2

)exp(4

)](exp[4

kik

ij kji rki

k

k

m

eirrki

k

k

m

eiv

)0(2

22

2

2 4)exp()(

qqqk

ii

k

q

qk

m

erkivk

t

),0(2

222

2

2 4)exp()(

kqqqk

iip

k

q

qk

m

erkivk

t

So the electric field is

The acceleration of the ith electron is calculated from the force acting on it from all other particles(electrons and ions)

It can derive

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.

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.

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ki

Mii

i m

Tkdfvkrkirkivk 222 )()()exp()exp()(

22Dkk TnekD

22 4

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

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.

with

Therefore, whether a plasma behaves collectively or like an assembly of individual particles depends on the wavelengths of the fluctuations.

Debye wave number

DL