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CHAPTER # 1 INTRODUCTION TO PLASMA Plasma is one of the four fundamental states of matter. OCCURRENCE OF PLASMAS IN NATURE: It has often been said that 99% of the matter in the universe is in the plasma state; that is, in the form of an electrified gas with the atoms dissociated into positive ions and negative electrons. In our everyday lives, encounters with plasmas are limited to only few examples i.e. the flash of lightning bolt, the soft glow of the Aurora Borealis, the conducting gas inside a fluorescent tube or neon sign, and the slight amount of ionization in a rocket exhaust. It would seem that we live in the 1% of the universe in which plasma do not occur naturally. The reason for this can be seen from Sahaโ€™s equation which tells us the amount of ionization to be expected in a gas with thermal equilibrium. 3

๐‘›๐‘– ๐‘‡ โ„2 โˆ’๐‘ข๐‘–โ„ ๐พ๐‘‡ โ‰ˆ 2.4 ร— 1021 ๐‘’ ๐‘›๐‘› ๐‘›๐‘– Here ๐‘›๐‘– and ๐‘›๐‘› are respectively the density (numbers per m3) of ionized atoms and of neutral atoms. T is the gas temperature and K is the Boltzmannโ€™s constant, and ๐‘ข๐‘– is the ionization energy of the gas. As the temperature is raised, the degree of ionization remains low until ๐‘ข๐‘– is only a few times KT. The ๐‘›๐‘– /๐‘›๐‘› rises abruptly and the gas is in a plasma state. Further increase in temperature makes ๐‘›๐‘› less than ๐‘›๐‘– and the plasma eventually becomes fully ionized. Where KT is thermal energy.

CREATION OF PLASMA Plasma can be created by heating a gas or subjecting it to a strong electromagnetic field with laser or microwave generator. This increases or decreases the number of electrons creating positive or negative charged particles called ions and is accompanied by the dissociation of molecular bonds if present. DEFINITION OF PLASMA: Any ionized gas cannot be called plasma. Of course there is always small degree of ionization in any gas. A useful definition can be; A plasma is a quasineutral gas of charged and neutral particles which exhibit collective behavior. The word quasi-neutrality was first used by Irving Langmuir and Levi Tanks in 1929 (density of electrons = density of ions, is called quasi-neurality). Quasineutrality describes the apparent charge neutrality of plasma overall. Since the electrons are very mobile, plasma are excellent conductors of electricity, and any changes that develop are readily neutralized.

Course: Experimental Plasma Physics

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If ๐‘›๐‘– >> ๐‘›๐‘› , then we have a plasma state. Collective behavior means each particle in plasma interacts simultaneously with many others. These collective interactions are about ten times more important than binary collisions in ordinary gas. A macroscopic force applied to a neutral gas such as from a loud speaker generating sound waves is transmitted to the individual atoms by collisions. The situation is totally different in a plasma which has charged particles. As these charges move around, they can generate local concentration of positive or negative charge which gives rise to electric field. Motion of charge also generates current and hence magnetic field. These fields effect the motion of other charged particles far away. Also when we produce negative or positive potential then particles rush towards that it is called collective behavior. In plasma, ions are massive than electrons therefore electrons are more energetic than ions Examples of Plasma: ๏‚ท ๏‚ท ๏‚ท ๏‚ท ๏‚ท ๏‚ท ๏‚ท ๏‚ท ๏‚ท

Gases in discharge tube (fluorescent lamps and neon signs) Welding arcs Lightening Auroras The upper atmosphere (ionosphere) Stars and the sun Interstellar gas clouds The fireball made by a nuclear weapon Comet tails

CONCEPT OF TEMPERATURE A gas in thermal equilibrium has particles of all velocities and the most probable distribution of these velocities is known as Maxwellian distribution. For simplicity, consider a gas in which particles can move only in one dimension. The one dimensional Maxwellian distribution is given by ๐‘“(๐‘ข) =

1 โˆ’ ๐‘š๐‘ข2โ„ 2 ๐พ๐‘‡ ๐ด๐‘’ 1

Where ๐‘“๐‘‘๐‘ข is the number of particles per m3 with velocity between ๐‘ข & ๐‘ข + ๐‘‘๐‘ข, 2 ๐‘š๐‘ฃ 2 is the KE and K is Boltzmannโ€™s constant. ๐พ = 1.38 ร— 10โˆ’23 ๐ฝ/๐พ The density โ€˜nโ€™, or number of particles per m3 is given by โˆž

๐‘› = โˆซ ๐‘“(๐‘ข)๐‘‘๐‘ข โˆ’โˆž 1

As ๐‘“(๐‘ข) =

Hisham Shah

โˆ’ ๐‘š๐‘ข2โ„ 2 ๐พ๐‘‡ , ๐ด๐‘’

so

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3 โˆž

๐‘› = โˆซ ๐ด๐‘’

โˆ’๐‘š๐‘ข2โ„ 2๐พ๐‘‡

๐‘‘๐‘ข

โˆ’โˆž 2

Using substitution, let ๐‘ฅ = ๐‘š๐‘ข โ„2๐พ๐‘‡ 2๐พ๐‘‡

๏ƒฐ ๐‘ข=โˆš

๐‘š

1

๐‘ฅ2

2๐พ๐‘‡ 1

๏ƒฐ ๐‘‘๐‘ข = โˆš

๐‘š

1

. 2 ๐‘ฅ โˆ’2 ๐‘‘๐‘ฅ

Putting values in integral โˆž

2๐พ๐‘‡ 1 โˆ’1 ๐‘› = โˆซ ๐ด๐‘’ โˆ’๐‘ฅ โˆš . ๐‘ฅ 2 ๐‘‘๐‘ฅ ๐‘š 2 โˆ’โˆž

Taking constant terms out of integral โˆž

1 ๐ด 2๐พ๐‘‡ ๐‘›= โˆš โˆซ ๐‘’ โˆ’๐‘ฅ ๐‘ฅ โˆ’2 ๐‘‘๐‘ฅ 2 ๐‘š โˆ’โˆž

โˆž

1 ๐ด 2๐พ๐‘‡ ๐‘›= โˆš . 2 โˆซ ๐‘’ โˆ’๐‘ฅ . ๐‘ฅ โˆ’2 ๐‘‘๐‘ฅ 2 ๐‘š 0

โˆž

1 2๐พ๐‘‡ ๐‘› = ๐ดโˆš โˆซ ๐‘’ โˆ’๐‘ฅ ๐‘ฅ โˆ’2 ๐‘‘๐‘ฅ ๐‘š 0

Using Gamma function as โˆž

โˆซ ๐‘ฅ ๐‘›โˆ’1 . ๐‘’ โˆ’๐‘ฅ ๐‘‘๐‘ฅ = ฮ“(n) 0

So using this โˆž

1 2๐พ๐‘‡ ๐‘› = ๐ดโˆš โˆซ ๐‘ฅ โˆ’2 . ๐‘’ โˆ’๐‘ฅ ๐‘‘๐‘ฅ ๐‘š 0

โˆž

1 2๐พ๐‘‡ ๐‘› = ๐ดโˆš โˆซ ๐‘ฅ 2โˆ’1 . ๐‘’ โˆ’๐‘ฅ ๐‘‘๐‘ฅ ๐‘š 0

๐‘› = ๐ดโˆš

2๐พ๐‘‡ 1 ฮ“( โ„2) ๐‘š

As ฮ“(1โ„2) = โˆš๐œ‹, so

Course: Experimental Plasma Physics

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๐‘› = ๐ดโˆš

2๐พ๐‘‡ 2๐œ‹๐พ๐‘‡ โˆš๐œ‹ = ๐ด โˆš ๐‘š ๐‘š ๐‘š 2๐œ‹๐พ๐‘‡

=> ๐ด = ๐‘›โˆš Which is the relation of โ€˜Aโ€™ to the density โ€˜nโ€™.

Now we can compute the average kinetic energy of the particles in this distribution as ๐ธ๐‘Ž๐‘ฃ = As ๐‘“(๐‘ข) = ๐ด๐‘’

โˆ’๐‘š๐‘ข2โ„ 2๐พ๐‘‡ ,

โˆž 1 โˆซโˆ’โˆž 2 ๐‘š๐‘ข2 ๐‘“(๐‘ข)๐‘‘๐‘ข โˆž

โˆซโˆ’โˆž ๐‘“(๐‘ข)๐‘‘๐‘ข

so

๐ธ๐‘Ž๐‘ฃ =

โˆ’๐‘š๐‘ข2 ( ) โˆž 1 2 โˆซโˆ’โˆž 2 ๐‘š๐‘ข ๐ด๐‘’ 2๐พ๐‘‡ ๐‘‘๐‘ข โˆ’๐‘š๐‘ข2 ( ) โˆž โˆซโˆ’โˆž ๐ด๐‘’ 2๐พ๐‘‡ ๐‘‘๐‘ข

Taking constant terms out of integral 2

๐ธ๐‘Ž๐‘ฃ

โˆ’๐‘š๐‘ข โˆž 1 2 ( 2๐พ๐‘‡ ) ๐‘š๐ด ๐‘ข ๐‘’ ๐‘‘๐‘ข โˆซ โˆ’โˆž =2 โˆ’๐‘š๐‘ข2 โˆž ( 2๐พ๐‘‡ ) ๐ด โˆซโˆ’โˆž ๐‘’ ๐‘‘๐‘ข

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2 As ๐ธ = 2 ๐‘š๐‘‰๐‘กโ„Ž , also ๐ธ = ๐พ๐‘‡, so

2 2 ๐‘š๐‘‰๐‘กโ„Ž = 2๐พ๐‘‡ => ๐‘‰๐‘กโ„Ž =

โˆ’๐‘š๐‘ข2 ( ) ๐‘’ 2๐พ๐‘‡

=

2๐พ๐‘‡ 2๐พ๐‘‡ => ๐‘‰๐‘กโ„Ž = โˆš ๐‘š ๐‘š

โˆ’๐‘ข2 2๐พ๐‘‡ ๐‘’ โ„๐‘š

=

๐‘ข2 โˆ’ 2 ๐‘’ ๐‘‰๐‘กโ„Ž

Now ๐‘ข2

๐ธ๐‘Ž๐‘ฃ

๐‘ข2

โˆ’ 2 โˆž 1 2 ๐‘‰๐‘กโ„Ž ๐‘‘๐‘ข ๐‘š ๐‘ข ๐‘’ โˆซ โˆ’โˆž =2 ๐‘ข2 โˆž โˆ’๐‘‰ 2 โˆซโˆ’โˆž ๐‘’ ๐‘กโ„Ž ๐‘‘๐‘ข

๐‘ข

Let ๐‘‰ 2 = ๐‘ฆ 2 => ๐‘ฆ = ๐‘‰ , differentiation gives ๐‘‰๐‘กโ„Ž ๐‘‘๐‘ฆ = ๐‘‘๐‘ข ๐‘กโ„Ž

๐‘กโ„Ž

๐ธ๐‘Ž๐‘ฃ

2 โˆž 1 2 ๐‘š โˆซโˆ’โˆž ๐‘ฆ 2 ๐‘‰๐‘กโ„Ž . ๐‘’ โˆ’๐‘ฆ . ๐‘‰๐‘กโ„Ž ๐‘‘๐‘ฆ 2 = โˆž 2 โˆซโˆ’โˆž ๐‘’ โˆ’๐‘ฆ . ๐‘‰๐‘กโ„Ž ๐‘‘๐‘ฆ

As Vth is constant with respect to integral so taking it out.

Hisham Shah

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5

๐ธ๐‘Ž๐‘ฃ

2 2 โˆž โˆž 1 1 2 2 ๐‘š๐‘‰๐‘กโ„Ž . ๐‘‰๐‘กโ„Ž โˆซโˆ’โˆž ๐‘ฆ 2 . ๐‘’ โˆ’๐‘ฆ ๐‘‘๐‘ฆ 2 ๐‘š๐‘‰๐‘กโ„Ž . 2 โˆซ0 ๐‘ฆ 2 . ๐‘’ โˆ’๐‘ฆ ๐‘‘๐‘ฆ 2 = = โˆž โˆž 2 2 ๐‘‰๐‘กโ„Ž โˆซโˆ’โˆž ๐‘’ โˆ’๐‘ฆ ๐‘‘๐‘ฆ 2 โˆซ0 ๐‘’ โˆ’๐‘ฆ ๐‘‘๐‘ฆ

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Let ๐‘ฆ 2 = ๐‘ฅ => ๐‘ฆ = โˆš๐‘ฅ, differentiation gives ๐‘‘๐‘ฆ = 2 ๐‘ฅ

๏ƒฐ

โˆ’1โ„ 2 ๐‘‘๐‘ฅ.

So

1 2 โˆž โˆ’๐‘ฅ 1 โˆ’1โ„2 ๐‘š๐‘‰ ๐‘ฅ. ๐‘’ ๐‘‘๐‘ฅ โˆซ ๐‘กโ„Ž 0 2๐‘ฅ ๐ธ๐‘Ž๐‘ฃ = 2 โˆž 1 โˆ’1 โˆซ0 ๐‘’ โˆ’๐‘ฅ 2 ๐‘ฅ โ„2 ๐‘‘๐‘ฅ 1 2 1 โˆž 1โ„2 โˆ’๐‘ฅ ๐‘š๐‘‰ . โˆซ0 ๐‘ฅ .๐‘’ ๐‘‘๐‘ฅ ๐‘กโ„Ž ๐ธ๐‘Ž๐‘ฃ = 2 1 โˆž2 โˆ’1 โˆซ ๐‘ฅ โ„2 .๐‘’ โˆ’๐‘ฅ ๐‘‘๐‘ฅ 2 0

๏ƒฐ ๐ธ๐‘Ž๐‘ฃ =

1 2 โˆž 1โ„2 โˆ’๐‘ฅ ๐‘š๐‘‰๐‘กโ„Ž โˆซ0 ๐‘ฅ .๐‘’ ๐‘‘๐‘ฅ 2 โˆž โˆ’1 โˆซ0 ๐‘ฅ โ„2 .๐‘’ โˆ’๐‘ฅ ๐‘‘๐‘ฅ

Using gamma function as โˆž

โˆซ ๐‘ฅ ๐‘›โˆ’1 . ๐‘’ โˆ’๐‘ฅ ๐‘‘๐‘ฅ = ฮ“(n) 0

In numerator: ๐‘› โˆ’ 1 = 1โ„2 => ๐‘› = 1โ„2 + 1 = 3โ„2 Similarly for denominator: ๐‘› โˆ’ 1 = โˆ’1โ„2 => ๐‘› = โˆ’1โ„2 + 1 = 1โ„2 ๐ธ๐‘Ž๐‘ฃ

1 2 โˆž 3โ„2โˆ’1 โˆ’๐‘ฅ ๐‘š๐‘‰๐‘กโ„Ž . ๐‘’ ๐‘‘๐‘ฅ โˆซ0 ๐‘ฅ 2 = โˆž 1 โˆซ0 ๐‘ฅ โ„2โˆ’1 . ๐‘’ โˆ’๐‘ฅ ๐‘‘๐‘ฅ

๏ƒฐ ๐ธ๐‘Ž๐‘ฃ =

1 2 ๐‘š๐‘‰๐‘กโ„Ž ฮ“(3/2) 2

ฮ“(1/2)

As ฮ“(๐‘› + 1) = ๐‘›ฮ“(๐‘›) 3

1

In numerator: 2 = 2 + 1 3 1 1 ฮ“ ( ) = ฮ“ ( + 1) = ฮ“(1โ„2) 2 2 2 ๐ธ๐‘Ž๐‘ฃ

1 1 2 1 ๐‘š๐‘‰ . ฮ“( ๐‘กโ„Ž 2 2) =2 1 ฮ“(2)

1

Now ฮ“(2) will be cancelled out, so ๐ธ๐‘Ž๐‘ฃ = 2 As ๐‘‰๐‘กโ„Ž =

2๐พ๐‘‡ ๐‘š

1 1 1 2 2 ๐‘š๐‘‰๐‘กโ„Ž . = ๐‘š๐‘‰๐‘กโ„Ž 2 2 4

, so

Course: Experimental Plasma Physics

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๐ธ๐‘Ž๐‘ฃ =

1 2๐พ๐‘‡ 1 ๐‘š = ๐พ๐‘‡ 4 ๐‘š 2

For three dimensional case, ๐ธ๐‘Ž๐‘ฃ =

3 ๐พ๐‘‡ 2

1

The general result is that ๐ธ๐‘Ž๐‘ฃ equals 2๐พ๐‘‡ per degree of freedom. Since โ€˜Tโ€™ and โ€˜๐ธ๐‘Ž๐‘ฃ โ€™ are so closely related, it is customary in plasma physics to give temperature in units of energy. For ๐พ๐‘‡ = 1 ๐‘’๐‘‰ = 1.6 ร— 10โˆ’19 ๐ฝ, we have ๐พ๐‘‡ = 1.6 ร— 10โˆ’19 ๐ฝ ๐‘‡=

1.6 ร— 10โˆ’19 ๐ฝ 1.6 = ร— 104 = 11,600 ๐พ โˆ’23 1.38 ร— 10 ๐ฝ/๐พ 1.38

Thus the conversion factor is 1 ๐‘’๐‘‰ = 11,600 ๐พ Now by a 2 eV plasma, we mean that KT = 2 eV , or ๐ธ๐‘Ž๐‘ฃ = 3 ๐‘’๐‘‰ in 3D

DEBYE SHIELDING The term Debye shielding and polarization of plasma are interchangeable A fundamental characteristic of the behavior of a plasma is its ability to shield out electric potentials that are applied to it. Suppose we tried to put an electric field inside a plasma by inserting two charged balls connected to a battery. The balls would attract particles of the opposite charges, and almost immediately a cloud of ions would surround the negative ball and cloud of electrons would surround the positive ball. In case of cold plasma, there would be as many charges in the cloud as in the ball, the shielding would be perfect and no electric field will be present in the body of the plasma outside the clouds. If the temperature is finite, then particles at edge of cloud where electric field is weak have enough thermal energy to escape from the electrostatic potential well and shielding is not complete. Potentials of the order of ๐พ๐‘‡/๐‘’ can leak into plasma and cause finite electric fields to exist there.

Hisham Shah

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7

When Temperature is Finite Consider a plasma of a finite temperature which contains large number of electrons and ions. Assume that the densities of both electrons and ions are initially equal but ions and electrons are not in thermal equilibrium with each other and having different temperature Ti & Te. Since ions and electrons have random thermal motion, thermally induced perturbations about the equilibrium will cause small, transient variation of electrostatic potential ฯ•. From Poisson equation โˆ‡2 ๐œ™ = โˆ’

๐œŒ => ๐œŒ = โˆ’๐œ€๐‘œ โˆ‡2 ๐œ™ ๐œ€๐‘œ

As ions and electrons density is ๐œŒ = โˆ’๐‘ž(๐‘›๐‘– โˆ’ ๐‘›๐‘’ ) ๐œŒ = +๐‘ž(๐‘›๐‘– โˆ’ ๐‘›๐‘’ ) โˆ’๐œ€๐‘œ โˆ‡2 ๐œ™ = +๐‘ž(๐‘›๐‘– โˆ’ ๐‘›๐‘’ ) ๐œ€๐‘œ โˆ‡2 ๐œ™ = โˆ’๐‘’(๐‘›๐‘– โˆ’ ๐‘›๐‘’ ) For ๐‘›๐‘’ =? Consider pressure of gas ๐‘ƒ = ๐‘›๐‘’ ๐พ๐‘‡ As total force is ๐น = ๐น๐‘š โˆ’ ๐น๐‘ As ๐น๐‘š = ๐‘ž๐ธ and ๐น๐‘ = โˆ†๐‘ƒ/๐‘›๐‘’ ๐น = ๐‘ž๐ธ โˆ’ โˆ†๐‘ƒ/๐‘›๐‘’ If ๐น = 0, then ๐‘ž๐ธ โˆ’

โˆ†๐‘ƒ =0 ๐‘›๐‘’

As ๐‘ƒ = ๐‘›๐‘’ ๐พ๐‘‡, so ๐‘ž๐ธ โˆ’

โˆ†(๐‘›๐‘’ ๐พ๐‘‡) =0 ๐‘›๐‘’

๐‘ž๐ธ โˆ’

๐พ๐‘‡โˆ†(๐‘›๐‘’ ) =0 ๐‘›๐‘’

=> ๐‘›๐‘’ ๐‘ž๐ธ = ๐พ๐‘‡โˆ†๐‘›๐‘’ Or

Course: Experimental Plasma Physics

8

1 ๐‘ž๐ธ โˆ†๐‘›๐‘’ = ๐‘›๐‘’ ๐พ๐‘‡ Integrating both sides ๐‘›๐‘’

โˆซ ๐‘›โˆž

1 ๐‘ž๐ธ โˆ†๐‘›๐‘’ = โˆซ ๐‘‘๐‘› ๐‘›๐‘’ ๐พ๐‘‡

๐‘ž๐ธ ๐‘›๐‘’ ln |๐‘›๐‘’ | |๐‘› = ๐‘› โˆž ๐พ๐‘‡ If ๐‘› = 1, then ๐‘™๐‘› (

๐‘›๐‘’ ๐‘ž๐ธ )= ๐‘›โˆž ๐พ๐‘‡

๐‘ž๐ธ ๐‘›๐‘’ = ๐‘’ ๐พ๐‘‡ ๐‘›โˆž ๐‘ž๐ธ

๐‘›๐‘’ = ๐‘›โˆž ๐‘’ ๐พ๐‘‡ For electron ๐‘ž = โˆ’๐‘’, and ๐ธ = โˆ’โˆ‡๐œ™ ๐‘’โˆ‡๐œ™

๐‘›๐‘’ = ๐‘›โˆž ๐‘’ ๐พ๐‘‡ ๐‘Ž๐‘›๐‘‘ ๐‘›๐‘– = ๐‘›โˆž As ๐œ€๐‘œ โˆ‡2 ๐œ™ = โˆ’๐‘’(๐‘›๐‘– โˆ’ ๐‘›๐‘’ ) Putting values ๐‘’โˆ‡๐œ™

๐œ€๐‘œ โˆ‡2 ๐œ™ = โˆ’๐‘’(๐‘›โˆž โˆ’ ๐‘›โˆž ๐‘’ ๐พ๐‘‡ ) ๐‘’โˆ‡๐œ™

๐œ€๐‘œ โˆ‡2 ๐œ™ = ๐‘›โˆž ๐‘’(๐‘’ ๐พ๐‘‡ โˆ’ 1) Using ๐‘’ ๐‘ฅ = 1 + ๐‘ฅ +

๐‘ฅ2 2!

+โ‹ฏ ๐œ€๐‘œ

๐‘‘2๐œ™ ๐‘’๐œ™ ๐‘’ 2 ๐œ™ 2 = ๐‘› ๐‘’(1 + + + โ‹ฏ) โˆž ๐‘‘๐‘ฅ 2 ๐พ๐‘‡ ๐พ 2 ๐‘‡ 2

๐‘’ 2 ๐œ™2

๐‘’๐œ™

If ๐พ๐‘‡ โ‰ช 1, then ๐พ2 ๐‘‡ 2 โ‰ˆ 0, so neglecting higher powers. ๐‘‘2๐œ™ ๐‘’๐œ™ ๐‘›โˆž ๐‘’ 2 ๐œ€๐‘œ 2 = ๐‘›โˆž ๐‘’ ( ) = ๐œ™ ๐‘‘๐‘ฅ ๐พ๐‘‡ ๐พ๐‘‡ ๐‘‘ 2 ๐œ™ ๐‘›โˆž ๐‘’ 2 = ๐œ™ ๐‘‘๐‘ฅ 2 ๐œ€๐‘œ ๐พ๐‘‡ ๐‘‘ 2 ๐œ™ ๐‘›โˆž ๐‘’ 2 โˆ’ ๐œ™=0 ๐‘‘๐‘ฅ 2 ๐œ€๐‘œ ๐พ๐‘‡ As ๐œ†2 =

๐œ€๐‘œ ๐พ๐‘‡ ๐‘›โˆž ๐‘’ 2

Hisham Shah

, so

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๐ท2 ๐œ™ โˆ’ (๐ท2 โˆ’

1 ๐œ™=0 ๐œ†2

1 )๐œ™ = 0 ๐œ†2

๐ท=ยฑ

1 ๐œ† โˆ’|๐‘ฅ|

General solution to this equation will be ๐œ™ = ๐œ™๐‘œ ๐‘’ ๐œ†๐ท DEBYE LENGTH ๐œ€ ๐พ๐‘‡

๐‘œ As ๐œ†๐ท = โˆš ๐‘›๐‘’ 2

๏ƒฐ ๐œ†๐ท = โˆš

๐œ€๐‘œ ๐พ ๐‘’2

.โˆš

๐‘‡ ๐‘›

Putting values of ๐œ€๐‘œ , ๐พ and ๐‘’ 2 , we get 1

๐‘‡ 2 ๐œ†๐ท = 69 ( ) ๐‘š; T in K ๐‘› 1

๐พ๐‘‡ 2 ๐œ†๐ท = 7430 ( ) ๐‘š; KT in eV ๐‘› Now we can define quasineutrality. If the dimensions โ€˜Lโ€™ of a system are much larger than โ€˜๐œ†๐ท โ€™, then whenever local concentration of charge arise or external potential are introduced these are shielded out in a distance short compared with โ€˜Lโ€™. The plasma is quasineutral that is neutral enough so that one can take ๐‘›๐‘– โ‰ˆ ๐‘›๐‘’ โ‰ˆ ๐‘›, where โ€˜nโ€™ is common density called plasma density, but not so neutral that all interesting electromagnetic forces vanish. A criterion for an ionized gas to be plasma is that it should be dense enough that โ€˜๐œ†๐ท โ€™ is much smaller than โ€˜Lโ€™. ๐œ†๐ท โ‰ช ๐ฟ If there are only one or two particles in the sheath region, Debye shielding would not be a statistically valid concept. We can compute the number of particles in Debye sphere as 4 ๐‘๐ท = ๐‘› ๐œ‹๐œ†3๐ท = 1.38 ร— 106 ๐‘‡ 3โ„2 โ„๐‘›1โ„2 3 In addition to ๐œ†๐ท โ‰ช ๐ฟ, collective behavior requires ๐‘๐ท โ‰ซ 1 The three conditions a plasma must satisfy are therefore 1. ๐œ†๐ท โ‰ช ๐ฟ 2. ๐‘๐ท โ‰ซ 1 3. ๐œ”๐œ > 1

Course: Experimental Plasma Physics

10

CHAPTER # 2 COLD PLASMA GENERATION Electric discharge is the basic method used for the generation, excitation and sustaining the cold plasmas. If the voltage is sufficiently high, electric breakdown occurs in the gas and an ionized state is formed. The methods including direct current (DC) discharges, radio frequency (RF) discharges, microwave (MW) discharges, Electron Cyclotron Resonance (ECR) discharges are most common methods for the generation, excitation and sustaining the cold plasma.

DC GLOW DISCHARGES A simple glow discharge is produced by applying direct current between two conducting electrodes inserted into a gas chamber at low pressure while a high impedance power supply provides electric field between two electrodes as shown in figure.

The DC glow discharges are easily realized in the pressure range of 10-2-10 mbar. The distance โ€˜dโ€™ between electrodes in the pressure โ€˜Pโ€™ in the discharge tube is found to satisfy relation ๐‘‰๐‘ =

๐ถ1 (๐‘ƒ๐‘‘) ๐ถ2 + ๐‘™๐‘›(๐‘๐‘‘)

Which is called Paschenโ€™s law, โ€˜C1โ€™ & โ€˜C2โ€™ are constants which depend upon nature of gas, โ€˜Vbโ€™ is breakdown voltage, or the minimum threshold voltage required to produce the glow discharge. It is evident from Paschenโ€™s law that no discharge is easily realized if the pressure is neither too low nor much high and similarly the distance between electrodes is neither too large nor too small. When voltage across the electrodes is small, few charge carriers are just collected and the current flowing within discharge tube is very small. With the increase of voltage the available charge carriers gain energy from the electric field between two electrodes and current starts to increase exponentially.

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Initially due to low energy of electrons, collisions is elastic one, meanwhile the electrons continue to gain energy between collisions until they attain sufficient energy to cause ionization of targets through inelastic collision. The newly formed electron ion pairs accelerated towards the anode and cathode cause electrons emission. The current in discharge tube increases exponentially.

RADIO FREQUENCY DISCHARGE A discharge generated by DC power source exhibit some serious disadvantages. For example ions and electrons gain energy and strike with the electrodes causing significant damage and sputtering of materials. The sputtered material is added into the plasma as impurity that significantly changes the characteristic of plasma. Further, discharges in some gases may result in depositing films of some insulating compounds. Therefore, the DC discharge is extinguished when surface of electrodes exposed to plasma becomes insulating. To overcome this problem, one may apply alternating voltage of high frequency across the electrodes. If an ion is moving towards momentarily cathode, the polarity of the applied electric field is reversed as it approaches to the electrode. Therefore the charged particles keep oscillating between the electrodes without colliding and sputtering of electrodes materials. The frequencies used in the high frequency discharges are in the range of radio transmission. That is why it is called RF discharges. ๐œ”

The RF discharges may be operated in the frequency range of ๐‘“ = 2๐œ‹ โ‰ˆ 1 โˆ’ 100 ๐‘€๐ป๐‘ง However the high RF power from generator may cause significant noise in the radio receivers in the vicinity of laboratory. To overcome this problem, a frequency of 13.56 MHz is fixed for cold plasma generation. If one uses RF frequency other than this particular value, one must arrange appropriate shielding of the experiment.

Course: Experimental Plasma Physics

12

If โ€˜Lโ€™ represents dimensions of the experimental chamber and < ๐‘ฃ >๐‘‘๐‘– is average drift velocity of ions, the critical ion transition frequency may be defined as, ๐‘“๐‘๐‘– =

< ๐‘‰ >๐‘‘๐‘– 2๐ฟ

It is obvious that frequency of applied electric field between electrodes should be higher than the critical ion frequency, to avoid the collision of ions with electrodes. To avoid electrons striking the electrodes, the frequency of alternating electric field should be much higher. ๐‘“๐‘๐‘’ =

< ๐‘‰ >๐‘‘๐‘’ 2๐ฟ

Voltage required to initiate and maintain the RF discharge is much lower than it is required to maintain DC glow discharge. The RF plasma discharge can be created at pressure as below as 10-3 mbar because efficiency of mixing collisions is enhanced by oscillations. Consider an electron oscillating along x-axis in an AC field E of amplitude Eo and angular frequency โ€˜ฯ‰โ€™. ๐ธ = ๐ธ๐‘œ ๐‘๐‘œ๐‘ ๐œ”๐‘ก From equation of electron motion ๐‘š๐‘’ ๐‘ฅฬˆ = โˆ’๐‘’๐ธ๐‘œ ๐‘๐‘œ๐‘ ๐œ”๐‘ก ๐‘ฅฬˆ =

โˆ’๐‘’๐ธ๐‘œ ๐‘๐‘œ๐‘ ๐œ”๐‘ก ๐‘š๐‘’

๐‘ฅฬ‡ =

โˆ’๐‘’๐ธ๐‘œ ๐‘ ๐‘–๐‘›๐œ”๐‘ก ๐‘š๐‘’ ๐œ”

Integration gives

Again integrating ๐‘ฅ=

๐‘’๐ธ๐‘œ ๐‘๐‘œ๐‘ ๐œ”๐‘ก โ†’ (๐‘Ž) ๐‘š๐‘’ ๐œ” 2

As electron energy โ€œWโ€ is given by ๐‘Š=

1 ๐‘š ๐‘ฅฬ‡ 2 โ†’ (๐‘) 2 ๐‘’

Equation (a) and (b) show that field strength and frequency are important in determining electron motion and their energies.

Hisham Shah

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MICROWAVE PLASMA Plasma generation using microwave is widely employed in many applications. A characteristic feature of microwave is the wavelength which is normally comparable to the dimensions of plasma apparatus(2.45 GHz; ฮป = 12.24 cm). The frequency of 2.45 GHz is commonly used for industrial or home heating applications which make suitable power supplies readily available. MW plasma & RF plasmas are similar but differ only in the large of frequencies. In collisionless plasma, the maximum amplitude โ€œxโ€ of electron oscillating in MW frequency range is ๐‘ฅ < 10โˆ’3 ๐‘๐‘š which can be written as ๐‘ฅ=

๐‘’๐ธ ๐‘š๐‘’ ๐œ” 2

Corresponding maximum energy acquired by an electron during one cycle can be written as ๐‘Š= ๐‘ฅฬ‡ =

1 ๐‘š ๐‘ฅฬ‡ 2 2 ๐‘’

โˆ’๐‘’๐ธ๐‘œ ๐‘ ๐‘–๐‘›๐œ”๐‘ก ๐‘š๐‘’ ๐œ”

So 1 โˆ’๐‘’๐ธ๐‘œ ๐‘ ๐‘–๐‘›๐œ”๐‘ก 2 ๐‘Š = ๐‘š๐‘’ ( ) 2 ๐‘š๐‘’ ๐œ” 1

๐‘’ 2 ๐ธ๐‘œ2 sin2 ๐œ”๐‘ก

๏ƒฐ ๐‘Š = ๐‘š๐‘’ ( ๏ƒฐ๐‘Š=

2 ๐‘š๐‘’2 ๐œ”2 2 2 2 ๐‘’ ๐ธ๐‘œ sin ๐œ”๐‘ก

)

2๐‘š๐‘’ ๐œ”2

So ๐‘Š=

1 โˆ’ ๐‘๐‘œ๐‘ 2๐œ”๐‘ก ) ๐‘’ 2 ๐ธ๐‘œ2 (1 โˆ’ ๐‘๐‘œ๐‘ 2๐œ”๐‘ก) 2 = โ†’ (๐ด) 2๐‘š๐‘’ ๐œ” 2 4๐‘š๐‘’ ๐œ” 2

๐‘’ 2 ๐ธ๐‘œ2 (

๐‘๐‘œ๐‘ 2๐œƒ = ๐‘๐‘œ๐‘  2 ๐œƒ โˆ’ ๐‘ ๐‘–๐‘›2 ๐œƒ ๏ƒฐ ๐‘๐‘œ๐‘ 2๐œƒ = 1 โˆ’ 2 ๐‘ ๐‘–๐‘›2 ๐œƒ ๏ƒฐ 2 ๐‘ ๐‘–๐‘›2 ๐œƒ = 1 โˆ’ ๐‘๐‘œ๐‘ 2๐œƒ

Equation (A) predicts that the corresponding energy gain by an electron during one cycle is about 0.03 ๐‘’๐‘‰ that is too small to sustain plasma.

๏ƒฐ ๐‘ ๐‘–๐‘›2 ๐œƒ =

1โˆ’๐‘๐‘œ๐‘ 2๐œƒ 2

Therefore MW discharges are more difficult to sustain at lower pressures (< 1 torr) than DC or RF discharges. In a collisional discharge case, the power density at constant electric field is given by ๐‘ƒ๐‘ฃ =

๐‘›๐‘’ ๐‘’ 2 ๐ธ๐‘œ2 ๐‘ฃ ( 2 ) 2๐‘š๐‘’ ๐‘ฃ + ๐œ” 2

Course: Experimental Plasma Physics

14

This equation shows that Pv becomes maximum at ๐œ” = ๐œˆ. The absorption of MW power is thus a function of collision frequency โ€œvโ€ of electrons with heavy species and is dependent on the pressure of the gas used. A microwave plasma reactor consists in principle of a MW power supply, a circulator, the applicator, and the plasma load. In MW plasma magnitude of electric field can vary within reactor which has dimensions of the same order of magnitude as wavelength. In typical MW plasmas, the strength of the electric fields is about E = 30 vโ„cm

ELECTRON CYCLOTRON RESONANCE PLASMAS At low temperatures, MW discharges are difficult to sustain hence ECR technique is generally used to execute and sustain plasma. ECR plasmas are typical example of MW plasma in the presence of magnetic fields. ๐‘’๐ต

In ECR technique, the applied magnetic field Bo of such a value the EC frequency ๐‘Š๐‘๐‘’ = ๐‘š

๐‘’

resonate with the frequency of source. Where โ€˜Bโ€™ is magnetic field in Tesla. The radius โ€˜๐‘Ÿ๐‘™ โ€™ called larmourโ€™s radius is given by ๐‘Ÿ๐‘™ =

๐‘š๐‘‰โŠฅ 1 2๐‘ŠโŠฅ โˆš = ๐‘’๐ต ๐‘’๐ต ๐‘š

Where โ€œmโ€ is mass of charges particle & โ€˜๐‘‰โŠฅโ€™ is the velocity component of particle normal to magnetic field line & ๐‘ŠโŠฅ is energy component corresponding to normal component of velocity. The plasma excited in the presence of an external magnetic field which satisfies the resonance condition is called ECR plasma. In the absence of an external magnetic field, the electric field of the incident wave can penetrate the plasma if ๐œ” > ๐œ”๐‘ , where ๐œ”๐‘ is the electron plasma frequency. 1

๐‘›๐‘’ ๐‘’ 2 2 ๐œ”๐‘ = ( ) ๐‘š๐‘œ โˆˆ๐‘œ Which sets upper limit for density ๐‘›๐‘’ โ‰ค

๐œ”2 โˆˆ๐‘œ ๐‘š๐‘’ = ๐‘›๐‘ ๐‘’2

Where โ€˜๐‘›๐‘ โ€™ is called critical density

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CHAPTER # 3 PLASMA DIAGNOSTICS LANGMUIRE PROBE: AN INTRODUCTION Electric or Langmuir probes are among the most basic plasma diagnostic components. They are generally simple in construction and use, relatively inexpensive and robust enough to withstand considerable heat fluxes. Langmuir probe technique was introduced by Irving Langmuir and his colleagues in 1920s. It can provide measurement of basic cold plasma parameters such as electron temperature โ€˜๐‘‡๐‘’ โ€™, electron number density โ€˜๐‘›๐‘’ โ€™, plasma potential โ€˜๐‘‰๐‘ โ€™, and floating potential โ€˜๐‘‰๐‘“ โ€™. A Langmuir probe is a metallic insulated wire (electrode) except its tip which is exposed of plasma. Generally the tip of probe is several millimeters long and less than one mm in diameter. For cold plasma diagnostic it is placed near the center of the interelectrode gap and is constructed so that it would be oriented to be either parallel or perpendicular to the electrode plane. The probe material should have high melting point and usually tungsten or platinum are used. The material by which probe is insulated should be chemically inert at high temperature. It can either be quartz or a vacuum compatible ceramics. The current โ€˜Iโ€™ of the probe is measured as a function of applied voltage and I-V characteristic of the probe is obtained. Data obtained from I-V characteristic is used to evaluate the electron temperature and density.

A schematic diagram for a single Langmuir probe

Course: Experimental Plasma Physics

16

WORKING OF LANGMUIR PROBE The probe is inserted into the plasma reactor through an electrically insulated seal. Seal can be fixed or can permit changes in the position of the probe. As measurements are made by inserting a probe in the plasma, so this technique is called an in-situ intrusive diagnostic method that perturbs the plasma locally. Due to this local plasma perturbation, the physical conditions of the plasma are changed. The probe affects the plasma particle distribution and energy by changing the electric field. The magnitude of these disturbances depends on the dimensions of the probe and properties of plasma. Langmuir or electric probe that satisfies the condition ๐‘Ÿ > ๐œ†๐‘‘ , where โ€˜rโ€™ is probe radius and โ€˜๐œ†๐‘‘ โ€™ is called thin sheath probe. For a single Langmuir probe, the voltage โ€˜Vโ€™ is referenced to a large metal surface in the plasma chamber itself. For a double Langmuir probe, the voltage is applied between the two electrodes both insulated from reactor. If plasma is placed in magnetic field measurements are interpreted with considerable difficulty. Same is true if positive ions are present. If there is no reference electrode a single probe is useless in such cases double probe circuits are used. A double probe is composed of two identically tipped probes separated by a fixed distance. The tips of the probes have to be far enough from each other so that the plasma sheaths of probes would not overlap but closed enough to sample the same region of plasma. Both probes are insulated from ground, float with plasma and are unaffected by changes in โ€˜Vโ€™. THE SINGLE LANGMUIR PROBE CHARACTERISTICS The typical I-V characteristics of a single Langmuir probe are presented in figure. The resulting probes I-V characteristic curve regions depends on whether the electron current or ion current dominates. The I-V characteristics of a probe can be divided into three regions, the electrons saturation region, transition region and ion saturation region. Conventionally the current flowing from probe to plasma is defined positive. So flow of electrons from plasma to probe constitute positive current. When probe is inserted, electrons being more mobile move to the probe and collected by it. The probe current due to ions is almost negligible. The magnitude of electron current depends mainly on electron number density and the average energy, the temperature. This part of curve is called electron saturation region. The continuous collection of electrons by probe makes it more and more negative. Transition region represents this behavior. When probe is charged sufficiently negative that electrons arriving at probe are repelled back and their contribution in probe current becomes equal to that of ions. This is called an ion saturation region.

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The plasma potential โ€˜Vplโ€™ is defined as the potential at which electrons arriving near probes are collected and probe current equal to electronic current. The floating potential is the potential of probe at which electrons and ion densities to the probe equals ๐ฝ๐‘’ = ๐ฝ๐‘–

MEASUREMENT OF PLASMA TEMPERATURE AND DENSITY WITH LANGMUIR PROBE Electrostatic probe is a convenient and reliable tool for plasma diagnostics suitable for measuring parameters of cold plasmas i.e. electron temperature and plasma density, which are important parameters in plasma processing. To use the discharges in different applications, it is essential to have information about plasma electron temperature and density. It is because the efficiency of the processes in the plasma and their reaction rates are generally dependent on the density of the charged particles and their energies.

PLASMA ELECTRON TEMPERATURE The electron current density in the transition region of the I-V characteristic can be written as โˆ’๐‘’๐‘‰

๐ฝ = ๐ฝ๐‘œ ๐‘’ ๐พ๐‘‡๐‘’ โ†’ (1) Where โ€˜Joโ€™ is random current density in the plasma ๐พ๐‘‡๐‘’ โƒ—โƒ—โƒ—๐‘’ = ๐‘›๐‘’ ๐‘’โˆš ๐ฝ๐‘œ = ๐‘›๐‘’ ๐‘’๐‘‰ 2๐œ‹๐‘š๐‘’ โˆ’๐‘’๐‘‰

As ๐ฝ = ๐ฝ๐‘œ ๐‘’ ๐พ๐‘‡๐‘’ โˆ’๐‘’๐‘‰

๏ƒฐ ๐ด๐ฝ = ๐ด๐ฝ๐‘œ ๐‘’ ๐พ๐‘‡๐‘’

โˆ’๐‘’๐‘‰

๏ƒฐ ๐‘™๐‘›(๐ด๐ฝ) = ๐‘™๐‘› (๐ด๐ฝ๐‘œ ๐‘’ ๐พ๐‘‡๐‘’ ) Course: Experimental Plasma Physics

18 ๐ผ

since ๐ฝ = ๐ด => ๐ผ = ๐ฝ๐ด, similarly ๐ผ๐‘œ = ๐ฝ๐‘œ ๐ด, so โˆ’๐‘’๐‘‰

๐‘™๐‘› ๐ผ = ๐‘™๐‘› ๐ผ๐‘œ + ๐‘™๐‘› (๐‘’ ๐พ๐‘‡๐‘’ ) ๏ƒฐ ๐‘™๐‘› ๐ผ = ๐‘™๐‘› ๐ผ๐‘œ โˆ’

๐‘’๐‘‰ ๐พ๐‘‡๐‘’ ๐‘’๐‘‰

๏ƒฐ ๐‘™๐‘› ๐ผ โˆ’ ๐‘™๐‘› ๐ผ๐‘œ = โˆ’ ๏ƒฐ ๐‘™๐‘› ๐ผ๐‘œ โˆ’ ๐‘™๐‘› ๐ผ = ๏ƒฐ ๐‘™๐‘›

๐ผ๐‘œ ๐ผ

=

๐พ๐‘‡๐‘’ ๐‘’๐‘‰

๐พ๐‘‡๐‘’

๐‘’๐‘‰ ๐พ๐‘‡๐‘’

The total probe current โ€œIโ€ is the difference between electron and ion current. Therefore ๐ผ = ๐ผ๐‘’ โˆ’ ๐ผ๐‘–

๏ƒฐ ๐‘™๐‘›๐ผ๐‘œ โˆ’ ln ๐ผ =

๐‘’๐‘‰ ๐พ๐‘‡๐‘’

๏ƒฐ ๐‘™๐‘›๐ผ๐‘œ โˆ’ ln(๐ผ๐‘’ โˆ’ ๐ผ๐‘– ) =

๐‘’๐‘‰ ๐พ๐‘‡๐‘’

For electrons ๐‘™๐‘›๐ผ๐‘’ =

๐‘’๐‘‰ ๐พ๐‘‡๐‘’

๐‘‘ ๐‘’ ๐‘™๐‘›๐ผ๐‘’ = ๐‘‘๐‘ฃ ๐พ๐‘‡๐‘’ ๐‘‘๐‘ฃ ๐พ๐‘‡๐‘’ = ๐‘‘๐‘™๐‘›๐ผ๐‘’ ๐‘’ ๐พ๐‘‡๐‘’ โˆ†(๐‘™๐‘›๐ผ๐‘’ ) โˆ’1 =[ ] ๐‘’ โˆ†๐‘ฃ

PLASMA ELECTRON DENSITY For the determination of plasma electron density we will proceed as follows. At the plasma potential the probe current will be dominantly due to electrons. Therefore 1 ๐ผ๐‘ ๐‘’ = โˆ’๐‘’๐ด(ฮ“๐‘– โˆ’ ฮ“๐‘’) โ‰ˆ ๐‘’๐ด๐‘ ฮ“๐‘’ = ๐‘’๐ด๐‘ ๐‘›๐‘’ โƒ—โƒ—โƒ— ๐‘ฃ๐‘’ โ†’ (๐ด) 4 Where โ€˜๐ด๐‘ โ€™ is the probe area, โ€˜ฮ“๐‘–โ€™ is the flux of ions and โ€˜ฮ“๐‘’โ€™ is the flux of electrons arriving at the probe. The probe would thus emit a net positive current. From the elementary gas kinetic theory, the number of particles of a given species crossing unit area per unit time is 1 ฮ“ = ๐‘›๐‘ฃฬ… 4

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Where โ€˜๐‘ฃฬ… โ€™ is mean speed that may be found as follows โˆž

๐‘ฃฬ… = โˆซ ๐‘ฃ๐‘“๐‘š (๐‘ฃ)๐‘‘ 3 ๐‘ฃ โˆ’โˆž

Where โ€˜๐‘“๐‘š โ€™ is the Maxwellian velocity distribution function. In 3-D, it can be written as 3

2

๐‘š 2 โˆ’๐‘š๐‘ฃ ๐‘“๐‘š = ( ) ๐‘’ 2๐พ๐‘‡๐‘’ 2๐œ‹๐พ๐‘‡๐‘’

The integral above can be solved in spherical coordinates more easily. Since volume of each spherical shell is ๐‘‘ 3 ๐‘‰ = 4๐œ‹๐‘ฃ 2 ๐‘‘๐‘ฃ, so โˆž

๐‘ฃ = โˆซ ๐‘ฃ๐‘“๐‘š (๐‘ฃ)๐‘‘ 3 (๐‘ฃ) โˆ’โˆž

As ๐‘“๐‘š = (

3 2

๐‘š 2๐œ‹๐พ๐‘‡๐‘’

) ๐‘’

โˆ’๐‘š๐‘ฃ2 2๐พ๐‘‡๐‘’ ,

โˆž

and ๐‘‘3 ๐‘‰ = 4๐œ‹๐‘ฃ 2 ๐‘‘๐‘‰. 3

3

2

โˆž

2

โˆ’๐‘š๐‘ฃ ๐‘š 2 โˆ’๐‘š๐‘ฃ ๐‘š 2 ) ๐‘’ 2๐พ๐‘‡๐‘’ . 4๐œ‹๐‘ฃ 2 ๐‘‘๐‘ฃ = 4๐œ‹ ( ) โˆซ ๐‘ฃ 2 ๐‘’ 2๐พ๐‘‡๐‘’ . ๐‘ฃ๐‘‘๐‘ฃ ๐‘ฃฬ…๐‘’ = โˆซ ๐‘ฃ. ( 2๐œ‹๐พ๐‘‡๐‘’ 2๐œ‹๐พ๐‘‡๐‘’ โˆ’โˆž

Let ๐‘ฅ

=

โˆ’โˆž

๐‘š 2๐พ๐‘‡๐‘’ 3

โˆž

๐‘ฅ 2 2 ๐‘ฃฬ…๐‘’ = 4๐œ‹ ( ) โˆซ ๐‘ฃ 2 ๐‘’ โˆ’๐‘ฅ๐‘ฃ . ๐‘ฃ๐‘‘๐‘ฃ ๐œ‹ โˆ’โˆž

Let ๐‘ฅ๐‘ฃ 2 = ๐‘ฆ =>

๐‘‘๐‘ฆ ๐‘‘๐‘ฃ

= 2๐‘ฅ๐‘ฃ =>

๐‘‘๐‘ฆ 2๐‘ฅ

= ๐‘ฃ๐‘‘๐‘ฃ , so 3

โˆž

๐‘ฅ 2 ๐‘ฆ ๐‘‘๐‘ฆ ๐‘ฃฬ…๐‘’ = 4๐œ‹ ( ) โˆซ ๐‘’ โˆ’๐‘ฆ ๐œ‹ ๐‘ฅ 2๐‘ฅ โˆ’โˆž

3

๏ƒฐ ๐‘ฃฬ…๐‘’ =

โˆž ๐‘ฅ 2 1 4๐œ‹ ( ) 2 โˆซโˆ’โˆž ๐‘ฆ๐‘’ โˆ’๐‘ฆ ๐‘‘๐‘ฆ ๐œ‹ 2๐‘ฅ

๏ƒฐ ๐‘ฃฬ…๐‘’ =

โˆž ๐‘ฅ 2 1 4๐œ‹ ( ) 2 2 โˆซ0 ๐‘ฆ๐‘’ โˆ’๐‘ฆ ๐‘‘๐‘ฆ ๐œ‹ 2๐‘ฅ

๏ƒฐ ๐‘ฃฬ…๐‘’ =

๐‘ฅ 2 1 โˆž 4๐œ‹ ( ) 2 โˆซ0 ๐‘ฆ๐‘’ โˆ’๐‘ฆ ๐‘‘๐‘ฆ ๐œ‹ ๐‘ฅ

3

3

โˆž

By using Gamma function as โˆซ0 ๐‘ฆ๐‘’ โˆ’๐‘ฆ ๐‘‘๐‘ฆ = 1, so 3

1 1 ๐‘ฅ 2 1 4 ๐‘ฃฬ…๐‘’ = 4๐œ‹ ( ) 2 = 4๐œ‹ โˆ’2 ๐‘ฅ โˆ’2 = ๐œ‹ ๐‘ฅ โˆš๐‘ฅ๐œ‹

Course: Experimental Plasma Physics

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As ๐‘ฅ =

๐‘š 2๐พ๐‘‡๐‘’

, so

๐‘ฃฬ…๐‘’ =

4

2๐พ๐‘‡๐‘’ 2๐พ๐‘‡๐‘’ = 4โˆš ๐‘š ๐‘š๐œ‹ โˆš๐œ‹ โˆš

Using this result in equation (A)

1 ๐ผ๐‘ ๐‘’ = ๐‘’๐ด๐‘ ๐‘›๐‘’ โƒ—โƒ—โƒ— ๐‘ฃ๐‘’ 4 1

2๐พ๐‘‡๐‘’

4

๐‘š๐œ‹

๏ƒฐ ๐ผ๐‘ ๐‘’ = ๐‘’๐ด๐‘ ๐‘›๐‘’ (4โˆš ๏ƒฐ ๐‘›๐‘’ =

๐ผ๐‘ ๐‘’ ๐‘’๐ด๐‘

)

๐‘š๐œ‹

โˆš2๐พ๐‘‡

๐‘’

Where โ€˜Apโ€™ is probe area, e is the magnitude of electronic charge, โ€˜mโ€™ is electron mass and โ€˜Kโ€™ is Boltzmannโ€™s constant. By measuring the electron saturation current and electron temperature, the electron density may be readily evaluated.

PLASMA SPECTROSCOPY It is the study of spectra of atoms and molecules. A spectrum is a chart or a graph that shows the intensity of light being emitted over a range of energies. Spectra can be produced for any energy of light from which useful information can be collected. In the field of plasma diagnostics, plasma spectroscopy has evolved as a very informative tool to study characteristics of different species. These characteristics include temperature, particle density, and history of plasma etc.

ATOMIC ABSORPTION SPECTROSCOPY When a beam of light passes through a material then intensities of certain wavelengths are found reduced and this phenomenon is attributed to absorption. Three basic objectives to study spectroscopy of the absorption of light are ๏‚ท ๏‚ท ๏‚ท

To learn which wavelengths are absorbed How much radiation is absorbed Why a radiation is absorbed

The basic instrument used for atomic absorption spectroscopy include a light source, a monochromator to isolate the specific wavelength of light, a detector, some electrons to treat signal, and a data display.

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ATOMIC EMISSION SPECTROSCOPY Atomic emission spectroscopy is used to study the spectra emitted from the excited atoms or ions especially in optical range. In plasma emission spectroscopy, the exciting medium/agent is plasma. In emission spectroscopy, outer orbital electrons in the atoms become excited from their ground state to higher energy state. After a short lifetime, the excited electrons return to the ground state, simultaneously electromagnetic radiations are emitted. The emitted radiations are analyzed by means of monochromator/spectrograph which separates various wavelengths. PES is similar to optical emission spectroscopy in principles but differs in mechanism. In OES the excitation is done by electric arc or spark mechanism in the temperature range 3000-5000 degrees kelvin. On reaching the high temperature (7000-9000 K) plasma, all type of molecular bonds break. Free atoms or ions are produced which emit their characteristics spectra. Plasma spectroscopy techniques are prepared to collect information about plasma compared to other methods because these techniques used the photons emitted from plasma and photons emissions cause negligible perturbations. Secondly, the information is more reliable quantitatively as well as qualitatively.

OPTICAL EMISSION SPECTROSCOPY OES is the spectral analysis of light emitting from plasma in optical range. It is most widely used technique to investigate glow discharges. By measuring the wavelength and intensities of emitted spectral lines one can identify the neutral particles and ions present in plasma. Although optical methods are non-intrusive, they have been of limited utility in plasmas because of selectivity and sensitivity issues. For example: OES is limited to species that emit characteristic light which is small fraction of total number of species present in the system. The most intense radiation is emitted from the species through de-excitation from first excited state E1 to ground state E0. Since every species (atoms, molecules, ion radicals) have certain energy levels, therefore, each emits a characteristic spectral line of frequency. ๐œˆ10 =

๐ธ1 โˆ’ ๐ธ๐‘œ โ„Ž

๐œ†10 =

โ„Ž๐‘ ๐ธ1 โˆ’ ๐ธ๐‘œ

And wavelength

The major tools employed in optical emission spectroscopy includes monochromator and photomultiplier tube.

Course: Experimental Plasma Physics

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MONOCHROMATOR A monochromator can be used as a wavelength analyzer for unknown source of light and secondly as a source of known wavelength. A monochromator consists of and entrance slit, a dispersion device, a focusing system and an exit slit. A dispersing medium may be a prism or diffraction grating. The dispersive medium separates different wavelengths in different directions. Grating is used extensively as compared to prism. The wavelength of monochromator varies with the choice of grating. As grating is rotated, a different wavelength is selected to focus onto the exit slit.

The dispersion of monochromator is defined as the power to spread different wavelengths. Dispersion is of two types i.e. linear and angular dispersion. โˆ†๐œƒ

Mathematically, angular dispersion is โˆ†๐œ†, where โ€˜โˆ†๐œƒโ€™ is difference in emerging angles of two

rays having difference โ€˜โˆ†๐œ†โ€™ in wavelengths. It is expressed in degrees or radians per angstrom. โˆ†๐‘ฅ

For linear dispersion โˆ†๐œ†, where โ€˜โˆ†๐‘ฅโ€™ is distance between two spectral lines. Linear dispersion is expressed in millimeter per angstrom. PHTOTOMULTIPLIER TUBE It is a sensitive device to detect very faint light signals and convert an extremely weak light signal into a detectable electrical signal without adding a large amount of noise. It consist of a photocathode and a series of dynodes in an evacuated enclosure. The photons that strike the photo emissive cathode emit electrons due to photoelectric effect. From there, the photoelectrons are accelerated towards the nearest electrode by means of potential difference. The photoelectrons eject secondary electrons from the first anode. This anode act as a cathode for next electrode and the process continues through several stages. The cascading effect creates 105 to 107 electrons for each photon hitting the first cathode, depending upon the number of dynodes and the accelerating voltage. Finally, this amplified signal can be recorded on a computer.

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DEDUCTIONS OF PLASMA PARAMETER WITH OES OES can be used as diagnostic tool for plasma, electron plasma temperature and plasma electron density can be found with this technique. ELECTRON PLASMA TEMPERATURE A commonly employed convenient method of temperature determines its two-line-emission ratio method, which yields electronic excitation temperature โ€˜๐‘‡๐‘’๐‘ฅ โ€™ that can be equated to electron temperature โ€˜๐‘‡๐‘’ โ€™. The pair of lines which have common lower level is chosen for calculation of โ€˜๐‘‡๐‘’๐‘ฅ๐‘ โ€™. For a system in thermodynamic equilibrium at temperature โ€˜Tโ€™, the relation which holds well is called Boltzmannโ€™s equation. ๐‘๐ฝ ๐‘”๐ฝ โˆ’โˆ†๐ธโ„๐พ๐‘‡ = ๐‘’ โ†’ (๐ด) ๐‘๐พ ๐‘”๐พ Where ๐‘๐ฝ and ๐‘๐พ are population densities of upper and lower level J and K along with ๐‘”๐ฝ and ๐‘”๐พ as their statistical weights respectively. The level โ€˜rโ€™ is common lower level for the upper โ€˜Jโ€™ and โ€™Kโ€™ levels.

The spectral line intensity corresponding to a transition from an upper level โ€˜Jโ€™ to a lower level โ€˜rโ€™ is ๐ผ๐ฝ๐‘Ÿ = ๐‘๐ฝ ๐ด๐ฝ๐‘Ÿ โ„Ž๐‘ฃ๐ฝ๐‘Ÿ = ๐‘๐ฝ ๐ด๐ฝ๐‘Ÿ

โ„Ž๐‘ โ†’ (1) ๐œ†๐ฝ๐‘Ÿ

Similarly for K ๐ผ๐พ๐‘Ÿ = ๐‘๐พ ๐ด๐พ๐‘Ÿ โ„Ž๐‘ฃ๐พ๐‘Ÿ = ๐‘๐พ ๐ด๐พ๐‘Ÿ

โ„Ž๐‘ โ†’ (2) ๐œ†๐พ๐‘Ÿ

From equation (1) ๐‘๐ฝ =

๐ผ๐ฝ๐‘Ÿ ๐œ†๐ฝ๐‘Ÿ ๐ด๐ฝ๐‘Ÿ โ„Ž๐‘

๐‘๐พ =

๐ผ๐พ๐‘Ÿ ๐œ†๐พ๐‘Ÿ ๐ด๐พ๐‘Ÿ โ„Ž๐‘

Similarly from (2)

So ๐‘๐ฝ ๐ผ๐ฝ๐‘Ÿ ๐œ†๐ฝ๐‘Ÿ ๐ด๐พ๐‘Ÿ โ„Ž๐‘ ๐ผ๐ฝ๐‘Ÿ ๐œ†๐ฝ๐‘Ÿ ๐ด๐พ๐‘Ÿ = . = โ†’ (๐ต) ๐‘๐พ ๐ด๐ฝ๐‘Ÿ โ„Ž๐‘ ๐ผ๐พ๐‘Ÿ ๐œ†๐พ๐‘Ÿ ๐ผ๐พ๐‘Ÿ ๐œ†๐พ๐‘Ÿ ๐ด๐ฝ๐‘Ÿ Course: Experimental Plasma Physics

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Comparing (A) and (B) ๐‘”๐ฝ โˆ’โˆ†๐ธโ„๐พ๐‘‡ ๐ผ๐ฝ๐‘Ÿ ๐œ†๐ฝ๐‘Ÿ ๐ด๐พ๐‘Ÿ ๐‘’ = ๐‘”๐พ ๐ผ๐พ๐‘Ÿ ๐œ†๐พ๐‘Ÿ ๐ด๐ฝ๐‘Ÿ Taking natural log ๐‘™๐‘› (

๐‘”๐ฝ โˆ’โˆ†๐ธโ„๐พ๐‘‡ ๐ผ๐ฝ๐‘Ÿ ๐œ†๐ฝ๐‘Ÿ ๐ด๐พ๐‘Ÿ ๐‘’ ) = ๐‘™๐‘› ( ) ๐‘”๐พ ๐ผ๐พ๐‘Ÿ ๐œ†๐พ๐‘Ÿ ๐ด๐ฝ๐‘Ÿ

๐‘”๐ฝ (๐ธ๐ฝ โˆ’ ๐ธ๐พ ) ๐ผ๐ฝ๐‘Ÿ ๐œ†๐ฝ๐‘Ÿ ๐ด๐พ๐‘Ÿ ๐‘™๐‘› ( ) โˆ’ = ๐‘™๐‘› ( ) ๐‘”๐พ ๐พ๐‘‡ ๐ผ๐พ๐‘Ÿ ๐œ†๐พ๐‘Ÿ ๐ด๐ฝ๐‘Ÿ (๐ธ๐พ โˆ’ ๐ธ๐ฝ ) ๐ผ๐ฝ๐‘Ÿ ๐œ†๐ฝ๐‘Ÿ ๐ด๐พ๐‘Ÿ ๐‘”๐พ = ๐‘™๐‘› ( ) ๐พ๐‘‡ ๐ผ๐พ๐‘Ÿ ๐œ†๐พ๐‘Ÿ ๐ด๐ฝ๐‘Ÿ ๐‘”๐ฝ ๐พ๐‘‡ =

(๐ธ๐พ โˆ’ ๐ธ๐ฝ ) ๐ผ๐ฝ๐‘Ÿ ๐œ†๐ฝ๐‘Ÿ ๐ด๐พ๐‘Ÿ ๐‘”๐พ ๐‘™๐‘› ( ) ๐ผ๐พ๐‘Ÿ ๐œ†๐พ๐‘Ÿ ๐ด๐ฝ๐‘Ÿ ๐‘”๐ฝ

Where ๐ผ๐พ๐‘Ÿ , ๐œ†๐พ๐‘Ÿ , ๐ด๐พ๐‘Ÿ and ๐‘”๐พ are total intensity (integrated over profile), the wavelength, the transition probability and the statistical weight, respectively of like โ€˜Kโ€™ with โ€ฒ๐ธ๐พ โ€ฒ its excitation energy. The corresponding quantities for other line โ€˜Jโ€™ are ๐ผ๐ฝ๐‘Ÿ , ๐œ†๐ฝ๐‘Ÿ, ๐ด๐ฝ๐‘Ÿ and ๐‘”๐ฝ . If we take K=1 and J=2, the above equation takes the form ๐พ๐‘‡ =

(๐ธ1 โˆ’ ๐ธ2 ) ๐ด ๐‘”๐ผ๐œ† ๐‘™๐‘› ( 1 1 2 2 ) ๐ด2 ๐‘”2 ๐ผ1 ๐œ†1

ELECTRON PLASMA DENSITY The electron number density can be determined from the measurement of the relative intensities of atomic and ionic lines by using Boltzmann and Sahaโ€™s equations.

(2๐œ‹๐‘š๐พ๐‘‡)3โ„2 2๐ด+ ๐‘”+ ๐œ†0 ๐ผ 0 โˆ’(๐ธ +โˆ’๐ธ0 +๐ธ0 โˆ’โˆ†๐ธ0 )โ„๐พ๐‘‡ ๐‘’๐‘ฅ ๐‘– ๐‘– ๐‘›๐‘’ = ( )( 0 0 + + )๐‘’ 3 โ„Ž ๐ด ๐‘” ๐œ† ๐ผ Where (0,+) represent the neutral and singly ionized atom. Tex = excitation temperature E = energy of emitting level Ei0 = ionization energy of neutral atom โˆ†Ei0 = lowering of ionization energy

The great probabilist Mark Kac (1914-1984) once gave a lecture at Caltech, with Feynman in the audience. When Kac finished, Feynman stood up and loudly proclaimed, "If all mathematics disappeared, it would set physics back precisely one week." To that outrageous comment, Kac shot back with that yes, he knew of that week; it was "Precisely the week in which God created the world."

Hisham Shah

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CHAPTER # 4 PINCH EFFECT The constriction of a plasma through which a large electric current is flowing, caused by the attractive force of the currentโ€™s own magnetic field. Consider an infinite cylindrical column of conducting fluid with an axial current density J and a resulting azimuthal magnetic induction โ€˜Bโ€™. The ๐ฝ ร— ๐ต force acting on a plasma forces the column to contract radially. This radial constriction of the plasma column is known as the pinch effect. There are two types of pinch effect 1. EQUILIBRIUM PINCH When plasma is compressed radially, the plasma number density and the temperature increase. The plasma kinetic pressure counteracts to hinder the constriction of the plasma column. Whereas the magnetic force acts to confine plasma. When these counteracting forces are balanced, a steady state condition results in which the plasma is mainly confined within a certain radius โ€˜Rโ€™ which remains constant in this. This situation is called equilibrium pinch. 2. DYNAMIC PINCH When the self-magnetic pressure exceeds the plasma kinetic pressure the column radius changes with time resulting in a situation known as dynamic pinch. As charged particles are present in plasma having own magnetic field known as selfmagnetic pressure. First we investigate the equilibrium pinch and afterwards the dynamic pinch.

EQUILIBRIUM PINCH For simplicity the current density, the magnetic field and plasma magnetic pressure are assumed only on the distance from cylindrical axis. For steady state conditions, none of the variable changes with time. Since the system is cylindrically symmetric, only radial component must be considered. For equilibrium pinch kinetic and magnetic pressures are equal, then rate of change of total pressure equals to the product of current density and magnetic field. ๐‘‘๐‘ƒ(๐‘Ÿ) = โˆ’๐ฝ๐‘ง (๐‘Ÿ)๐ต๐œƒ (๐‘Ÿ) โ†’ (1) ๐‘‘๐‘Ÿ Course: Experimental Plasma Physics

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Now total enclosed current ๐ผ๐‘ง (๐‘Ÿ) is ๐‘Ÿ

๐ผ๐‘ง (๐‘Ÿ) = โˆซ ๐ฝ๐‘ง (๐‘Ÿ)2๐œ‹๐‘Ÿ๐‘‘๐‘Ÿ 0

We can write ๐‘‘๐ผ๐‘ง (๐‘Ÿ) = 2๐œ‹๐‘Ÿ๐ฝ๐‘ง (๐‘Ÿ) ๐‘‘๐‘Ÿ ๐ฝ๐‘ง (๐‘Ÿ) =

1 ๐‘‘๐ผ๐‘ง (๐‘Ÿ) . โ†’ (2) 2๐œ‹๐‘Ÿ ๐‘‘๐‘Ÿ

Ampereโ€™s law in integral form relates ๐ต๐œƒ (๐‘Ÿ) to the total enclosed current. As ๐ต๐œƒ (๐‘Ÿ) =

๐œ‡๐‘œ ๐ผ (๐‘Ÿ) โ†’ (3) 2๐œ‹๐‘Ÿ ๐‘ง

Using value of ๐ผ๐‘ง (๐‘Ÿ) we get ๐‘Ÿ

๐‘Ÿ

๐œ‡๐‘œ ๐œ‡๐‘œ ๐ต๐œƒ (๐‘Ÿ) = . โˆซ ๐ฝ๐‘ง (๐‘Ÿ). 2๐œ‹๐‘Ÿ๐‘‘๐‘Ÿ = . โˆซ ๐ฝ๐‘ง (๐‘Ÿ). ๐‘Ÿ๐‘‘๐‘Ÿ 2๐œ‹๐‘Ÿ ๐‘Ÿ 0

0

Using (2) and (3) in (1), we get ๐‘‘๐‘ƒ(๐‘Ÿ) = โˆ’๐ฝ๐‘ง (๐‘Ÿ)๐ต๐œƒ (๐‘Ÿ) ๐‘‘๐‘Ÿ

๏ƒฐ ๏ƒฐ ๏ƒฐ ๏ƒฐ

๐‘‘๐‘ƒ(๐‘Ÿ) ๐‘‘๐‘Ÿ ๐‘‘๐‘ƒ(๐‘Ÿ)

=( =

โˆ’1 ๐‘‘๐ผ๐‘ง (๐‘Ÿ)

2๐œ‹๐‘Ÿ โˆ’๐œ‡๐‘œ

4๐œ‹2 ๐‘Ÿ 2

๐‘‘๐‘Ÿ ๐‘‘๐‘ƒ(๐‘Ÿ) 4๐œ‹ 2 ๐‘Ÿ 2 ๐‘‘๐‘Ÿ ๐‘‘๐‘ƒ(๐‘Ÿ) 4๐œ‹ 2 ๐‘Ÿ 2 ๐‘‘๐‘Ÿ

.

๐‘‘๐‘Ÿ

๐ผ๐‘ง (๐‘Ÿ).

๐œ‡

๐‘œ ๐ผ๐‘ง (๐‘Ÿ)) ) (2๐œ‹๐‘Ÿ

๐‘‘๐ผ๐‘ง (๐‘Ÿ) ๐‘‘๐‘Ÿ

= โˆ’๐œ‡๐‘œ ๐ผ๐‘ง (๐‘Ÿ). =

๐‘‘๐ผ๐‘ง (๐‘Ÿ)

๐‘‘๐‘Ÿ โˆ’ ( ๐œ‡๐‘œ ๐ผ๐‘ง2 (๐‘Ÿ)) ๐‘‘๐‘Ÿ 2 ๐‘‘

1

Integrating this equation from ๐‘Ÿ = 0 to ๐‘Ÿ = ๐‘… ๐‘…

๐‘…

๐‘Ÿ 2 ๐‘‘๐‘ƒ(๐‘Ÿ) ๐‘‘ 1 2 4๐œ‹ โˆซ ๐‘‘๐‘Ÿ = โˆ’ โˆซ ( ๐œ‡๐‘œ ๐ผ๐‘ง2 (๐‘Ÿ)) ๐‘‘๐‘Ÿ ๐‘‘๐‘Ÿ ๐‘‘๐‘Ÿ 2 0

0

๐‘…

๐‘…

๐‘‘๐‘ƒ(๐‘Ÿ) ๐‘‘ ๐‘‘๐‘ƒ(๐‘Ÿ) 1 4๐œ‹ 2 [๐‘Ÿ 2 โˆซ ๐‘‘๐‘Ÿ โˆ’ โˆซ ( ๐‘Ÿ 2 ) (โˆซ ๐‘‘๐‘Ÿ) ๐‘‘๐‘Ÿ] = โˆ’ ๐œ‡๐‘œ ๐ผ02 ๐‘‘๐‘Ÿ ๐‘‘๐‘Ÿ ๐‘‘๐‘Ÿ 2 0

4๐œ‹ 2 [๐‘Ÿ 2 . ๐‘ƒ(๐‘Ÿ) |

0

๐‘… 0

Hisham Shah

๐‘…

1 โˆ’ โˆซ 2๐‘Ÿ. ๐‘ƒ(๐‘Ÿ)๐‘‘๐‘Ÿ] = โˆ’ ๐œ‡๐‘œ ๐ผ02 2 0

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27

4๐œ‹ 2 ๐‘Ÿ 2 ๐‘ƒ(๐‘Ÿ) |

๐‘… 0

๐‘…

1 โˆ’ 4๐œ‹ โˆซ 2๐œ‹๐‘Ÿ. ๐‘ƒ(๐‘Ÿ)๐‘‘๐‘Ÿ = โˆ’ ๐œ‡๐‘œ ๐ผ02 2 0

Where ๐ผ0 = ๐ผ๐‘ง (๐‘…) is total current flowing through the entire cross section of plasma column. Considering plasma column to be confined to the range 0 โ‰ค ๐‘Ÿ < ๐‘…, it follows that ๐‘ƒ(๐‘Ÿ) is zero for ๐‘Ÿ โ‰ฅ ๐‘… and finite for 0 โ‰ค ๐‘Ÿ < ๐‘…. So as ๐‘ƒ(๐‘Ÿ) = 0, ๐‘…

1 โˆ’4๐œ‹ โˆซ 2๐œ‹๐‘Ÿ๐‘ƒ(๐‘Ÿ)๐‘‘๐‘Ÿ = โˆ’ ๐œ‡๐‘œ ๐ผ02 2 0

Or ๐‘…

8๐œ‹ ๐ผ๐‘œ2 = โˆซ 2๐œ‹๐‘Ÿ๐‘ƒ(๐‘Ÿ)๐‘‘๐‘Ÿ โ†’ (4) ๐œ‡๐‘œ 0

Now if the partial pressures of electrons and ions are governed by ideal gas law, then ๐‘ƒ๐‘’ (๐‘Ÿ) = ๐‘›(๐‘Ÿ)๐พ๐‘‡๐‘’ ๐‘ƒ๐‘– (๐‘Ÿ) = ๐‘›(๐‘Ÿ)๐พ๐‘‡๐‘– The total pressure will be ๐‘ƒ(๐‘Ÿ) = ๐‘ƒ๐‘’ (๐‘Ÿ) + ๐‘ƒ๐‘– (๐‘Ÿ) ๐‘ƒ(๐‘Ÿ) = ๐‘›(๐‘Ÿ)๐พ๐‘‡๐‘’ + ๐‘›(๐‘Ÿ)๐พ๐‘‡๐‘– ๐‘ƒ(๐‘Ÿ) = ๐‘›(๐‘Ÿ)๐พ(๐‘‡๐‘’ + ๐‘‡๐‘– ) Using this in equation (4) we get ๐‘…

8๐œ‹ โˆซ 2๐œ‹๐‘Ÿ๐‘›(๐‘Ÿ)๐พ(๐‘‡๐‘’ + ๐‘‡๐‘– )๐‘‘๐‘Ÿ = ๐ผ๐‘œ2 โ†’ (4) ๐œ‡๐‘œ 0

๐‘…

๐ผ๐‘œ2

8๐œ‹ = ๐พ(๐‘‡๐‘’ + ๐‘‡๐‘– ) โˆซ 2๐œ‹๐‘Ÿ๐‘›(๐‘Ÿ)๐‘‘๐‘Ÿ ๐œ‡๐‘œ 0

๐‘…

As ๐‘๐‘™ = โˆซ0 2๐œ‹๐‘Ÿ๐‘›(๐‘Ÿ)๐‘‘๐‘Ÿ is the number of particles per unit length of plasma column, so ๐ผ๐‘œ2 =

8๐œ‹ ๐พ(๐‘‡๐‘’ + ๐‘‡๐‘– )๐‘๐‘™ ๐œ‡๐‘œ

This relation is called Bennet relation. It gives the total current that must be discharged through the plasma column in order to confine a plasma at a specified temperature and a given number of particles โ€˜๐‘๐‘™ โ€™ per unit length. The current required for the confinement of hot plasma is usually very large. Course: Experimental Plasma Physics

28

๏ƒฐ ๐“๐จ ๐จ๐›๐ญ๐š๐ข๐ง ๐ญ๐ก๐ž ๐ซ๐š๐๐ข๐š๐ฅ ๐๐ข๐ฌ๐ญ๐ซ๐ข๐›๐ฎ๐ญ๐ข๐จ๐ง ๐จ๐Ÿ ๐(๐ซ)๐ข๐ง ๐ญ๐ž๐ซ๐ฆ๐ฌ ๐จ๐Ÿ ๐๐›‰ (๐ซ) From Maxwellโ€™s equation โˆ‡ ร— B = ฮผo J In cylindrical coordinates with radial dependence, we have 1๐‘‘ [๐‘Ÿ๐ต๐œƒ (๐‘Ÿ)] = ๐œ‡๐‘œ ๐ฝ๐‘ง (๐‘Ÿ) ๐‘Ÿ ๐‘‘๐‘Ÿ 1 ๐‘‘ ๐‘‘ [๐‘Ÿ ๐ต๐œƒ (๐‘Ÿ) + ๐ต๐œƒ (๐‘Ÿ) ๐‘Ÿ] = ๐œ‡๐‘œ ๐ฝ๐‘ง (๐‘Ÿ) ๐‘Ÿ ๐‘‘๐‘Ÿ ๐‘‘๐‘Ÿ ๐‘‘ ๐ต๐œƒ (๐‘Ÿ) ๐‘‘ ๐ต๐œƒ (๐‘Ÿ) + ๐‘Ÿ = ๐œ‡๐‘œ ๐ฝ๐‘ง (๐‘Ÿ) ๐‘‘๐‘Ÿ ๐‘Ÿ ๐‘‘๐‘Ÿ ๐ฝ๐‘ง (๐‘Ÿ) =

1 ๐‘‘ 1 ๐ต๐œƒ (๐‘Ÿ) ๐ต๐œƒ (๐‘Ÿ) + ๐œ‡๐‘œ ๐‘‘๐‘Ÿ ๐œ‡๐‘œ ๐‘Ÿ

As ๐‘‘๐‘ƒ(๐‘Ÿ) = โˆ’๐ฝ๐‘ง (๐‘Ÿ)๐ต๐œƒ (๐‘Ÿ) ๐‘‘๐‘Ÿ Using value of ๐ฝ๐‘ง (๐‘Ÿ)

๏ƒฐ ๏ƒฐ

๐‘‘๐‘ƒ(๐‘Ÿ) ๐‘‘๐‘Ÿ ๐‘‘๐‘ƒ(๐‘Ÿ) ๐‘‘๐‘Ÿ

= โˆ’[ = โˆ’[

1 ๐‘‘

๐ต๐œƒ (๐‘Ÿ) +

๐œ‡๐‘œ ๐‘‘๐‘Ÿ ๐ต๐œƒ (๐‘Ÿ) ๐‘‘ ๐œ‡๐‘œ

๐‘‘๐‘Ÿ

1 ๐ต๐œƒ (๐‘Ÿ) ๐œ‡๐‘œ

] ๐ต๐œƒ (๐‘Ÿ)

๐‘Ÿ ๐ต๐œƒ2 (๐‘Ÿ)

๐ต๐œƒ (๐‘Ÿ) +

๐œ‡๐‘œ ๐‘Ÿ

]

Multiply and divide by 2๐‘Ÿ 2 and taking ๐œ‡๐‘œ common

๏ƒฐ ๏ƒฐ

๐‘‘๐‘ƒ(๐‘Ÿ) ๐‘‘๐‘Ÿ ๐‘‘๐‘ƒ(๐‘Ÿ) ๐‘‘๐‘Ÿ

=โˆ’ =โˆ’

2๐‘Ÿ 2 1 2๐‘Ÿ 2 ๐œ‡๐‘œ 1 2๐‘Ÿ 2 ๐œ‡๐‘œ

[๐ต๐œƒ (๐‘Ÿ)

๐‘‘ ๐‘‘๐‘Ÿ

๐ต๐œƒ (๐‘Ÿ) +

[2๐‘Ÿ 2 ๐ต๐œƒ (๐‘Ÿ)

The term in bracket can be written as

๐‘‘ ๐‘‘๐‘Ÿ

๐‘‘ ๐‘‘๐‘Ÿ

๐ต๐œƒ2 (๐‘Ÿ) ๐‘Ÿ

]

๐ต๐œƒ (๐‘Ÿ) + 2๐‘Ÿ๐ต๐œƒ2 (๐‘Ÿ)]

๐‘Ÿ 2 ๐ต๐œƒ2 (๐‘Ÿ).

As ๐‘‘ 2 2 ๐‘‘ ๐‘Ÿ ๐ต๐œƒ (๐‘Ÿ) = ๐‘Ÿ 2 . 2๐ต๐œƒ (๐‘Ÿ) ๐ต๐œƒ (๐‘Ÿ) + ๐ต๐œƒ2 (๐‘Ÿ). 2๐‘Ÿ ๐‘‘๐‘Ÿ ๐‘‘๐‘Ÿ So ๐‘‘๐‘ƒ(๐‘Ÿ) 1 ๐‘‘ 2 2 =โˆ’ 2 ๐‘Ÿ ๐ต๐œƒ (๐‘Ÿ) ๐‘‘๐‘Ÿ 2๐‘Ÿ ๐œ‡๐‘œ ๐‘‘๐‘Ÿ Integrating this equation from ๐‘Ÿ = 0 to a general radius ๐‘Ÿ ๐‘Ÿ

๐‘Ÿ

๐‘‘๐‘ƒ(๐‘Ÿ) 1 ๐‘‘ โˆซ = โˆ’ 2 โˆซ ๐‘Ÿ 2 ๐ต๐œƒ2 (๐‘Ÿ)๐‘‘๐‘Ÿ ๐‘‘๐‘Ÿ 2๐‘Ÿ ๐œ‡๐‘œ ๐‘‘๐‘Ÿ 0

Hisham Shah

0

Download: phylib.wordpress.com

29 ๐‘Ÿ

๐‘Ÿ

1 1 ๐‘‘ ๐‘ƒ(๐‘Ÿ) | = โˆ’ โˆซ 2 ๐‘Ÿ 2 ๐ต๐œƒ2 (๐‘Ÿ)๐‘‘๐‘Ÿ 2๐œ‡๐‘œ ๐‘Ÿ ๐‘‘๐‘Ÿ 0 0 ๐‘Ÿ

1 1 ๐‘‘ ๐‘ƒ(๐‘Ÿ) โˆ’ ๐‘ƒ(0) = โˆ’ โˆซ 2 ๐‘Ÿ 2 ๐ต๐œƒ2 (๐‘Ÿ)๐‘‘๐‘Ÿ โ†’ (๐ด) 2๐œ‡๐‘œ ๐‘Ÿ ๐‘‘๐‘Ÿ 0

Since for ๐‘Ÿ = ๐‘… we have ๐‘ƒ(๐‘…) = 0, so ๐‘…

1 1 ๐‘‘ ๐‘ƒ(0) = โˆซ 2 ๐‘Ÿ 2 ๐ต๐œƒ2 (๐‘Ÿ)๐‘‘๐‘Ÿ 2๐œ‡๐‘œ ๐‘Ÿ ๐‘‘๐‘Ÿ 0

Using this in equation A ๐‘…

๐‘Ÿ

1 1 ๐‘‘ 1 1 ๐‘‘ ๐‘ƒ(๐‘Ÿ) = โˆซ 2 ๐‘Ÿ 2 ๐ต๐œƒ2 (๐‘Ÿ)๐‘‘๐‘Ÿ โˆ’ โˆซ 2 ๐‘Ÿ 2 ๐ต๐œƒ2 (๐‘Ÿ)๐‘‘๐‘Ÿ โ†’ (๐ด) 2๐œ‡๐‘œ ๐‘Ÿ ๐‘‘๐‘Ÿ 2๐œ‡๐‘œ ๐‘Ÿ ๐‘‘๐‘Ÿ 0

0

0

๐‘…

1 1 ๐‘‘ 1 1 ๐‘‘ ๐‘ƒ(๐‘Ÿ) = [โˆซ 2 ๐‘Ÿ 2 ๐ต๐œƒ2 (๐‘Ÿ)๐‘‘๐‘Ÿ + โˆซ 2 ๐‘Ÿ 2 ๐ต๐œƒ2 (๐‘Ÿ)๐‘‘๐‘Ÿ] 2๐œ‡๐‘œ ๐‘Ÿ ๐‘‘๐‘Ÿ 2๐œ‡๐‘œ ๐‘Ÿ ๐‘‘๐‘Ÿ 0

0

๐‘…

๐‘Ÿ

๐‘…

We can write โˆซ๐‘Ÿ + โˆซ0 = โˆซ๐‘Ÿ , so ๐‘…

1 1 ๐‘‘ ๐‘ƒ(๐‘Ÿ) = โˆซ 2 ๐‘Ÿ 2 ๐ต๐œƒ2 (๐‘Ÿ)๐‘‘๐‘Ÿ 2๐œ‡๐‘œ ๐‘Ÿ ๐‘‘๐‘Ÿ ๐‘Ÿ

The average pressure ๐‘ƒฬ… inside the cylinder can be related to the total current ๐ผ๐‘œ and column radius R as ๐‘…

1 ๐‘ƒฬ… = โˆซ(2๐œ‹๐‘Ÿ)๐‘ƒ(๐‘Ÿ)๐‘‘๐‘Ÿ ๐œ‹๐‘… 2 0

๐‘…

1 ๐‘ƒฬ… = 2๐œ‹ โˆซ ๐‘Ÿ๐‘ƒ(๐‘Ÿ)๐‘‘๐‘Ÿ ๐œ‹๐‘… 2 0

๐‘…

2 ๐‘Ÿ2 ๐‘… ๐‘‘๐‘ƒ(๐‘Ÿ) ๐‘ƒฬ… = 2 [๐‘ƒ(๐‘Ÿ). | โˆ’ โˆซ ( โˆซ ๐‘Ÿ๐‘‘๐‘Ÿ) ๐‘‘๐‘Ÿ] ๐‘… 2 ๐‘‘๐‘Ÿ 0 0 As ๐‘ƒ(๐‘…) = 0 and ๐‘ƒ(0) = 0 ๐‘…

2 1 ๐‘‘๐‘ƒ(๐‘Ÿ) ๐‘ƒฬ… = 2 [โˆ’ โˆซ ๐‘Ÿ 2 ๐‘‘๐‘Ÿ] ๐‘… 2 ๐‘‘๐‘Ÿ 0

Course: Experimental Plasma Physics

30 ๐‘…

1 ๐‘‘๐‘ƒ(๐‘Ÿ) ๐‘ƒฬ… = โˆ’ 2 [โˆซ ๐‘Ÿ 2 ๐‘‘๐‘Ÿ] ๐‘… ๐‘‘๐‘Ÿ 0

As

๐‘‘๐‘ƒ(๐‘Ÿ) ๐‘‘๐‘Ÿ

=โˆ’

1

๐‘‘

2๐‘Ÿ 2 ๐œ‡๐‘œ ๐‘‘๐‘Ÿ

๐‘Ÿ 2 ๐ต๐œƒ2 (๐‘Ÿ) ๐‘…

1 1 ๐‘‘ 2 2 ๐‘ƒฬ… = 2 โˆซ ๐‘Ÿ 2 . 2 (๐‘Ÿ ๐ต๐œƒ (๐‘Ÿ)) ๐‘‘๐‘Ÿ ๐‘… 2๐‘Ÿ ๐œ‡๐‘œ ๐‘‘๐‘Ÿ 0

๐‘…

1 ๐‘‘ ๐‘ƒฬ… = โˆซ (๐‘Ÿ 2 ๐ต๐œƒ2 (๐‘Ÿ)) ๐‘‘๐‘Ÿ 2 2๐œ‡๐‘œ ๐‘… ๐‘‘๐‘Ÿ 0

๐‘ƒฬ… = ๐‘ƒฬ… =

๐‘… 1 2 2 (๐‘Ÿ) ๐‘Ÿ ๐ต | ๐œƒ 2๐œ‡๐‘œ ๐‘…2 0

1 (๐‘… 2 ๐ต๐œƒ2 (๐‘…) โˆ’ 0) 2๐œ‡๐‘œ ๐‘… 2 ๐‘ƒฬ… =

As ๐ต๐œƒ (๐‘…) =

๐œ‡๐‘œ ๐ผ๐‘œ 2๐œ‹๐‘…

=> ๐ต๐œƒ2 (๐‘…) = ๐‘ƒฬ… =

๐ต๐œƒ2 (๐‘…) 2๐œ‡๐‘œ

๐œ‡๐‘œ2 ๐ผ๐‘œ2 4๐œ‹2 ๐‘… 2

1 ๐œ‡๐‘œ2 ๐ผ๐‘œ2 ๐œ‡๐‘œ ๐ผ๐‘œ2 . 2 2 = 2 2 โ†’ (๐ต) 2๐œ‡๐‘œ 4๐œ‹ ๐‘… 8๐œ‹ ๐‘…

This result shows that the average kinetic pressure in this equilibrium plasma column is balanced by the magnetic pressure at the boundary. As an example, consider the case in which the current density ๐ฝ๐‘ง (๐‘Ÿ) is constant for ๐‘Ÿ < ๐‘…. ๐ผ Taking ๐ฝ๐‘ง = ๐‘œโ„๐œ‹๐‘… 2 As ๐‘Ÿ

๐œ‡๐‘œ ๐ต๐œƒ (๐‘Ÿ) = โˆซ ๐ฝ๐‘ง (๐‘Ÿ)๐‘Ÿ๐‘‘๐‘Ÿ ๐‘… 0

๐‘Ÿ

๐œ‡๐‘œ ๐ผ๐‘œ ๐ต๐œƒ (๐‘Ÿ) = โˆซ ๐‘Ÿ๐‘‘๐‘Ÿ ๐‘Ÿ ๐œ‹๐‘… 2 0

๐ต๐œƒ (๐‘Ÿ) =

๐œ‡๐‘œ ๐ผ๐‘œ ๐‘Ÿ 2 ๐œ‡๐‘œ ๐ผ๐‘œ ๐‘Ÿ ๐œ‡๐‘œ ๐ผ๐‘œ ๐‘Ÿ = = (๐‘Ÿ < ๐‘…) ๐‘Ÿ ๐œ‹๐‘… 2 2 ๐œ‹๐‘… 2 2 2๐œ‹๐‘… 2

As

Hisham Shah

Download: phylib.wordpress.com

31 ๐‘…

1 1 ๐‘‘ ๐‘ƒ(๐‘Ÿ) = โˆซ 2 ๐‘Ÿ 2 ๐ต๐œƒ2 (๐‘Ÿ)๐‘‘๐‘Ÿ 2๐œ‡๐‘œ ๐‘Ÿ ๐‘‘๐‘Ÿ ๐‘Ÿ

So ๐‘…

๐‘…

1 1 ๐‘‘ ๐œ‡2๐ผ2๐‘Ÿ 2 ๐œ‡๐‘œ2 ๐ผ๐‘œ2 1 1 ๐‘‘ 2 ๐‘œ ๐‘œ => ๐‘ƒ(๐‘Ÿ) = โˆซ 2 (๐‘Ÿ . 2 4 ) ๐‘‘๐‘Ÿ = . 2 4 โˆซ 2 ๐‘Ÿ 4 ๐‘‘๐‘Ÿ 2๐œ‡๐‘œ ๐‘Ÿ ๐‘‘๐‘Ÿ 4๐œ‹ ๐‘… 2๐œ‡๐‘œ 4๐œ‹ ๐‘… ๐‘Ÿ ๐‘‘๐‘Ÿ ๐‘Ÿ

๐‘Ÿ

๐‘…

๐‘…

๐œ‡๐‘œ ๐ผ๐‘œ2 1 ๐œ‡๐‘œ ๐ผ๐‘œ2 ๐œ‡๐‘œ ๐ผ๐‘œ2 ๐‘Ÿ 2 ๐‘… ๐œ‡๐‘œ ๐ผ๐‘œ2 => ๐‘ƒ(๐‘Ÿ) = 2 4 โˆซ 2 4๐‘Ÿ 3 ๐‘‘๐‘Ÿ = 2 4 โˆซ ๐‘Ÿ๐‘‘๐‘Ÿ = 2 4 | = 2 4 (๐‘… 2 โˆ’ ๐‘Ÿ 2 ) 8๐œ‹ ๐‘… ๐‘Ÿ 2๐œ‹ ๐‘… 2๐œ‹ ๐‘… 2 4๐œ‹ ๐‘… ๐‘Ÿ ๐‘Ÿ ๐‘Ÿ ๐œ‡๐‘œ ๐ผ๐‘œ2 ๐‘… 2 ๐‘Ÿ 2 ๐œ‡๐‘œ ๐ผ๐‘œ2 ๐‘Ÿ2 => ๐‘ƒ(๐‘Ÿ) = 2 2 ( 2 โˆ’ 2 ) = 2 2 (1 โˆ’ 2 ) 4๐œ‹ ๐‘… ๐‘… ๐‘… 4๐œ‹ ๐‘… ๐‘… In this case the axial pressure ๐‘ƒ(0) is twice the average pressure given in (B). ๐‘ƒฬ… = ๐‘ƒ(๐‘Ÿ) =

๐œ‡๐‘œ ๐ผ๐‘œ2 8๐œ‹ 2 ๐‘… 2

๐œ‡๐‘œ ๐ผ๐‘œ2 ๐‘Ÿ2 (1 โˆ’ ) 4๐œ‹ 2 ๐‘… 2 ๐‘…2

2๐œ‡๐‘œ ๐ผ๐‘œ2 ๐‘Ÿ2 ๐‘ƒ(๐‘Ÿ) = 2 2 (1 โˆ’ 2 ) 8๐œ‹ ๐‘… ๐‘… ๐‘ƒ(๐‘Ÿ) = 2๐‘ƒฬ… (1 โˆ’

๐‘Ÿ2 ) ๐‘…2

This equation shows that axial pressure ๐‘ƒ(๐‘Ÿ) is twice the average pressure.

THE BENNET PINCH W. H. Bennet, the discoverer of pinch effect, investigated a special model of the equilibrium longitudinal pinch in which the radial distribution of various quantities are such that the drift velocity of plasma particle is constant throughout the column cross section. Ion mass is much larger than the electron mass, so drift velocity of ion is much smaller than that of electrons and therefore neglected in a first approximation. We consider the current density to be given by, ๐ฝ(๐‘Ÿ) = โˆ’๐‘’๐‘›(๐‘Ÿ)๐‘ข๐‘’ Since the applied electric field is in 2 directions, we have ๐ฝ(๐‘Ÿ) = ๐ฝ๐‘ง (๐‘Ÿ)๐‘งฬ…, and ๐‘ข๐‘’ โˆ’ ๐‘ข๐‘’๐‘ง ๐‘งฬ‚ , where ๐‘ข๐‘’๐‘ง is positive and constant, independent of r, therefore, ๐ฝ๐‘ง (๐‘Ÿ) = ๐‘’๐‘›(๐‘Ÿ)๐‘ข๐‘’๐‘ง As ๐‘๐‘’ (๐‘Ÿ) + ๐‘๐‘– (๐‘Ÿ) = ๐‘(๐‘Ÿ) = ๐‘›(๐‘Ÿ)๐พ(๐‘‡๐‘’ + ๐‘‡๐‘– ) And ๐‘‘๐‘ƒ(๐‘Ÿ) = โˆ’๐ฝ๐‘ง (๐‘Ÿ)๐ต๐œƒ (๐‘Ÿ) ๐‘‘๐‘Ÿ Course: Experimental Plasma Physics

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Using values or ๐‘ƒ(๐‘Ÿ) and ๐ฝ๐‘ง (๐‘Ÿ), we get ๐พ(๐‘‡๐‘’ + ๐‘‡๐‘– )

๐‘‘ ๐‘›(๐‘Ÿ) = โˆ’๐‘’๐‘›(๐‘Ÿ)๐‘ข๐‘’๐‘ง ๐ต๐œƒ (๐‘Ÿ) ๐‘‘๐‘Ÿ

Multiplying both sides by ๐‘Ÿโ„๐‘›(๐‘Ÿ)๐พ(๐‘‡ + ๐‘‡ ) ๐‘’ ๐‘– ๐‘Ÿ ๐‘‘ ๐‘Ÿ . ๐พ(๐‘‡๐‘’ + ๐‘‡๐‘– ) ๐‘›(๐‘Ÿ) = . (โˆ’๐‘’๐‘›(๐‘Ÿ)๐‘ข๐‘’๐‘ง ๐ต๐œƒ (๐‘Ÿ)) ๐‘›(๐‘Ÿ)๐พ(๐‘‡๐‘’ + ๐‘‡๐‘– ) ๐‘‘๐‘Ÿ ๐‘›(๐‘Ÿ)๐พ(๐‘‡๐‘’ + ๐‘‡๐‘– ) ๐‘Ÿ ๐‘‘ โˆ’๐‘’๐‘ข๐‘’๐‘ง . ๐‘›(๐‘Ÿ) = . ๐‘Ÿ(๐ต๐œƒ (๐‘Ÿ)) ๐‘›(๐‘Ÿ) ๐‘‘๐‘Ÿ ๐พ(๐‘‡๐‘’ + ๐‘‡๐‘– ) Differentiating both sides w.r.t โ€œrโ€ ๐‘‘ ๐‘Ÿ ๐‘‘ โˆ’๐‘’๐‘ข๐‘’๐‘ง ๐‘‘ [๐‘Ÿ๐ต๐œƒ (๐‘Ÿ)] [ . ๐‘›(๐‘Ÿ)] = ๐‘‘๐‘Ÿ ๐‘›(๐‘Ÿ) ๐‘‘๐‘Ÿ ๐พ(๐‘‡๐‘’ + ๐‘‡๐‘– ) ๐‘‘๐‘Ÿ From equation โ€ฆ 1๐‘‘ [๐‘Ÿ๐ต๐œƒ (๐‘Ÿ)] = ๐œ‡๐‘œ ๐ฝ๐‘ง (๐‘Ÿ) ๐‘Ÿ ๐‘‘๐‘Ÿ And as ๐ฝ๐‘ง (๐‘Ÿ) = ๐‘’๐‘›(๐‘Ÿ)๐‘ข๐‘’๐‘ง ๐‘‘ [๐‘Ÿ๐ต๐œƒ (๐‘Ÿ)] = ๐‘Ÿ๐œ‡๐‘œ ๐‘’๐‘›(๐‘Ÿ)๐‘ข๐‘’๐‘ง ๐‘‘๐‘Ÿ Using this result, we get 2 ๐‘‘ ๐‘Ÿ ๐‘‘ ๐‘’ 2 ๐‘ข๐‘’๐‘ง ๐œ‡๐‘œ [ . ๐‘›(๐‘Ÿ)] โˆ’ [ ] ๐‘Ÿ๐‘›(๐‘Ÿ) = 0 โ†’ (๐ถ) ๐‘‘๐‘Ÿ ๐‘›(๐‘Ÿ) ๐‘‘๐‘Ÿ ๐พ(๐‘‡๐‘’ + ๐‘‡๐‘– )

The solution of this nonlinear differential eqn gives radial dependence of number density โ€˜๐‘›(๐‘Ÿ)โ€™. Bennet obtained the solution of this nonlinear equation subjected to the boundary condition that ๐‘›(๐‘Ÿ) is symmetric about z-axis, where ๐‘Ÿ = 0 [

๐‘‘๐‘›(๐‘Ÿ) ] =0 ๐‘‘๐‘Ÿ ๐‘Ÿ=0

Solution of eq C subjected to boundary condition is knows as Bennet distribution and is given by ๐‘›(๐‘Ÿ) =

๐‘›0 (1 + ๐‘›๐‘œ ๐‘๐‘Ÿ 2 )2

Where ๐‘›๐‘œ = ๐‘›(0) which is number density on the axis, and ๐‘=

Hisham Shah

2 ๐œ‡๐‘œ ๐‘’ 2 ๐‘ข๐‘’๐‘ง 8๐พ(๐‘‡๐‘’ + ๐‘‡๐‘– )

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33

The Bennet distribution shows that the particles are present upto density but since โ€˜๐‘›(๐‘Ÿ)โ€™ falls off very rapidly with increasing values of โ€˜rโ€™. We obtain number of particles ๐‘๐‘™ (๐‘…) per unit length contained a cylindrical column of radius โ€˜Rโ€™. ๐‘…

๐‘…

๐‘๐‘™ (๐‘…) = โˆซ ๐‘›(๐‘Ÿ)2๐œ‹๐‘Ÿ๐‘‘๐‘Ÿ = 2๐œ‹๐‘›๐‘œ โˆซ 0

0

๐‘Ÿ ๐‘‘๐‘Ÿ (1 + ๐‘›๐‘œ ๐‘๐‘Ÿ 2 )2

Let ๐‘ก = 1 + ๐‘›๐‘œ ๐‘๐‘Ÿ 2 ๐‘‘๐‘ก ๐‘‘๐‘ก = 2๐‘›๐‘œ ๐‘๐‘Ÿ => 2๐‘Ÿ๐‘‘๐‘Ÿ = ๐‘‘๐‘Ÿ ๐‘›๐‘œ ๐‘ ๐‘…

๐‘๐‘™ (๐‘…) = ๐œ‹๐‘›๐‘œ โˆซ 0

2๐‘Ÿ ๐‘‘๐‘Ÿ (1 + ๐‘›๐‘œ ๐‘๐‘Ÿ 2 )2

1+๐‘›๐‘œ ๐‘๐‘Ÿ 2

๐‘๐‘™ (๐‘…) = ๐œ‹๐‘›๐‘œ

โˆซ 1

๐‘๐‘™ (๐‘…) =

๐œ‹ ๐‘

1 1 ๐‘‘๐‘ก 2 ๐‘ก ๐‘›๐‘œ ๐‘

1+๐‘›๐‘œ ๐‘๐‘Ÿ 2

โˆซ

๐‘ก โˆ’2 ๐‘‘๐‘ก

1

2 ๐œ‹ ๐‘ก โˆ’1 1 + ๐‘›๐‘œ ๐‘๐‘Ÿ ๐‘๐‘™ (๐‘…) = | ๐‘ โˆ’1 1 ๐œ‹ ๐‘๐‘™ (๐‘…) = โˆ’ [(1 + ๐‘›๐‘œ ๐‘๐‘… 2 )โˆ’1 โˆ’ (1)โˆ’1 ] ๐‘

๐œ‹ 1 ๐œ‹ โˆ’๐‘›๐‘œ ๐‘๐‘… ๐‘๐‘™ (๐‘…) = โˆ’ [ โˆ’ 1] = โˆ’ [ ] 2 ๐‘ (1 + ๐‘›๐‘œ ๐‘๐‘… ) ๐‘ (1 + ๐‘›๐‘œ ๐‘๐‘… 2 ) Particles are present upto infinity, total number of particles per unit length can be obtained by taking ๐‘… โ†’ โˆž

Course: Experimental Plasma Physics

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๐œ‹ 1 ๐œ‹ ๐‘๐‘™ (๐‘…) = โˆ’ [ โˆ’ 1] = ๐‘ โˆž ๐‘ If we let and denote the fraction of the number of particles per unit length that is contained in a cylinder of radius R, that is ๐›ผ=

๐‘๐‘™ (๐‘…) ๐‘ = ๐‘ (๐‘…) ๐‘๐‘™ (โˆž) ๐œ‹ ๐‘™

๐›ผ=

๐‘ ๐‘›๐‘œ ๐œ‹๐‘… 2 ๐œ‹ (1 + ๐‘›๐‘œ ๐‘๐‘… 2 )

๐›ผ(1 + ๐‘›๐‘œ ๐‘๐‘… 2 ) = ๐‘๐‘›๐‘œ ๐‘… 2 ๐›ผ = ๐‘๐‘›๐‘œ ๐‘… 2 โˆ’ ๐›ผ๐‘›๐‘œ ๐‘๐‘… 2 ๐›ผ = ๐‘๐‘›๐‘œ ๐‘… 2 (1 โˆ’ ๐›ผ) ๐›ผ = ๐‘๐‘›๐‘œ ๐‘… 2 1โˆ’๐›ผ Taking under root ๐›ผ 1โˆ’๐›ผ

โˆš๐‘๐‘›๐‘œ ๐‘… = โˆš

Therefore 90% of plasma particles are confined within the cylindrical plasma column of radius โ€˜Rโ€™ that is ๐›ผ = 0.9 0.9 0.9 =โˆš = โˆš9 = 3 โˆš๐‘๐‘›๐‘œ ๐‘… = โˆš 1 โˆ’ 0.9 0.1 If we assume arbitrary that a plasma is confined within a cylindrical surface of radius โ€˜Rโ€™. If 90% of particles are within this cylindrical column, then this radius must satisfy the above equation.

INSTABILITIES IN A PINCHED PLASMA COLUMN Although it is possible to achieve an equilibrium state for plasma confinement with the pinch effect, this equilibrium style is not stable. The growth of instabilities is the reason why it is difficult to sustain reasonably long lived pinched plasma in the laboratory. In the following discussion of instabilities we shall consider a perfectly diamagnetic plasma column confined by a static magnetic field. Since the plasma is perfectly diamagnetic, there is no magnetic field and consequently no magnetic pressure inside the plasma column. The plasma kinetic pressure is assumed to be uniform inside the plasma and vanishes outside it.

Hisham Shah

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35

In the equilibrium state, the magnetic pressure of the plasma surface โ€˜๐‘ƒ๐‘š๐‘œ โ€™ must be equal to the kinetic pressure โ€˜Pโ€™ of plasma. ๐‘ƒ = ๐‘ƒ๐‘š๐‘œ =

๐ต๐œƒ2 2๐œ‡๐‘œ

Where โ€˜๐ต๐œƒ โ€™ is magnitude of magnetic flux density at the plasma surface. This situation of a sharp plasma is an idealized one and is difficult to create in laboratory.

THE SAUSAGE INSTABILITY Consider equilibrium state of the pinched plasma column is distributed by a wave perturbation with the crests and troughs on the surface of plasma column. We shall consider that the plasma is constricted in some locations and expanded at others in such a way that its volume doesnโ€™t change consequently the uniform kinetic pressure of plasma is left unchanged. At the locations where radius has decreased in relation to equilibrium value, the magnetic pressure at the constricted plasma surface radially inward, thus enhancing the constrictor. At the locations where radius has become larger than the equilibrium value, the plasma kinetic pressure will be larger than magnetic pressure at the expanded plasma surface and will force the surface radially outwards. Therefore the troughs will become deeper and crests higher. So the equilibrium state is unstable. When the constrictor reach the axis, the column appears like a surface of sausages, for this reason this type of instability is known as sausage instability.

Course: Experimental Plasma Physics

36

The sausage instability can be inhibited by a longitudinal magnetic field applied inside the plasma column. This longitudinal magnetic field can be produced by passing a current through a solenoidal coil wound around the column. When sausage distortion starts to grow the longitudinal magnetic field lines are compressed at the constrictions causing increase in total pressure inside plasma at the location where radius has increased thus decreasing the total internal pressure. If the radius โ€˜rโ€™ of the column at the column at the constriction is decreased by an amount โ€˜drโ€™, considering magnetic flux ๐œ™๐‘š = ๐ต๐‘ง ๐œ‹๐‘Ÿ 2 ๐‘‘๐œ™๐‘š ๐‘‘๐ต๐‘ง = ๐œ‹๐‘Ÿ 2 + ๐ต๐‘ง . 2๐œ‹๐‘Ÿ = 0 ๐‘‘๐‘Ÿ ๐‘‘๐‘Ÿ ๐‘‘๐œ™๐‘š = ๐œ‹๐‘Ÿ 2 ๐‘‘๐ต๐‘ง + ๐ต๐‘ง . 2๐œ‹๐‘Ÿ๐‘‘๐‘Ÿ = 0 ๐œ‹๐‘Ÿ 2 ๐‘‘๐ต๐‘ง = โˆ’๐ต๐‘ง . 2๐œ‹๐‘Ÿ๐‘‘๐‘Ÿ ๐‘‘๐ต๐‘ง = โˆ’2๐ต๐‘ง

๐‘‘๐‘Ÿ โ†’ (1) ๐‘Ÿ

Consequently the corresponding internal magnetic pressure increases as ๐‘ƒ๐‘ง =

๐ต๐‘ง2 1 => ๐‘‘๐‘ƒ๐‘ง = ๐ต ๐‘‘๐ต 2๐œ‡๐‘œ ๐œ‡๐‘œ ๐‘ง ๐‘ง

Using ๐‘‘๐ต๐‘ง from (1) ๐‘‘๐‘ƒ๐‘ง =

1 ๐‘‘๐‘Ÿ 1 ๐‘‘๐‘Ÿ ๐ต๐‘ง (โˆ’2๐ต๐‘ง ) = โˆ’ 2๐ต๐‘ง2 ๐œ‡๐‘œ ๐‘Ÿ ๐œ‡๐‘œ ๐‘Ÿ

Considering now the azimuthal flux density โ€˜๐ต๐œƒ โ€™. From Ampereโ€™s law, ๐‘Ÿ๐ต๐œƒ (๐‘Ÿ) = ๐‘๐‘œ๐‘›๐‘ ๐‘ก๐‘Ž๐‘›๐‘ก ๐‘Ÿ๐‘‘๐ต๐œƒ (๐‘Ÿ) + ๐ต๐œƒ (๐‘Ÿ)๐‘‘๐‘Ÿ = 0 ๐‘‘๐ต๐œƒ (๐‘Ÿ) = โˆ’๐ต๐œƒ (๐‘Ÿ)

๐‘‘๐‘Ÿ โ†’ (3) ๐‘Ÿ

As ๐‘ƒ๐œƒ =

๐ต๐œƒ2 ๐‘‘๐‘ƒ๐œƒ 2๐ต๐œƒ ๐ต๐œƒ => = => ๐‘‘๐‘ƒ๐œƒ = ๐‘‘๐ต๐œƒ 2๐œ‡๐‘œ ๐‘‘๐ต๐œƒ 2๐œ‡๐‘œ ๐œ‡๐‘œ

Using ๐‘‘๐ต๐œƒ from eq (3) ๐‘‘๐‘ƒ๐œƒ =

๐ต๐œƒ ๐‘‘๐‘Ÿ (โˆ’๐ต๐œƒ (๐‘Ÿ) ) ๐œ‡๐‘œ ๐‘Ÿ

๐ต๐œƒ2 (๐‘Ÿ) ๐‘‘๐‘Ÿ ๐‘‘๐‘ƒ๐œƒ = ๐œ‡๐‘œ ๐‘Ÿ From (2) and (4) we can say that ๐‘‘๐‘ƒ๐‘ง > ๐‘‘๐‘ƒ๐œƒ . Condition in order to plasma column be stable 1 against sausage distortion or ๐ต๐‘ง2 > 2 ๐ต๐œƒ2 Hisham Shah

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37

THE KINK INSTABILITY The kink distortion consists of a perturbation in the form of a bend or kink in the column, but with the disturbed column maintaining its uniform circular cross-section.

Usually there may be several kinks along the column length. In the neighborhood of the column, where kink has developed the magnetic field lines are brought closer together on convex side. Therefore the changes in the external magnetic pressure are in such a way as to accentuate the distortion still. Thus the type of distortion is therefore unstable.

FAMOUS QUOTES ON FAILURES We should not get afraid of failing; we should embrace failure with open arms. Itโ€™s only through failure that we learn. Here are some of my favorite quotes on failure. โ€œIโ€™ve missed more than 9,000 shots in my career. Iโ€™ve lost almost 300 games. 26 times Iโ€™ve been trusted to take the game's winning shot and missed. Iโ€™ve failed over and over and over again in my life and that's why I succeed.โ€

โ€“ Michael Jordan, NBA Hall of Famer โ€œShe turned him down because he had no prospects, he grew up to become an oil baron and the don of Wall Street.โ€

โ€“ John D. Rockefeller, Standard Oil โ€“ Soichiro Honda, Honda Motors

โ€œHe was turned down for an engineering job by Toyotaโ€

"When everything seems to be going against you, remember that the airplane takes off against the wind, not with it." โ€“ Henry Ford, Ford Motors โ€œItโ€™s important to question your sanity because at the point which you stop questioning your sanity, youโ€™re probably insane.โ€ โ€“ Elon Musk: Tesla, SpaceX, SolarCity, PayPal, OpenAI, Hyperloop โ€œDonโ€™t be embarrassed by your failures, learn from them and start again.โ€ โ€“ Richard Branson, Virgin "I have not failed. Iโ€™ve just found 10,000 ways that wonโ€™t work."

โ€“ Thomas Edison

โ€œIf youโ€™re not embarrassed by first version of your product, youโ€™ve launched too late.โ€ โ€“ Reid Hoffman โ€œYou have to be willing to be misunderstood if youโ€™re going to innovate.โ€

โ€“ Jeff Bezos, Amazon

Course: Experimental Plasma Physics

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CHAPTER #5 PHYSICAL CONDITIONS FOR THERMONUCLEAR REACTIONS RATES OF THERMONUCLEAR REACTIONS (2.15-2.18) Consider a binary reaction in a system containing n1 nuclei per cm3 of one reacting species and n2 nuclei per cm3 of the other. To determine the rate at which the two nuclear species interact, it may be supposed that the nuclei of the first kind form a stationary lattice within which the nuclei of second kind move at random with a constant velocity ๐‘ฃ cm/sec, equal to the relative velocity of the nuclei. The rate of thermonuclear energy production is readily obtained from the reaction rate โ€œR 12โ€ for a system of two species of nuclei of number densities n1 and n2 and is given by ๐‘…12 = ๐‘›1 ๐‘›2 < ฯƒ๐‘ฃ > interactionsโ„cm3 . ๐‘  โ†’ (1) If the reaction occurs between the two nuclei of the same kind, e.g. two deuterons, so that n1 and n2 are equal, the expression for nuclear reaction rate, represented by R12 becomes 1 ๐‘…11 = ๐‘›2 < ฯƒ๐‘ฃ > interactionsโ„cm3 . ๐‘  โ†’ (2) 2 Where โ€˜nโ€™ is number of reactant nuclei per cm3. In order that each interaction between identical nuclei should not be counted twice, a factor of ยฝ is introduced in above equation. If the velocity distribution is maxwellian, the equation for the distribution in terms of relative velocity is obtained upon substituting the reduced mass โ€˜Mโ€™ of the interacting nuclei for the individual masses. Thus 3

2

๐‘€ 2 (โˆ’๐‘€๐‘ฃ )๐‘ฃ 2 ๐‘‘๐‘ฃ ๐‘‘๐‘› = ๐‘› ( ) ๐‘’ 2๐‘˜๐‘‡ โ†’ (3) 2๐œ‹๐‘˜๐‘‡ Where dn is the number of particles whose velocities relative to that of a given particle lie in the range from ๐‘ฃ ๐‘ก๐‘œ ๐‘ฃ + ๐‘‘๐‘ฃ. Hence it follows that ฬ…ฬ…ฬ…ฬ… = ฯƒ๐‘‰

โˆž โˆซ0 ฯƒ๐‘ฃ๐‘‘๐‘› โˆž โˆซ0 ๐‘‘๐‘›

=

๐‘€๐‘ฃ 2 (โˆ’ ) โˆž 2๐‘˜๐‘‡ ๐‘ฃ 2 ๐‘‘๐‘ฃ ฯƒ๐‘ฃ๐‘’ โˆซ0 ๐‘€๐‘ฃ 2 โˆž (โˆ’ 2๐‘˜๐‘‡ ) 2 ๐‘’ ๐‘ฃ ๐‘‘๐‘ฃ โˆซ0

โ†’ (4)

The denominator of the above equation is โˆž

โˆซ๐‘’ 0

Hisham Shah

(โˆ’

๐‘š๐‘ฃ 2 ) 2๐‘˜๐‘‡ ๐‘ฃ 2 ๐‘‘๐‘ฃ

3

2๐‘˜๐‘‡ 2 โˆš๐œ‹ =( ) ๐‘€ 4

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39 3 โˆž

2

๐‘€๐‘ฃ ๐‘€ 2 (โˆ’ ) ฬ…ฯƒ๐‘ฃ ฬ…ฬ…ฬ… = ( ) โˆซ ฯƒ๐‘’ 2๐‘˜๐‘‡ ๐‘ฃ 3 ๐‘‘๐‘ฃ โ†’ (5) โˆš๐œ‹ 2๐‘˜๐‘‡

4

0

The integral in equation (5) can be evaluated by changing the variable. Since nuclear cross sections are always determined and expressed as a function of the energy of the bombarding particle, the bombarded particle being essentially at rest in the target, the actual velocity of bombarding nucleus is also its relative velocity. Hence if โ€˜Wโ€™ is the actual energy, in the laboratory system, of the bombarding nucleus of mass โ€˜mโ€™, then 1

1 2๐‘Š 2 ๐‘Š = ๐‘š๐‘ฃ 2 ; ๐‘‰ = ( ) 2 ๐‘š 1

๏ƒฐ

๐‘‘๐‘ฃ ๐‘‘๐‘Š

1 2๐‘Š โˆ’2 2

= ( 2

๏ƒฐ ๐‘‘๐‘ฃ = (

)

๐‘š

๐‘š 2๐‘Š

)

=(

๐‘š

1 2 1

๐‘š

๏ƒฐ ๐‘ฃ ๐‘‘๐‘ฃ = ( ๏ƒฐ ๐‘ฃ 3 ๐‘‘๐‘ฃ =

2๐‘Š 2

๐‘€ 2๐‘Š ๐‘š2

2๐‘Š

)

1 2 1

๐‘š

๐‘‘๐‘Š

3

3

๐‘š

) (

๐‘€ 2๐‘Š

)

1 2 1

๐‘š

๐‘‘๐‘Š

๐‘‘๐‘Š

Now eq (5) implies that 3 โˆž

2

๐‘€๐‘ฃ 2๐‘Š ๐‘€ 2 (โˆ’ ) ฬ…ฬ…ฬ…ฬ… ฯƒ๐‘‰ = ( ) โˆซ ฯƒ๐‘’ 2๐‘˜๐‘‡ ๐‘‘๐‘Š ๐‘š2 โˆš๐œ‹ 2๐‘˜๐‘‡

4

0

๏ƒฐ ฬ…ฬ…ฬ…ฬ… ฯƒ๐‘‰ = ๏ƒฐ ฬ…ฬ…ฬ…ฬ… ฯƒ๐‘‰ =

4

(

3 2

๐‘€

โˆš๐œ‹ 2๐พ๐‘‡ 8

(

๐‘€

โˆš๐œ‹ 2๐พ๐‘‡

) . 3 2

) .

2

โˆž

2๐‘Š

(โˆ’๐‘€ โ„2๐พ๐‘‡ ) ๐‘š ๐‘Š ๐‘‘๐‘Š โˆซ ฯƒ๐‘’ ๐‘š2 0 1

โˆž

๐‘€๐‘Š

(โˆ’ ) โˆซ ฯƒ๐‘’ ๐‘š๐พ๐‘‡ ๐‘Š ๐‘‘๐‘Š ๐‘š2 0

Where ฯƒ in the integrand is the cross section for a bombarding nucleus of mass โ€˜mโ€™ and energy โ€˜Wโ€™.

NUCLEAR FUSION REACTIONS (2.21) Thermonuclear fusion reaction offers and inexhaustible source of energy for the future. In this process two light nuclei combine to form a heavier one, the total final mass being slightly less than the total initial mass. The mass difference โˆ†๐‘š appears as energy โ€˜Eโ€™ according to Einsteinโ€™s famous equation ๐ธ = โˆ†๐‘š๐‘ 2 . It is believed that such a source of energy will provide easy, cheap and relatively radiative free energy for our future needs. There are two possible ways to obtain fusion. 1. An energetic light nuclei beam may be directed to stationary nuclei in solid or gaseous form. The beam and target nuclei undergo fusion reactions and this process is called

Course: Experimental Plasma Physics

40

beam target mechanism. However, this technique does not work, because most of deuterons lose energy by scattering before undergoing a fusion reaction. 2. The light nucleic gas may be heated to sufficiently high temperature and confined for sufficiently long time. The gas obtains the Maxwellian distribution. The nuclei undergo fusion reactions and this process is known as thermonuclear fusion. To achieve thermonuclear fusion energy, one must solve two problems ๏‚ท ๏‚ท

Produce and heat a plasma fuel to thermonuclear fusion temperatures Confine it long enough to produce more fusion energy than expanded in heating and containing the fuel.

These twin requirements are quantified by a relation known as Lawsonโ€™s criterion. The reactions of highest interest are controlled thermonuclear fusion are as follows 2 1D 1D

+ 1D2 โ†’

2

2 1D

+ 1D2 โ†’ + 1T 3 โ†’

2He 1T

3

+ 0n1 + 3.27 MeV

3

+ 1H1 + 4.03 MeV

4

+ 0n1 + 17.6 MeV

2He

Here the first two reactions are respectively the neutron branch and the proton branch of the DD-reaction. The tritium produced in the proton branch or obtained in other way as explained above, can then react, at a considerably faster rate, with deuteron nuclei in the D-T-reaction. Among all these nuclear fusion reactions, the deuterium-tritium 1D2 + 1T 3 โ†’ 2He4 + 1 4 0n has large cross sections even at relatively low (~10 ๐พ๐‘’๐‘‰) temperature, and has large Qvalue as compared to 1D2 + 1D2 โ†’ reactions. For those reasons. CHARGED PARTICLE ENERGY (2.24-2.25) To access how much energy will be carried by the end products, one can simply use the conservation laws of energy and momentum. If m and mโ€™ are the masses of particles produced in a given fusion reaction and V and Vโ€™ are their respective velocities, then the energy carried by the end products can be written by using conservative laws. ๐‘š๐‘ฃ = ๐‘šโ€ฒ ๐‘ฃ โ€ฒ 2

2

๏ƒฐ ๐‘š2 ๐‘ฃ 2 = ๐‘šโ€ฒ ๐‘ฃ โ€ฒ 2 ๏ƒฐ ๐‘š(๐‘š๐‘ฃ 2 ) = ๐‘šโ€ฒ (๐‘šโ€ฒ ๐‘ฃ โ€ฒ ) ๏ƒฐ ๐‘š๐‘ž = ๐‘šโ€ฒ๐‘žโ€ฒ Where q and qโ€™ are the energies carried by the particles of masses m and mโ€ฒ. The total energy release of the nuclear reaction is Q. then As ๐‘š๐‘ž = ๐‘šโ€ฒ ๐‘ž โ€ฒ

๏ƒฐ ๐‘žโ€ฒ =

๐‘š๐‘ž ๐‘šโ€ฒ

๏ƒฐ ๐‘„=๐‘ž+

Hisham Shah

๐‘š๐‘ž ๐‘šโ€ฒ

=

๐‘šโ€ฒ ๐‘ž+ ๐‘š๐‘ž ๐‘šโ€ฒ

=

๐‘ž(๐‘šโ€ฒ + ๐‘š) ๐‘šโ€ฒ

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41

๏ƒฐ ๐‘ž=

๐‘šโ€ฒ ๐‘„ ๐‘šโ€ฒ +๐‘š

Similarly ๐‘ž =

๐‘šโ€ฒ ๐‘ž โ€ฒ +๐‘žโ€ฒ๐‘š ๐‘š

=

๐‘žโ€ฒ(๐‘š+๐‘šโ€ฒ) ๐‘š

๐‘š๐‘„

๏ƒฐ ๐‘ž โ€ฒ = ๐‘š+๐‘šโ€ฒ; where Q is the total energy released in a typical fusion reaction. Using this result, one can re-write the reactions as 2 1D

+ 1D2 โ†’

2He

3

3

2 1D

+ 1D2 โ†’

1T

2 1D

+ 1T 3 โ†’

2He

2

+ 2He3 โ†’

1D

(0.82 MeV) + 0n1 (2.45 MeV)

(1.01 MeV) + 1H1 (3.02 MeV) 4

2He

(3.5 MeV) + 0n1 (14.1 MeV)

4

(3.6 MeV) + 1H1 (14.7 MeV)

Notice that in thermonuclear fusion reaction high energy neutrons are emitted which would practically escape from reacting system and would deposit their energy elsewhere. Only energy of charged particles will be retained within reaction region. THE LAWSON CRITERION The usual Lawson criterion can be obtained by balancing the fusion energy release against the energy by investment in bringing the fuel to the required thermonuclear fusion temperatures and the energy loss through Bremsstrahlung and Cyclotron radiations. ๐ธ๐‘“๐‘ข๐‘ ๐‘–๐‘œ๐‘› = ๐ธ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘š๐‘Ž๐‘™ + ๐ธ๐‘Ÿ๐‘Ž๐‘‘ The fusion energy release can be expressed in terms of fusion reactions rate and confinement time โ€˜๐œโ€™as ๐ธ๐‘“๐‘ข๐‘ ๐‘–๐‘œ๐‘›

๐‘›2 = < ฯƒ๐‘‰ > ๐‘„๐œ 4

Where < ฯƒ๐‘‰ > is the Maxwellian averaged reaction rate parameter, Q is the fusion energy released per fusion reaction and ๐œ is the plasma confinement time. We have assumed equimolar density of the fusion fuel. The thermal energy assuming ideal gas behavior of the plasma is given by 3 3 ๐ธ๐‘กโ„Ž๐‘’๐‘Ÿ๐‘š๐‘Ž๐‘™ = ๐‘›๐พ๐‘‡๐‘– + ๐‘›๐พ๐‘‡๐‘’ = 3๐‘›๐พ๐‘‡ 2 2 Where we assume ๐‘‡๐‘’ = ๐‘‡๐‘– = ๐‘‡ If we balance the fusion energy released against thermal energy and neglect radiation energy loss for simplicity, then we have ๐‘›2 < ฯƒ๐‘‰ > ๐‘„๐œ > 3๐‘›๐พ๐‘‡ 4 ๐‘›๐œ >

12๐‘›๐พ๐‘‡ < ฯƒ๐‘‰ > ๐‘„

Course: Experimental Plasma Physics

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It is evident from the above expression that ๐‘›๐œ product is a function of temperature alone. At suitable temperature, the ๐‘›๐œ criterion becomes nฯ„ > 1014 secโ„cm3 for DT reactions nฯ„ > 1016 secโ„cm3 for DD reactions The criterion provides confinement time โ€˜๐œโ€™ for a given number density โ€˜mโ€™ of the plasma. In Magnetic Confinement Fusion, ๐‘› ๐‘–๐‘  ~(1015 โˆ’ 1016 )๐‘๐‘šโˆ’3 and therefore the confinement time is (0.1 โˆ’ 10)๐‘ ๐‘’๐‘

2.47: Radiation losses from Plasma (Bremsstrahlung) 2.77: Cyclotron radiation

ASSIGNMENT QUESTIONS Q No: 1 (2.3-2.4) (a) Lets we have two nuclei of charge ๐‘1 ๐‘’ ๐‘Ž๐‘›๐‘‘ ๐‘2 ๐‘’. The separation between the nuclei โ‰ˆ 5 Fermi. So the energy required to overview the Coulomb repulsion so that the fusion can occur is given by ๐ธ=

๐‘! ๐‘2 ๐‘’ 2 ๐‘…

As ๐‘… = 5 ๐น๐‘’๐‘Ÿ๐‘š๐‘– = 5 โˆ— 1013 ๐‘๐‘š and ๐‘’ = 4.8 ร— 10โˆ’10 esu (statcoulomb) ๐ธ=

๐‘! ๐‘2 (4.8 ร— 10โˆ’10 )2 = ๐‘1 ๐‘2 . 4.608 โˆ— 10โˆ’7 ๐‘’๐‘Ÿ๐‘”๐‘  5 ร— 1013

1 ๐‘’๐‘Ÿ๐‘” =

1 ๐‘’๐‘‰ 1.602 ร— 10โˆ’12

4.608 ร— 10โˆ’7 ๐ธ = ๐‘1 ๐‘2 ๐‘’๐‘‰ 1.602 ร— 10โˆ’12 ๏ƒฐ ๐ธ = ๐‘1 ๐‘2 ร— 2.876 โˆ— 105 ๐‘’๐‘‰ ๏ƒฐ ๐ธ = ๐‘1 ๐‘2 ร— 0.2876 โˆ— 106 ๐‘’๐‘‰ ๏ƒฐ ๐ธ = ๐‘1 ๐‘2 ร— 0.2876๐‘€๐‘’๐‘‰ For hydrogen isotopes ๐‘1 = ๐‘2 = 1 ๏ƒฐ ๐ธ = 0.2876๐‘€๐‘’๐‘‰ This is the required amount of energy to surround the Coulombโ€™s barrier (b) โ€œOne atomic mass unit (1 amu) is equal to the 1/16 of the mass of 8๐‘‚16 โ€. that is 1 1 1 ๐‘Ž๐‘ก๐‘œ๐‘š๐‘–๐‘ ๐‘š๐‘œ๐‘™๐‘’ ๐‘œ๐‘“ 8๐‘‚16 1 16 ๐‘” ๐‘œ๐‘“ 8๐‘‚16 16 1 ๐‘Ž๐‘š๐‘ข = mass of 8๐‘‚ = = 16 16 ๐ด๐‘ฃ๐‘œ๐‘”๐‘Ž๐‘‘๐‘Ÿ๐‘œโ€ฒ ๐‘  ๐‘›๐‘ข๐‘š๐‘๐‘’๐‘Ÿ 16 6.02 ร— 1023 Hisham Shah

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43 1๐‘”

๏ƒฐ 1 ๐‘Ž๐‘š๐‘ข = 6.02โˆ—1023 = 0.166 ร— 10โˆ’23 ๐‘” Now ๐ธ = ๐‘š๐‘ 2 ๏ƒฐ ๐ธ = (0.166 ร— 10โˆ’23 )(3 ร— 1010 )2 ๐‘’๐‘Ÿ๐‘”๐‘  ๏ƒฐ ๐ธ = 0.166 ร— 10โˆ’23 โˆ— 9 ร— 1020 ๐‘’๐‘Ÿ๐‘”๐‘  ๏ƒฐ ๐ธ = 1.494 ร— 10โˆ’3 ๐‘’๐‘Ÿ๐‘”๐‘  1

As 1 ๐‘’๐‘Ÿ๐‘” = 1.602โˆ—10โˆ’12 ๐‘’๐‘‰ ๏ƒฐ ๐ธ=

1.494ร—10โˆ’3 1.602ร—10โˆ’12

๐‘’๐‘‰ = 0.9325 ร— 109 ๐‘’๐‘‰ = 932.5 ๐‘€๐‘’๐‘‰ 1 ๐‘Ž๐‘š๐‘ข = 932.5 ๐‘€๐‘’๐‘‰

Q No: 2 โ€œIt is the amount of energy released in a nuclear reactionโ€ Generally a reaction is written as ๐‘‹+๐‘Ž =๐‘Œ+๐‘+๐‘„ Where Q is the energy released in the reaction and is called Q-value of the reaction. (1) For D-D Reaction To calculate the โ€˜Qโ€™ value of this reaction, we proceed as follows 2 1D

+ P2 โ†’ โ†’

2He 1 1H

3

+ 0n1 + 1T 3

mass of 1D2 = 2.014743 amu mass of 2He3 = 3.016986 amu mass of 0n1 = 1.008987 amu mass of 1H1 = 1.008145 amu mass of 1T 3 = 3.017005 amu (I) Now โˆ†๐‘š = (๐‘š๐ท + ๐‘š๐ท ) โˆ’ (๐‘š๐ป๐‘’ 3 + ๐‘š๐‘› ) ๏ƒฐ โˆ†๐‘š = 4.029486 โˆ’ 4.025973 = 0.003513 ๐‘Ž๐‘š๐‘ข So ๐‘„ = โˆ†๐‘š๐‘ 2 ๏ƒฐ ๐‘„ = 0.003513 ๐‘Ž๐‘š๐‘ข. ๐‘ 2 = 0.003513 โˆ— 931.5๐‘€๐‘’๐‘‰ = 3.27 ๐‘€๐‘’๐‘‰ (II) โˆ†๐‘š = (๐‘š๐ท + ๐‘š๐ท ) โˆ’ (๐‘š๐ป 1 + ๐‘š ๐‘‡ 3 ) ๏ƒฐ โˆ†๐‘š = 4.029486 โˆ’ 4.02515 = 0.004336 ๐‘Ž๐‘š๐‘ข So ๐‘„ = โˆ†๐‘š๐‘ 2

Course: Experimental Plasma Physics

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๏ƒฐ ๐‘„ = 0.004336 ร— 931.5๐‘€๐‘’๐‘‰ = 4.03 ๐‘€๐‘’๐‘‰ To calculate the energy level shared among the reaction products ๐‘ž๐ป๐‘’ 3 =

๐‘š๐‘› ๐‘„ 1.008987 ร— 3.27 3.299 = = = 0.82 ๐‘€๐‘’๐‘‰ ๐‘š๐‘› + ๐‘š๐ป๐‘’ 3 1.008797 ร— 3.016986 4.025973

Now ๐‘ž๐‘› = ๐‘ž๐ป 1 =

๐‘š๐ป๐‘’ 3 ๐‘„ 3.016986 ร— 3.27 = = 2.45 ๐‘€๐‘’๐‘‰ ๐‘š๐‘› + ๐‘š๐ป๐‘’ 3 4.025973

๐‘š๐‘‡3 ๐‘„ 3.017005 ร— 4.03 12.15853015 = = = 3.02 ๐‘€๐‘’๐‘‰ ๐‘š๐ป 1 + ๐‘š ๐‘‡ 3 1.008145 + 3.017005 4.02515

And ๐‘š๐ป 1 ๐‘„ 1.008145 ร— 4.03 = = 1.01 ๐‘€๐‘’๐‘‰ ๐‘š๐ป 1 + ๐‘š ๐‘‡ 3 4.02515

๐‘ž๐‘‡3 = (2) D-T reaction

1D

2

+ T3 โ†’

2He

4

+ 0n1 (17.6 ๐‘€๐‘’๐‘‰)

Now โˆ†๐‘š = (๐‘š๐ท2 + ๐‘š ๐‘‡ ) โˆ’ (๐‘š๐ป๐‘’ 4 + ๐‘š๐‘› ) ๏ƒฐ โˆ†๐‘š = (2.014743 + 3.017005) โˆ’ (4.003874 + 1.008987) = 0.018887 ๐‘Ž๐‘š๐‘ข Now ๐‘„ = โˆ†๐‘š๐‘ 2 ๏ƒฐ ๐‘„ = 0.018887 โˆ— 931.5 ๐‘€๐‘’๐‘‰ = 17.58 ๐‘€๐‘’๐‘‰ And ๐‘ž๐ป๐‘’ 4 = ๐‘ž

1 0๐‘›

=

๐‘š๐‘› ๐‘„ 1.008987 ร— 17.6 = = 3.54 ๐‘€๐‘’๐‘‰ ๐‘š๐‘› + ๐‘š๐ป๐‘’ 4 5.012861 ๐‘š๐ป๐‘’ 4 ๐‘„ 4.003874 ร— 17.6 = = 14.05 ๐‘€๐‘’๐‘‰ ๐‘š๐‘› + ๐‘š๐ป๐‘’ 4 5.012861

(3) D-He3 reaction 1๐ท

2

+ 2๐ป๐‘’ 3 โ†’

2๐ป๐‘’

4

โ†’ 1๐ป1 (18.3 ๐‘€๐‘’๐‘‰)

โˆ†๐‘š = (๐‘š๐ท + ๐‘š๐ป๐‘’ 3 ) โˆ’ (๐‘š๐ป๐‘’ 4 + ๐‘š๐ป 1 ) โˆ†๐‘š = (2.014743 + 3.016986) โˆ’ (4.003874 + 1.008145) = 0.01971 ๐‘Ž๐‘š๐‘ข Now ๐‘„ = โˆ†๐‘š๐‘ 2 ๏ƒฐ ๐‘„ = 0.01971 โˆ— 931.5 ๐‘€๐‘’๐‘‰ = 18.35 ๐‘€๐‘’๐‘‰ Now

Hisham Shah

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45

๐‘ž๐ป๐‘’ 4 =

๐‘š๐ป 1 ๐‘„ 1.008145 ร— 18.3 = = 3.69 ๐‘€๐‘’๐‘‰ ๐‘š๐ป 1 + ๐‘š๐ป๐‘’ 4 5.012019

๐‘ž๐ป๐‘’ 1 =

๐‘š๐ป๐‘’ 4 ๐‘„ 4.003874 ร— 18.3 = = 14.65 ๐‘€๐‘’๐‘‰ ๐‘š๐ป 1 + ๐‘š๐ป๐‘’ 4 5.012019

(4) P-B11 reaction 1๐‘ƒ

2

+ 5๐ต11 โ†’ 2๐ป๐‘’ 4 โ†’ 4๐ต๐‘’ 8 (8.590 ๐‘€๐‘’๐‘‰) (8.682 ๐‘€๐‘’๐‘‰) โ†’ 3๐ป๐‘’ 4

(I) โˆ†๐‘š = (๐‘š๐‘ƒ + ๐‘š๐ต11 ) โˆ’ (๐‘š๐ป๐‘’ 4 + ๐‘š๐ต๐‘’ 8 ) ๏ƒฐ โˆ†๐‘š = (1.007593 + 11.009305) โˆ’ (1.002603 + 8.012053) = 0.002242 ๐‘Ž๐‘š๐‘ข

Now ๐‘„ = โˆ†๐‘š๐‘ 2 ๏ƒฐ ๐‘„ = 0.002242 โˆ— 931.5 ๐‘€๐‘’๐‘‰ = 2.08 ๐‘€๐‘’๐‘‰ (II) โˆ†๐‘š = (๐‘š๐‘ƒ + ๐‘š๐ต11 ) โˆ’ (๐‘š3๐ป๐‘’ 4 ) ๏ƒฐ โˆ†๐‘š = (1.007593 + 11.009305) โˆ’ 3(4.003874) = 0.008782 ๐‘Ž๐‘š๐‘ข Now ๐‘„ = โˆ†๐‘š๐‘ 2 ๏ƒฐ ๐‘„ = 0.008782 โˆ— 931.5 ๐‘€๐‘’๐‘‰ = 8.18 ๐‘€๐‘’๐‘‰ ๐‘š๐ต๐‘’ 8 ๐‘„ 8.012053 ร— 2.08 ๐‘ž๐ป๐‘’ 4 = = = 1.38 ๐‘€๐‘’๐‘‰ ๐‘š๐ต๐‘’ 8 + ๐‘š๐ป๐‘’ 4 4.003374 + 8.012053 ๐‘š๐ป๐‘’ 4 ๐‘„ 4.003374 ร— 2.08 ๐‘ž๐ต๐‘’ 8 = = = 0.69 ๐‘€๐‘’๐‘‰ ๐‘š๐ป๐‘’ + ๐‘š๐ต๐‘’ 12.015927 Q No: 3 (I)

To calculate ๐‘›๐œ fir D-D reactions at 100 KeV we will proceed as follows.

We know that ๐‘›๐œ =

12๐พ๐‘‡ < ๐œŽ๐œˆ > ๐‘Š

Where W is the Q-value for the reaction. < ฯƒฮฝ >DD for 100 KeV temperature = 4.5 โˆ— 10โˆ’23 m3 /sec 12 ร— 100๐พ๐‘’๐‘‰ 12 ร— 100 ร— 103 ๐‘’๐‘‰ ๐‘›๐œ = = 3 m3 โˆ’23 โˆ’23 m ร— 106 ๐‘’๐‘‰ 4.5 ร— 10 ร— 3.65 ๐‘€๐‘’๐‘‰ 16.42 ร— 10 sec sec

๏ƒฐ ๐‘›๐œ =

12ร—105 ๐‘’๐‘‰ ๐‘m3 16.42ร—10โˆ’17 ร—106 ๐‘’๐‘‰ sec

= 0.73 ร— 1016 secโ„cm3

๏ƒฐ ๐‘›๐œ โ‰ˆ 1016 secโ„cm3

Course: Experimental Plasma Physics

46

(II)

๐‘›๐œ for D-T reactions: ๐‘›๐œ =

12๐พ๐‘‡ < ๐œŽ๐œˆ >๐ท๐‘‡ ๐‘Š

Here < ๐œŽ๐œˆ >๐ท๐‘‡ = 7.0 โˆ— 10โˆ’22 ๐‘š3 โ„๐‘  and ๐‘Š = 17.6 ๐‘€๐‘’๐‘‰, so 12 ร— 100๐พ๐‘’๐‘‰ 12 ร— 105 ๐‘’๐‘‰ = m3 ๐‘m3 7 ร— 10โˆ’22 sec ร— 17.6 ๐‘€๐‘’๐‘‰ 7 ร— 10โˆ’16 sec ร— 17.6 ร— 106 ๐‘’๐‘‰ ๐‘›๐œ = 0.09 โˆ— 1015 ๐‘ ๐‘’๐‘ โ„๐‘๐‘š3 ๐‘›๐œ =

(III)

D-He3 ๐‘›๐œ =

12๐พ๐‘‡ < ๐œŽ๐œˆ > ๐‘Š

Here < ๐œŽ๐œˆ >= 1.0 ร— 10โˆ’22 ๐‘š3 โ„๐‘  and ๐‘Š = 18.3 ๐‘€๐‘’๐‘‰, so ๐‘›๐œ = (IV)

12 ร— 105 ๐‘’๐‘‰ = 0.65 ร— 1015 ๐‘ ๐‘’๐‘ โ„๐‘๐‘š3 3 ๐‘m 1 ร— 10โˆ’16 sec ร— 18.3 ร— 106 ๐‘’๐‘‰

P-Li6 ๐‘›๐œ =

12๐พ๐‘‡ < ๐œŽ๐œˆ > ๐‘Š

Here < ๐œŽ๐œˆ >= 0.8 โˆ— 10โˆ’23 ๐‘š3 โ„๐‘  and ๐‘Š = 4.023 ๐‘€๐‘’๐‘‰, so ๐‘›๐œ = (V)

12 ร— 105 ๐‘’๐‘‰ = 3.71 ร— 1016 ๐‘ ๐‘’๐‘ โ„๐‘๐‘š3 3 ๐‘m 0.8 ร— 10โˆ’17 sec ร— 4.023 ร— 106 ๐‘’๐‘‰

P-B reaction ๐‘›๐œ =

12๐พ๐‘‡ < ๐œŽ๐œˆ > ๐‘Š

Here < ๐œŽ๐œˆ >= 5.0 ร— 10โˆ’23 ๐‘š3 โ„๐‘  and ๐‘Š = 8.682 ๐‘€๐‘’๐‘‰, so 12 ร— 105 ๐‘’๐‘‰ ๐‘›๐œ = = 0.27 ร— 1016 ๐‘ ๐‘’๐‘ โ„๐‘๐‘š3 3 ๐‘m 5.0 ร— 10โˆ’17 sec ร— 8.682 ร— 106 ๐‘’๐‘‰

Hisham Shah

Download: phylib.wordpress.com

Experimental Plasma (Hisham Shah).pdf

energy. CREATION OF PLASMA. Plasma can be created by heating a gas or subjecting it to a strong electromagnetic field with. laser or microwave generator.

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