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Dr. Peter T. Gallagher Astrophysics Research Group

Trinity College Dublin

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o  “A plasma is a quasi-neutral gas consisting of positively and negatively charged particles (usually ions and electrons) which are subject to electric, magnetic and other forces, and which exhibit collective behaviour.” o  Schwartz, Owen and Burgess, “Astrophysical Plasmas”.

o  Langmuir in 1920s showed that an ionised gas can support oscillations, which resemble a jelly-like substance. He named it a “plasma”.

o  Scientific term for “plasma” was introduced in 1839 by Czech biologist to describe a jelly-like medium of cells.

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o  For ensemble of N particles of mass m and velocity u, average energy per particle is

o  In thermal equilibrium, particles have Maxwell-Boltzmann distribution of speeds.

o  Average KE can be calculated using

o  Integrating numerator by parts, and denominator using we have or in 3D,

!

E =12N

miui2

i=1

N

"

!

f (u) = N m2"kBT

e#1/ 2mu2 / kBT

!

E =1/2mu2 f (u)du

"#

+#

$f (u)du

"#

+#

$

!

e"a2x 2dx

"#

+#

$ = % /a

!

E =1/2kBT

!

E = 3/2kBT

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o  Using <E> = 3/2 kBT, average KE of a gas at1 K is ~2.07 x 10-23 J.

o  Plasma temperature often given in electron Volts (eV), where 1 eV = 1.602 x 10-19 Coulombs (C).

o  For electron or proton, |q| = 1.602 x 10-19 C, therefore

o  Note: Typically, energy corresponding to kBT used for temperature, where kBT = 1 eV = 1.602 x 10-19 J.

o  Q. Show that a 0.5 eV plasma corresponds to a temperature of 5,800 K (solar photosphere).

!

E =1/2mu2

2q eV

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o  Density range: >30 orders of magnitude. Temperature range: ~10 orders.

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o  Consider plasma of equal number of positive and negative charges. Overall, plasma is neutral, so ne = ni = n

o  Now displace group of electrons by !x.

o  Charge separation gives E, which accelerates electrons towards initial position. Electrons overshoot equilibrium position.

o  Using Newton’s Law, (1)

o  Displacement sets up E across distance L, similar to parallel plate capacitor.

o  Charge per unit area of slab is " = -ne!x => E = " / #0 = -ne!x / #0 !

med2"xdt 2

= eE

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o  Therefore, Eqn. (1) can be written

where

o  Plasma oscillations are result of plasma trying to maintain charge neutrality.

o  Plasma frequency commonly written where ne is in cm-3.

o  In Solar System, fp ranges from hundreds of MHz (in solar corona) to <1 kHz (near outer planets).

o  What is plasma frequency of Earth’s ionospere? What implication does this have for (i) short wave radio communications and (ii) radio astronomy?

!

f p =" p

2#= 9000 ne Hz

!

d 2"xdt 2

= #ne2

me$0"x = #% p

2"x

!

" p

2 =ne2

me#0Electron Plasma

Frequency

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o  In a partially ionized gas where collisions are important, plasma oscillations can only develop if mean free time between collisions ($c) is long compared with the oscillation period ($p=1/%p).

o  That is, $c >> $p or $c /$p >> 1 Plasma Criterion #1

o  Above is a criterion for an ionized gas to be considered a plasma.

o  Behaves like neutral gas is criterion is not true.

o  Plasma oscillations can be driven by natural thermal motions of electrons (E=1/2kBTe). Work by displacement of electron by !x is and using E = -ne!x / #0

!

W = Fdx"

= eE(x)dx0

#x"

=e2n#x 2

2$0

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o  Equating work done by displacement with average energy in thermal agitation

o  The maximum distance an electron can travel is !xmax = &D, where

o  Gas is considered a plasma if length scale of system is larger than Debye Length:

&D<< L

o  Debye Length is spatial scale over which charge neutrality is violated by spontaneous fluctuations.

o  Debye Number is defined as ND = n [4'&D3/3]

!

e2n"x 2

2#0$12kBTe

!

"D =#0kBTe

e2nDebye Length

Plasma Criterion #2

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o  Plasmas do not contain strong electric fields as they reorganize to shield from them.

o  Plasma oscillations are excited to assert macroscopic neutrality.

o  If plasma subjected to external E, free charges redistribute so that plasma is shielded.

o  Suppose immerse test particle +Q within a plasma with ni = ne = n

o  At t = 0, electric potential is

o  As time progresses, electrons are attracted, while ions are repelled. As mi >> me neglect motion of ions.

o  At t >> 0, ne > n and a new potential is set up, with charge density: ( = e(ne – ni).

!

"(r) =14#$0

Qr

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o  New potential evaluated using Poisson’s equation:

o  In presence of potential, electron number density is

o  Subbing this into Poisson’s equation in spherical coordinates.

o  For |e)| << kBTe can us Taylor expansion: ex * 1 + x =>

where &D is the Debye shielding length.

!

" 2#(r) = $%&0

= $e(ne $ ni )

&0

!

ne (r) = ne"e#(r)/kBTe

!

1r 2ddr

r 2 d"dr

# $ %

& ' ( = )

en*0

e)e" /kBTe )1+ , -

. / 0

!

1r 2ddr

r 2 d"dr

# $ %

& ' ( )

ne2

*0kBTe

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, -

.

/ 0 "(r) =

11D

2"(r)

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o  Solution to previous is

o  As r -> 0, potential is that of a free charge in free space, but for r >> &D potential falls exponentially.

o  Coloumb force is long range in free space, but only extends to Debye length in plasma.

o  For positive test charge, shielding cloud contains excess of electrons.

o  Recall

=> size of shielding cloud increases as electron temperature as electrons can overcome Coulomb attraction. Also, &D is smaller for denser plasma because more electrons available to populate shielding cloud.

!

"(r) =14#$0

Qr

%

& '

(

) * e+r /,D

!

"D =#0kBTe

e2n

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o  Useful numerical expression for Debye length:

where Te is in K and n is in m-3.

!

"D # 69 Te /n

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o  The typical number of particles in a Deybe sphere is given by the plasma parameter:

o  Defined as ND in Inan & Gollowski and usually given as argument of Coulomb Logarithm (log(!)).

o  If !<<1, the Debye sphere is sparsely populated, corresponding to a strongly coupled plasma.

o  Strongly coupled plasmas tend to be cold and dense, whereas weakly coupled tend to be diffuse and hot.

!

" = 4#n$D

3

=1.38%106Te

3 / 2

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o  Neutral particles have quite small collision cross sections. As Coulomb force is long range, charged particles are more frequent.

o  A collisional plasma is one where &mfp << L

where L is the observational length scale and &mfp = 1/"n is the mean free path.

o  The effective Coulomb cross-section is " = ' rc2

o  An electron will be affected by a neighbouring ion if the Coulomb potential is of the order of the electron thermal energy:

o  At T = 106 K, " ~ 10-22 m2, which is much larger than the geometric nuclear cross-section of 10-33 m2.

!

e2

4"#0rc$32kBT

=>% = "e2

6"kB#0

&

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o  In terms of the plasma parameter, the collision frequency (" = n e v) is

where ln(+) is the Coulomb logarithm. Used as ! is large, but 10 < ln(+) <30.

o  In a weakly coupled plasma, , << %p => collisions to not effect plasma oscillations

o  More rigorously, it can be shown that

o  Thus, diffuse, high temperature plasmas tend to be collisionless. See page 156 of Inan & Golkowski.

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* From Plasma Physics by R. Fitzpatrick

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