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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
+
, -
.
/ 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.
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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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