!"#$%&'()%"*#%*+,-./-*+01.2(.*3+456789* !"#$%&"'()(*(+,(-.'/#(012#"3$'(
Dr. Peter T. Gallagher Astrophysics Research Group Trinity College Dublin
:0-#*2.*-*;,-./-<* o “A plasma is a quasi-neutral gas consisting of positively and negatively charged particles (usually ions and electrons) which are subject to electric, magnetic and other forces, and which exhibit collective behaviour.” o Schwartz, Owen and Burgess, “Astrophysical Plasmas”.
o Langmuir in 1920s showed that an ionised gas can support oscillations, which resemble a jelly-like substance. He named it a “plasma”.
o Scientific term for “plasma” was introduced in 1839 by Czech biologist to describe a jelly-like medium of cells. +,-./-.*2"*=-#'$>*-"&*?>(0"%,%@1*
,&-)'-*)+.'(+) /&("('$) !"#$%&#'()%#"&*+)
4"6'2'6+)
0'+$()1("*&%$*)1#'+2'+) 3#'+2')45+)
A2>$-$(01*%B*;,-./-*;0>"%/>"-*
3#'+2')37$-"2$-')
,8-9#$:1'(;%#$) <8+.(8=&;"-) ?"#.@2'--) !";"-) >&-%;"-) AB&';"-)
!"2$-.+)"C)?"#.@2'--) AB&';"-)
,8-9#$)D&8*) !&#;1#$)) !E<) D&8*+) C-.2(*+-$-/>#>$.D*E;>>&F*G">$@1*-"&*?>/;>$-#'$>* o For ensemble of N particles of mass m and velocity u, average energy per particle is N 1 2 E = "miui 2N i=1 o In thermal equilibrium, particles have Maxwell-Boltzmann distribution of speeds.
m 2 #1/ 2mu / kBT ! f (u) = N e 2"kBT o Average KE can be calculated using +# 1/2mu2 f (u)du ! $"# E = +# f (u)du $"#
+# 2 2 o Integrating numerator by parts, and denominator using e"a x dx = % /a $"# we have E =1 /2k T or in 3D, ! B E = 3/2kBT ! ! !
C-.2(*+-$-/>#>$.D*E;>>&F*G">$@1*-"&*?>/;>$-#'$>*
-23 o Using
1/2mu2 E = eV 2q o Note: Typically, energy corresponding to kBT used for temperature, where k T = 1 eV = 1.602 x 10-19 J. B ! o Q. Show that a 0.5 eV plasma corresponds to a temperature of 5,800 K (solar photosphere). H-"@>*%B*;,-./-*;-$-/>#>$.** o Density range: >30 orders of magnitude. Temperature range: ~10 orders.
+,-./-*I.(2,,-)%".*-"&*+,-./-*J$>K'>"(1* o Consider plasma of equal number of positive and negative charges. Overall, plasma is neutral, so ne = ni = n o Now displace group of electrons by !x. o Charge separation gives E, which accelerates electrons towards initial position. Electrons overshoot equilibrium position.
d 2"x o Using Newton’s Law, m = eE (1) e dt 2 o Displacement sets up E across distance L, similar to parallel plate capacitor. ! o Charge per unit area of slab is " = -ne!x => E = " / #0 = -ne!x / #0 +,-./-*I.(2,,-)%".*-"&*+,-./-*J$>K'>"(1*
2 2 d "x ne 2 o Therefore, Eqn. (1) can be written 2 = # "x = #% p "x dt me$0
2 where 2 ne " = Electron Plasma p m # Frequency ! e 0 o Plasma oscillations are result of plasma trying to maintain charge neutrality.
" p o Plasma frequency commonly! written f p = = 9000 ne Hz -3 2# where ne is in cm .
In Solar System, f ranges from hundreds of MHz (in solar corona) to <1 kHz (near o p ! 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?
+,-./-*L$2#>$2-* o In a partially ionized gas where collisions are important, plasma oscillations can only develop if mean free time between collisions ($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
#x = "0 eE(x)dx e 2n#x 2 =
2$0
! +,-./-*L$2#>$2-* o Equating work done by displacement with average energy in thermal agitation e 2n"x 2 1 $ kBTe
2#0 2 o The maximum distance an electron can travel is !xmax = &D, where # k T ! " = 0 B e Debye Length D e 2n o Gas is considered a plasma if length scale of system is larger than Debye Length:
! &D<< L Plasma Criterion #2 o Debye Length is spatial scale over which charge neutrality is violated by spontaneous fluctuations.
3 o Debye Number is defined as ND = n [4'&D /3]
M>N1>*E02>,&2"@* 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
1 Q o At t = 0, electric potential is "(r) =
4#$0 r 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). M>N1>*E02>,&2"@* o New potential evaluated using Poisson’s equation:
2 % e(n $ n ) " #(r) = $ = $ e i
&0 &0 o In presence of potential, electron number density is "e#(r)/k T ! B e ne (r) = ne o Subbing this into Poisson’s equation in spherical coordinates.
1 d # 2 d"& en+ )e"/k T . ! r = ) e B e )1 2 % ( - 0 r dr $ dr ' *0 , /
x o For |e)| << kBTe can us Taylor expansion: e * 1 + x =>
! 2 1 d # 2 d"& + ne . 1 r (r) (r) 2 % ( ) - 0" = 2 " r dr $ dr ' ,* 0 kBTe / 1D
where &D is the Debye shielding length. !
M>N1>*E02>,&2"@* o Solution to previous is % 1 Q( +r /, "(r) = ' *e D & 4#$0 r )
o As r -> 0, potential is that of a free charge in free space, but for r >> &D potential falls exponentially.! o Coloumb force is long range in free space, but only extends to Debye length in plasma. o For positive test charge, shielding cloud contains excess of electrons.
# k T o Recall " = 0 B e D e 2n => size of shielding cloud increases as electron temperature as electrons can
overcome Coulomb attraction. Also, &D is smaller for denser plasma because more !electrons available to populate shielding cloud. M>N1>*E02>,&2"@* o Useful numerical expression for Debye length:
"D # 69 Te /n -3 where Te is in K and n is in m .
!
?0>*+,-./-*+-$-/>#>$* o The typical number of particles in a Deybe sphere is given by the plasma parameter:
3 " = 4#n$D 1.38 %10 6 T 3 / 2 = e n1/ 2 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. ?0>*+,-./-*+-$-/>#>$*
L%,,2.2%".* 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.
2 o The effective Coulomb cross-section is " = ' rc o An electron will be affected by a neighbouring ion if the Coulomb potential is of the order of the electron thermal energy: e 2 3 $ kBT
4"#0rc 2
2 & e 2 ) 1 => % = "( + 2 ' 6"kB#0 * T o At T = 106 K, " ~ 10-22 m2, which is much larger than the geometric nuclear cross- section of 10-33 m2. ! L%,,2.2%".* o In terms of the plasma parameter, the collision frequency (" = n e v) is
$ p ln& " # 64% &
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
,3 / 2 4 2$ p & k T ) " # ( B + ln- 64%ne ' me * o Thus, diffuse, high temperature plasmas tend to be collisionless. See page 156 of Inan & Golkowski. !
GO-/;,>.*%B*P>1*;,-./-*;-$-/>#>$.*
* From Plasma Physics by R. Fitzpatrick =>O#*Q>(#'$>D*E2"@,>R;-$)(,>*S%)%"*
3#'+2')37$-"2$-')
,8-9#$:1'(;%#$) <8+.(8=&;"-) ?"#.@2'--) !";"-) >&-%;"-) AB&';"-)
!"2$-.+)"C)?"#.@2'-) AB&';"-)
,8-9#$)D&8*) !&#;1#$)) !E<) >#&8*+)