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## +,-./-.*2"*=-#'\$>*-"&*?>(0"%,%@1*

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## A2>\$-\$(01*%B*;,-./-*;0>"%/>"-*

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### C-.2(*+-\$-/>#>\$.D*E;>>&F*G">\$@1*-"&*?>/;>\$-#'\$>*

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,

i

i 2 i=1 N

### "

! f(u)=N m 2"kBTe #1/ 2mu2/k BT ! E = 1/2mu 2 f (u)du "# +#

f (u)du "# +#

! e"a2x2dx "# +#

### \$

= %/a ! E =1/2kBT ! E =3/2kBT

B

-23

-19

-19

B

B

-19

!

E =1/2mu 2

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## +,-./-*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.

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

med 2

"x dt2 =eE

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## +,-./-*I.(2,,-)%".*-"&*+,-./-*J\$>K'>"(1*

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?

! fp = "p 2# =9000 ne Hz ! d2 "x dt2 =# ne2 me

0 "x=#

p 2 "x

p 2

2

e

### #

0 Electron Plasma Frequency

## +,-./-*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 = 0 eE(x)dx #x " =e 2 n#x2 2\$0

(6)

## +,-./-*L\$2#>\$2-*

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]

2

2

0

D

0

B

e

2

### n

Debye Length Plasma Criterion #2

## 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

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)= 1 4#\$0 Q r

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## M>N1>*E02>,&2"@*

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 ! 1 r2 d dr r 2d" dr # \$ % & ' ( =)en *0 e)e"/kBTe )1 + , - ./0 ! 1 r2 d dr r 2d" dr # \$ % & ' () ne 2 *0kBTe + , - . / 0"(r)=11 D 2 "(r)

## M>N1>*E02>,&2"@*

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)= 1 4#\$0 Q r % & ' ( ) *e+r/,D ! "D= #0kBTe e2 n

(8)

## M>N1>*E02>,&2"@*

o  Useful numerical expression for Debye length:

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

!

"D#69 Te/n

## ?0>*+,-./-*+-\$-/>#>\$*

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.

D 3

6

e 3 / 2

1/ 2

(9)

## 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. 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 \$ 3 2kBT =>% =" e 2 6"kB#0 & ' ( ) * + 2 1 T2

(10)

## L%,,2.2%".*

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.

! " # \$p 64% ln& &

p 4

e

B

e

,3 / 2

(11)

## =>O#*Q>(#'\$>D*E2"@,>R;-\$)(,>*S%)%"*

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