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### 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

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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*

### =

### 1

### 2N

*m*

*i*

*u*

*i*2

*i*=1

*N*

### "

!*f*(

*u*)=

*N*

*m*2"

*k*#1/ 2mu2

_{B}Te_{/}

_{k}*BT*!

*E*=

*1/2mu*2

*f (u)du*"# +#

### $

*f (u)du*"# +#

### $

!*e*"

*a*2

*x*2

*"# +#*

_{dx}### $

= %/*a*!

*E*=1/2

*k*!

_{B}T*E*=3/2

*k*

_{B}T**C-.2(*+-$-/>#>$.D*E;>>&F*G">$@1*-"&*?>/;>$-#'$>***

### o

### Using

*<E> = 3/2 k*

_{B}*T*

### , 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 **

*k*

_{B}*T *

### used for temperature, where

*k*

_{B}*T *

### = 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* =1/2*mu*
2

**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 n_{e }= n_{i} = 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}*

*m _{e}d*
2

"*x*
*dt*2 =*eE*

**+,-./-*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 n_{e} is in cm-3_{. }

o In Solar System, f*p 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
!
*d*2
"x
*dt*2 =#
*ne*2
*me*

### $

0 "x=#### %

*p*2 "x

### !

### "

*p*2

### =

*ne*

2
*m*

_{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=1/2k*BTe*). Work by displacement of electron by *!x is and using E = -ne!x / #0*

!
*W* = "*Fdx*
= 0 *eE*(*x*)*dx*
#*x*
"
=*e*
2
*n*#*x*2
2$0

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

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

o The maximum distance an electron can travel is *!x _{max} = &_{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 N_{D} = n[4'&_{D}3_{/3] }

### !

*e*

2
*n*

### "

*x*

2
### 2

### #

0### $

### 1

### 2

*kBTe*

### !

### "

*D*

### =

### #

0*k*

*B*

*T*

*e*

*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 n*i = ne = n *

o At t = 0, electric potential is

o As time progresses, electrons are attracted, while ions are repelled. As m*i >> m*e

neglect motion of ions.

o At t >> 0, n*e > n and a new potential is set up, with charge density: *

*( = e(n _{e} – n_{i}). *
!
"(

*r*)= 1 4#$0

*Q*

*r*

**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*)| << k _{B}T_{e }* can us Taylor expansion: e

*x*

_{*}_{ 1 + x => }* where & _{D}* is the Debye shielding length.

!
"2
#*(r)*=$%
&0
=$*e(ne*$*ni*)
&0
!
*ne*(*r*)=*ne*
"*e*#(*r*)/*kBTe*
!
1
*r*2
*d*
*dr* *r*
2*d*"
*dr*
#
$
% &
'
( =)*en*
*0
*e*)*e*"/*kBTe*
)1
+
,
- ._{/}0
!
1
*r*2
*d*
*dr* *r*
2*d*"
*dr*
#
$
% &
'
() *ne*
2
*0*kBTe*
+
,
- .
/
0"(*r*)=_{1}1
*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*=
#0*kBTe*
*e*2
*n*

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

o Useful numerical expression for Debye length:

where T* _{e} 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 N* _{D }*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

### %

### 10

6*T*

*e*3 / 2

*n*

1/ 2
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**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

_{ m}2

_{, which is much larger than the geometric nuclear }

cross-section of 10-33_{ m}2_{. }
!
*e*2
4"#0*rc*
$ 3
2*kBT*
=>% =" *e*
2
6"*kB#*0
&
'
( )
*
+
2
1
*T*2

**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&
&

### !

### "

### #

### 2

### $

*p*4

### 64

### %

*n*

*e*

*k*

*B*

*T*

*m*

*e*

### &

### '

### (

### )

### *

### +

,3 / 2### ln

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