Pukhov Alexander Introduction to Plasma Theory

72 Pages·2.192 MB·English

Introduction to Plasma Theory

By Alexander Pukhov, WS2010/2011

Literature.

1. Introduction to plasma physics: R.J.Goldston and P.H.Rutherford (IoPP, Bristol,
England, 1995)
2. Fundamentals of Plasma Physics: Paul M. Bellan (ISBN 0521821169, 609 p., Cambridge
University Press, Cambridge, UK, 2006)
3. The theory of plasma waves: T.H.Stix, 1st edition (McGrow-Hill, N.Y., 1962).
4. Physics of fully ionized gases: LSpitzer, Jr., 1st edition (Interscience, N.Y., 1956)
5. Plasma physics: R.A.Cairns (Blackie, Glasgow, Scotland, 1985).
6. И.А.Котельников, Г.В.Ступаков, Лекции по физике плазмы (НГУ, Новосибирск,
Россия).
7. Principles of Plasma Physics: Krall, Trivelpiece (San Francisco Pr., 1986).
8. The physics of laser-plasma interactions: W. Kruer (Addison-Wesley, 1988).

Lecture 1. Definition

Plasma is a quasi-neutral gas of charged particles. The most general case: electrons and
positively charged ions. Plasma may contain neutral atoms. In this case plasma is called partially
or incompletely ionized. Otherwise plasma is completely ionized.

The term ―plasma‖ was introduced in the work of Langmuir and Tonks in 1929 when they
studied processes in electronic lamps filled with ionized gases. Now we call this case discharge
in low pressure gas. The natural example is lightning.

Modern plasma physics emerged in 1950-ies, when the idea of thermonuclear reactor was put
forward. In turn, this activity was started by the H-bombs developed in the U.S. and the U.S.S.R.
in 1952 and 1953. It was quickly recognized, however, that the fusion energy is unlikely to be
useful in the nearest future and is not of military use. The fusion energy works were declassified
in 1958. For the thermonuclear reaction to work, one needs temperatures of several 10 keV (100
millions K).

Fusion progress was slow through most of the 1960‘s, but by the end of that decade the
empirically developed Russian tokamak configuration began producing plasmas with parameters
far better than the lackluster results of the previous two decades. By the 1970‘s and 80‘s many
tokamaks with progressively improved performance were constructed and at the end of the 20th
century fusion break-even had nearly been achieved in tokamaks. International agreement was
reached in the early 21st century to build the International Thermonuclear Experimental Reactor
(ITER), a break-even tokamak designed to produce 500 megawatts of fusion output power.
Non-tokamak approaches to fusion have also been pursued with varying degrees of success ;
many involve magnetic confinement schemes related to that used in tokamaks. In contrast to
fusion schemes based on magnetic confinement, inertial confinement schemes were also
developed in which high power lasers or similarly intense power sources bombard millimeter
diameter pellets of thermonuclear fuel with ultra-short, extremely powerful pulses of strongly
focused directed energy. The intense incident power causes the pellet surface to ablate and in so
doing, act like a rocket exhaust pointing radially outwards from the pellet. The resulting radially
inwards force compresses the pellet adiabatically, making it both denser and hotter ; with
sufficient adiabatic compression, fusion ignition conditions are predicted to be achieved.
Simultaneous with the fusion effort, there has been an equally important and extensive study of
space plasmas. Measurements of near-Earth space plasmas such as the aurora and the
ionosphere have been obtained by ground-based instruments since the late 19th century. Space
plasma research was greatly stimulated when it became possible to use spacecraft to make
routine in situ plasma measurements of the Earth‘s magnetosphere, the solar wind, and the
magnetospheres of other planets. Additional interest has resulted from ground-based and
spacecraft measurements of topologically complex, dramatic structures sometimes having
explosive dynamics in the solar corona. Using radio telescopes, optical telescopes, Very Long
Baseline Interferometry and most recently the Hubble and Spitzer spacecraft, large numbers of
astrophysical jets shooting out from magnetized objects such as stars, active galactic nuclei, and
black holes have been observed. Space plasmas often behave in a manner qualitatively similar to
laboratory plasmas, but have a much grander scale.
Since the 1960‘s an important effort has been directed towards using plasmas for space
propulsion. Plasma thrusters have been developed ranging from small ion thrusters for
spacecraft attitude correction to powerful magnetoplasmadynamic thrusters that –given an
adequate power supply – could be used for interplanetary missions. Plasma thrusters are now in
use on some spacecraft and are under serious consideration for new and more ambitious
spacecraft designs.
Starting in the late 1980‘s a new application of plasma physics appeared – plasma processing – a
critical aspect of the fabrication of the tiny, complex integrated circuits used in modern
electronic devices. This application is now of great economic importance.
In the 1990‘s studies began on dusty plasmas. Dust grains immersed in a plasma can become
electrically charged and then act as an additional charged particle species. Because dust grains
are massive compared to electrons or ions and can be charged to varying amounts, new physical
behavior occurs that is sometimes an extension of what happens in regular plasma and
sometimes altogether new. In the 1980‘s and 90‘s there has also been investigation of non-
neutral plasmas; these mimic the equations of incompressible hydrodynamics and so provide a
compelling analog computer for problems in incompressible hydrodynamics. Both dusty
plasmas and non-neutral plasmas can also form bizarre strongly coupled collective states where
the plasma resembles a solid (e.g., forms quasi-crystalline structures). Another application of
non-neutral plasmas is as a means to store large quantities of positrons.
In addition to the above activities there have been continuing investigations of industrially
relevant plasmas such as arcs, plasma torches, and laser plasmas. In particular, approximately
40% of the steel manufactured in the United States is recycled in huge electric arc furnaces
capable of melting over 100 tons of scrap steel in a few minutes. Plasma displays are used for flat
panel televisions and of course there are naturally occurring terrestrial plasmas such as lightning.

The term plasma is not limited to the most common electron-ion case. One talks about electron-
positron plasmas, quark-gluon plasmas. Semiconductors contain plasma consisting of electrons
and holes.

It is convenient to measure plasma temperature in eV, 1 eV=11600K.

Let us consider plasma with temperature T and density n. When the density is high, quantum
effects may become important. When the density is low, plasma is classical.

In plasma with temperature T, the characteristic momentum of particles is p  2mT the
corresponding de Broile wavelength is
h
 ~
B
2mT

If  is small compared to the interparticle distance n-1/3, plasma is classical. Thus, to have
B
classical plasma we need high temperatures:

h2n2/3
T 
2m

This condition fails first for electrons.

Another important characteristic is the relative role of electrostatic interaction of particles and
the kinetic energy of particles.

The mean electrostatic energy is
e2
W ~
E 2n1/3

The mean kinetic energy is T. Thus, if

T e2n1/3

One speaks about ideal plasma, where Coulomb collisions are negligible. Otherwise, plasma is
non-ideal.

For quantum or degenerate plasma, where

2n2/3
T 
2m

the criterion for ideal plasma changes, because the mean kinetic energy of plasma electrons is
not their temperature anymore. Rather, the mean electron energy is defined by the inter-particle
distance p~/x, so that

2n2/3
W ~
K 2m

On the order of magnitude this is the Fermi energy of a degenerate e-gas. Thus, for the quantum
plasma to be ideal, its density must be high:

3
me2 
nn  
*  2 

In the plane (n,T) we define then four regions of (1) classical ideal plasma; (2) classical non-
ideal; (3) quantum non-ideal; (4) quantum ideal plasmas.
All four regions have the one common point at
3
me2 
n~ n   a3 6,751024cm3
*  2  B
1 me4
T ~T  e2n1/3  Ry13,6eV
* 2 * 22

Fig. 1.1
1- ideal classic plasma; 1‘ – low temperature (<10 eV); 1‘‘- high temperature (>>10eV); 2-
non-ideal classic; 3-non-ideal quantum; 4 – ideal quantum plasma.

Plasma quasi-neutrality.

Charge fluctuations generate electrostatic potential:

24en

Rough estimate gives

2/l2

Where l is the characteristic fluctuation scale. Thus,

4enl2

On the other hand, the characteristic potential energy e cannot be larger than the mean kinetic
energy of particles T. Thus,
n T
~
n 4ne2l2

Introducing the Debye radius

T
r 
D 4ne2
We claim that plasma is quasi-neutral on distances much larger than the Debye radius.

n r2
~ D
n l2

If the plasma size is comparable with r , then it is not plasma, but rather just a heap of charged
D
particles.

Characteristic time scale for the charge fluctuations is

1/2 1/2 1/2
r  T  m  m 
t ~ D ~    ~ 
v 4ne2  T  4ne2 
Te

The inverse value is called plasma frequency

1/2
4ne2 
  
p  m 

Let us show how this frequency appears in electrodynamics equations.

 EcB4j
t
jenv
md veE
t

Assuming no magnetic fields, B=0, and n<<n, we get in the linear approximation

 E4env
t
md veE
t

Taking time derivative of the second equation, we arrive at

d2v2v0
t p

That is an oscillator equation with the eigenfrequency  .
p
Lecture 2. Debye schielding.

Potential around an external charge embedded into plasma decays much faster than the Coulomb
law would give in vacuum. Particles are distributed according to the Boltzmann law:

 e
nn exp 
0  T 

If we suppose e<<T, we can Taylor develop the exponent:

 e e
exp 1
 T  T

and we get

 1 1 
nn n n e  
i e 0 T T 
 
i e

Then, the Poisson Eq. is


24en
r2
D
(2.1)
1  1 1 
4e2n   
r2 0T T 
 
D i e

We look for spherically symmetrical solutions of (2.1):

1 d d 
r2 
r2 dr dr r2
D

Its solution vanishing at infinity is

q  r 
 exp  (2.2)
 
r r
 
D
Thus the potential in plasma is exponentially shielded.
Fig. 2.1 Debye shielding vs vacuum Coulomb.

Returning back to the expression (2.2) we see that the potential is a sum of the Coulomb
potential of the charge and that of the surrounding plasma cloud. The latter can be found as the
difference

q  r   q
scr  r exp r 1 r , r<<rD (2.3)
  D   D

This is potential created by the whole plasma at the charge position. Thus, the interaction energy
charge-plasma is

q2
q  (2.4)
scr r
D

In the same way one finds electrostatic interaction energy of all charges per unit plasma volume:

1 1
w  Z e2n (2.5)
2 r s s
s D

If the ions are single charged, then

ne2
w (2.6)
r
D

Energy per single particle is then W =w/2n:
E
e2 T
W   (2.7)
E 2r 6N
D D

4
where N  r3n is the number of particles in the Debye sphere.
D 3 D

The ideal plasma condition of the electrostatic energy minimum reduces to

N 1
D

Lecture 3. Thermodynamic equilibrium, Saha formula

The most natural way of plasma producing is to heat up a gas to high temperatures. How does
the ionization degree depend on the temperature?

For simplicity we consider atomic hydrogen plasma. Statistics tells us that the probability to find
an electron in a particular energy state  is
k

  
w  Aexp k  (3.1)
k  T 

Bounded hydrogen levels are:

me4
  (3.2)
k 22k2

For simplicity we take into account only the ground state of hydrogen with =-I, I=Ry=13.6 eV.

One finds the factor A demanding

w 1 (3.3)
k
k

This gives

1
  I    
A exp  exp k  (3.4)
 T  0  T 
k

Because the unbound levels are continuous, we have to integrate:

d3pd3r
  (3.5)
23
Then,

Where v = V/N = 1/n .is the volume per electron. Introducing de Broile wavelength
e e
  2/ mT we find
B

  I  1 
Aexp   (3.6)
 
 T  3n 
B e

The probability to find a neutral atom is then

eI/T
w  (3.7)
a eI/T 1/3n
B e

The probability to find a free electron is

1/3n
w 1w  B e (3.8)
i a eI/T 1/3n
B e

Their relation is

n eI/T
i  (3.9)
n 3n
a B e

Taking into account the level degeneracy we come to the Saha formula

n n g g eI/T
i e  i e  K(T) (3.10)
n g 3
a a B

K(T) is the equilibrium constant.
Let us define the ionization level :
N
 i
N
a0

and we denote n  N /V the density of neutral atoms. Then
a0 a0

n n n n (1)n
e i a0, a a0

From the Saha formula we find for the ionization rate

2 g g eI/T
 i e  K(T)
1 g 3
a B