PHYSICS OF PLASMAS
In this course we shall study the basics of plasma physics, with
on waves and instabilities.
TEXT: Required: Chen, Francis: Introduction
to Plasma Physics and Nuclear
Volume I, (Plenum)
Supplementary: Lea, Susan: Mathematics
for Physicists (Brooks/Cole)
We shall cover almost the entire text during the semester, with a
amount of supplementary material. Please give me feedback as we go
I want to know when you are having difficulties so that we can clear
Part of the class assignment is a term paper in
which you can explore some topic in Plasma Physics in more depth than
can get into in class. The paper may be a review paper based on a
search, or it may involve original calculations or a computer project.
Please begin to think about your paper right away.
In the problems I am looking for an honest
and a steady growth in capability. Some of the problems have solutions
in the back of the book. Simply copying these solutions will not help
learn. Try to use the solutions only as a guide if you really get
(It takes will power!)
Supplementary texts: There are many interesting and helpful
books on Plasma Physics. I especially recommend:
- Electrodynamics of particles and plasmas: Clemmow and
- Principles of Plasma Physics: Krall and Trivelpiece (Out
but is in the library.)
- Introduction to plasma physics: Nicholson. (Wiley).
- The framework of plasma physics: Hazeltine and
- Introduction to Plasma Physics: Goldston and
- The Physics of Plasmas: Boyd
and Sanderson, Cambridge, 2003
- Fundamentals of Plasma Physics:
Bellan, CUP 2006
- Physics of Space plasmas.
George Parkes, Wetsview, 2004
Midterm (in class): Thursday April 19th
|% of paper grade
||Thursday, February 22
|Abstract and outline:
||Tuesday March 20
||Tuesday April 24
||Tuesday May 8
||Tuesday May 22
||May 15, 24
Class schedule (Rough outline - subject to change)
More about the paper.
The paper will take the form of a literature review, or a summary
of your own work, if you did an original project.
If you choose to do a literature review, I expect to see sources
the refereed journals that are of a technical nature. Avoid the
to stick with Scientific American and the Web! This is a graduate
course, and I expect to see a paper at the appropriate level, with
If you do an original calculation, I expect to see more detail than
would be typical in a journal article. Show me what you
If you use a computer program, include the source code as an appendix.
Your topic statement should include:
- A title.
- A one or two paraagraph discussion of the topic.
- At least one major reference.
The international thermonuclear reactor.
Wave-wave interactions and their effect on inertial fusion.
Conditions in the coronal plasma as determined by x-ray spectroscopy.
An analysis of the Kruskal-Schwarzschild instability using an energy
Propagation of whistlers in the Earth's upper atmosphere.
Determination of the intergalactic magnetic field.
Magnetic field reconnection.
(Don't know what all those things are? Look them up!)
Abstract and outline.
Abstract: A one or two paragraph summary of your paper.
Outline: A list of sections and subsections, with a one or two
sentence description of the contents of each.
A list of references. (You should have a dozen or so by now.)
An almost complete paper, typed, double-spaced, 10-20 pages. (A paper
contains original work may be shorter than a review paper.) A few
may be left to discover. Your reference list should be essentially
Art work may need a final polish. Paper must include:
- Author name (you!)
- Body (divided into sections and subsections as necessary)
- List of references
- Figures and figure captions where appropriate.
Research should now be complete. You should have responded to comments
made on the first draft. Reference list must be complete, art work
be in final form.
Your best effort, including response to comments on the second draft.
should be in a form that you would be ready to send to a journal or
to your supervisor.
Have clear, legible viewgraphs or power-point slides (if you use them)
that show at most one or two big ideas. One clear diagram is worth 6
cluttered ones. Have an outline of your talk written out (some people
to use index cards). Time your presentation beforehand to make sure the
important stuff is included. Remember: your audience knows less about
topic than you do - go slowly and explain everything!
There are several styles for citing references. Within the body of your
paper you may use a number to label your sources in the order they
Then you list the references by number at the end of the paper. The
used by the Astrophysical Journal (and most astrophysics journals) is
cite the reference by Author and year (Lea, 2007). At the end of the
you list the references alphabetically by author. Convention dictates
more than three authors become et. al. (Lea et.al.,
In the reference list, the entry should be:
Lea, S.M., 2007, Journal of irreproducible results, 15,
Here the 15 is the volume number and the 75 is the page number.
To cite references on the web, give the author's name, title of the
work in quotes, title of complete work (if appropriate) in italics, the
complete URL, and the date of the document or of your visit to the
See the following site for more info:
A word of warning: The web is not refereed! Look carefully at
the credibility of your site. A NASA site probably has reliable
but Joe Bloe's home page may not. Use some common sense.
A paper with only (or even predominantly) web site references
is not acceptable.
I expect to see some papers from the professional literature as well.
Here is a useful reference
on how to write a good paper.
Problem Set #1 Due: Thursday February 1
These problems will get you used to the orders of magnitude encountered
in laboratory and space plasmas. Use a log-log scale for the
- Chen 1.3 (pg 12)
Include the following additional astrophysical plasmas, extending the
axes as appropriate:
Chen 1-11 (page 17)
Compute the magnitude of the shielded point source potential
(5) in Debye notes) and numerically compare the result with the
unshielded potential at distances 0.1, 1 and 5 times the Debye length
the charge. Comment.
- 8) cluster of galaxies: ne = 103 m-3,
T = 10 keV.
- 9) Supernova remnant: ne = 105 m-3,
= 1 keV
- 10) Solar interior: ne = 1032 m-3,
- 11) Solar corona: ne = 1014 m-3,
Problem set #2 Due: Thursday February 8th
- Chen 2-1 (pg 25) Also compute omegac, and include
e) An electron near the surface of an accreting neutron star: E = 10
keV, B = 108 T
- Chen 2-7
- Chen 2-8. (pg 35)
- Chen 2-12 (pg 35)
- Chen 2-13 (pg 49)
Problem Set 3 Due: Thursday February 15th
- Chen 2-16
- Chen 2-17
- Refer to problem Chen 2-8. Calculate the bounce frequency for the
protons and 30 keV electrons. (You'll need to use the formula for field
strength in a dipole field.) The last part involves some numerical
Problem Set 4 Due: Thursday February 22nd
- Read carefully sections 3.3.1 to 3.3.5. In a steady state there
time dependence (partial time derivatives identically zero). Consider
3-44 for a steady state. Assume isotropic pressure, E = B = 0, but
gravitational field. All variations are 1-dimensional.
Show that the distribution function on page 9 of Chen,
f(u) = Aexp(-[(1/2)mu2 + q(phi)]/kT)
satisfies the collisionless Vlasov equation.
Chen 3-1 (page 58)
Chen 3-7 (page 74)
- (i) Write down a modified version of 3-44 appropriate to these
(Hint: gravitational force replaces Lorentz force.
- (ii) Derive Bernoulli's equation.
Problem Set 5 Due: Thursday March 1st
- Calculate the plasma frequency for the plasmas in problem 1-3 of
Problem set # 1) plus, a laser fusion pellet with n = 5x1022
- Chen 4-1 (page 81)
- Chen 4-2 (page 86)
- Chen 4-3
Problem set 6 Due: Thursday March 8th
- Chen 4-6 (page 94)
- Chen 4-8 (pg 108)
- Chen 4-9 (pg 120)
- Chen problem 4-10 (pg 120)
- Chen 4-13
Problem Set 7 Due: Thursday March 15th.
- Chen 4-16 (pg 135)
- Chen 4-19
- Chen 4-21 (pg 135)
- Chen 4-24 (Be careful. Chen's answer is not correct.)
- Chen 4-25
Problem Set 8 Due: Thursday March 22nd
- Chen 4-26 (page 148)
- Chen 4-30
- Chen 4-32
- Chen 4-34
- Chen 4-45
Problem set 9 Due: Thursday March 29th
- On a plot of frequency omega versus wave number k, draw
relations for all the waves we have studied. Label each wave and
its nature ( EM or electrostatic).
- Draw a graph of refractive index squared versus omega. Choose a
ordering of the plasma frequency and the cyclotron frequency (ie omegap
less than or greater than omegac) Mark all relevant
(eg omegaR and omegaL). Show the dispersion
for all the waves we have studied. Use different colors or otherwise
the waves that propagate across B from the waves that propagate
along B. Label all the waves. Estimate where the relations for
propagating at an intermediate angle to B would lie, and sketch them
(You should have a total of 4 waves, including plasma oscillations.
T = 0 throughout.)
- Chen 5-1 (page 175-176)
- Chen 5-2
Problem Set 10 Due Thursday April 5th
- In the interstellar medium the electron density is about 10-2
cm-3, B is about 3x10-6 Gauss and the density of
neutral hydrogen atoms is about 1 cm-3. The temperature is
100 K and is the same for electrons and ions. If a typical interstellar
cloud has dimension r approximately 10 pc = 3x1019
how long does it take for the magnetic field to escape from a cloud if
the only means of escape is electron and ion diffusion across B? Assume
cylindrical symmetry, and neglect velocities along B. The collision
section for collisions with neutrals is about 10-16 cm2.
(This result is relevant to the problem of star formation.)
- Here is another exercise in ambipolar diffusion across B.
ionized plasma in a conducting box which has finite dimensions in the x
and z directions, but which is effectively infinite in the y
direction. The magnetic field is in the z direction.
Chen 5-7 (pg 195/6)
- a) Write the 2-d continuity equation appropriate to this
- b) Neglecting mobility across field lines, show that the
may be written:
partial dn/dt = Dparallel partial d2n/dz2
+ Dperp partial d2n/dx2
and find expressions for Dparallel and Dperp.
Compare your results with field free diffusion and the
results we found
in class for a different case of ambipolar diffusion. Comment.
Problem Set 11 Due: Thursday April 19th
- Chen 5-14
- When we derived the fluid equations we used the definition of
Pij = mn<(vi-ui)(vj-uj)>
where u is the mean fluid velocity <v> and
the average over the distribution function:
<x> = (1/n) (integral) x f(v)
For the single fluid MHD case, we define the pressure to be:
PMHD,ij = (sum over e and i) mn<(vi-Vi)(vj-Vj)>
where V = (Mu+ + mu-)/(M+m),
= <v> for the ions, and u- = <v>
for the electrons.
Show that the difference between the two expressions Pe
+ Pi and PMHD is exactly balanced
the difference between the convective derivative terms niMu+
· gradu+ + ne mu-
· gradu- and
(M+m)nV · gradV, and thus that our
form of the MHD equations is exact.
- Referring to problem 1 of the
previous set above, calculate the
for this plasma (the interstellar medium) (ignoring the neutrals) and
calculate the time for the
field to decay. Repeat the calculation for a stellar interior ( n = 1024
cm-3, T = 107 K, R = 1010 cm).
- Chen 5-17 (pg 197)
Problem set 12 Due: Thursday April 26th
- Using the MHD equations, derive the dispersion relation for MHD
at an arbitrary angle theta to the B field.
- a) Show that there are three waves, and sketch the phase speed
as a function of theta.
- b) Find the velocity eigenvectors for the three waves. Show
(the Alfven wave) is always transverse, while the other two are neither
longitudinal nor transverse.
- c) Evaluate the phase velocities and eigenvectors of the mixed
v A > v S, and show that one wave has v along
mostly along B) and one has v almost perpendicular to B (Alfven type).
What happens when vS > vA?
- d) Show how these waves fit on the dispersion diagram (n2
omega) that you drew in problem set 9.
Problem Set 13 Due Thusday May 3rd
- Chen 6-6 (pg 214)
- Chen 6-9 (pg 215)
Problem Set 14 Due Thursday May 17th
- Chen 7-1 (pg 263),
- Chen 7-3
- Using equation (10) from the Vlasov notes, find the
susceptibility for a two component plasma with protons at temperature Ti
and electrons at temperature Te. Hence find the
dielectric constant, and the normal mode frequencies. Show that
you obtain ion sound waves, and find their damping rate in terms of the
two temperatures. What condition is needed for the damping to be small?
In doing the integrals, make the following
and ω/k≪vth,e. To do the integral for the electrons
you may want to "undo" the integration by parts, (use eqn 9).