PHYSICS OF PLASMAS

SUSAN LEA

TEXT: Required: Chen, Francis: Introduction to Plasma Physics and Nuclear fusion, Volume I, (Plenum)
Supplementary:  Lea, Susan: Mathematics for Physicists  (Brooks/Cole)

We shall cover almost the entire text during the semester, with a small amount of supplementary material. Please give me feedback as we go along: I want to know when you are having difficulties so that we can clear them up.

Part of the class assignment is a term paper in which you can explore some topic in Plasma Physics in more depth than we can get into in class. The paper may be a review paper based on a literature search, or it may involve original calculations or a computer project. Please begin to think about your paper right away.

Grading:

In the problems I am looking for an honest effort, 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 you learn. Try to use the solutions only as a guide if you really get stuck. (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 Dougherty, Addison Wesley.
• Principles of Plasma Physics: Krall and Trivelpiece (Out of Print, but is in the library.)
• Introduction to plasma physics: Nicholson. (Wiley).
• The framework of plasma physics: Hazeltine and Waelbroeck, Cambridge.
• Introduction to Plasma Physics: Goldston and Rutherford, IOP 1995
• The Physics of Plasmas: Boyd and Sanderson, Cambridge,  2003
• Fundamentals of Plasma Physics: Bellan, CUP 2006
• Physics of Space plasmas.  George Parkes, Wetsview, 2004
  

Important Dates:

Paper

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 from the refereed journals that are of a technical nature.  Avoid the temptation to stick with Scientific American and the Web!  This is a graduate physics course, and I expect to see a paper at the appropriate level, with technical detail included.

If you do an original calculation, I expect to see more detail than would be typical in a journal article.  Show me what you did!  If you use a computer program, include the source code as an appendix.

Topic statement.

• A title.
• A one or two paraagraph discussion of the topic.
• At least one major reference.

Abstract and outline.

First draft.

• Title
• Author name (you!)
• Abstract
• Introduction
• Body (divided into sections and subsections as necessary)
• Conclusion
• List of references
• Figures and figure captions where appropriate.

Second draft.

Final paper:

Class presentation.

Citing references

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 site. See the following site for more info:
http://ia.juniata.edu/citation/cse/cse05.htm

A word of warning: The web is not refereed! Look carefully at the credibility of your site. A NASA site probably has reliable information, 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 SETS

Problem Set #1 Due: Thursday February 1

1. Chen 1.3 (pg 12)

Include the following additional astrophysical plasmas, extending the axes as appropriate:
• 8) cluster of galaxies: n_e = 10³ m^-3, T = 10 keV.
• 9) Supernova remnant: n_e = 10⁵ m^-3, T = 1 keV
• 10) Solar interior: n_e = 10³² m^-3, T = 1 keV
• 11) Solar corona: n_e = 10¹⁴ m^-3, T = 100 eV.
2. Chen 1-5
3. Chen 1-8,
4. Chen 1-11 (page 17)
5. Compute the magnitude of the shielded point source potential (equation (5) in Debye notes) and numerically compare the result  with the unshielded potential at distances 0.1, 1 and 5 times the Debye length from the charge.  Comment.

Problem set #2 Due: Thursday February 8th

1. Chen 2-1 (pg 25) Also compute omega_c, and include

e) An electron near the surface of an accreting neutron star: E = 10 keV, B = 10⁸ T
2. Chen 2-7
3. Chen 2-8. (pg 35)
4. Chen 2-12 (pg 35)
5. Chen 2-13 (pg 49)

Problem Set 3 Due: Thursday February 15th

1. Chen 2-16
2. Chen 2-17
3. Refer to problem Chen 2-8. Calculate the bounce frequency for the 1 eV 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 calculations.

Problem Set 4 Due: Thursday February 22nd

1. Read carefully sections 3.3.1 to 3.3.5. In a steady state there is no explicit time dependence (partial time derivatives identically zero). Consider equation 3-44 for a steady state. Assume isotropic pressure, E = B = 0, but non-zero gravitational field. All variations are 1-dimensional.
• (i) Write down a modified version of 3-44 appropriate to these assumptions. (Hint: gravitational force replaces Lorentz force.
• (ii) Derive Bernoulli's equation.
2. Show that the distribution function on page 9 of Chen, f(u) = Aexp(-[(1/2)mu² + q(phi)]/kT) satisfies the collisionless Vlasov equation.
3. Chen 3-1 (page 58)
4. Chen 3-7 (page 74)

Problem Set 5 Due: Thursday March 1st

1. Calculate the plasma frequency for the plasmas in problem 1-3 of Chen (see Problem set # 1) plus, a laser fusion pellet with n = 5x10²² cm^-3.
2. Chen 4-1 (page 81)
3. Chen 4-2 (page 86)
4. Chen 4-3

Problem set 6 Due: Thursday March 8th

1. Chen 4-6 (page 94)
2. Chen 4-8 (pg 108)
3. Chen 4-9 (pg 120)
4. Chen problem 4-10 (pg 120)
5. Chen 4-13

Problem Set 7 Due: Thursday March 15th.

1. Chen 4-16 (pg 135)
2. Chen 4-19
3. Chen 4-21 (pg 135)
4. Chen 4-24 (Be careful. Chen's answer is not correct.)
5. Chen 4-25

Problem Set 8 Due: Thursday March 22nd

1. Chen 4-26 (page 148)
2. Chen 4-30
3. Chen 4-32
4. Chen 4-34
5. Chen 4-45

Problem set 9 Due: Thursday March 29th

1. On a plot of frequency omega versus wave number k, draw the dispersion relations for all the waves we have studied. Label each wave and indicate its nature ( EM or electrostatic).  
   
2. Draw a graph of refractive index squared versus omega. Choose a particular ordering of the plasma frequency and the cyclotron frequency (ie omega_p less than or greater than omega_c) Mark all relevant frequencies, (eg omega_R and omega_L). Show the dispersion relations for all the waves we have studied. Use different colors or otherwise distinguish the waves that propagate across B from the waves that propagate along B. Label all the waves. Estimate where the relations for waves propagating at an intermediate angle to B would lie, and sketch them in. (You should have a total of 4 waves, including plasma oscillations. Assume T = 0 throughout.)
3. Chen 5-1 (page 175-176)
4. Chen 5-2

Problem Set 10 Due Thursday April 5th

1. 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 about 100 K and is the same for electrons and ions. If a typical interstellar cloud has dimension r approximately 10 pc = 3x10¹⁹ cm, 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 cross section for collisions with neutrals is about 10^-16 cm². (This result is relevant to the problem of star formation.)
2. Here is another exercise in ambipolar diffusion across B. Consider a weakly 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.
• a) Write the 2-d continuity equation appropriate to this problem.
• b) Neglecting mobility across field lines, show that the continuity equation may be written:
partial dn/dt = D_parallel partial d²n/dz² + D_perp partial d²n/dx² and find expressions for D_parallel and D_perp.
• c) Compare your results with field free diffusion and the results we found in class for a different case of ambipolar diffusion. Comment.
3. Chen 5-7 (pg 195/6)
4. Chen 5-8
5. Chen 5-9

Problem Set 11 Due: Thursday April 19th

1. Chen 5-14
2. When we derived the fluid equations we used the definition of pressure:
P_ij = mn<(v_i-u_i)(v_j-u_j)> where u is the mean fluid velocity <v> and <> denotes the average over the distribution function: <x> = (1/n) (integral) x f(v) d³v. For the single fluid MHD case, we define the pressure to be: P_MHD,ij = (sum over e and i) mn<(v_i-V_i)(v_j-V_j)> where V = (Mu₊ + mu_-)/(M+m), u₊ = <v> for the ions, and u_- = <v> for the electrons.
Show that the difference between the two expressions P_e + P_i and P_MHD is exactly balanced by the difference between the convective derivative terms n_iMu₊ · gradu₊ + n_e mu_- · gradu_- and
(M+m)nV · gradV, and thus that our form of the MHD equations is exact.
3. Referring to problem 1 of the previous set above, calculate the resistivity 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).
4. Chen 5-17 (pg 197)

Problem set 12 Due: Thursday April 26th

1. Using the MHD equations, derive the dispersion relation for MHD waves propagating at an arbitrary angle theta to the B field.
• a) Show that there are three waves, and sketch the phase speed of the waves as a function of theta.
• b) Find the velocity eigenvectors for the three waves. Show that one wave (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 waves when v _A > v _S, and show that one wave has v along B (or mostly along B) and one has v almost perpendicular to B (Alfven type). What happens when v_S > v_A?
• d) Show how these waves fit on the dispersion diagram (n² versus omega) that you drew in problem set 9.

Problem Set 13 Due Thusday May 3rd

1. Chen 6-6 (pg 214)
2. Chen 6-9 (pg 215)

Problem Set 14 Due Thursday May 17th

1. Chen 7-1 (pg 263),
2. Chen 7-3
3.   Using equation (10) from the Vlasov notes, find the susceptibility for a two component plasma with protons at temperature T_i and electrons at temperature T_e.  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 assumptions:  ω/k≫v_th,i=√(k_BT_i/M)  and  ω/k≪v_th,e. To do the integral for the electrons you may want to "undo" the integration by parts, (use eqn 9).