plasma physics

A plasma is a quasineutral gas of charged and neutral particles which exhibit collective behavior¹.

This w/u will attempt to summarize some important aspects of the physics of plasmas. No detailed derivations will be attempted. This is intended as a brief introduction to plasma physics.

Conditions
In saying that the plasma is quasineutral we mean that the number of ions and electrons are approximately equal (which would simplify certain equations) but that electromagnetic forces still exist there. The ions may be fully or partially stripped of their electrons. The plasma exhibits collective behaviour due to the long-range Coulomb interaction between charged particles.

Our swirling mass of ions and electrons should satisfy three basic conditions before it will satisfy the above definition of a plasma-

1. λ_D<<L
2. N_D>>1
3. ωτ>1

where λ_D is the Debye length, L is the length scale of the plasma, N_D is the number of particles in a Debye sphere, ω is the approximate frequency of plasma oscillations and τ is the mean time between collisions in the plasma. Don't worry, these terms will all now be explained.

If spheres of opposite charge are lowered into the plasma what happens? A swarm of particles of opposite charge will be attracted to each sphere (as described by Anark). The rest of the plasma will be almost unaffected by the charged spheres. This process is known as Debye Shielding. The cancelling sheath of charges form a Debye sphere whose radius is the Debye length λ_D given by

λ_D=(ε_oKT_e/ne²)^1/2   m

where ε_o is the electric permittivity in a vaccuum, K is the Boltzmann constant, T_e is the electron temperature, n is the density of the plasma and e is the electronic charge. The Debye length increases with temperature² as more random movement of the shielding electrons will allow more electic potential from the external source to seep through. One wishes to keep the Debye length as small as possible (compared to the dimensions of the plasma) so that the plasma will be shielded from any external fields and maintain its quasineutrality (the electon density n_e should almost equal the ion density n_i). Furthermore, we require a proper swarm of particles, not one or two, to surround the foreign charged object. Hence, condition two.

Electrons are much less massive than ions so they move much faster. Consider a bunch of electrons pulled apart from a group of ions. The electrons will naturally speed back to the positively charged ions. The ions will move so little in this time that we can consider them to be fixed. When the electons reach the location of the ions their momentum will carry them through (described well here) . They will execute a simple harmonic motion about the ion location at the plasma frequency ω_p given by

ω_p=(ne²/ε_om)^1/2   rad s^-1

The plasma frequency is the fundamental frequency of the plasma and we require this to be higher than the collisional frequency. This is to prevent the collective effects peculiar to a plasma from being drowned out. Condition three is a mathematical statement of this requirement.

Single particle motion
To begin an analysis of whats going in a plasma, its best to start with the simplest case- that of a charged particle moving through an electomagnetic field. In reality, the particle itself will have some influence over the field it experiences. Nevertheless, some particle drift effects will be apparent in this simple analysis. Let us start with the equation of motion of a particle of charge q and mass m in an electric field E, a magnetic field B moving at a velocity v

m(dv/dt)= q(E+vXB)

First consider a particle moving through a uniform magnetic field only. The acceleration it experiences will be perpendicular to both the field and its velocity, i.e. it will execute a circular motion about the magnetic field line. This motion will be at the cylcotron frequency and described by the Larmor radius. Since this acceleration has no component along the magnetic field line, the particle will move freely in this direction. In other words, it will gyrate about the magnetic field. It is often useful to disregard the gyration of the particle about the field (the Larmor motion) and consider only the guiding center motion.

Introducing a finite, uniform electric field introduces a drift into this gyrating motion. This is given by

v_E=EXB/B²

Considering the effects of a non-uniformities in the magnetic and electric fields lead to further drifts- Curvature drift due to curved magnetic fields, Grad-B drift, Polarization drift due to a time varying electic field and a drift due to spatial non-uniformities in the electric field.

Also falling out of this analysis are three adiabatic invariants. These are quantities that remain constant when a periodic system is given a gentle push. The three quantities are the magnetic moment μ, the longitudinal invariant J and the total magnetic flux enclosed by the trajectory of the particle Φ. The magnetic moment is given by

μ=mv_⊥²/2B

where v_⊥ is the component of the velocity of the particle perpendicular to the magnetic field. As the particle moves to an area of increasingly strong magnetic field, the conservation of μ implies that v_⊥ increases. Conservation of energy implies that the total velocity of the particle is kept constant. Thus the velocity component along B (v_//) must decrease until it reaches zero, at which point the particle is reflected back along the field. This is known as the magnetic mirror effect. This is the basis for such devices as the Marshall magnetic mirror beam-plasma device (a concept propulsion device for a spacecraft).

Kinetic Equations
To move to the next level of detail it is necessary to account for the motion of all the particles in a statistical fashion. Kinetic theory describes the plasma in terms of a distribution function f(r,v,t). This function measures the probability that a particle will have a certain displacement and velocity at a given moment. The Vlasov equation describes the evolution of the distribution function of a collisionless plasma while the Fokker-Planck equation includes a collisional term.

δf/δt + v.δf/δr + (q/m)(E + vxB).(δf/δv)= (δf/δt)_col

where (δf/δt)_col is the change in the distribution function due to collsions. Other forms of the Fokker-Planck equation are obtained by averaging out the fast gyration about the magnetic field (i.e. the Larmor motion). The drift kinetic equation and the gyro-kinetic equation are important examples used in the study of certain plasma instabilities⁴.Kinetic theory is used in tokamak plasmas to calculate the neutral beam slowing down time and particle collision times. The Braginskii equations calculate transport processes (such as heat flux) from collisional kinetic theory. Finally, the theory helps with calculations of the plasma resistivity.

Fluid approximation
In a plasma collisions are infrequent and collective effects distinguish it from a normal fluid. However, a fluid approximation works suprisingly well. This may be due to the magnetic field in a plasma limiting the transverse velocity (i.e. v_⊥) in a similar way to collisions limiting the velocity in a fluid. The fluid equations are obtained by integrating the kinetic equations. Each particle species (electrons and various types of ions) will have its own system of fluid equations, the equation of motion being-

nm(δv/δt + v(div.v))= -∇p + qn(E+vXB)

where n, m, v and p are the density, mass, velocity and scalar pressure of one particular species. For a derivation go here. This is similar to the Navier-Stokes equation in hydrodynamics except collision terms have been dropped and, of course, there is an electromagnetic term.

On comparing the equation of motion in the fluid approximation and in the single particle approach one can see similarities. Indeed, the fluid approximation also predicts the EXB drift. In addition, the fluid approximation predicts a diamagnetic drift due to the pressure gradient.

v_D= -(∇pXB)/qnB²

Magnetohydrodynamics / Plasma Instabilities
MHD simplifies the analysis still further by assuming that the plasma is composed of a single fluid. The behaviour of the ions and electrons are subsumed into one set of equations; the equations of magnetohydrodynamics. The equation of motion is now reduced to

ρ(dv/dt)= jxB - ∇p

where ρ is the plasma density and j is the plasma current density.

Though this model is a greatly simplified picture of the plasma, it is very useful in describing plasma instabilities. Such instabilities arise from gradients in the current or pressure together with certain magnetic field curvatures. They may either be ideal or resistive depending on whether they depend on the plasma having a finite resistivity or not. For MHD generated pictures of plasma instabilities in stellerators and tokamaks see note 5.

Of course we are also interested in studying a plasma that is in equilibrium. Since this requires the plasma be stationery, the MHD condition is

jXB = ∇p

This is merely a statement that the tendency of the particles to escape (due to the pressure gradient) is held in check by the magnetic force. This is the starting point for the field of plasma equilibrium (see tokamak magnetic equilibrium).

Plasma Waves
Electromagnetic waves are a consequence of Maxwell's equations. In a plasma we get the following dispersion equation

K.E=E+(i/ε_oω)j

where K is the dielectric tensor and i is √-1 (i.e. this is a complex number). From this general equation it is possible to find the linear response of the plasma to different waves⁶. The most fundamental plasma oscillation, described previously, is at the plasma frequency ω_p. Transverse electromagnetic waves, Alfvén waves and magnetosonic waves are three of the most basic plasma waves (the latter two are derived from the ideal MHD model). There are many more beyond the scope of this w/u (standard cop out phrase).

Even in a collisionless plasma the electromagnetic waves are damped. This is due to a process of energy exchange between the wave and the particles known as Landau Damping. It is analagous to a surfer slowing down the surfed wave. The particles must be travelling at the wave phase velocity for this damping effect to kick in.

Magnetic Reconnection
The magnetic flux map of the plasma often has areas, called magnetic islands where the field is oppositely directed. When magnetic reconnection occurs, the flux topology suddenly changes to one of a lower energy state⁷. The resulting energy is imparted to the plasma particles. This process lies behind such events as solar flares and aurorae⁸. However, it is a poorly understood mechanism, particulary where there is no resistivity to facilitate the transition.

Confinement
The plasma in a nebula is at such a low temperature that its not going anywhere fast. At higher temperatures one needs another force to keep the plasma from expanding endlessly. The Sun is so massive that its gravitational force suffices. On Earth, it is usual to store the plasma in some form of magnetic bottle. Various magnetic fields must be applied to cancel out the particle drifts (mentioned above).

A toroidal device in which is there is a helical magnetic field is the favoured way. The three main types are the tokamak, stellerator and the reverse field tokamak. Plasmas are also confined in linear devices such as a magnetic mirror or a z-pinch. However, its difficult to stop the slippery plasma from getting away and improvements must still be made in the confinement of high temperature plasmas. I'm not going to go into the physics of plasmas in these devices right here.

Summary
Plasma physics is a massive subject and there is a lot more interesting physics that I could discuss. However, the w/u is already looking fairly long. I've briefly described the basic conditions that should exist in a plasma. Drifts in single particle motion have been introduced. Kinetic theory logically led to the fluid equations and then magnetohydrodynamics. These physical pictures of the plasma help to explain whats going on in there, such as plasma instabilities and magnetic reconnection events. Suggestions and questions are encouraged.

Notes and sources
1. 'Plasma Physics and Controlled Fusion' by Francis F. Chen
2. The temperature of a plasma is generally measured in eV
3. http://www.spacetransportation.com/ast/abstracts/9D_Schneider.html
4. 'Tokamaks' by John Wesson
5. http://www.ornl.gov/fed/mhd/mhd.html
6. http://sol.physics.usyd.edu.au/sosp9/node1.html
7. http://w3.pppl.gov/~mrx/reconnection.html. This is the website of the Magnetic Reconnection Experiment
8. http://www.physicstoday.org/pt/vol-54/iss-10/p16.html