Professor G' D' Earle

1
A Primer on Plasma Physics and Arc Discharges
2
Contents

• Symbol Table
• Types of Discharges
• What Is Plasma?
• Producing and Sustaining Plasmas
• Mathematical Description of Plasmas
• Basic Characteristics of Plasmas
• Measurement Techniques
• References

The top figure shows a laboratory plasma produced
by a hollow cathode discharge in a vacuum
chamber. The bottom figure shows the aurora
borealis - a plasma produced naturally in the
upper atmosphere.
3
Symbol Table

• This table defines common parameters used
  throughout this primer.

4
Types of Discharges

• Discharges can be classified into two broad
  categories self-sustaining and non
  self-sustaining.
• Glow discharges and arc discharges fall into the
  self-sustaining category, as do corona discharge
  phenomena.
• Raizer (1991) classifies discharges into 12
  different types

5
Types of Discharges

• Glow discharges are generally characterized by
• Low pressures, lt 10 Torr
• Currents ranging from 1 ?A to 100 mA
• Weak ionization, 10-6 to 10-8 ions per neutral
  particle
• Energies less than 1 eV (11,600 K)
• Arc discharges typically have
• Pressures up to 1 atmosphere (760 Torr)
• Unlimited current, usually exceeding 1 A, bright
  emission of visible light
• 0.1-10 ionization
• High temperatures, capable of melting containment
  vessels
• Arc discharges are fast - the gas typically
  becomes 1 ionized in 0.1 ?s - 1 ms, with
  current levels growing by several orders of
  magnitude on the same timescale.

6
Types of Discharges - Emitter Example

• Quiescent electron emission and arc discharges
  are both easily created near sharp edges that are
  held at different potentials, because the
  potential gradients are largest there.
• This result can be rigorously derived from
  Maxwells equations, but a heuristic
  understanding may be obtained from the diagram
  below, which shows a device known as a microtip
  emitter.
• Microtip emitters can be run continuously in
  vacuum for long periods when operated carefully,
  but their geometries are also appropriate for
  producing arc discharges.

When a large potential difference is created
between two closely spaced surfaces the electric
field between them can become very large - this
can produce an arc discharge in which an ionized
(plasma) path forms between the surfaces and
carries current between them. The result is the
microscopic equivalent of a lightning
discharge. The close up view of the emitter tip
(below left) illustrates how this can happen -
electrons congregate on the surface of the tip
because of the proximity of the positively biased
(gate) electrode. These electrons all repel each
other. On the lateral edges of the emitter tip
these repulsive forces lie in the plane of the
surface, but near the atomically sharp tip the
net force will be perpendicular to the surface.
As a result the electrons near the tip are more
easily emitted into the surrounding air, creating
the arc discharge.
7
The Paschen Curve (1)

• There is a preferred pressure range for arc
  discharges between closely spaced electrodes.
  The important quantities are the product of
  ambient pressure and electrode separation, along
  with the potential difference between the
  electrodes.
• If the pressure is too low the mean free path in
  the gas becomes too large for efficient
  collisional ionization by the free electrons.
• If the pressure is too high the mean time between
  collisions is very short, so the ions and
  electrons quickly recombine.
• The Paschen curve illustrates where the
  conditions are best suited to formation of arc
  discharges.

8
The Paschen Curve (2)

• The Paschen curves shown in these two plots
  illustrate the voltage differences at which arc
  breakdown occurs for various gases and electrode
  separations.
• In both cases these voltages are plotted vs. the
  product of the ambient pressure (p in Torr) and
  the separation distance (d in cm) between the two
  electrodes.

9
Practical Uses of Discharges

• Ionization sources for mass spectrometry,
  electron beam generation, X-ray sources, etc.
• Arc welders
• Lighting
• Sterilization of medical instruments
• Plasma processing of materials, such as
  semiconductor dry etching
• Cold pasteurization of food

10
What Is Plasma?

• Plasma is a gaseous state of matter in which
  enough free energy is present to produce
  physically significant numbers of free ions and
  electrons.
• These gas populations may be considered to be
  immersed in each other as co-located fluids.
• The result is a gas with multiple interacting
  constituents, which may include
• Neutral particles (perhaps with spatially or
  temporally varying molecular or atomic
  composition)
• Electrons
• One or more positive ion species
• One or more negative ion species

11
What Is Plasma?

• Physically significant numbers of charged
  species means enough ions and electrons to
  exhibit Coulomb forces at least comparable to the
  other forces within the fluid.
• The charged particles interact through these
  Coulomb forces (FqE), which act over a distance.
• The effect of small angle Coulomb collisions in a
  plasma always dominate the effects of large angle
  collisions.
• The chart shows the enormous range of particle
  densities and temperatures over which plasmas can
  exist.

12
Producing and Sustaining Plasmas

• Due to random thermal interactions, any neutral
  gas at finite temperature will contain a few free
  electrons and ions at any given time.
• To create and sustain a plasma, energy must be
  available to accelerate this small cohort of free
  charged particles so that they collisionally
  ionize other neutral particles with sufficient
  frequency to overcome immediate chemical
  recombination.
• Common techniques for producing laboratory
  plasmas include
• Capacitive discharges - Free electrons are
  accelerated by a strong DC electric field
  sustained in vacuum between two flat plates held
  at different potentials.
• Inductive discharges - Radio frequency fields
  drive current in a coil system adjacent to (or
  immersed within) a gas in a vacuum chamber.
• Thermionic emission - Large currents are driven
  through filaments made of tungsten or some other
  metal. At sufficiently high filament
  temperatures electrons will be emitted from the
  metallic surface.
• Hollow cathode discharges - Gas is introduced
  through a short heated tube into a vacuum
  chamber. When the tube is heated to high
  temperatures the combination of photochemical and
  thermal processes can produce weakly ionized
  plasmas.
• Photochemical discharge - Short wavelength UV
  light irradiates a gas and stimulates ionization.
  Due to large optical depths in rarefied gases
  this technique generally produces weak, low
  temperature plasmas.

13
The Saha Equation

• The density of ions relative to the density of
  neutral particles in a gas is given by the Saha
  equation, where T is the neutral temperature in
  eV (1 eV11,600 K), the ion and neutral densities
  ni and nn are in m-3, and U is the potential
  energy of the least tightly bound electron in the
  system, also called the ionization energy

14
Plasma Ionization Percentage

• The chart below shows the fractional ionization
  of an atomic hydrogen gas, which has an
  ionization energy U13.6 eV. Here ntninn, and
  the fractional ionization is given by ?ni/nt.
  The numerical labels on the curves indicate the
  values of log(nt). For reference, log(nt)25 at
  standard atmospheric temperature and pressure.

15
Mathematical Description of Plasmas

• Kinetic theory equations are necessary for a
  rigorous description of plasmas, but for many
  applications the fluid equations are sufficiently
  accurate, and much simpler to solve.
• The three equations below apply independently to
  each species in the plasma, with the subscript s
  representing electrons, each ion species, and
  each neutral gas species.
• The Q and L terms in the continuity equation
  stand for chemical (or other) production and
  loss, respectively. Other variables are as
  defined in the symbol table.
• For neutral gases the E and VxB terms on the
  right side of equation (2) disappear, and a term
  representing neutral fluid viscosity must be
  added. Note - viscosity is not a well defined
  quantity for plasmas.

16
Mathematical Descriptions of Plasmas

• In combination with the four Maxwell equations of
  electromagnetism, the equations on the previous
  slide comprise a coupled, nonlinear, closed
  system.
• Various limiting cases can sometimes be solved
  analytically, but due to the vector nature of the
  variables, the number of scalar equations that
  result, and the complex geometries likely to be
  encountered in practical situations,
  computer-based solutions are necessary for most
  cases of practical importance.
• Note the similarity of the three equations on the
  previous slide to the three coupled equations of
  neutral fluid mechanics.
• This similarity implies that all of the wave
  modes and instabilities present in ordinary
  fluids will also exist in plasmas.
• However, the addition of the Coulomb and Lorentz
  terms, along with the existence of multiple
  co-located species (ions, electrons, and
  neutrals) in even the simplest plasmas, suggests
  that many more wave modes and instabilities will
  exist in the plasma domain.
• In fact, plasmas are the richest medium yet
  discovered for waves, resonances, and
  instabilities.
• The amazingly diverse mechanisms for energy
  dissipation that exist within plasmas help to
  explain why the plasma confinement necessary for
  controlled nuclear fusion is such an elusive
  goal. In fact, the only method known to be
  capable of containing high temperature plasmas in
  order to sustain long-lived fusion reactions is a
  strong gravitational potential well (stars).

17
Basic Characteristics of Plasmas

• Plasmas are quasi-neutral at low frequencies, so
  to a very good approximation the macroscopic ion
  and electron number densities (particles/m3) will
  be equal. This characteristic holds because the
  electrostatic forces created in a plasma become
  quite large if the charge densities are unequal,
  so restoring forces quickly reinforce the
  quasi-neutral condition.
• Plasmas can be described and parameterized by
  several characteristic frequencies. The most
  important of these are the plasma frequency, the
  gyro-frequency, and the upper-hybrid resonance
  frequency.
• These frequencies are usually easy to observe in
  a plasma. They are often used to determine the
  plasma density and the strength of the magnetic
  field, since the electron plasma frequency is
  proportional to the square root of the number
  density, and the gyro-frequency is proportional
  to the magnetic field strength.

18
Basic Characteristics - Plasma Frequency

• Consider a slab of plasma in which the electrons
  are (somehow) displaced laterally from the ions.
  Assume for simplicity that there is no background
  magnetic field. On short time scales the massive
  ions can be considered stationary relative to the
  more mobile electrons.
• A strong electric field will develop to pull the
  electrons back toward the ions. As the electrons
  move back they will overshoot the equilibrium,
  and an oscillation will be set up. This
  oscillation occurs at the electron plasma
  frequency, wpe. The ions undergo a similar
  oscillation, but the ion plasma frequency wpi is
  always much lower and can generally be neglected.

If q is the charge and m is the species mass then
the plasma frequency is given by
19
Basic Characteristics - Gyro Frequency

• If there is a magnetic field within the plasma
  then the random thermal motions induce Lorentz
  forces (qV x B) that create circular motions of
  the ions and electrons in planes perpendicular to
  the magnetic field lines.
• The radii of these orbits are called the Larmor
  radii (R, RL or rL), and the frequency of the
  oscillation is called the cyclotron frequency ?c.
  Electrons have a smaller Larmor radius and a
  larger cyclotron frequency than ions.
• The figure below shows the helical motion of a
  particle that has non-zero parallel and
  perpendicular components of velocity with respect
  to the magnetic field. The central axis of the
  helix defines the locus of the guiding center
  of the particles trajectory.

20
Basic Characteristics - Cutoffs and Resonances

• The fundamental resonance frequency of a plasma
  immersed in a magnetic field occurs at the upper
  hybrid frequency, which is the square root of the
  sum of the squares of the electron plasma and
  cyclotron frequencies
• ?UH2 ?pe2 ?ce2
• Electromagnetic energy incident on the plasma at
  this frequency will be strongly absorbed at this
  resonance.
• In contrast, the plasma frequency represents a
  cutoff for wave propagation. Electromagnetic
  energy incident on a plasma is strongly reflected
  at this plasma frequency cutoff. This phenomenon
  is used by ham radio operators and
  over-the-horizon radars to achieve long-distance
  propagation in the AM radio band, using
  reflection from the ionosphere.

21
Basic Characteristics - Magnetic Confinement

• The ratio of kinetic pressure to magnetic
  pressure is defined as ?.
• ? nKT/(B2/2?o)
• In high ? plasmas the particles can distort the
  shape of the ambient magnetic field, while in low
  ? plasmas the magnetic field controls the
  distribution of plasma. Magnetic fields are
  therefore useful for constraining plasma motions
  if ? ?? 1.
• The surface of the sun is a high ? environment,
  while examples of low ? plasmas include
  florescent lights and the ionosphere.

The left panel shows a diffuse plasma emitted
from a hollow cathode source (bright spot at
top). The right panel shows the same plasma
after a magnetic field is imposed along the
vertical axis in the figure.
22
Basic Characteristics - Adiabatic Invariants

• There are several adiabatic invariants describing
  plasma motion, but only one is robust enough to
  be generally useful. This is the magnetic moment
  ?.
• The magnetic flux (?) enclosed by the Larmor
  radius of the particle orbit can be written as a
  constant times the magnetic moment, as shown
  below. As a result, ? is also an adiabatic
  invariant of the motion provided that
• The magnetic field strength and geometry do not
  change over a gyro period
• No collisions occur within a gyro orbit to modify
  the particles trajectory.

23
Basic Characteristics - Adiabatic Invariants

• Since ? is conserved, a particle moving parallel
  to a converging magnetic field will gain
  perpendicular energy as the field strength
  increases. Since the total energy is conserved
  the particles speed along B must decrease.
  Eventually the parallel velocity will reach zero
  and the particle will be mirrored - in effect
  reflected from the region of converging field
  lines. Particles with different initial energies
  will mirror at different points along the
  converging field.

The bottom panel shows the field lines associated
with a magnetic bottle geometry, which can be
easily created with Helmholtz coils. As plasma
ions and electrons move into the constricted
region (s2) they encounter the equivalent of a
potential gradient, which causes the magnetic
mirroring effect.
24
Basic Characteristics - Debye Shielding

• If a test charge qT is introduced into a plasma,
  the ions and electrons in the plasma will
  redistribute themselves according to the sign of
  the charge.
• The Debye length (?D) is simply the e-folding
  distance for the potential field ?(r) that forms
  around such an object. This potential sheath
  region effectively shields the rest of the
  plasma from the test charge, so that only the
  particles near the test charge react to its
  presence.
• Ions and electrons have different Debye lengths,
  owing to their greatly different masses (
  corresponding mobility). The overall Debye
  length is a combination of the two (see below),
  but in practice the ion Debye length is
  negligible if Te gtgt Ti.
• Cs is the thermal sound speed in the plasma, so
  an alternative definition of the Debye length is
  the distance a particle traveling at the thermal
  speed moves in the time associated with the
  electron plasma frequency oscillation.
• A plasma can be quantitatively defined as a gas
  in which the number of charged particles
  occupying a sphere with radius equal to the Debye
  length is much greater than one.

25
Basic Characteristics - Guiding Center Drifts

• Particles in plasma can be tracked by following
  the motions of their guiding centers, with the
  under-standing that the gyro-motions are
  superimposed on these guiding center
  trajectories.
• If a body force F is applied to a magnetized
  plasma, the effect on the charged particles is to
  move with a drift velocity given by w(FxB)/qB2.
• The formulas at the right give the most important
  guiding center drifts for ions and electrons in a
  plasma.
• Note that if an electric field is imposed on the
  plasma, the ions and electrons will drift
  together in the same direction, so no net current
  will result.

26
Basic Characteristics - Diffusion

• Plasma particles undergo diffusion in response to
  density or temperature gradients, but the
  classical understanding of diffusion is modified
  in this case because of the electrostatic forces
  between ions and electrons.
• In general the electrons have much larger thermal
  speeds (vs2KTs/ms), so they will diffuse faster
  than the ions. This sets up an electric field
  that retards the electrons and speeds diffusion
  of the ions.
• The net effect is called ambipolar diffusion,
  where the diffusion coefficient is a hybrid value
  involving the electron temperature and the ion
  mass and collision rate (note that Te is
  generally much larger than Ti).
• In the equations below, lmfp is the mean free
  path between collisions.

27
Basic Characteristics - Collisions

• Collision frequencies in plasmas are quite
  difficult to define quantitatively because of the
  wide range temperatures and densities at which
  plasmas exist, coupled with the fact that each
  species has three different collision
  frequencies
• Collisions with neutrals
• Collisions with electrons
• Collisions with ions
• The simplest case to quantify is the collision
  frequency between charged species and neutrals,
  which is the most important collision term in low
  temperature, low density plasmas.
• In this special case we obtain the result
• ?sn nn?s(KTs/ms)1/2
• where ?s is the scattering cross section, the
  subscript s refers to the charged species, and
  the subscript n refers to the neutral species.
• For electrons the scattering cross section is
  approximately 4x10-15 cm2 at all temperatures
  below 3000 K, while for ions it varies from
  about 50x10-15 cm2 at 10 K to about 5x10-15 cm2
  at 1000 K.

28
Waves in Plasmas

• A very large number of wave modes exist in a
  plasma.
• The dispersion relations (? vs. k) for the most
  important of these are given in the list shown
  here.
• The electromagnetic ion modes are analogous to
  waves on a plucked string, where the string in
  this case is a magnetic field line.
• The Alfven velocity in these ion wave equations
  is defined by vA2 B2/(nimi?o).
• The study of plasma waves becomes very complex,
  because the various modes can couple to one
  another under proper conditions.

29
Waves in Plasmas

• For mathematical simplicity the fast propagating
  electromagnetic waves are often considered to
  comprise four separate modes, called the X, O, R,
  and L mode waves.
• The X and O modes propagate perpendicular to B,
  while the R and L modes propagate along B.
• The actual ray-path of a wave is the
  superposition of the X, O, R, and L modes at any
  point in space.
• Because of the free charge in a plasma, the phase
  velocities of the fast electromagnetic modes can
  exceed the speed of light.
• The dispersion relation that describes the
  propagation of all four fast modes is known as
  the Appleton-Hartree equation.

30
Appleton-Hartree Equation

• Define ? to be the propagation angle relative to
  the magnetic field (so ?0 implies propagation
  parallel to B).
• Define nck/? as the index of refraction, where c
  is the speed of light in vacuum, k is the
  wavenumber, and ? is the wave frequency.
• The equations below then describe the propagation
  of all four of the fast electromagnetic modes.

31
Waves in Plasmas

• A CMA diagram can be used to help visualize or
  predict the wave modes that will be excited in
  various conditions.
• Consider the magnetic field that permeates the
  plasma to lie along the Y-axis.
• A given frequency is represented as a point on
  the plot. The diagram in the region containing
  the point shows the wave modes that exist in that
  frequency regime, and the shapes of the figures
  give the relative values of the phase velocities
  in each direction relative to B.
• Recall that waves are reflected at cutoffs, and
  absorbed at resonances.

32
Measurement Techniques

• A Langmuir probe is the oldest and simplest form
  of in-situ plasma diagnostic. It is simply a
  wire inserted into the plasma that can be biased
  to some potential.
• The diagram on the left (below) shows a set up
  in which such a biased probe is inserted into a
  plasma. The graph on the right shows the current
  collected by the probe wire as a function of its
  potential. The sense of the Y-axis label is such
  that positive values of probe current correspond
  to electrons being collected on the probe.
• The probe voltage at which no net current is
  collected is called the floating potential (Vf),
  and the point labeled Vp is the plasma potential
  (which is difficult to measure in practice
  because the curve doesnt generally show such a
  sharp break point there).
• The slope of the curve between Vf and Vp is
  inversely related to the electron temperature.
• The electron current greatly exceeds the ion
  current because of the higher thermal velocity.
• The relationship Vp-Vf 2-3 KTe/q generally
  holds.
• Fitting a theoretical function to such a Langmuir
  curve allows estimation of the plasma density and
  electron temperature.

33
Plasma Measurement Techniques

• In addition to Langmuir probes, measurements in
  plasmas can be made using the following
  techniques
• Coherent and incoherent scatter (radar-like
  techniques) to measure Te, Ti, N, and Vs.
• Optical resonance and absorption techniques to
  determine plasma densities, temperatures, and
  Doppler shifts (line of sight motion).
• Interferometry to measure average line of sight
  density.
• Energetic particle detectors and thermal plasma
  detectors to measure the distribution functions
  vs. energy.

34
Concluding Comments

• This primer has attempted to present in a compact
  form the basic principles of plasma physics, with
  no attempt at derivations.
• Many interesting but esoteric phenomena, such as
  plasma stability, Landau damping of waves, and
  soliton formation, have been omitted.
• Much more information is readily available on
  all of the topics and phenomena described herein.
• The reference list on the next slide provides a
  listing of source materials that can
  significantly extend the basic introductory
  material presented here.

35
References

• Bellan, Paul M., Fundamentals of Plasma Physics,
  Cambridge University Press, 2006.
• Bittencourt, J. A., Fundamentals of Plasma
  Physics, Springer Science and Business Media,
  LLC, 2004.
• Chandrasekhar, S., Plasma Physics, University of
  Chicago Press, 1960.
• Chen, Francis F., Introduction to Plasma Physics,
  Plenum Press, 1974.
• Goeckner, M., G. D. Earle, L. Overzet, and J.
  Maynard, Electron confinement on magnetic field
  lines, IEEE Trans. On Plasma Science, 33, 436,
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• Gurnett, D. A., and A. Bhattacharjee,
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• Ichimaru, S. Basic Principles of Plasma Physics -
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• Johnson, E. E., R. I. Desourdis, G. D. Earle, S.
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  House, Boston, 1997.
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• Nasser, E., Fundamentals of Gaseous Ionization
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• National Research Council, Plasma Science From
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• Nicholson, D., Introduction to Plasma Theory,
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• Raizer, Yuri P., Gas Discharge Physics,
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• Schmidt, George, Physics of High Temperature
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