II' Plasma Physics Fundamentals

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II. Plasma Physics Fundamentals

• 4. The Particle Picture
• 5. The Kinetic Theory
• 6. The Fluid Description of Plasmas
• 7. Waves in Plasma

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7. Waves in Plasmas

• 7.1 Electrostatic Waves in Non-Magnetized
• Plasmas
• 7.2 Electrostatic Waves in Magnetized Plasmas
• 7.3 Electromagnetic Waves in Plasmas

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7.3 Electromagnetic Waves in Plasmas

• 7.3.1 Electromagnetic Waves Refresher
• 7.3.2 E.M. Waves in a Non-Magnetized Plasma
• 7.3.3 E.M. Waves in a Magnetized Plasma
• 7.3.4 Hydromagnetic (Alfven) Waves
• 7.3.5 Magnetosonic Waves

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7.3.1 Electromagnetic Waves Refresher

• The equation for the e.m. waves can be deducted
  from the Maxwell equations by taking the curl of
  Amperes law in vacuum (j0)

• Since div H0 and substituting the curl E with
  the Faradays law it is found

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Electromagnetic Waves Refresher (II)

• The vector H then satisfies the Helmholtz wave
  equation
• In a perfectly similar fashion it can be derived
  a wave equation for E

• It can be verified by substitution that a
  solution of the Helmholtz equation is
  exp(ikx-iwt) for w/kc

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Electromagnetic Waves Refresher (III)

• The electromagnetic field E, H propagates in
  vacuum with phase velocity w/kc
• The relation w/kc constitutes the dispersion
  relation for electromagnetic waves in vacuum
• Since the e.m. field of the wave must satisfy the
  Maxwell equations and r0 (in vacuum there are no
  charges), if Ew is the vector of wave electric
  field amplitude, it will be (for k along the x
  axis)

that implies
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Electromagnetic Waves Refresher (IV)

• The electric field of an e.m. wave then must not
  have a component along the propagation direction
  k (must be perpendicular to k)
• By taking the curl of the electric field
  component of the wave and using Faradays law it
  can be easily shown that the magnetic field also
  lies in a plane perpendicular to k ( and
  direction perpendicular to both k and E)

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7.3.2 Electromagnetic Waves in a Plasma

• In a plasma there will be current carriers,
  therefore the curl of Amperes law is

• By taking the curl of Faradays law

and eliminating the curl of H
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Electromagnetic Waves in a Plasma (II)

• If a wave solution of the form exp(kr-wt) is
  assumed it can be written (De0E)

• By recalling that an e.m. must be transverse (kE
  0) and that c21/(m0e0) it follows

• In order to estimate the current the ions are
  considered fixed (good approximation for high
  frequencies) and the current is carried by
  electrons with density n0 and velocity u

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Electromagnetic Waves in a Plasma (III)

• The electron equation of motion is

• The motion of the electrons here is the
  self-consistent solution of u, E, B (E and B are
  not external imposed field like in the particle
  trajectory calculations)
• A first-order form of the equation of motion is
  then

then
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Electromagnetic Waves in a Plasma (IV)

• Finally, substituting the expression of j in

it is found

• that is the dispersion relation for e.m. waves
  in a plasma (without external magnetic field)
• The phase velocity is always greater than c while
  the group velocity is always less than c

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Electromagnetic Waves in a Plasma (V)

• For a given frequency w the dispersion relation

• gives a particular k or wavelength (k2p/l) for
  the wave propagation
• If the frequency is raised up to wwp then it
  must be k0. This is the cutoff frequency
  (conversely, cutoff densitywill be the value that
  makes wp equal to w)
• For even larger densities, or simply wltwp there
  is no real k that satisfies the dispersion
  relation and the wave cannot propagate through
  the plasma

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Electromagnetic Waves in a Plasma (VI)

• When k becomes imaginary the wave is attenuated
• The spatial part of the wave can be written as

where d is the skin depth defined as
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7.3.3 E.M. Waves in a Magnetized Plasma

• The case of an e.m. wave perpendicular to an
  external magnetic field B0 is considered
• If the wave electric field is parallel to B0 the
  same derivation as for non magnetized plasma can
  be applied (essentially because the first-order
  electron equation of motion is not affected by
  B0)
• The the wave is called ordinary wave and the
  dispersion relation in this case is still

z
E
B0
k
y
x
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E.M. Waves in a Magnetized Plasma (II)

• The case of the wave electric field perpendicular
  to B0 requires both x and y components of E since
  the wave becomes elliptically polarized

z
E
B0
k
y
x

• A linearized (first-order) form of the equation
  electron equation of motion is then

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E.M. Waves in a Magnetized Plasma (III)

• The wave equation now must keep the longitudinal
  electric field kEkEx

or

• By solving for the separate x and y components a
  dispersion relation for the extraordinary wave is
  found as

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E.M. Waves in a Magnetized Plasma (IV)

• The case of the wave vector parallel to B0 also
  requires both x and y components of E

z
k
E
B0
y
x

• The same derivation as for the extraordinary wave
  can be used by simply by changing the direction
  of k

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E.M. Waves in a Magnetized Plasma (V)

• The resulting dispersion relation is

• or the choice of sign distinguish between a
  right-hand circular polarization (R-wave) and a
  left hand circular polarization (L-wave)
• The R-wave has a resonance corresponding to the
  electron Larmor frequency in this case the wave
  looses energy by accelerating the electrons along
  the Larmor orbit
• It can be shown that the L-wave has a resonance
  in correspondence to the ion Larmor frequency

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7.3.4 Hydromagnetic (Alfven) Waves

• This case considers still the wave vector
  parallel to B0 but includes both electrons and
  ion motions and current j and electric field E
  perpendicular to B0

z
k
B0
E,j
y
x

• The solution neglects the electron Larmor orbits,
  leaving only the ExB drift and considers
  propagation frequencies much smaller than the ion
  cyclotron frequency

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Hydromagnetic (Alfven) Waves (II)

• The dispersion relation for the hydromagnetic
  (Alfven) waves can be derived as

• where r is the mass density
• It can be shown that the denominator is the
  relative dielectric constant for low-frequency
  perpendicular motion in the plasma
• The dispersion relation for Alfven waves gives
  the phase velocity of e.m. waves in the plasma
  considered as a dielectric medium

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Hydromagnetic (Alfven) Waves (III)

• In most laboratory plasmas the dielectric
  constant is much larger than unity, therefore,
  for hydromagnetic waves,

• where vA is the Alfven velocity
• The Alfven velocity can be considered the
  velocity of the perturbations of the magnetic
  lines of force due to the wave magnetic field in
  the plasma
• Under the approximations made the fluid and the
  field lines oscillate as they were glued
  together

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7.3.5 Magnetosonic Waves

• This case considers the wave vector perpendicular
  to B0 and includes both electrons and ion motions
  (low-frequency waves) with E perpendicular to B0

z
k
B0
E
y
x

• The solution includes the pressure gradient in
  the (fluid) equation of motion since the
  oscillating ExB0 drifts will cause compressions
  in the direction of the wave

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Magnetosonic Waves (II)

• For frequencies much smaller than the ion
  cyclotron frequency the dispersion relation for
  magnetosonic waves can be derived as

• where vs is the sound speed in the plasma
• The magnetosonic wave is an ion-acoustic wave
  that travels perpendicular to the magnetic field
• Compressions and rarefactions are due to the ExB0
  drifts

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Magnetosonic Waves (III)

• In the limit of zero magnetic field the
  ion-acoustic dispersion relation is recovered
• In the limit of zero temperature the sound speed
  goes to zero and the wave becomes similar to an
  Alfven wave