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In [[plasma physics]], an '''electromagnetic electron wave''' is a [[Waves in plasmas|wave]] in a [[plasma (physics)|plasma]] which has a [[magnetic field]] component and in which primarily the [[electron]]s oscillate.
 
In an unmagnetized plasma, an electromagnetic electron wave is simply a [[light]] wave modified by the plasma. In a magnetized plasma, there are two modes perpendicular to the field, the O and X modes, and two modes parallel to the field, the R and L waves.
 
==Cut-off frequency and critical density==
In an unmagnetized plasma in the high frequency or low density limit, i.e. for <math>\omega >> (4\pi n_ee^2/m_e)^{1/2}</math>
or
<math>n_e << m_e\omega^2\,/\,4\pi e^2</math>,
the wave speed is the [[speed of light]] in vacuum. As the density increases, the [[phase velocity]] increases and the [[group velocity]] decreases until the '''[[cut-off frequency]]''' where the light frequency is equal to the plasma frequency. This density is known as the '''critical density''' for the [[angular frequency]] ω of that wave and is given by <ref name=Chen>{{cite book|last=Chen|first=Francis|title=Introduction to Plasma Physics and Controlled Fusion, Volume 1|publisher=Plenum Publishing Corporation|year=1984|page=116|edition=2nd|isbn=0-306-41332-9}}</ref>
 
:<math>n_c = \frac{\varepsilon_o\,m_e}{e^2}\,\omega^2</math>.
 
If the critical density is exceeded, the plasma is called '''over-dense'''.
 
In a magnetized plasma, except for the O wave, the cut-off relationships are more complex.
 
==O wave==
The '''O wave''' is the "ordinary" wave in the sense that its [[dispersion relation]] is the same as that in an unmagnetized plasma. It is [[Plane polarization|plane polarized]] with
'''E'''<sub>1</sub> || '''B'''<sub>0</sub>.
It has a cut-off at the [[plasma frequency]].
 
==X wave==
The '''X wave''' is the "extraordinary" wave because it has a more complicated dispersion relation. It is partly transverse (with '''E'''<sub>1</sub>⊥'''B'''<sub>0</sub>)
and partly longitudinal. As the density is increased, the phase velocity rises from ''c'' until the cut-off at ω<sub>R</sub> is reached. As the density is further increased, the wave is evanescent until the resonance at the upper hybrid frequency ω<sub>h</sub>. Then it can propagate again until the second cut-off at ω<sub>L</sub>. The cut-off frequencies are given by <ref name=Chen2>{{cite book|last=Chen|first=Francis|title=Introduction to Plasma Physics and Controlled Fusion, Volume 1|publisher=Plenum Publishing Corporation|year=1984|page=127|edition=2nd|isbn=0-306-41332-9}}</ref>
:<math>\omega_R = \frac{1}{2}\left[ \omega_c + (\omega_c^2+4\omega_p^2)^{1/2} \right]</math>
:<math>\omega_L = \frac{1}{2}\left[ -\omega_c + (\omega_c^2+4\omega_p^2)^{1/2} \right]</math>
where <math>\omega_c</math> is the [[electron cyclotron resonance]] frequency, and <math>\omega_p</math> is the electron [[plasma frequency]].
 
==R wave and L wave==
The '''R wave''' and the '''L wave''' are right-hand and left-hand circularly polarized, respectively. The R wave has a cut-off at ω<sub>R</sub> (hence the designation of this frequency) and a resonance at ω<sub>c</sub>. The L wave has a cut-off at ω<sub>L</sub> and no resonance. R waves at frequencies below ω<sub>c</sub>/2 are also known as '''whistler modes'''. <ref name=Chen3>{{cite book|last=Chen|first=Francis|title=Introduction to Plasma Physics and Controlled Fusion, Volume 1|publisher=Plenum Publishing Corporation|year=1984|page=131|edition=2nd|isbn=0-306-41332-9}}</ref>
 
==Dispersion relations==
The [[dispersion relation]] can be written as an expression for the frequency (squared), but it is also common to write it as an expression for the [[index of refraction]] ''ck''/ω (squared).
 
{| class="wikitable"
|+ Summary of electromagnetic electron waves
|-
! conditions !! dispersion relation !! name
|-
| <math>\vec B_0=0</math> || <math>\omega^2=\omega_p^2+k^2c^2</math> || light wave
|-
| <math>\vec k\perp\vec B_0,\ \vec E_1\|\vec B_0</math> || <math>\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2}{\omega^2}</math> || O wave
|-
| <math>\vec k\perp\vec B_0,\ \vec E_1\perp\vec B_0</math> || <math>\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2}{\omega^2}\,
\frac{\omega^2-\omega_p^2}{\omega^2-\omega_h^2}</math> || X wave
|-
| <math>\vec k\|\vec B_0</math> ([[Polarization (waves)|right circ. pol.]])|| <math>\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2/\omega^2}{1-(\omega_c/\omega)}</math> || R wave (whistler mode)
|-
| <math>\vec k\|\vec B_0</math> ([[Polarization (waves)|left circ. pol.]])|| <math>\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2/\omega^2}{1+(\omega_c/\omega)}</math> || L wave
|}
 
==References==
{{Reflist|35em}}
 
==See also==
* [[Appleton-Hartree equation]]
* [[List of plasma (physics) articles]]
 
{{DEFAULTSORT:Electromagnetic Electron Wave}}
[[Category:Waves in plasmas]]

Revision as of 21:12, 31 January 2014

In plasma physics, an electromagnetic electron wave is a wave in a plasma which has a magnetic field component and in which primarily the electrons oscillate.

In an unmagnetized plasma, an electromagnetic electron wave is simply a light wave modified by the plasma. In a magnetized plasma, there are two modes perpendicular to the field, the O and X modes, and two modes parallel to the field, the R and L waves.

Cut-off frequency and critical density

In an unmagnetized plasma in the high frequency or low density limit, i.e. for ω>>(4πnee2/me)1/2 or ne<<meω2/4πe2, the wave speed is the speed of light in vacuum. As the density increases, the phase velocity increases and the group velocity decreases until the cut-off frequency where the light frequency is equal to the plasma frequency. This density is known as the critical density for the angular frequency ω of that wave and is given by [1]

nc=εomee2ω2.

If the critical density is exceeded, the plasma is called over-dense.

In a magnetized plasma, except for the O wave, the cut-off relationships are more complex.

O wave

The O wave is the "ordinary" wave in the sense that its dispersion relation is the same as that in an unmagnetized plasma. It is plane polarized with E1 || B0. It has a cut-off at the plasma frequency.

X wave

The X wave is the "extraordinary" wave because it has a more complicated dispersion relation. It is partly transverse (with E1B0) and partly longitudinal. As the density is increased, the phase velocity rises from c until the cut-off at ωR is reached. As the density is further increased, the wave is evanescent until the resonance at the upper hybrid frequency ωh. Then it can propagate again until the second cut-off at ωL. The cut-off frequencies are given by [2]

ωR=12[ωc+(ωc2+4ωp2)1/2]
ωL=12[ωc+(ωc2+4ωp2)1/2]

where ωc is the electron cyclotron resonance frequency, and ωp is the electron plasma frequency.

R wave and L wave

The R wave and the L wave are right-hand and left-hand circularly polarized, respectively. The R wave has a cut-off at ωR (hence the designation of this frequency) and a resonance at ωc. The L wave has a cut-off at ωL and no resonance. R waves at frequencies below ωc/2 are also known as whistler modes. [3]

Dispersion relations

The dispersion relation can be written as an expression for the frequency (squared), but it is also common to write it as an expression for the index of refraction ck/ω (squared).

Summary of electromagnetic electron waves
conditions dispersion relation name
B0=0 ω2=ωp2+k2c2 light wave
kB0,E1B0 c2k2ω2=1ωp2ω2 O wave
kB0,E1B0 c2k2ω2=1ωp2ω2ω2ωp2ω2ωh2 X wave
kB0 (right circ. pol.) c2k2ω2=1ωp2/ω21(ωc/ω) R wave (whistler mode)
kB0 (left circ. pol.) c2k2ω2=1ωp2/ω21+(ωc/ω) L wave

References

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See also

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