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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.
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| 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.
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| ==Cut-off frequency and critical density==
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| 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>
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| or
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| <math>n_e << m_e\omega^2\,/\,4\pi e^2</math>,
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| 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>
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| :<math>n_c = \frac{\varepsilon_o\,m_e}{e^2}\,\omega^2</math>.
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| If the critical density is exceeded, the plasma is called '''over-dense'''.
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| In a magnetized plasma, except for the O wave, the cut-off relationships are more complex.
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| ==O wave==
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| 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
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| '''E'''<sub>1</sub> || '''B'''<sub>0</sub>.
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| It has a cut-off at the [[plasma frequency]].
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| ==X wave==
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| 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>)
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| 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>
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| :<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]].
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| ==R wave and L wave==
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| 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>
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| ==Dispersion relations==
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| 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).
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| {| class="wikitable"
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| |+ Summary of electromagnetic electron waves
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| |-
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| ! conditions !! dispersion relation !! name
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| |-
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| | <math>\vec B_0=0</math> || <math>\omega^2=\omega_p^2+k^2c^2</math> || light wave
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| |-
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| | <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
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| |-
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| | <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}\,
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| \frac{\omega^2-\omega_p^2}{\omega^2-\omega_h^2}</math> || X wave
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| |-
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| | <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)
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| |-
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| | <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
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| |}
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| ==References==
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| {{Reflist|35em}}
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| ==See also==
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| * [[Appleton-Hartree equation]]
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| * [[List of plasma (physics) articles]]
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| {{DEFAULTSORT:Electromagnetic Electron Wave}}
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| [[Category:Waves in plasmas]]
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The author is known by the name of Numbers Lint. Hiring is her working day job now and she will not alter it anytime quickly. What I love performing is performing ceramics but I haven't produced a dime with it. Puerto Rico is where he's always been living but she needs to move because of her family members.
Here is my blog :: http://www.pinaydiaries.com