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| {{Unreferenced|date=December 2009}}
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| In [[plasma physics]], an '''upper hybrid oscillation''' is a mode of [[oscillation]] of a magnetized [[Plasma (physics)|plasma]]. It consists of a [[longitudinal wave|longitudinal]] motion of the electrons perpendicular to the [[magnetic field]] with the [[dispersion relation]]
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| :<math>
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| \omega^2 = \omega_{pe}^2 + \omega_{ce}^2 + 3 k^2 v_{\mathrm{e,th}}^2
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| </math>,
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| where (in [[Centimetre gram second system of units|cgs units]])
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| :<math>\omega_{pe} = (4\pi n_ee^2/m_e)^{1/2}</math> | |
| is the electron [[plasma frequency]], and
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| :<math>\omega_{ce} = eB/m_ec</math>
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| is the electron [[cyclotron frequency]]. | |
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| This oscillation is closely related to the [[plasma oscillation]] found in unmagnetized plasmas or parallel to the magnetic field, where the ω<sub>pe</sub> term arises from the electrostatic Coulomb restoring force and the 3''k''²''v''<sub>e,th</sub>² term arises from the restoring force of electron pressure. In the upper hybrid oscillation there is an additional restoring force due to the [[Lorentz force]]. Consider a [[plane wave]] where all perturbed quantities vary as exp(''i''(''kx''-ω''t'')). If the displacement in the direction of propagation is δ<sub>x</sub>, then
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| :''v''<sub>x</sub> = -''i''ωδ
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| :''f''<sub>y</sub> = ''nev''<sub>x</sub>''B''<sub>z</sub>/''c'' = -''i''ω(''neB''<sub>z</sub>/''c'')δ | |
| :''v''<sub>y</sub> = -''f''<sub>y</sub>/''i''ω''nm'' = (''eB''<sub>z</sub>/''mc'')δ
| |
| :''f''<sub>x</sub> = -''nev''<sub>y</sub>''B''<sub>z</sub>/''c'' = -(''nm'')(''eB''<sub>z</sub>/''mc'')²δ
| |
| :''a''<sub>x</sub> = -ω<sub>ce</sub>²δ
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| Thus the perpendicular magnetic field effectively provides a [[harmonic oscillator|harmonic]] restoring force with a frequency ω<sub>ce</sub>, explaining the third term in the dispersion relation. The particle orbits (or fluid trajectories) are [[ellipse]]s in the plane perpendicular to the magnetic field, elongated in the direction of propagation.
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| The frequency of long wavelength oscillations is a "hybrid", or mix, of the electron plasma and electron cyclotron frequencies,
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| :ω<sub>h</sub>² = ω<sub>pe</sub>² + ω<sub>ce</sub>², | |
| and is known as the '''upper hybrid frequency'''. There are also a lower hybrid frequency and [[lower hybrid oscillation]]s.
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| For propagation at angles oblique to the magnetic field, two modes exist simultaneously. If the plasma frequency is higher than the cyclotron frequency, then the upper hybrid oscillation transforms continuously into the plasma oscillation. The frequency of the other mode varies between the cyclotron frequency and zero. Otherwise, the frequency of the mode related to the upper hybrid oscillation remains above the cyclotron frequency, and the mode related to the plasma oscillation remains below the plasma frequency. In particular, the frequencies are given by
| |
| :<math>
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| \omega^2 = (1/2)\omega_h^2\,\left(
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| 1 \pm \sqrt{
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| 1 - \left(
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| \frac{\cos\theta}{\omega_h^2/2 \omega_c\omega_p}
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| \right)^2
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| }
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| \right)
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| </math> | |
| | |
| ==See also==
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| * [[Plasma oscillation]]
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| * [[Lower hybrid oscillation]]
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| * [[List of plasma (physics) articles]]
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| | |
| {{DEFAULTSORT:Upper Hybrid Oscillation}}
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| [[Category:Waves in plasmas]] | |
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