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Alfvén wave

From Wikipedia, the free encyclopedia

An Alfvén wave, named after Hannes Alfvén, is a type of magnetohydrodynamic wave.

[edit] Definition

An Alfvén wave in a plasma is a low-frequency traveling oscillation of the ions and the magnetic field (by low frequency we mean that the wave frequency is well below the ion cyclotron frequency). The ion mass density provides the inertia and the magnetic field line tension provides the restoring force.

The wave propagates in the direction of the magnetic field, although waves exist at oblique incidence and smoothly change into the magnetosonic wave when the propagation is perpendicular to the magnetic field.

The motion of the ions and the perturbation of the magnetic field are in the same direction and transverse to the direction of propagation. The wave is dispersionless.

[edit] Alfvén velocity

The low-frequency permittivity $\epsilon\,$ of a magnetized plasma is given by

$\epsilon = 1 + \frac{c^2 \mu_0 \rho}{B^2}~,$

where $B\,$ is the magnetic field strength, $c\,$ is the speed of light, $\mu_0\,$ is the permeability of the vacuum, and $\rho = \Sigma n_s m_s\,$ is the total mass density of the charged plasma particles. Here, $s\,$ goes over all plasma species, both electrons and (few types of) ions.

Therefore, the velocity of an electromagnetic wave in such a medium is

$v = c/\sqrt{\epsilon} = \frac{c}{\sqrt{1 + \frac{c^2 \mu_0 \rho}{B^2}}}~,$

$v = \frac{v_A}{\sqrt{1 + v_A^2/c^2}}~,$

where

$v_A = \frac{B}{\sqrt{\mu_0 \rho}}$

is the Alfvén velocity. If $v_A \ll c$ , then $v \approx v_A$ . On the other hand, when $v_A \approx c$ , then $v \approx c$ . That is, at high field or low density, the velocity of the Alfvén wave approaches the speed of light, and the Alfvén wave becomes an ordinary electromagnetic wave.

Neglecting the contribution of the electrons to the mass density and assuming that there is a single ion species, we get

$v_A = \frac{B}{\sqrt{\mu_0 n_i m_i}}~~$ in SI

$v_A = \frac{B}{\sqrt{4 \pi n_i m_i}}~~$ in CGS

$\qquad \ \approx (2.18\times10^{11}\,\mbox{cm/s})\,(m_i/m_p)^{-1/2}\,(n_i/{\rm cm}^{-3})^{-1/2}\,(B/{\rm gauss})$

where $n_i\,$ is the ion number density and $m_i\,$ is the ion mass.

[edit] Relativistic case

The general Alfvén wave velocity is defined by ^[1]

$v = \frac{c}{\sqrt{1 + \frac{e + P}{2 P_m}}}~,$

where

$e\,$ is the total energy density of plasma particles, $P\,$ is the total plasma pressure, and $P_m = B^2/2\mu_0\,$ is the magnetic field pressure. In the non-relativistic limit $P \ll e \approx \rho c^2$ , and we immediately get the expression from the previous section.

[edit] History

How this phenomenon became understood

1942: Alfvén suggests the existence of electromagnetic-hydromagnetic waves in a paper published in Nature.
1949: Laboratory experiments by S. Lundquist produce such waves in magnetized mercury, with a velocity that approximated Alfvén's formula.
1949: Enrico Fermi uses Alfvén waves in his theory of cosmic rays. According to Alex Dessler in a 1970 Science journal article, Fermi had heard a lecture at the University of Chicago, Fermi nodded his head exclaiming "of course" and the next day, the physics world said "of course".
1950: Alfvén publishes the first edition of his book, Cosmical Electrodynamics, detailing hydromagnetic waves, and discussing their application to both laboratory and space plasmas.
1952: Additional confirmation appears in experiments by Winston Bostick and Morton Levine with ionized helium
1954: Bo Lehnert produces Alfvén waves in liquid sodium
1958: Eugene Parker suggests hydromagnetic waves in the interstellar medium
1958: Berthold, Harris, and Hope detect Alfvén waves in the ionosphere after the Argus nuclear test, generated by the explosion, and traveling at speeds predicted by Alfvén formula.
1958: Eugene Parker suggests hydromagnetic waves in the Solar corona extending into the Solar wind.
1959: D. F. Jephcott produces Alfvén waves in a gas discharge
1960: Coleman, et al., report the measurement of Alfvén waves by the magnetometer aboard the Pioneer and Explorer satellites
1960: Sugiura suggests evidence of hydromagnetic waves in the Earth's magnetic field
1966: R.O.Motz generates and observes Alfven waves in mercury
1970 Hannes Alfvén wins the 1970 Nobel Prize in physics for "fundamental work and discoveries in magneto-hydrodynamics with fruitful applications in different parts of plasma physics"
1973: Eugene Parker suggests hydromagnetic waves in the intergalactic medium
1974: Hollweg suggests the existence of hydromagnetic waves in interplanetary space
1974: Ip and Mendis suggests the existence of hydromagnetic waves in the coma of Comet Kohoutek.
2007: Tomczyk, et al., report the detection of Alfvén waves in images of the solar corona with the Coronal Multi-Channel Polarimeter (CoMP) instrument at the National Solar Observatory, New Mexico.
2007: Alfvén wave discoveries appear in articles by Jonathan Cirtain and colleagues, Takenori J. Okamoto and colleagues, and Bart De Pontieu and colleagues. De Pontieu’s team also shows that the energy associated with the waves is sufficient to heat the corona and accelerate the solar wind. These results appear in a special collection of 10 articles, by scientists in Japan, Europe and the United States, in the 7 December issue of the journal Science.

[edit] Further reading

Related research papers

Alfvén, H. "Cosmic Plasma". Holland. 1981.
Alfvén, H. "Existence of electromagnetic-hydrodynamic waves", Nature (1942) Vol. 150, pp. 405
Berthold, W. K.; Harris, A. K.; Hope, H. J., "World-Wide Effects of Hydromagnetic Waves Due to Argus" (1960), Journal of Geophysical Research, Vol. 65, p.2233
Bostick, Winston H.; Levine, Morton A., "Experimental Demonstration in the Laboratory of the Existence of Magneto-Hydrodynamic Waves in Ionized Helium", Physical Review (1952), vol. 87, Issue 4, pp. 671–671
Coleman, P. J., Jr.; Sonett, C. P.; Judge, D. L.; Smith, E. J., "Some Preliminary Results of the Pioneer V Magnetometer Experiment", Journal of Geophysical Research (1960), Vol. 65, p.1856
Dessler, A. J., "Swedish iconoclast recognized after many years of rejection and obscurity," Science (1970) , vol. 170, p. 604
Fermi, E., "On the Origin of the Cosmic Radiation", Physical Review (1949), vol. 75, Issue 8, pp. 1169–1174
Hollweg, J. V., "Hydromagnetic waves in interplanetary space", Astronomical Society of the Pacific, Publications (1974), vol. 86, Oct. 1974, p. 561-594.
Ip, W.-H.; Mendis, D. A., "The cometary magnetic field and its associated electric currents", Icarus (1975), vol. 26, Dec. 1975, p. 457-461.
Jephcott, D.F., "Alfvén waves in a gas discharge", Nature, (1959) vol.183, p.1653
Lehnert, Bo, "Magneto-Hydrodynamic Waves in Liquid Sodium", Physical Review (1954), vol. 94, Issue 4, pp. 815–824
Lundquist, S., "Experimental Investigations of Magneto-Hydrodynamic Waves", Physical Review (1949), vol. 76, Issue 12, pp. 1805–1809
Otani, N. F., "Application of Nonlinear Dynamical Invariants in a Single Electromagnetic Wave to the Study of the Alfvén-Ion-Cyclotron Instability", Physics of Fluids 31, 1456-1464 (1988).
Parker, E. N.,

"Suprathermal Particle Generation in the Solar Corona", Astrophysical Journal (1958), vol. 128, p.677
"Hydromagnetic Waves and the Acceleration of Cosmic Rays", Physical Review (1955), vol. 99, Issue 1, pp. 241-253
"Extragalactic Cosmic Rays and the Galactic Magnetic Field", Astrophysics and Space Science (1973), Vol. 24, p.279

Silberstein, M., and N. F. Otani, "Computer simulation of Alfvén waves and double layers along auroral magnetic field lines", Journal of Geophysical Research 99, 6351-6365 (1994). (PDF)
Sugiura, Masahisa, "Some Evidence of Hydromagnetic Waves in the Earth's Magnetic Field", Physical Review Letters (1961), vol. 6, Issue 6, pp. 255–257
Cramer, N. F., and S. V. Vladimirov, "Alfvén Waves in Dusty Interstellar Clouds". PASA, 14 (2).
Otani, N. F., "The Alfvén ion-cyclotron instability, simulation theory and techniques". Journal of Computational Physics 78, 251-277 (1988).
Falceta-Gonçalves, D. and Jatenco-Pereira, V., "The Effects of Alfvén Waves and Radiation Pressure in Dust Winds of Late-Type Stars". Astrophysical Journal, 576, 976 (2002).
Motz, R.O., "Alfven Wave Generation in a Spherical System", Physics of Fluids, 9, 411-412, (1966)
S. Tomczyk, S. W. McIntosh, S. L. Keil, P. G. Judge, T. Schad, D. H. Seeley and J. Edmondson, "Waves in the Solar Corona", Science Magazine, Vol. 317. no. 5842, pp. 1192–1196, (2007)

[edit] External links

Related websites

Mysterious Solar Ripples Detected Dave Mosher 2 September 2007 Space.com
EurekAlert! notification of 7 December 2007 Science special issue
EurekAlert! notification: "Scientists find solution to solar puzzle"

[edit] References

^ Gedalin M., "Linear waves in relativistic anisotropic magnetohydrodynamics", Phys. Rev. E 47, 4354 (1993)

Retrieved from "http://en.wikipedia.org/wiki/Alfv%C3%A9n_wave"

Magnetosonic wave

Longitudinal wave

From Wikipedia, the free encyclopedia

Jump to: navigation, search

Longitudinal waves are waves that have same direction of oscillations or vibrations along or parallel to their direction of travel, which means that the oscillations of the medium (particle) is in the same direction or opposite direction as the motion of the wave. Mechanical longitudinal waves have been also referred to as compressional waves or compression waves.

Plane pressure wave

Representation of the propagation of a longitudinal wave on a 2d grid (empirical shape)

[hide]

1 Non-electromagnetic
- 1.1 Sound waves
- 1.2 Pressure waves
2 Electromagnetic
3 See also
4 References
5 Further reading
6 External links

[edit] Non-electromagnetic

Examples of longitudinal waves include sound waves (alternation in pressure, particle displacement, or particle velocity propagated in an elastic material) and seismic P-waves (created by earthquakes and explosions).

[edit] Sound waves

In the case of longitudinal harmonic sound waves, the frequency and wavelength can be described with the formula

$y(x,t) = y_0 \sin\Bigg( \omega \left(t-\frac{x}{c} \right) \Bigg)$

where:

y is the displacement of the point on the traveling sound wave (y is a function of x and t);
x is the distance the point has traveled from the wave's source;
t is the time elapsed;
$y 0$ is the amplitude of the oscillations,
c is the speed of the wave; and
ω is the angular frequency of the wave.

The quantity x/c is the time that the wave takes to travel the distance x.

The ordinary frequency $f$ , in hertz, of the wave can be found using

$f = \frac{\omega}{2 \pi}.$

For sound waves, the amplitude of the wave is the difference between the pressure of the undisturbed air and the maximum pressure caused by the wave.

Sound's propagation speed depends on the type, temperature and pressure of the medium through which it propagates.

[edit] Pressure waves

In an elastic medium with rigidity, a harmonic pressure wave oscillation has the form,

$y(x,t)\, = y_0 \cos(k x - \omega t +\phi)$

where:

y₀ is the amplitude of displacement,
k is the wavenumber,
x is distance along the axis of propagation,
ω is angular frequency,
t is time, and
φ is phase difference.

The force acting to return the medium to its original position is provided by the medium's bulk modulus.^[1]

Video of a longitudinal wave

[edit] Electromagnetic

Maxwell's equations lead to the prediction of electromagnetic waves in a vacuum, which are transverse (in that the electric fields and magnetic fields vary perpendicularly to the direction of propagation).^[2] However, in a plasma or a confined space, there can exist waves which are either longitudinal or transverse, or a mixture of both.^[2]^[3] In plasma waves, there exists some examples and these plasma waves can occur in the situation of force-free magnetic fields.

In the early development of electromagnetism there was some suggesting that longitudinal electromagnetic waves existed in a vacuum. After Heaviside's attempts to generalize Maxwell's equations, Heaviside came to the conclusion that electromagnetic waves were not to be found as longitudinal waves in "free space" or homogeneous media.^[4] But it should be stated that Maxwell's equations do lead to the appearance of longitudinal waves under some circumstances in either plasma waves or guided waves. Basically distinct from the "free-space" waves, such as those studied by Hertz in his UHF experiments, are Zenneck waves.^[5] The longitudinal mode of a resonant cavity is a particular standing wave pattern formed by waves confined in a cavity. The longitudinal modes correspond to the wavelengths of the wave which are reinforced by constructive interference after many reflections from the cavity's reflecting surfaces. Recently, Haifeng Wang et al. proposed a method that can generate longitudinal electromagnetic (light) wave in free space, and this wave can propagate without divergence for a few wavelengths.^[6]

[edit] See also

[edit] References

^ Weisstein, Eric W., "P-Wave". Eric Weisstein's World of Science.
^ ^a ^b David J. Griffiths, Introduction to Electrodynamics, ISBN 0-13-805326-X
^ John D. Jackson, Classical Electrodynamics, ISBN 0-471-30932-X.
^ Heaviside, Oliver, "Electromagnetic theory". Appendices: D. On compressional electric or magnetic waves. Chelsea Pub Co; 3rd edition (1971) 082840237X
^ Corum, K. L., and J. F. Corum, "The Zenneck surface wave", Nikola Tesla, Lightning observations, and stationary waves, Appendix II. 1994.
^ Haifeng Wang, Luping Shi, Boris Luk'yanchuk, Colin Sheppard and Chong Tow Chong, "Creation of a needle of longitudinally polarized light in vacuum using binary optics," Nature Photonics, Vol.2, pp 501-505, 2008

[edit] Further reading

Varadan, V. K., and Vasundara V. Varadan, "Elastic wave scattering and propagation". Attenuation due to scattering of ultrasonic compressional waves in granular media - A.J. Devaney, H. Levine, and T. Plona. Ann Arbor, Mich., Ann Arbor Science, 1982.
Schaaf, John van der, Jaap C. Schouten, and Cor M. van den Bleek, "Experimental Observation of Pressure Waves in Gas-Solids Fluidized Beds". American Institute of Chemical Engineers. New York, N.Y., 1997.
Krishan, S, and A A Selim, "Generation of transverse waves by non-linear wave-wave interaction". Department of Physics, University of Alberta, Edmonton, Canada.
Barrow, W. L., "Transmission of electromagnetic waves in hollow tubes of metal", Proc. IRE, vol. 24, pp. 1298-1398, October 1936.
Russell, Dan, "Longitudinal and Transverse Wave Motion". Acoustics Animations, Kettering University Applied Physics.
Longitudinal Waves, with animations "The Physics Classroom"

[edit] External links

Websites

Retrieved from "http://en.wikipedia.org/wiki/Longitudinal_wave"

Permittivity

From Wikipedia, the free encyclopedia

Jump to: navigation, search

This article needs additional citations for verification.
Please help improve this article by adding reliable references. Unsourced material may be challenged and removed. (November 2008)

Permittivity is a physical quantity that describes how an electric field affects, and is affected by, a dielectric medium, and is determined by the ability of a material to polarize in response to the field, and thereby reduce the total electric field inside the material. Thus, permittivity relates to a material's ability to transmit (or "permit") an electric field.

It is directly related to electric susceptibility. For example, in a capacitor, an increased permittivity allows the same charge to be stored with a smaller electric field (and thus a smaller voltage), leading to an increased capacitance.

In SI units, permittivity is measured in farads per meter (F/m). The constant value ε₀ is known as the electric constant or the permittivity of free space, and has the value $ε 0$ ≈ 8.854 187 817… × 10⁻¹² F/m or A²s⁴ kg⁻¹m⁻³ in SI base units.

[hide]

1 Explanation
2 Vacuum permittivity
3 Relative permittivity
4 Permittivity in media
5 Measurement
6 See also
7 References
8 Further reading
9 External links

[edit] Explanation

In electromagnetism, the electric displacement field D represents how an electric field E influences the organization of electrical charges in a given medium, including charge migration and electric dipole reorientation. Its relation to permittivity in the very simple case of linear, homogeneous, isotropic materials with "instantaneous" response to changes in electric field is

$\mathbf{D}=\varepsilon \mathbf{E}$

where the permittivity ε is a scalar. If the medium is anisotropic, the permittivity is a second rank tensor.

In general, permittivity is not a constant, as it can vary with the position in the medium, the frequency of the field applied, humidity, temperature, and other parameters. In a nonlinear medium, the permittivity can depend on the strength of the electric field. Permittivity as a function of frequency can take on real or complex values.

In SI units, permittivity is measured in farads per meter (F/m). The displacement field D is measured in units of coulombs per square meter (C/m²), while the electric field E is measured in volts per meter (V/m). D and E describe the interaction between charged objects. D is related to the charge densities associated with this interaction, while E is related to the forces and potential differences.

[edit] Vacuum permittivity

Main article: vacuum permittivity

Vacuum permittivity $\varepsilon_{0}$ (also called permittivity of free space or the electric constant) is the ratio D/E in free space.

$\begin{align} \varepsilon_0 & \stackrel{\mathrm{def}}{=}\ \frac{1}{c_0^2\mu_0} = \frac{1}{35950207149.4727056\pi}\ \frac\text{F}\text{m} \text{ (farads per meter) } \\[8pt] & \approx 8.8541878176\ldots\times 10^{-12}\ \frac\text{F}\text{m} \end{align}$

(or A²s⁴kg⁻¹m⁻³), where

c₀ is the speed of light in free space,^[1]

μ 0

is the magnetic constant.

Constants c₀ and μ₀ are defined in SI units to have exact numerical values (see NIST), shifting responsibility of experiment to the determination of the meter and the ampere. (The approximation in the second value of ε₀ above stems from π being an irrational number.) The electric constant ε₀ also appears in Coulomb's law as a part of the Coulomb force constant, 1/(4πε₀), which expresses the force between two unit charges separated by unit distance in vacuum.

[edit] Relative permittivity

Main article: relative permittivity

The linear permittivity of a homogeneous material is usually given relative to that of free space, as a relative permittivity $\varepsilon_{r}$ (also called dielectric constant, although this sometimes only refers to the static, zero-frequency relative permittivity). In an anisotropic material, the relative permittivity may be a tensor. The actual permittivity is then calculated by multiplying the relative permittivity by $\varepsilon_{0}$ :

$\varepsilon = \varepsilon_r \varepsilon_0 = (1+\chi_e)\varepsilon_0$

where

$\,\chi_e$ is the electric susceptibility of the material.

[edit] Permittivity in media

In the common case of isotropic media, D and E are parallel vectors and $\varepsilon$ is a scalar, but in general anisotropic media this is not the case and $\varepsilon$ is a rank-2 tensor (causing birefringence). The permittivity $\varepsilon$ and permeability $μ$ of a medium together determine the phase velocity v of electromagnetic radiation through that medium:

$\varepsilon \mu = \frac{1}{v^2}.$

When an external electric field is applied to a real medium, a current flows. The total current within the medium consists of two parts: a conduction and a displacement current. The displacement current can be thought of as the elastic response of the material to the applied electric field. As the magnitude of the externally applied electric field is increased, an increasing amount of energy is stored in the electric displacement field within the material. If the electric field is subsequently decreased, the material will release the stored electrostatic energy. The displacement current reflects the resulting change in electrostatic energy stored within the material. The electric displacement can be separated into a vacuum contribution and one arising from the material by

$\mathbf{D} = \varepsilon \mathbf{E} = \varepsilon_0 \mathbf{E} + \mathbf{P} = \varepsilon_{0} \mathbf{E} + \varepsilon_0 \chi\mathbf{E} = \varepsilon_0 \mathbf{E} \left( 1 + \chi \right),$

where

P is the polarization of the medium

χ

its electric susceptibility.

It follows that the relative permittivity and susceptibility of a sample are related: $\varepsilon_r = \chi + 1.$

[edit] Complex permittivity

A dielectric permittivity spectrum over a wide range of frequencies. ε′ and ε″ denote the real and the imaginary part of the permittivity, respectively. Various processes are labeled on the image: ionic and dipolar relaxation, and atomic and electronic resonances at higher energies. From the Dielectric spectroscopy page [1] of the research group of Dr. Kenneth A. Mauritz.

As opposed to the response of a vacuum, the response of normal materials to external fields generally depends on the frequency of the field. This frequency dependence reflects the fact that a material's polarization does not respond instantaneously to an applied field. The response must always be causal (arising after the applied field) which can be represented by a phase difference. For this reason permittivity is often treated as a complex function (since complex numbers allow specification of magnitude and phase) of the frequency of the applied field $ω$ , $\varepsilon \rightarrow \widehat{\varepsilon}(\omega)$ . The definition of permittivity therefore becomes

$D_0 e^{-i \omega t} = \widehat{\varepsilon}(\omega) E_0 e^{-i \omega t},$

where

D 0

and

E 0

are the amplitudes of the displacement and electrical fields, respectively,

i is the imaginary unit, i ² = −1.

It is important to realise that the choice of sign for time-dependence dictates the sign convention for the imaginary part of permittivity. The signs used here correspond to those commonly used in physics, whereas for the engineering convention one should reverse all imaginary quantities.

The response of a medium to static electric fields is described by the low-frequency limit of permittivity, also called the static permittivity $\varepsilon_s$ (also $\varepsilon_{DC}$ ):

$\varepsilon_s = \lim_{\omega \rightarrow 0} \widehat{\varepsilon}(\omega).$

At the high-frequency limit, the complex permittivity is commonly referred to as ε_∞. At the plasma frequency and above, dielectrics behave as ideal metals, with electron gas behavior. The static permittivity is a good approximation for altering fields of low frequencies, and as the frequency increases a measurable phase difference $δ$ emerges between D and E. The frequency at which the phase shift becomes noticeable depends on temperature and the details of the medium. For moderate fields strength ( $E 0$ ), D and E remain proportional, and

$\widehat{\varepsilon} = \frac{D_0}{E_0}e^{i\delta} = |\varepsilon|e^{i\delta}.$

Since the response of materials to alternating fields is characterized by a complex permittivity, it is natural to separate its real and imaginary parts, which is done by convention in the following way:

$\widehat{\varepsilon}(\omega) = \varepsilon'(\omega) + i\varepsilon''(\omega) = \frac{D_0}{E_0} \left( \cos\delta + i\sin\delta \right).$

where

$\varepsilon''$ is the imaginary part of the permittivity, which is related to the dissipation (or loss) of energy within the medium.

$\varepsilon'$ is the real part of the permittivity, which is related to the stored energy within the medium.

The complex permittivity is usually a complicated function of frequency ω, since it is a superimposed description of dispersion phenomena occurring at multiple frequencies. The dielectric function $\varepsilon(\omega)$ must have poles only for frequencies with positive imaginary parts, and therefore satisfies the Kramers–Kronig relations. However, in the narrow frequency ranges that are often studied in practice, the permittivity can be approximated as frequency-independent or by model functions.

At a given frequency, the imaginary part of $\widehat{\varepsilon}$ leads to absorption loss if it is positive (in the above sign convention) and gain if it is negative. More generally, the imaginary parts of the eigenvalues of the anisotropic dielectric tensor should be considered.

In the case of solids, the complex dielectric function is intimately connected to band structure. The primary quantity that characterize the electronic structure of any crystalline material is the probability of photon absorption, which is directly related to the imaginary part of the optical dielectric function ε(ω). The optical dielectric function is given by the fundamental expression:^[2]

$\varepsilon(\omega)=1+\frac{8\pi^2e^2}{m^2}\sum_{c,v}\int W_{cv}(E) \left[ \varphi (\hbar \omega - E)-\varphi( \hbar \omega +E) \right ] \, dx.$

In this expression, W_cv(E) represents the product of the Brillouin zone-averaged transition probability at the energy E with the joint density of states,^[3]^[4] J_cv(E); φ is a broadening function, representing the role of scattering in smearing out the energy levels.^[5] In general, the broadening is intermediate between Lorentzian and Gaussian;^[6]^[7] for an alloy it is somewhat closer to Gaussian because of strong scattering from statistical fluctuations in the local composition on a nanometer scale.

[edit] Classification of materials

Materials can be classified according to their permittivity and conductivity, σ. Materials with a large amount of loss inhibit the propagation of electromagnetic waves. In this case, generally when $\frac{\sigma}{\omega\varepsilon}\gg1$ , we consider the material to be a good conductor. Dielectrics are associated with lossless or low-loss materials, where $\frac{\sigma}{\omega\varepsilon}\ll1$ . Those that do not fall under either limit are considered to be general media. A perfect dielectric is a material that has no conductivity, thus exhibiting only a displacement current. Therefore it stores and returns electrical energy as if it were an ideal capacitor.

[edit] Lossy medium

In the case of lossy medium, i.e. when the conduction current is not negligible, the total current density flowing is:

$J_\text{tot} = J_c + J_d = \sigma E - i \omega \varepsilon E = -i \omega \widehat{\varepsilon} E$

where

σ is the conductivity of the medium;

ε is the real part of the permittivity.

$\widehat{\varepsilon}$ is the complex permittivity

The size of the displacement current is dependent on the frequency ω of the applied field E; there is no displacement current in a constant field.

In this formalism, the complex permittivity is defined as:

$\widehat{\varepsilon} = \varepsilon + i \frac{\sigma}{\omega}$

In general, the absorption of electromagnetic energy by dielectrics is covered by a few different mechanisms that influence the shape of the permittivity as a function of frequency:

Relaxation effects associated with permanent and induced molecular dipoles. At low frequencies the field changes slowly enough to allow dipoles to reach equilibrium before the field has measurably changed. For frequencies at which dipole orientations cannot follow the applied field due to the viscosity of the medium, absorption of the field's energy leads to energy dissipation. The mechanism of dipoles relaxing is called dielectric relaxation and for ideal dipoles is described by classic Debye relaxation.

Resonance effects, which arise from the rotations or vibrations of atoms, ions, or electrons. These processes are observed in the neighborhood of their characteristic absorption frequencies.

The above effects often combine to cause non-linear effects within capacitors. For example, dielectric absorption refers to the inability of a capacitor that has been charged for a long time to completely discharge when briefly discharged. Although an ideal capacitor would remain at zero volts after being discharged, real capacitors will develop a small voltage, a phenomenon that is also called soakage or battery action. For some dielectrics, such as many polymer films, the resulting voltage may be less than 1-2% of the original voltage. However, it can be as much as 15 - 25% in the case of electrolytic capacitors or supercapacitors.

[edit] Quantum-mechanical interpretation

In terms of quantum mechanics, permittivity is explained by atomic and molecular interactions.

At low frequencies, molecules in polar dielectrics are polarized by an applied electric field, which induces periodic rotations. For example, at the microwave frequency, the microwave field causes the periodic rotation of water molecules, sufficient to break hydrogen bonds. The field does work against the bonds and the energy is absorbed by the material as heat. This is why microwave ovens work very well for materials containing water. There are two maxima of the imaginary component (the absorptive index) of water, one at the microwave frequency, and the other at far ultraviolet (UV) frequency. It should be noted that both of these resonances are at higher frequencies than the operating frequency of microwave ovens.

At moderate frequencies, the energy is too high to cause rotation, yet too low to affect electrons directly, and is absorbed in the form of resonant molecular vibrations. In water, this is where the absorptive index starts to drop sharply, and the minimum of the imaginary permittivity is at the frequency of blue light (optical regime). This is why sunlight does not damage water-containing organs such as the eye.^[8]

At high frequencies (such as UV and above), molecules cannot relax, and the energy is purely absorbed by atoms, exciting electron energy levels. Thus, these frequencies are classified as ionizing radiation.

While carrying out a complete ab initio (that is, first-principles) modelling is now computationally possible, it has not been widely applied yet. Thus, a phenomenological model is accepted as being an adequate method of capturing experimental behaviors. The Debye model and the Lorentz model use a 1st-order and 2nd-order (respectively) lumped system parameter linear representation (such as an RC and an LRC resonant circuit).

[edit] Measurement

This section requires expansion.

The dielectric constant of a material can be found by a variety of static electrical measurements. The complex permittivity is evaluated over a wide range of frequencies by using different variants of dielectric spectroscopy, covering nearly 21 orders of magnitude from 10⁻⁶ to 10¹⁵ Hz. Also, by using cryostats and ovens, the dielectric properties of a medium can be characterized over an array of temperatures. In order to study systems for such diverse exciting fields, a number of measurement setups are used, each adequate for a special frequency range.

Various microwave measurement techiques are outlined in Chen et al..^[9] Typical errors for the Hakki-Coleman method employing a puck of material between conducting planes are about 0.3%.^[10]

Low-frequency time domain measurements (10⁻⁶−10³ Hz)
Low-frequency frequency domain measurements (10⁻⁵−10⁶ Hz)
Reflective coaxial methods (10⁶−10¹⁰ Hz)
Transmission coaxial method (10⁸−10¹¹ Hz)
Quasi-optical methods (10⁹−10¹⁰ Hz)
Fourier-transform methods (10¹¹−10¹⁵ Hz)

At infrared and optical frequencies, a common technique is ellipsometry.

[edit] See also

[edit] References

^ Current practice of international standards organizations such as NIST and BIPM is to use c₀, rather than c, to denote the speed of light in vacuum according to ISO 31. In the original Recommendation of 1983, the symbol c was used for this purpose. See NIST Special Publication 330, Appendix 2, p. 45 .
^ Peter Y. Yu, Manuel Cardona (2001). Fundamentals of Semiconductors: Physics and Materials Properties. Berlin: Springer. p. 261 ff. ISBN 3540254706. http://books.google.com/books?id=W9pdJZoAeyEC&pg=PA261&dq=%22joint+density+of+states%22&lr=&as_brr=0&sig=alCOU9kdaDyMOFD4s9pHbydOWsI.
^ José García Solé, Jose Solé, Luisa Bausa, (2001). An introduction to the optical spectroscopy of inorganic solids. Wiley. Appendix A1, pp, 263ff. ISBN 0470868856. http://books.google.com/books?id=c6pkqC50QMgC&pg=PA263&dq=%22joint+density+of+states%22&lr=&as_brr=0&sig=s7Bo8OFEwMCPRVoRYfVccMKA15A#PPA264,M1.
^ John H. Moore, Nicholas D. Spencer (2001). Encyclopedia of chemical physics and physical chemistry. Taylor and Francis. p. 105 ff. ISBN 0750307986. http://books.google.com/books?id=Pn2edky6uJ8C&pg=PA108.
^ Solé and Bausa. p. 10 ff. ISBN 3540254706. http://books.google.com/books?id=c6pkqC50QMgC&pg=PA10&dq=optic+absorption+%22line+broadening%22&lr=&as_brr=0&sig=12Vm3PaKN31XoUB5pm10bt0WBqc.
^ Hartmut Haug, Stephan W. Koch (1994). Quantum Theory of the Optical and Electronic Properties of Semiconductors. World Scientific. p. 196 ff. ISBN 9810218648. http://books.google.com/books?id=Ab2WnFyGwhcC&pg=PA197&dq=broadening+%22band+edge+absorption%22&lr=&as_brr=0&sig=LxWWmJXMo9rwK69TV6zx6zQdoOQ#PPA196,M1.
^ Manijeh Razeghi (2006). Fundamentals of Solid State Engineering. Birkhauser. p. 383 ff. ISBN 0387281525. http://books.google.com/books?id=6x07E9PSzr8C&pg=PA383&dq=broadening+%22band+edge+absorption%22&lr=&as_brr=0&sig=poIPUqw_N5bvjNeBzM3-3GW2pZA#PPA383,M1.
^ Braun, Charles L.; Smirnov, Sergei N. (1993), "Why is water blue?", Journal of Chemical Education 70 (8): 612, http://www.dartmouth.edu/~etrnsfer/water.htm
^ Linfeng Chen, V. V. Varadan, C. K. Ong, Chye Poh Neo (2004). "Microwave theory and techniques for materials characterization". Microwave electronics. Wiley. p. 37. ISBN 0470844922. http://books.google.com/books?id=2oA3po4coUoC&pg=PA37.
^ Mailadil T. Sebastian (2008). Dielectric Materials for Wireless Communication. Elsevier. p. 19. ISBN 0080453309. http://books.google.com/books?id=eShDR4_YyM8C&pg=PA19.

[edit] Further reading

Theory of Electric Polarization: Dielectric Polarization, C.J.F. Böttcher, ISBN 0-444-41579-3
Dielectrics and Waves edited by A. von Hippel, Arthur R., ISBN 0-89006-803-8
Dielectric Materials and Applications edited by Arthur von Hippel, ISBN 0-89006-805-4.

[edit] External links

Electromagnetism, a chapter from an online textbook

Retrieved from "http://en.wikipedia.org/wiki/Permittivity"

Cyclotron

From Wikipedia, the free encyclopedia

(Redirected from Cyclotron frequency)

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A modern Cyclotron for radiation therapy

For other uses, see Cyclotron (disambiguation).

A cyclotron is a type of particle accelerator. Cyclotrons accelerate charged particles using a high-frequency, alternating voltage (potential difference). A perpendicular magnetic field causes the particles to spiral almost in a circle so that they re-encounter the accelerating voltage many times.

Ernest Lawrence, of the University of California, Berkeley, is credited with the development of the cyclotron in 1929, though others had been working along similar lines at the time.^{[citation
needed]}

[hide]

1 How the cyclotron works
2 Uses of the cyclotron
3 Problems solved by the cyclotron
4 Advantages of the cyclotron
5 Limitations of the cyclotron
6 Mathematics of the cyclotron
- 6.1 Non-relativistic
- 6.2 Relativistic
7 Related technologies
8 See also
9 External links

[edit] How the cyclotron works

Diagram of cyclotron operation from Lawrence's 1934 patent.

Beam of electrons moving in a circle. Lighting is caused by excitation of gas atoms in a bulb.

The electrodes shown at the right would be in the vacuum chamber, which is flat, in a narrow gap between the two poles of a large magnet.

In the cyclotron, a high-frequency alternating voltage applied across the "D" electrodes (also called "dees") alternately attracts and repels charged particles. The particles, injected near the center of the magnetic field, accelerate only when passing through the gap between the electrodes. The perpendicular magnetic field (passing vertically through the "D" electrodes), combined with the increasing energy of the particles forces the particles to travel in a spiral path.

With no change in energy the charged particles in a magnetic field will follow a circular path. In the cyclotron, energy is applied to the particles as they cross the gap between the dees and so they are accelerated (at the typical sub-relativistic speeds used) and will increase in mass as they approach the speed of light. Either of these effects (increased velocity or increased mass) will increase the radius of the circle and so the path will be a spiral.

(The particles move in a spiral, because a current of electrons or ions, flowing perpendicular to a magnetic field, experiences a perpendicular force. The charged particles move freely in a vacuum, so the particles follow a spiral path.)

The radius will increase until the particles hit a target at the perimeter of the vacuum chamber. Various materials may be used for the target, and the collisions will create secondary particles which may be guided outside of the cyclotron and into instruments for analysis. The results will enable the calculation of various properties, such as the mean spacing between atoms and the creation of various collision products. Subsequent chemical and particle analysis of the target material may give insight into nuclear transmutation of the elements used in the target.

[edit] Uses of the cyclotron

For several decades, cyclotrons were the best source of high-energy beams for nuclear physics experiments; several cyclotrons are still in use for this type of research.

Cyclotrons can be used to treat cancer. Ion beams from cyclotrons can be used, as in proton therapy, to penetrate the body and kill tumors by radiation damage, while minimizing damage to healthy tissue along their path.

Cyclotron beams can be used to bombard other atoms to produce short-lived positron-emitting isotopes suitable for PET imaging.

[edit] Problems solved by the cyclotron

60-inch cyclotron, circa 1939, showing a beam of accelerated ions (likely protons or deuterons) escaping the accelerator and ionizing the surrounding air causing a blue glow. This phenomenon of air ionization is analogous to the one responsible for producing the "blue flash" infamously noted by witnesses of criticality accidents. Though the effect is often mistaken for Cherenkov radiation, this is not the case.

The cyclotron was an improvement over the linear accelerators that were available when it was invented. A linear accelerator (also called a linac) accelerates particles in a straight line through an evacuated tube (or series of such tubes placed end to end). A set of electrodes shaped like flat donuts are arranged inside the length of the tube(s). These are driven by high-power radio waves that continuously switch between positive and negative voltage, causing particles traveling along the center of the tube to accelerate. In the 1920s, it was not possible to get high frequency radio waves at high power, so either the accelerating electrodes had to be far apart to accommodate the low frequency or more stages were required to compensate for the low power at each stage. Either way, higher-energy particles required longer accelerators than scientists could afford.

Modern linacs use high power Klystrons and other devices able to impart much more power at higher frequencies. But before these devices existed, cyclotrons were cheaper than linacs.

Cyclotrons accelerate particles in a spiral path. Therefore, a compact accelerator can contain much more distance than a linear accelerator, with more opportunities to accelerate the particles.

[edit] Advantages of the cyclotron

Cyclotrons have a single electrical driver, which saves both money and power, since more expense may be allocated to increasing efficiency.
Cyclotrons produce a continuous stream of particles at the target, so the average power is relatively high.
The compactness of the device reduces other costs, such as its foundations, radiation shielding, and the enclosing building.

[edit] Limitations of the cyclotron

The magnet portion of a large cyclotron. The gray object is the upper pole piece, routing the magnetic field in two loops through a similar part below. The white canisters held conductive coils to generate the magnetic field. The D electrodes are contained in a vacuum chamber that was inserted in the central field gap.

The spiral path of the cyclotron beam can only "synch up" with klystron-type (constant frequency) voltage sources if the accelerated particles are approximately obeying Newton's Laws of Motion. If the particles become fast enough that relativistic effects become important, the beam gets out of phase with the oscillating electric field, and cannot receive any additional acceleration. The cyclotron is therefore only capable of accelerating particles up to a few percent of the speed of light. To accommodate increased mass the magnetic field may be modified by appropriately shaping the pole pieces as in the isochronous cyclotrons, operating in a pulsed mode and changing the frequency applied to the dees as in the synchrocyclotrons, either of which is limited by the diminishing cost effectiveness of making larger machines. Cost limitations have been overcome by employing the more complex synchrotron or linear accelerator, both of which have the advantage of scalability, offering more power within an improved cost structure as the machines are made larger.

[edit] Mathematics of the cyclotron

[edit] Non-relativistic

The centripetal force is provided by the transverse magnetic field B, and the force on a particle travelling in a magnetic field (which causes it to be angularly displaced, i.e spiral) is equal to Bqv. So,

$\frac{mv^2}{r} = Bqv$

(Where m is the mass of the particle, q is its charge, B the magnetic field strength, v is its velocity and r is the radius of its path.)

The speed at which the particles enter the cyclotron due to a potential difference, V.

$v = \sqrt{\frac{2Vq}{m}}$

Therefore,

$\frac{v}{r} = \frac{Bq}{m}$

v/r is equal to angular velocity, ω, so

$\omega = \frac{Bq}{m}$

And since the angular frequency is

ω = 2π f

Therefore,

$f = \frac{Bq}{2\pi m}$

But this is for one complete loop and cyclotron must switch twice every cycle, therefore

$f_c = \frac{Bq}{\pi m}$

A pair of "dee" electrodes with loops of coolant pipes on their surface at the Lawrence Hall of Science. The particle exit point may be seen at the top of the upper dee, where the target would be positioned

This shows that for a particle of constant mass, the frequency does not depend upon the radius of the particle's orbit. As the beam spirals out, its frequency does not decrease, and it must continue to accelerate, as it is travelling more distance in the same time. As particles approach the speed of light, they acquire additional mass, requiring modifications to the frequency, or the magnetic field during the acceleration. This is accomplished in the synchrocyclotron.

[edit] Relativistic

The radius of curvature for a particle moving relativistically in a static magnetic field is

$r = \frac{\gamma m v}{q B}$

where

$\gamma=\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}}$ the Lorentz factor

Note that in high-energy experiments energy, E, and momentum, p, are used rather than velocity, and both measured in units of energy. In that case one should use the substitution,

$\frac{E}{p} = v$

where this is in Natural units

The relativistic cyclotron frequency is

$f=f_c\sqrt{1-\left(\frac{v}{c}\right)^2}$ ,

where

f c

is the classical frequency, given above, of a charged particle with velocity

v

circling in a magnetic field.

The rest mass of an electron is 511 keV/c^2, so the frequency correction is 1% for a magnetic vacuum tube with a 5.11 keV/c^2 direct current accelerating voltage. The proton mass is nearly two thousand times the electron mass, so the 1% correction energy is about 9 MeV, which is sufficient to induce nuclear reactions.

An alternative to the synchrocyclotron is the isochronous cyclotron, which has a magnetic field that increases with radius, rather than with time. The de-focusing effect of this radial field gradient is compensated by ridges on the magnet faces which vary the field azimuthally as well. This allows particles to be accelerated continuously, on every period of the radio frequency, rather than in bursts as in most other accelerator types. This principle that alternating field gradients have a net focusing effect is called strong focusing. It was obscurely known theoretically long before it was put into practice.

[edit] Related technologies

The spiraling of electrons in a cylindrical vacuum chamber within a transverse magnetic field is also employed in the magnetron, a device for producing high frequency radio waves (microwaves).

The Synchrotron moves the particles through a path of constant radius, allowing it to be made as a pipe and so of much larger radius than is practical with the cyclotron and synchrocyclotron. The larger radius allows the use of numerous magnets, each of which imparts angular momentum and so allows particles of higher velocity (mass) to be kept within the bounds of the evacuated pipe. The magnetic field strength of each of the bending magnets is increased as the particles gain energy in order to keep the bending angle constant.

[edit] See also

[edit] External links

U.S. Patent 1,948,384 -- Method and apparatus for the acceleration of ions
Indiana University Cyclotron Facility MPRI treats first patient using robotic gantry system.
"The 88-Inch Cyclotron at LBNL"
"The NSCL at Michigan State University" Home of coupled K500 and K1200 cyclotrons; the K500, being the first superconducting cyclotron and the K1200 currently the most powerful in the world.
Rutgers Cyclotron and "Building a Cyclotron on a Shoestring" Tim Koeth, now a graduate student at Rutgers University, built a 12-inch 1 MeV cyclotron as an undergraduate project, which is now used for a senior-level undergraduate and a graduate lab course.
"Cyclotron java applet"
"Resonance Spectral Analysis with a Homebuilt Cyclotron" an experiment done by Fred M. Niell, III his senior year of high school (1994-95) with which he won the overall grand prize in the ISEF.
Relativistic accelerator physics PDF
Wired news article about a neighborhood cyclotron in Anchorage, Alaska
web site of company IBA
The Cyclotron Kids A group of high schooled students building their own 2.3 MeV cyclotron for experimentation.

Retrieved from "http://en.wikipedia.org/wiki/Cyclotron"

Magnetohydrodynamics

From Wikipedia, the free encyclopedia

Jump to: navigation, search

Magnetohydrodynamics (MHD) (magnetofluiddynamics or hydromagnetics) is the academic discipline which studies the dynamics of electrically conducting fluids. Examples of such fluids include plasmas, liquid metals, and salt water. The word magnetohydrodynamics (MHD) is derived from magneto- meaning magnetic field, and hydro- meaning liquid, and -dynamics meaning movement. The field of MHD was initiated by Hannes Alfvén^[1], for which he received the Nobel Prize in Physics in 1970.

The idea of MHD is that magnetic fields can induce currents in a moving conductive fluid, which create forces on the fluid, and also change the magnetic field itself. The set of equations which describe MHD are a combination of the Navier-Stokes equations of fluid dynamics and Maxwell's equations of electromagnetism. These differential equations have to be solved simultaneously, either analytically or numerically. MHD is a continuum theory and as such it cannot treat kinetic phenomena, i.e. those in which the existence of discrete particles or of a non-thermal velocities distribution, are important.^{[citation
needed]}

[hide]

1 Ideal and resistive MHD
2 Structures in MHD systems
3 MHD waves
4 Extensions to magnetohydrodynamics
5 Applications
6 History
7 In fiction
8 See also
9 Notes
10 References

[edit] Ideal and resistive MHD

MHD Simulation of the Solar Wind

The simplest form of MHD, Ideal MHD, assumes that the fluid has so little resistivity that it can be treated as a perfect conductor. This is the limit of infinite magnetic Reynolds number. In ideal MHD, Lenz's law dictates that the fluid is in a sense tied to the magnetic field lines. To explain, in ideal MHD a small rope-like volume of fluid surrounding a field line will continue to lie along a magnetic field line, even as it is twisted and distorted by fluid flows in the system. The connection between magnetic field lines and fluid in ideal MHD fixes the topology of the magnetic field in the fluid -- for example, if a set of magnetic field lines are tied into a knot, then they will remain so as long as the fluid/plasma has negligible resistivity. This difficulty in reconnecting magnetic field lines makes it possible to store energy by moving the fluid or the source of the magnetic field. The energy can then become available if the conditions for ideal MHD break down, allowing magnetic reconnection that releases the stored energy from the magnetic field.

[edit] Ideal MHD equations

The ideal MHD equations consist of the continuity equation, the momentum equation, and Ampere's Law in the limit of no electric field and no electron diffusivity, and a temperature evolution equation. As with any fluid description to a kinetic system, a closure approximation must be applied to highest moment of the particle distribution equation. This is often accomplished with approximations to the heat flux through a condition of adiabaticity or isothermality.

[edit] Applicability of ideal MHD to plasmas

Ideal MHD is only strictly applicable when:

The plasma is strongly collisional, so that the time scale of collisions is shorter than the other characteristic times in the system, and the particle distributions are therefore close to Maxwellian.
The resistivity due to these collisions is small. In particular, the typical magnetic diffusion times over any scale length present in the system must be longer than any time scale of interest.
We are interested in length scales much longer than the ion skin depth and Larmor radius perpendicular to the field, long enough along the field to ignore Landau damping, and time scales much longer than the ion gyration time (system is smooth and slowly evolving).

[edit] The importance of resistivity

In an imperfectly conducting fluid the magnetic field can generally move through the fluid following a diffusion law with the resistivity of the plasma serving as a diffusion constant. This means that solutions to the ideal MHD equations are only applicable for a limited time for a region of a given size before diffusion becomes too important to ignore. One can estimate the diffusion time across a solar active region (from collisional resistivity) to be hundreds to thousands of years, much longer than the actual lifetime of a sunspot -- so it would seem reasonable to ignore the resistivity. By contrast, a meter-sized volume of seawater has a magnetic diffusion time measured in milliseconds.

Even in physical systems which are large and conductive enough that simple estimates of the Lundquist number suggest that we can ignore the resistivity, resistivity may still be important: many instabilities exist that can increase the effective resistivity of the plasma by factors of more than a billion. The enhanced resistivity is usually the result of the formation of small scale structure like current sheets or fine scale magnetic turbulence, introducing small spatial scales into the system over which ideal MHD is broken and magnetic diffusion can occur quickly. When this happens, magnetic reconnection may occur in the plasma to release stored magnetic energy as waves, bulk mechanical acceleration of material, particle acceleration, and heat. Magnetic reconnection in highly conductive systems is important because it concentrates energy in time and space, so that gentle forces applied to a plasma for long periods of time can cause violent explosions and bursts of radiation.

When the fluid cannot be considered as completely conductive, but the other conditions for ideal MHD are satisfied, it is possible to use an extended model called resistive MHD. This includes an extra term in Ampere's Law which models the collisional resistivity. Generally MHD computer simulations are at least somewhat resistive because their computational grid introduces a numerical resistivity.

[edit] The importance of kinetic effects

Another limitation of MHD (and fluid theories in general) is that they depend on the assumption that the plasma is strongly collisional (this is the first criterion listed above), so that the time scale of collisions is shorter than the other characteristic times in the system, and the particle distributions are Maxwellian. This is usually not the case in fusion, space and astrophysical plasmas. When this is not the case, or we are interested in smaller spatial scales, it may be necessary to use a kinetic model which properly accounts for the non-Maxwellian shape of the distribution function. However, because MHD is relatively simple and captures many of the important properties of plasma dynamics it is often qualitatively accurate and is almost invariably the first model tried.

Effects which are essentially kinetic and not captured by fluid models include double layers, Landau damping, a wide range of instabilities, chemical separation in space plasmas and electron runaway.

[edit] Structures in MHD systems

Schematic view of the different current systems which shape the Earth's magnetosphere

In many MHD systems most of the electric current is compressed into thin nearly-two-dimensional ribbons termed current sheets. These can divide the fluid into magnetic domains, inside of which the currents are relatively weak. Current sheets in the solar corona are thought to be between a few meters and a few kilometers in thickness, which is quite thin compared to the magnetic domains (which are thousands to hundreds of thousands of kilometers across). Another example is in the Earth's magnetosphere, where current sheets separate topologically distinct domains, isolating most of the Earth's ionosphere from the solar wind.

Further information: Magnetosphere particle motion

[edit] MHD waves

See also: Waves in plasmas and Category:Waves in plasmas

The wave modes derived using MHD plasma theory are called magnetohydrodynamic waves or MHD waves. In general there are three MHD wave modes:

Pure (or oblique) Alfvén wave
Slow MHD wave
Fast MHD wave

All these waves have constant phase velocities for all frequencies, and hence there is no dispersion. At the limits when the angle a between the wave propagation vector k and magnetic field B is either 0 (180) or 90 degrees, the wave modes are called^[2]:

name	type	propagation	phase velocity	association	medium	other names
Sound wave	longitudinal	$\vec k\\|\vec B$	adiabatic sound velocity	none	compressible, nonconducting fluid
Alfvén wave	transverse	$\vec k\\|\vec B$	Alfvén velocity	$B$		shear Alfvén wave, the slow Alfvén wave, torsional Alfvén wave
Magnetosonic wave	longitudinal	$\vec k\perp\vec B$		$B$ , $E$		compressional Alfvén wave, fast Alfvén wave, magnetoacoustic wave

The MHD oscillations will be damped if the fluid is not perfectly conducting but has a finite conductivity, or if viscous effects are present.

MHD waves and oscillations are a popular tool for the remote diagnostics of laboratory and astrophysical plasmas, e.g. the corona of the Sun (Coronal seismology).

[edit] Extensions to magnetohydrodynamics

[edit] Resistive MHD

Resistive MHD describes magnetized fluids with finite electron diffusivity ( $\eta \neq 0$ ). This diffusivity leads to a breaking in the magnetic topology.

[edit] Extended MHD

Extended MHD describes a class of phenomena in plasmas that are higher order than resistive MHD, but which can adequately be treated with a single fluid description. These include the effects of Hall physics, electron pressure gradients, finite Larmor Radii in the particle gyromotion, and electron inertia.

[edit] Two-Fluid MHD

Two-Fluid MHD describes plasmas that include a non-negligible Hall electric field. As a result, the electron and ion momenta must be treated separately. This description is more closely tied to Maxwell's equations as an evolution equation for the electric field exists.

[edit] Hall MHD

In 1960, M. J. Lighthill criticized the applicability of ideal or resistive MHD theory for plasmas ^[3]. It concerned the neglect of the "Hall current term", a frequent simplification made in magnetic fusion theory. Hall-magnetohydrodynamics (HMHD) takes into account this electric field description of magnetohydrodynamics. The most important difference is that in the absence of field line breaking, the magnetic field is tied to the electrons and not to the bulk fluid. ^[4]

[edit] Collisionless MHD

MHD is also often used for collisionless plasmas. In that case the MHD equations are derived from the Vlasov equation. ^[5]

[edit] Applications

[edit] Geophysics

The fluid core of the Earth and other planets is theorized to be a huge MHD dynamo that generates the Earth's magnetic field due to the motion of liquid iron.

[edit] Astrophysics

MHD applies quite well to astrophysics since over 99% of baryonic matter content of the Universe is made up of plasma, including stars, the interplanetary medium (space between the planets), the interstellar medium (space between the stars), nebulae and jets. Many astrophysical systems are not in local thermal equilibrium, and therefore require an additional kinematic treatment to describe all the phenomena within the system (see Astrophysical plasma).

Sunspots are caused by the Sun's magnetic fields, as Joseph Larmor theorized in 1919. The solar wind is also governed by MHD. The differential solar rotation may be the long term effect of magnetic drag at the poles of the Sun, an MHD phenomenon due to the Parker spiral shape assumed by the extended magnetic field of the Sun.

Previously, theories describing the formation of the Sun and planets could not explain how the Sun has 99.87% of the mass, yet only 0.54% of the angular momentum in the solar system. In a closed system such as the cloud of gas and dust from which the Sun was formed, mass and angular momentum are both conserved. That conservation would imply that as the mass concentrated in the center of the cloud to form the Sun, it would spin up, much like a skater pulling their arms in. The high speed of rotation predicted by early theories would have flung the proto-Sun apart before it could have formed. However, magnetohydrodynamic effects transfer the Sun's angular momentum into the outer solar system, slowing its rotation.

Breakdown of ideal MHD (in the form of magnetic reconnection) is known to be the cause of solar flares, the largest explosions in the solar system. The magnetic field in a solar active region over a sunspot can become quite stressed over time, storing energy that is released suddenly as a burst of motion, X-rays, and radiation when the main current sheet collapses, reconnecting the field.

[edit] Engineering

MHD is related to engineering problems such as plasma confinement, liquid-metal cooling of nuclear reactors, and electromagnetic casting (among others).

In early 1990s, Mitsubishi built a boat, the 'Yamato,' which uses a magnetohydrodynamic drive, is driven by a liquid helium-cooled superconductor, and can travel at 15 km/h.

MHD power generation fueled by potassium-seeded coal combustion gas showed potential for more efficient energy conversion (the absence of solid moving parts allows operation at higher temperatures), but failed due to cost prohibitive technical difficulties.^[6]

In microfluidic devices, the MHD pump is so far the most effective for producing a continuous, nonpulsating flow in a complex microchannel design. It was used to implement a PCR protocol.

[edit] History

Michael Faraday

The first recorded use of the word magnetohydrodynamics is by Hannes Alfvén in 1942:

"At last some remarks are made about the transfer of momentum from the Sun to the planets, which is fundamental to the theory (§11). The importance of the magnetohydrodynamic waves in this respect is pointed out." ^[7]

The ebbing salty water flowing past London's Waterloo Bridge interacts with the Earth's magnetic field to produce a potential difference between the two river-banks. Michael Faraday tried this experiment in 1832 but the current was too small to measure with the equipment at the time,^[8] and the river bed contributed to short-circuit the signal. However, by the same process, Dr. William Hyde Wollaston was able to measure the voltage induced by the tide in the English Channel in 1851.^[9]

[edit] In fiction

The Hunt for Red October/Caterpillar drive
Clive Cussler's novel Valhalla Rising involved two ships, one surface and one submarine, that used MHD drives. Also, his novels with Brul: Dark Watch and Plague Ship which are part of the Oregon Files
Dale Brown's novel Silver Tower (1988) depicted Space Station Armstrong maintaining a small nuclear MHD reactor to power the Skybolt laser.
The game Outpost 2 featured a MHD generator powered by the planet's magnetic field.

[edit] See also

[hide] v • d • e Fusion power

Core topics	Nuclear fusion (Timeline) · Nuclear power · Nuclear reactor technology · Atomic nucleus · Fusion energy gain factor · Lawson criterion · Magnetohydrodynamics · Neutron · Plasma

[hide] v • d • e Nuclear fusion methods

Magnetic confinement	Tokamak · Spheromak · Stellarator · Reversed field pinch · Field-reversed configuration · Levitated Dipole

Inertial confinement	Laser-driven · Z-pinch · Bubble fusion (acoustic) · Fusor (electrostatic) · Magnetized target

Other forms	Muon-catalyzed · Pyroelectric · Migma · Polywell · Dense plasma focus

[hide]

Fusion experiments by confinement method

Magnetic

International	ITER

Asia/Australia	EAST (China) · JT-60 and the Large Helical Device (Japan) · KSTAR (South Korea) · H-1NF (Australia)

Europe	JET (UK) · Tore Supra and TFR (France) · ASDEX Upgrade and Wendelstein 7-X (Germany) · T-15 (Russia) · IGNITOR (Italy) · TCV (Switzerland) · MAST and START (UK)

USA	DIII-D · TFTR · NSTX · NCSX · UCLA ET · Alcator C-Mod · LDX · MST

Commercial	DEMO

Inertial

Laser

Asia	GEKKO XII (Japan)

Europe	HiPER · Asterix IV (Czech Republic) · LMJ and LULI2000 (France) · ISKRA (Russia) · Vulcan (UK)

USA	NIF · OMEGA · Nova · Novette · Nike · Shiva · Argus · Cyclops · Janus · Long path · 4 pi

Non-laser

USA	Z machine · PACER

International Fusion Materials Irradiation Facility

Waves in plasmas

From Wikipedia, the free encyclopedia

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Waves in plasmas are an interconnected set of particles and fields which propagates in a periodically repeating fashion. A plasma is a quasineutral, electrically conductive fluid. In the simplest case, it is composed of electrons and a single species of positive ions, but it may also contain multiple ion species including negative ions as well as neutral particles. Due to its electrical conductivity, a plasma couples to electric and magnetic fields. This complex of particles and fields supports a wide variety of waves.

[hide]

1 Terminology and classification
2 External links
3 References
4 See also

[edit] Terminology and classification

Waves in plasmas can be classified as electromagnetic or electrostatic according to whether or not there is an oscillating magnetic field. Applying Faraday's law of induction to plane waves, we find $\mathbf{k}\times\tilde{\mathbf{E}}=\omega\tilde{\mathbf{B}}$ , implying that an electrostatic wave must be purely longitudinal. An electromagnetic wave, in contrast, must have a transverse component, but may also be partially longitudinal.

Waves can be further classified by the oscillating species. In most plasmas of interest, the electron temperature is comparable to or larger than the ion temperature. This fact, coupled with the much smaller mass of the electron, implies that the electrons are much faster than the ions. An electron mode depends on the mass of the electrons, but the ions may be assumed to be infinitely massive, i.e. stationary. An ion mode depends on the ion mass, but the electrons are assumed to be massless and to redistribute themselves instantaneously according to the Boltzmann relation. Only rarely, e.g. in the lower hybrid oscillation, will a mode depend on both the electron and the ion mass.

The various modes can also be classified according to whether they propagate in an unmagnetized plasma or parallel, perpendicular, or oblique to the stationary magnetic field. Finally, for perpendicular electromagnetic electron waves, the perturbed electric field can be parallel or perpendicular to the stationary magnetic field.

Summary of elementary plasma waves
EM character	oscillating species	conditions	dispersion relation	name
electrostatic	electrons	$\vec B_0=0\ {\rm or}\ \vec k\\|\vec B_0$	$\omega^2=\omega_p^2+(3/2)k^2v_{th}^2$	plasma oscillation (or Langmuir wave)
	electrons	$\vec k\perp\vec B_0$	$\omega^2=\omega_p^2+\omega_c^2=\omega_h^2$	upper hybrid oscillation
	ions	$\vec B_0=0\ {\rm or}\ \vec k\\|\vec B_0$	$\omega^2=k^2v_s^2=k^2\frac{\gamma_eKT_e+\gamma_iKT_i}{M}$	ion acoustic wave
		$\vec k\perp\vec B_0$ (nearly)	$\omega^2=\Omega_c^2+k^2v_s^2$	electrostatic ion cyclotron wave
		$\vec k\perp\vec B_0$ (exactly)	$\omega^2=\omega_i^2=\Omega_c\omega_c$	lower hybrid oscillation
electromagnetic	electrons	$\vec B_0=0$	$\omega^2=\omega_p^2+k^2c^2$	light wave
		$\vec k\perp\vec B_0,\ \vec E_1\\|\vec B_0$	$\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2}{\omega^2}$	O wave
		$\vec k\perp\vec B_0,\ \vec E_1\perp\vec B_0$	$\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2}{\omega^2}\, \frac{\omega^2-\omega_p^2}{\omega^2-\omega_h^2}$	X wave
		$\vec k\\|\vec B_0$ (right circ. pol.)	$\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2/\omega^2}{1-(\omega_c/\omega)}$	R wave (whistler mode)
		$\vec k\\|\vec B_0$ (left circ. pol.)	$\frac{c^2k^2}{\omega^2}=1-\frac{\omega_p^2/\omega^2}{1+(\omega_c/\omega)}$	L wave
	ions	$\vec B_0=0$		none
		$\vec k\\|\vec B_0$	$\omega^2=k^2v_A^2$	Alfvén wave
		$\vec k\perp\vec B_0$	$\frac{\omega^2}{k^2}=c^2\, \frac{v_s^2+v_A^2}{c^2+v_A^2}$	magnetosonic wave