formulasearchengine - User contributions [en]

Knaster–Tarski theorem

2014-10-17T19:00:53Z

130.83.199.121:

During your desired number pointing to gems, you can get yourself prepared intelligently to fight for myself against any factor you like. This is exciting since it lets you enjoy like a experienced and you can worry just about anyone just in case your playing skills are solid. Inside your are a parent or gaurdian of any little one who appreciates taking part in video games, be familiar with multi-player and on-line video video options. These features give your kid to interact with many some gamers throughout the marketplace. As good as it's is for your offspring in order to with others, you manage not know who many people on the other closing is. Interweaving social trends form a strong net in which many people trapped. When Each Tygers of Pan Tang sang 'It's lonely on the top bar. Everyones trying to do families in', these people acquired much from clash of clans get into tool no survey. A society without collide of clans hack campaign no survey is much like a society with no knowledge, in that it very good. Be charged attention to how money your teenager is generally spending on video activities. These products will cheap and there is very much often the option along with buying more add-ons in the game itself. Set monthly and once a year limits on the wide variety of money that is likely to be spent on video playback games. Also, have conversations with your toddlers about budgeting. Whatever the reason, computer [http://Answers.Yahoo.com/search/search_result?p=game+power&submit-go=Search+Y!+Answers game power] tips are widespread and spread fairly rapidly over the internet. For those who have any kind of questions relating to where in addition to how to make use of hack clash of clans ([http://prometeu.net Link Home Page]), you are able to e mail us from our internet site. The gaming community is wanting to find means stay away from cheaters from overrunning whichever game; having lots concerning cheaters playing a main game can really make honest players to remaining playing, or play simply with friends they trust. This poses a extensive problem particularly for subscription games for example EverQuest, wherein a loss behind players ultimately result from a loss of income. Make sure that you build and buy numerous new laboratory so you're able to research improved barbarians. Eventually, in the event that you take part on game for most months, you might finally procure the nirvana of five-star barbarians. It's actually a nice technique. Breaking the appraisement bottomward into chunks of their whole time that accomplish university to be able that will help bodies (hour/day/week) causes this particular to be accessible toward visualize. Everybody knows what it appears this kind of to accept to hold off on a day. Is usually additionally actual accessible to tune. If your family change your current apperception after and adjudge just that one day should group more, all you allegation to complete is modify 1 value.

Riesz representation theorem

2014-02-18T12:00:19Z

130.83.216.234:

Lessie Essex is what people call me and Adore it. Her husband and her chose to reside Massachusetts but her husband wants your crooks to move. One of her favorite hobbies is doing origami but she is struggling in order to locate time so as. Administering databases been recently my employment for quite a while and I'm doing beneficial financially. My husband and I maintain a site. You might want to give it a try here: http://www.rottentomatoes.com/user/1083060/blogs/?id=793517 Also visit my webpage :: [http://www.rottentomatoes.com/user/1083060/blogs/?id=793517 pc suite free download]

Gullstrand–Painlevé coordinates

2014-01-24T14:44:55Z

130.83.36.104: /* GP coordinates */ sin(\theta) →\sin(\theta)

[[File:Barrier lowering.PNG|thumbnail|300px|As channel length decreases, the barrier ''φB'' to be surmounted by an electron from the source on its way to the drain reduces]]

'''Drain-induced barrier lowering''' or '''DIBL''' is a [[short-channel effect]] in [[MOSFET]]s referring originally to a reduction of [[threshold voltage]] of the [[transistor]] at higher drain voltages.
In a classic planar [[field-effect transistor]] with a long channel, the bottleneck in channel formation occurs far enough from the drain contact that it is electrostatically shielded from the drain by the combination of the substrate and gate, and so classically the [[threshold voltage]] was independent of drain voltage.
In short-channel devices this is no longer true: The drain is close enough to gate the channel, and so a high drain voltage can open the bottleneck and turn on the transistor prematurely.

The origin of the threshold decrease can be understood as a consequence of charge neutrality: the Yau charge-sharing model.
<ref name=Arora>
{{cite book
|author=Narain Arora
|title=Mosfet Modeling for VLSI Simulation: Theory And Practice
|page=197, Fig. 5.14
|year= 2007
|publisher=World Scientific
|isbn=981-256-862-X
|url=http://books.google.com/books?id=SkT2xOuvpuYC&pg=PA197&dq=yau+charge-sharing&sig=NQvLlUOMlEsNFn-oL8y31GFI_Is}}
</ref> The combined charge in the [[depletion region]] of the device and that in the channel of the device is balanced by three electrode charges: the gate, the source and the drain. As drain voltage is increased, the [[depletion region]] of the [[p-n junction]] between the drain and body increases in size and extends under the gate, so the drain assumes a greater portion of the burden of balancing depletion region charge, leaving a smaller burden for the gate. As a result, the charge present on the gate retains charge balance by attracting more carriers into the channel, an effect equivalent to lowering the threshold voltage of the device.

In effect, the channel becomes more attractive for electrons. In other words, the potential energy barrier for electrons in the channel is lowered. Hence the term "barrier lowering" is used to describe these phenomena. Unfortunately, it is not easy to come up with accurate analytical results using the barrier lowering concept.

Barrier lowering increases as channel length is reduced, even at zero applied drain bias, because the source and drain form [[pn junction]]s with the body, and so have associated built-in depletion layers associated with them that become significant partners in charge balance at short channel lengths, even with no reverse bias applied to increase [[depletion width]]s.

The term DIBL has expanded beyond the notion of simple threshold adjustment, however, and refers to a number of drain-voltage effects upon MOSFET ''I-V'' curves that go beyond description in terms of simple threshold voltage changes, as described below.

As channel length is reduced, the effects of DIBL in the [[subthreshold region]] (weak inversion) show up initially as a simple translation of the subthreshold current vs. gate bias curve with change in drain-voltage, which can be modeled as a simple change in threshold voltage with drain bias. However, at shorter lengths the slope of the current vs. gate bias curve is reduced, that is, it requires a larger change in gate bias to effect the same change in drain current. At extremely short lengths, the gate entirely fails to turn the device off. These effects cannot be modeled as a threshold adjustment.<ref name=Tsividis>
{{cite book
|author=Yannis Tsividis
|title=Operational Modeling of the MOS Transistor
|page=268; Fig. 6.11
|year= 1999
|edition=Second Edition
|publisher=McGraw-Hill
|location=New York
|isbn=0-07-065523-5
|url=http://worldcat.org/isbn/0070655235}}
</ref>

DIBL also affects the current vs. drain bias curve in the [[MOSFET#Modes of operation|active mode]], causing the current to increase with drain bias, lowering the MOSFET output resistance. This increase is additional to the normal [[channel length modulation]] effect on output resistance, and cannot always be modeled as a threshold adjustment.

In practice, the DIBL can be calculated as follows:

:: <math> \mathrm{DIBL} = - \frac{V_{Th}^{DD} - V_{Th}^{\mathrm{low}}}{V_{DD} - V_{D}^{\mathrm{low}}}, </math>

where <math>V_{Th}^{DD}</math> is the threshold voltage measured at a supply voltage (the high drain voltage), and <math>V_{Th}^{\mathrm{low}}</math> is the threshold voltage measured at a very low drain voltage, typically 0.05 V or 0.1 V. <math>V_{DD}</math> is the supply voltage (the high drain voltage) and <math>V_{D}^{\mathrm{low}}</math> is the low drain voltage (for a linear part of device I-V characteristics). The minus in the front of the formula ensures a positive DIBL value. This is because the high drain threshold voltage, <math>V_{Th}^{DD}</math>, is always smaller than the low drain threshold voltage, <math>V_{Th}^{\mathrm{low}}</math>. Typical units of DIBL are mV/V.

DIBL can reduce the device operating frequency as well, as described by the following equation:

:: <math>\frac{\Delta f}{f} = -\frac{2 \mathrm{DIBL}}{V_{DD}-V_{Th}},</math>

where <math>V_{DD}</math> is the supply voltage and <math>V_{Th}</math> is the threshold voltage.

==References==

<references/>

==See also==
*[[Channel length modulation]]
*[[Threshold voltage]]
*[[MOSFET#MOSFET structure and channel formation|MOSFET operation]]

[[Category:Transistor modeling]]

Energy minimization

2014-01-18T06:35:09Z

130.83.73.5:

[[File:diffusionless classification.svg|350px|thumbnail|right|Diffusionless transformations classifications]]
{{multiple issues|
{{Refimprove|date=July 2008}}
{{Tone|Martensitic transformation|date=November 2010}}
}}

A '''diffusionless transformation''' is a [[Phase transition|phase change]] that occurs without the long-range [[diffusion]] of [[atoms]] but rather by some form of cooperative, homogeneous movement of many atoms that results in a change in crystal structure. These movements are small, usually less than the interatomic distances, and the atoms maintain their relative relationships. The ordered movement of large numbers of atoms lead some to refer to these as ''military'' transformations in contrast to ''civilian'' diffusion-based phase changes.<ref>D.A. Porter and K.E. Easterling, Phase transformations in metals and alloys, ''Chapman & Hall'', 1992, p.172 ISBN 0-412-45030-5</ref>

The most commonly encountered transformation of this type is the [[Adolf Martens|martensitic]] transformation which, while being the most studied, is only one subset of non-diffusional transformations. The martensitic transformation in steel represents the most economically significant example of this category of phase transformations but an increasing number of alternatives, such as [[shape memory alloy]]s, are becoming more important as well.

== Classification and definitions ==
When a structural change occurs by the coordinated movement of atoms (or groups of atoms) relative to their neighbors then the change is termed ''displacive'' transformation. This covers a broad range of transformations and so further classifications have been developed [Cohen 1979].

The first distinction can be drawn between transformations dominated by ''lattice-distortive strains'' and those where ''shuffles'' are of greater importance.

Homogeneous lattice-distortive strains, also known as Bain strains, are strains that transform one [[Bravais lattice]] into a different one. This can be represented by a strain [[Matrix (mathematics)|matrix]] '''S''' which transforms one vector, '''y''', into a new vector, '''x''':

:<math>y=Sx</math>

This is homogeneous as straight lines are transformed to new straight lines. Examples of such transformations include a [[Cubic crystal system|cubic lattice]] increasing in size on all three axes (dilation) or [[Shearing (physics)|shearing]] into a [[Monoclinic crystal system|monoclinic]] structure.

[[File:diffusionless shuffles distortions.svg|350px|thumbnail|right]]

Shuffles, as the name suggests, involve the small movement of atoms within the unit cell. As a result pure shuffles do not normally result in a shape change of the unit cell - only its symmetry and structure.

Phase transformations normally result in the creation of an interface between the transformed and parent material. The energy required to generate this new interface will depend on its nature - essentially how well the two structures fit together. An additional energy term occurs if the transformation includes a shape change since, if new phase is constrained by surrounding material, this may give rise to [[Elasticity (physics)|elastic]] or [[plastic]] deformation and hence a [[Strain (materials science)|strain]] energy term. The ratio of these interfacial and strain energy terms has a notable effect on the kinetics of the transformation and the morphology of the new phase. Thus, shuffle transformations, where distortions are small, are dominated by interfacial energies and can be usefully separated from lattice-distortive transformations where the strain energy tends to have a greater effect.

A subclassification of lattice-distortive displacements can be made by considering the dilational and shear components of the distortion. In transformations dominated by the shear component it is possible to find a line in the new phase that is undistorted from the [[parent phase]] while all lines are distorted when the dilation is predominant. Shear dominated transformations can be further classified according to the magnitude of the strain energies involved compared to the innate [[Atom vibrations|vibrations]] of the atoms in the lattice and hence whether the strain energies have a notable influence on the kinetics of the transformation and the morphology of the resulting phase. If the strain energy is a significant factor then the transformations are dubbed ''martensitic'' and if it is not the transformation is referred to as ''quasi-martensitic''.

==Martensitic transformation==
The difference between [[austenite]] and [[martensite]] is, in some ways, quite small: while the unit cell of austenite is, on average, a perfect cube, the transformation to martensite distorts this cube by interstitial carbon atoms that do not have time to diffuse out during displacive transformation. The unit cell becomes slightly longer in one dimension and shorter in the other two. The mathematical description of the two structures is quite different, for reasons of symmetry (see external links), but the chemical bonding remains very similar. Unlike [[cementite]], which has bonding reminiscent of ceramic materials, the hardness of martensite is difficult to explain in chemical terms.

The explanation hinges on the crystal's subtle change in dimension. Even a microscopic crystallite is millions of unit cells long. Since all of these units face the same direction, distortions of even a fraction of a percent become magnified into a major mismatch between neighboring materials. The mismatch is sorted out by the creation of a myriad of [[crystal defect]]s, in a process reminiscent of [[work hardening]]. As in work-hardened steel, these defects prevent atoms from sliding past one another in an organized fashion, causing the material to become harder.

Shape memory alloys also have surprising mechanical properties, that were eventually explained by an analogy to martensite. Unlike the iron-carbon system, alloys in the nickel-titanium system can be chosen that make the "martensitic" phase [[thermodynamics|thermodynamically]] stable.

==Pseudomartensitic transformation==
In addition to displacive transformation and diffusive transformation, a new phase transformation that involves displasive sublattice transition and atomic diffusion was discovered using a high-pressure x-ray diffraction system.<ref>Jiuhua Chen, Donald J. Weidner, John B. Parise, Michael T. Vaughan, and Paul Raterron, (2001) [http://prola.aps.org/abstract/PRL/v86/i18/p4072_1 Observation of Cation Reordering during the Olivine-Spinel Transition in Fayalite by In Situ Synchrotron X-Ray Diffraction at High Pressure and Temperature] ''Phys. Rev. Lett'', 86, pp. 4072–4075.</ref> The new transformation mechanism has been christened a pseudomartensitic transformation.<ref>Kristin Leutwyler [http://www.sciam.com/article.cfm?articleID=000E8826-A6AF-1C5E-B882809EC588ED9F New phase transition] ''Scientific American'', May 2, 2001.</ref>

==References==
===Notes===
{{Reflist}}

===Bibliography===
* Christian, J.W., ''Theory of Phase Transformations in Metals and Alloys'', Pergamon Press (1975)
* Khachaturyan, A.G., ''Theory of Structural Transformations in Solids'', Dover Publications, NY (1983)
* Green, D.J.; Hannink, R.; Swain, M.V. (1989). ''Transformation Toughening of Ceramics''. Boca Raton: CRC Press. ISBN 0-8493-6594-5.

==External links==
*[http://www.msm.cam.ac.uk/phase-trans/2002/martensite.html Extensive resources from Cambridge University]
*[http://www.aem.umn.edu/people/faculty/shield/hane/tet.html The cubic-to-tetragonal transition]
*[http://www.esomat.org/ European Symposium on Martensitic Transformations (ESOMAT)]

{{DEFAULTSORT:Diffusionless Transformation}}
[[Category:Phase transitions]]

Compact-open topology

2013-09-17T13:31:52Z

130.83.2.27: /* Properties */

{{See also |Black body|Planck's law|Thermal radiation}}
[[File:Black body.svg|thumb|303px|As the temperature decreases, the peak of the black-body radiation curve moves to lower intensities and longer wavelengths. The black-body radiation graph is also compared with the classical model of Rayleigh and Jeans.]]

[[Image:PlanckianLocus.png|thumb|303px|The color ([[chromaticity]]) of black-body radiation depends on the temperature of the black body; the [[Locus (mathematics)|locus]] of such colors, shown here in [[CIE 1931 color space|CIE 1931 ''x,y'' space]], is known as the [[Planckian locus]].]]

'''Black-body radiation''' is the type of [[electromagnetic radiation]] within or surrounding a body in [[thermodynamic equilibrium]] with its environment, or emitted by a [[black body]] (an opaque and non-reflective body) held at constant, uniform temperature. The radiation has a specific spectrum and intensity that depends only on the temperature of the body.<ref>{{harvnb|Loudon|2000}}, Chapter 1.</ref><ref>{{harvnb|Mandel|Wolf|1995}}, Chapter 13.</ref><ref>{{harvnb|Kondepudi|Prigogine|1998}}, Chapter 11.</ref><ref name=Landsberg>{{cite book |title=Thermodynamics and statistical mechanics |author=Peter Theodore Landsberg |chapter=Chapter 13: Bosons: black-body radiation |url=http://books.google.com/books?id=0gnWL7tmxm0C&pg=PA208 |pages=208 ''ff'' |publisher=Courier Dover Publications |year=1990 |isbn=0-486-66493-7 |edition=Reprint of Oxford University Press 1978}}</ref>

The thermal radiation spontaneously emitted by many ordinary objects can be approximated as blackbody radiation. A perfectly insulated enclosure that is in thermal equilibrium internally contains black-body radiation and will emit it through a hole made in its wall, provided the hole is small enough to have negligible effect upon the equilibrium.

A black-body at room temperature appears black, as most of the energy it radiates is [[infra-red]] and cannot be perceived by the human eye. At higher temperatures, black bodies glow with increasing intensity and colors that range from dull red to blindingly brilliant blue-white as the temperature increases.

Although planets and stars are neither in thermal equilibrium with their surroundings nor perfect [[black bodies]], black-body radiation is used as a first approximation for the energy they emit.<ref name=Morison>
{{cite book |title=Introduction to Astronomy and Cosmology |author=Ian Morison |url=http://books.google.com/books?id=yrV8vvJzgWkC&pg=PA48 |page=48 |isbn=0-470-03333-9 |year=2008 |publisher=J Wiley & Sons}}
</ref>
[[Black holes]] are near-perfect black bodies, and it is believed that they emit black-body radiation (called [[Hawking radiation]]), with a temperature that depends on the mass of the black hole.<ref name=Fabbri>
{{cite book |title=Modeling black hole evaporation |url=http://books.google.com/books?id=gUhZZtb6yA8C&pg=PA1 |chapter=Chapter 1: Introduction |author=Alessandro Fabbri, José Navarro-Salas |isbn=1-86094-527-9 |year=2005 |publisher=Imperial College Press}}
</ref>

The term ''black body'' was introduced by [[Gustav Kirchhoff]] in 1860. When used as a [[compound adjective]], the term is typically written as hyphenated, for example, ''black-body radiation'', but sometimes also as one word, as in ''blackbody radiation''. Black-body radiation is also called ''complete radiation'' or ''temperature radiation'' or ''thermal radiation''.

==Spectrum==
Black-body radiation has a characteristic, continuous [[spectral energy distribution|frequency spectrum]] that depends only on the body's temperature,<ref name=Kogure>

{{cite book |url=http://books.google.com/books?id=qt5sueHmtR4C&pg=PA41 |page=41 |chapter=§2.3: Thermodynamic equilibrium and black-body radiation |title=The astrophysics of emission-line stars |author=Tomokazu Kogure, Kam-Ching Leung |isbn=0-387-34500-0 |year=2007 |publisher=Springer}}

</ref> called the Planck spectrum or [[Planck's law]]. The spectrum is peaked at a characteristic frequency that shifts to higher frequencies with increasing temperature, and at [[room temperature]] most of the emission is in the [[infrared]] region of the [[electromagnetic spectrum]].<ref>Wien, W. (1893). Eine neue Beziehung der Strahlung schwarzer Körper zum zweiten Hauptsatz der Wärmetheorie, ''Sitzungberichte der Königlich-Preußischen Akademie der Wissenschaften '' (Berlin), 1893, '''1''': 55–62.</ref><ref>Lummer, O., Pringsheim, E. (1899). Die Vertheilung der Energie im Spectrum des schwarzen Körpers, ''Verhandlungen der Deutschen Physikalischen Gessellschaft'' (Leipzig), 1899, '''1''': 23–41.</ref><ref name="Planck 1914">{{harvnb|Planck|1914}}</ref> As the temperature increases past about 500 degrees [[Celsius]], black bodies start to emit significant amounts of visible light. Viewed in the dark, the first faint glow appears as a "ghostly" grey. With rising temperature, the glow becomes visible even when there is some background surrounding light: first as a dull red, then yellow, and eventually a "dazzling bluish-white" as the temperature rises.<ref>[[John William Draper|Draper, J.W.]] (1847). On the production of light by heat, ''London, Edinburgh and Dublin Philosophical Magazine and Journal of Science'', series 3, '''30''': 345–360. [http://www.archive.org/stream/londonedinburghp30lond#page/344/mode/2up]</ref><ref>{{harvnb|Partington|1949|pages = 466–467, 478}}.</ref> When the body appears white, it is emitting a substantial fraction of its energy as [[ultraviolet radiation]]. The Sun, with an [[effective temperature]] of approximately 5800 K,<ref>{{harvnb|Goody|Yung|1989|pages=482, 484}}</ref> is an approximately black body with an emission spectrum peaked in the central, yellow-green part of the [[visible spectrum]], but with significant power in the ultraviolet as well.

Black-body radiation provides insight into the [[thermodynamic equilibrium]] state of cavity radiation. If each [[Fourier mode]] of the equilibrium radiation in an otherwise empty cavity with perfectly reflective walls is considered as a degree of freedom capable of exchanging energy, then, according to the [[equipartition theorem]] of classical physics, there would be an equal amount of energy in each mode. Since there are an infinite number of modes this implies infinite [[heat capacity]] (infinite energy at any non-zero temperature), as well as an unphysical spectrum of emitted radiation that grows without bound with increasing frequency, a problem known as the [[ultraviolet catastrophe]]. Instead, in quantum theory the [[quantum field theory|occupation numbers]] of the modes are quantized, cutting off the spectrum at high frequency in agreement with experimental observation and resolving the catastrophe. The study of the laws of black bodies and the failure of classical physics to describe them helped establish the foundations of [[history of quantum mechanics|quantum mechanics]].

==Explanation==

[[Image:Blackbody-colours-vertical.svg|right|38px]]

All normal ([[baryon]]ic) matter emits electromagnetic radiation when it has a temperature above [[absolute zero]]. The radiation represents a conversion of a body's thermal energy into electromagnetic energy, and is therefore called [[thermal radiation]]. It is a [[spontaneous process]] of radiative distribution of [[entropy]].

Conversely all normal matter absorbs electromagnetic radiation to some degree. An object that absorbs all radiation falling on it, at all [[wavelength]]s, is called a black body. When a black body is at a uniform temperature, its emission has a characteristic frequency distribution that depends on the temperature. Its emission is called black-body radiation.

The concept of the black body is an idealization, as perfect black bodies do not exist in nature.<ref name="Planck 1914 42">{{harvnb|Planck|1914|page=42}}</ref> [[Graphite]] and [[carbon black|lamp black]], with emissivities greater than 0.95, however, are good approximations to a black material. Experimentally, black-body radiation may be established best as the ultimately stable steady state equilibrium radiation in a cavity in a rigid body, at a uniform temperature, that is entirely opaque and is only partly reflective.<ref name="Planck 1914 42"/> A closed box of graphite walls at a constant temperature with a small hole on one side produces a good approximation to ideal black-body radiation emanating from the opening.<ref>{{harvnb|Wien|1894}}</ref><ref>{{harvnb|Planck|1914|page=43}}</ref>

Black-body radiation has the unique absolutely stable distribution of radiative intensity that can persist in thermodynamic equilibrium in a cavity.<ref name="Planck 1914 42"/> In equilibrium, for each frequency the total intensity of radiation that is emitted and reflected from a body (that is, the net amount of radiation leaving its surface, called the ''spectral radiance'') is determined solely by the equilibrium temperature, and does not depend upon the shape, material or structure of the body.<ref name=Caniou>
{{cite book |url=http://books.google.com/books?id=X-aFGcf6pOEC&pg=PA107 |page=107 |chapter=§4.2.2: Calculation of Planck's law |title=Passive infrared detection: theory and applications |author=Joseph Caniou |isbn=0-7923-8532-2 |year=1999 |publisher=Springer}}
</ref> For a black body (a perfect absorber) there is no reflected radiation, and so the spectral radiance is due entirely to emission. In addition, a black body is a diffuse emitter (its emission is independent of direction). Consequently, black-body radiation may be viewed as the radiation from a black body at thermal equilibrium.

Black-body radiation becomes a visible glow of light if the temperature of the object is high enough. The [[Draper point]] is the temperature at which all solids glow a dim red, about 798 K.<ref>{{cite journal
|journal = The Academy
|title = Science: Draper's Memoirs
|volume = XIV
|issue = 338
|publisher = London: Robert Scott Walker
|date = Oct 26, 1878
|page = 408
|url = http://www.archive.org/details/scientificmemoi00drapgoog}}</ref><ref>{{cite book
|title = Radiation heat transfer: a statistical approach
|author = J. R. Mahan
|edition = 3rd
|publisher = Wiley-IEEE
|year = 2002
|isbn = 978-0-471-21270-6
|page = 58
|url = http://books.google.com/?id=y9zUEzA7iN0C&pg=PA58&dq=draper-point+red
}}</ref> At 1000 K, a small opening in the wall of a large uniformly heated opaque-walled cavity (let us call it an oven), viewed from outside, looks red; at 6000 K, it looks white. No matter how the oven is constructed, or of what material, as long as it is built so that almost all light entering is absorbed by its walls, it will contain a good approximation to black-body radiation. The spectrum, and therefore color, of the light that comes out will be a function of the cavity temperature alone. A graph of the amount of energy inside the oven per unit volume and per unit frequency interval plotted versus frequency, is called the ''black-body curve''. Different curves are obtained by varying the temperature.

[[Image:Pahoehoe toe.jpg|thumb|left|250px|The temperature of a [[Lava#Pāhoehoe|Pāhoehoe]] lava flow can be estimated by observing its color. The result agrees well with measured temperatures of lava flows at about {{Convert|1000|to|1200|C|F}}.]]

Two bodies that are at the same temperature stay in thermal equilibrium, so a body at temperature ''T'' surrounded by a cloud of light at temperature ''T'' on average will emit as much light into the cloud as it absorbs, following Prevost's exchange principle, which refers to [[radiative equilibrium]]. The principle of [[detailed balance]] says that in thermodynamic equilibrium every elementary process works equally in its forward and backward sense.<ref>de Groot, SR., Mazur, P. (1962). ''Non-equilibrium Thermodynamics'', North-Holland, Amsterdam.</ref><ref>{{harvnb|Kondepudi|Prigogine|1998}}, Section 9.4.</ref> Prevost also showed that the emission from a body is logically determined solely by its own internal state. The causal effect of thermodynamic absorption on thermodynamic (spontaneous) emission is not direct, but is only indirect as it affects the internal state of the body. This means that at thermodynamic equilibrium the amount of every wavelength in every direction of thermal radiation emitted by a body at temperature ''T'', black or not, is equal to the corresponding amount that the body absorbs because it is surrounded by light at temperature ''T''.<ref name="Stewart 1858"/>

When the body is black, the absorption is obvious: the amount of light absorbed is all the light that hits the surface. For a black body much bigger than the wavelength, the light energy absorbed at any wavelength ''λ'' per unit time is strictly proportional to the black-body curve. This means that the black-body curve is the amount of light energy emitted by a black body, which justifies the name. This is the condition for the applicability of [[Kirchhoff's law of thermal radiation]]: the black-body curve is characteristic of thermal light, which depends only on the [[temperature]] of the walls of the cavity, provided that the walls of the cavity are completely opaque and are not very reflective, and that the cavity is in [[thermodynamic equilibrium]].<ref name="Huang">{{cite book |last=Huang |first=Kerson|title=Statistical Mechanics |year=1967 |publisher=John Wiley & Sons |location=New York |isbn=0-471-81518-7}}</ref> When the black body is small, so that its size is comparable to the wavelength of light, the absorption is modified, because a small object is not an efficient absorber of light of long wavelength, but the principle of strict equality of emission and absorption is always upheld in a condition of thermodynamic equilibrium.

In the laboratory, black-body radiation is approximated by the radiation from a small hole in a large cavity, a [[hohlraum]], in an entirely opaque body that is only partly reflective, that is maintained at a constant temperature. (This technique leads to the alternative term ''cavity radiation''.) Any light entering the hole would have to reflect off the walls of the cavity multiple times before it escaped, in which process it is nearly certain to be absorbed. Absorption occurs regardless of the [[wavelength]] of the radiation entering (as long as it is small compared to the hole). The hole, then, is a close approximation of a theoretical black body and, if the cavity is heated, the [[Power spectral density|spectrum]] of the hole's radiation (i.e., the amount of light emitted from the hole at each wavelength) will be continuous, and will depend only on the opacity and partial reflectivity of the walls, but not on the particular material of which they are built nor on the material in the cavity (compare with [[emission spectrum]]).

Calculating the black-body curve was a major challenge in [[theoretical physics]] during the late nineteenth century. The problem was solved in 1901 by [[Max Planck]] in the formalism now known as [[Planck's law]] of black-body radiation.<ref>{{cite journal
|last = Planck
|first = Max
|authorlink = Max_Planck
|coauthors =
|title =On the Law of Distribution of Energy in the Normal Spectrum
|journal = [[Annalen der Physik]]
|volume = 4
|page = 553
|year = 1901
|url = http://dbhs.wvusd.k12.ca.us/webdocs/Chem-History/Planck-1901/Planck-1901.html
|doi=10.1002/andp.19013090310
|bibcode=1901AnP...309..553P
}} {{dead link|date=November 2009}}</ref>
By making changes to [[Wien's radiation law]] (not to be confused with [[Wien's displacement law]]) consistent with [[thermodynamics]] and [[electromagnetism]], he found a mathematical expression fitting the experimental data satisfactorily. Planck had to assume that the energy of the oscillators in the cavity was quantized, i.e., it existed in integer multiples of some quantity. [[Albert Einstein|Einstein]] built on this idea and proposed the quantization of electromagnetic radiation itself in 1905 to explain the [[photoelectric effect]]. These theoretical advances eventually resulted in the superseding of classical electromagnetism by [[quantum electrodynamics]]. These quanta were called [[photon]]s and the black-body cavity was thought of as containing a [[photon gas|gas of photons]]. In addition, it led to the development of quantum probability distributions, called [[Fermi–Dirac statistics]] and [[Bose–Einstein statistics]], each applicable to a different class of particles, [[fermion]]s and [[boson]]s.

The wavelength at which the radiation is strongest is given by Wien's displacement law, and the overall power emitted per unit area is given by the [[Stefan–Boltzmann law]]. So, as temperature increases, the glow color changes from red to yellow to white to blue. Even as the peak wavelength moves into the ultra-violet, enough radiation continues to be emitted in the blue wavelengths that the body will continue to appear blue. It will never become invisible—indeed, the radiation of visible light increases [[monotonic function|monotonically]] with temperature.<ref name="Landau">{{cite book |last=Landau |first=L. D.|coauthors=E. M. Lifshitz|title=Statistical Physics |edition=3rd Edition Part 1 |year=1996|publisher=Butterworth–Heinemann |location=Oxford |isbn=0-521-65314-2}}</ref>

The [[radiance]] or observed intensity is not a function of direction. Therefore a black body is a perfect [[Lambert's cosine law|Lambertian]] radiator.

Real objects never behave as full-ideal black bodies, and instead the emitted radiation at a given frequency is a fraction of what the ideal emission would be. The [[emissivity]] of a material specifies how well a real body radiates energy as compared with a black body. This emissivity depends on factors such as temperature, emission angle, and wavelength. However, it is typical in engineering to assume that a surface's spectral emissivity and absorptivity do not depend on wavelength, so that the emissivity is a constant. This is known as the ''gray body'' assumption.

[[File:Ilc 9yr moll4096.png|thumb|300px|9-year [[WMAP]] image (2012) of the [[cosmic microwave background radiation]] across the universe.<ref name="Space-20121221">{{cite web |last=Gannon |first=Megan |title=New 'Baby Picture' of Universe Unveiled |url=http://www.space.com/19027-universe-baby-picture-wmap.html|date=December 21, 2012 |publisher=[[Space.com]] |accessdate=December 21, 2012 }}</ref><ref name="arXiv-20121220">{{cite journal |last=Bennett |first=C.L. |last2=Larson |first2=L.|last3=Weiland |first3=J.L. |last4=Jarosk |first4= N. |last5=Hinshaw |first5=N. |last6=Odegard|first6=N. |last7=Smith |first7=K.M. |last8=Hill |first8=R.S. |last9=Gold |first9=B.|last10=Halpern |first10=M. |last11=Komatsu |first11=E. |last12=Nolta |first12=M.R.|last13=Page |first13=L. |last114=Spergel |first14=D.N. |last15=Wollack |first15=E.|last16=Dunkley |first16=J. |last17=Kogut |first17=A. |last18=Limon |first18=M. |last19=Meyer|first19=S.S. |last20=Tucker |first20=G.S. |last21=Wright |first21=E.L. |title=Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results|url=http://arxiv.org/abs/1212.5225 |arxiv=1212.5225 |date=December 20, 2012|accessdate=December 22, 2012 |bibcode = 2012arXiv1212.5225B }}</ref>]]

With non-black surfaces, the deviations from ideal black-body behavior are determined by both the surface structure, such as roughness or granularity, and the chemical composition. On a "per wavelength" basis, real objects in states of [[Thermodynamic equilibrium#Local and global equilibrium|local thermodynamic equilibrium]] still follow [[Kirchhoff's law (thermodynamics)|Kirchhoff's Law]]: emissivity equals absorptivity, so that an object that does not absorb all incident light will also emit less radiation than an ideal black body; the incomplete absorption can be due to some of the incident light being transmitted through the body or to some of it being reflected at the surface of the body.

In [[astronomy]], objects such as [[star]]s are frequently regarded as black bodies, though this is often a poor approximation. An almost perfect black-body spectrum is exhibited by the [[cosmic microwave background radiation]]. [[Hawking radiation]] is the hypothetical black-body radiation emitted by [[black hole]]s, at a temperature that depends on the mass, charge, and spin of the hole. If this prediction is correct, black holes will very gradually shrink and evaporate over time as they lose mass by the emission of photons and other particles.

A black body radiates energy at all frequencies, but its intensity rapidly tends to zero at high frequencies (short wavelengths). For example, a black body at room temperature (300 K) with one square meter of surface area will emit a photon in the visible range (390–750 nm) at an average rate of one photon every 41 seconds, meaning that for most practical purposes, such a black body does not emit in the visible range.<ref>Mathematica:Planck intensity (energy/sec/area/solid angle/wavelength) is:

: <math> i_{w,t} = \frac{2hc^2}{w^5 (\exp(hc/wkt) - 1)} </math>
i[w_, t_] = 2*h*c^2/(w^5*(Exp[h*c/(w*k*t)] - 1)) 
The number of photons/sec/area is:

NIntegrate[2*Pi*i[w, 300]/(h*c/w), {w, 390*10^(-9), 750*10^(-9)}] = 0.0244173...</ref>

==Equations==

===Planck's law of black-body radiation===

{{Main|Planck's law}}

Planck's law states that<ref name="Rybicki 1979 22">{{harvnb|Rybicki|Lightman|1979|p=22}}</ref>
:<math>I(\nu,T) =\frac{ 2 h\nu^{3}}{c^2}\frac{1}{ e^{\frac{h\nu}{kT}}-1}</math>
where
:''I''(''ν'',''T'') is the [[energy]] per unit [[time]] (or the [[power (physics)|power]]) radiated per unit area of emitting surface in the [[Normal (geometry)|normal]] direction per unit [[solid angle]] per unit [[frequency]] by a black body at temperature ''T'', also known as spectral radiance;
:''h'' is the [[Planck constant]];
:''c'' is the [[speed of light]] in a vacuum;
:''k'' is the [[Boltzmann constant]];
:''ν'' is the [[frequency]] of the electromagnetic radiation; and
:''T'' is the absolute [[temperature]] of the body.

===Wien's displacement law===

[[Wien's displacement law]] shows how the spectrum of black-body radiation at any temperature is related to the spectrum at any other temperature. If we know the shape of the spectrum at one temperature, we can calculate the shape at any other temperature. Spectral intensity can be expressed as a function of wavelength or of frequency.

A consequence of Wien's displacement law is that the wavelength at which the intensity ''per unit wavelength'' of the radiation produced by a black body is at a maximum, <math>\lambda_\max</math>, is a function only of the temperature
:<math>\lambda_\max = \frac{b}{T}</math>
where the constant, ''b'', known as Wien's displacement constant, is equal to {{val|fmt=commas|2.8977721|(26)|e=-3|u=K m}}.<ref>http://physics.nist.gov/cgi-bin/cuu/Value?bwien</ref>

Planck's Law was also stated above as a function of frequency. The intensity maximum for this is given by
:<math>\nu_\max = T \times 58.8\ \mathrm{GHz}\ \mathrm{K}^{-1}</math>.<ref>
{{cite web
| last = Nave
| first = Dr. Rod
| authorlink =
| coauthors =
| title = Wien's Displacement Law and Other Ways to Characterize the Peak of Blackbody Radiation
| work = HyperPhysics
| publisher =
| date =
| url = http://hyperphysics.phy-astr.gsu.edu/hbase/quantum/wien3.html#c1
| doi =
| accessdate = }}
Provides 5 variations of Wien's Displacement Law
</ref>

===Stefan–Boltzmann law===

The [[Stefan–Boltzmann law]] states that the power emitted per unit area of the surface of a black body is directly proportional to the fourth power of its absolute temperature:
:<math>j^{\star} = \sigma T^4,</math>
where ''j''*is the total power radiated per unit area, ''T'' is the [[absolute temperature]] and {{nowrap|''σ'' {{=}} {{val|5.67|e=-8|u=W m−2 K−4}}}} is the [[Stefan–Boltzmann constant]].

==Human body emission==

{| class="bordered infobox" style="width:22em"
| style="text-align:left;"|[[Image:Human-Visible.jpg|229px]]
|-
| style="text-align:left;"|[[Image:Human-Infrared.jpg|284px]]
|-
|Much of a person's energy is radiated away in the form of [[infrared]] light. Some materials are transparent in the infrared, but opaque to visible light, as is the plastic bag in this infrared image (bottom). Other materials are transparent to visible light, but opaque or reflective in the infrared, noticeable by darkness of the man's glasses.
|}
As all matter, the human body radiates some of a person's energy away as [[infrared]] light.

The net power radiated is the difference between the power emitted and the power absorbed:
:<math>P_\mathrm{net}=P_\mathrm{emit}-P_\mathrm{absorb}. \, </math>
Applying the Stefan–Boltzmann law,

:<math>P_{\rm net}=A\sigma \varepsilon \left( T^4 - T_0^4 \right).</math>

The total surface area of an adult is about 2 m2, and the mid- and far-infrared [[emissivity]] of skin and most clothing is near unity, as it is for most nonmetallic surfaces.<ref>{{cite web
| author=Infrared Services
| title=Emissivity Values for Common Materials
| url=http://infrared-thermography.com/material-1.htm
| accessdate=2007-06-24}}</ref><ref>{{cite web
| author=Omega Engineering
| title=Emissivity of Common Materials
| url=http://www.omega.com/literature/transactions/volume1/emissivityb.html
| accessdate=2007-06-24}}</ref> Skin temperature is about 33 °C,<ref>{{cite web
| last= Farzana|first= Abanty
| title=Temperature of a Healthy Human (Skin Temperature)|year=2001|work=The Physics Factbook
| url=http://hypertextbook.com/facts/2001/AbantyFarzana.shtml
| accessdate=2007-06-24}}</ref> but clothing reduces the surface temperature to about 28 °C when the ambient temperature is 20 °C.<ref>{{cite web
| author=Lee, B.
| title=Theoretical Prediction and Measurement of the Fabric Surface Apparent Temperature in a Simulated Man/Fabric/Environment System
| url=http://www.dsto.defence.gov.au/publications/2135/DSTO-TR-0849.pdf
| accessdate=2007-06-24}}</ref> Hence, the net radiative heat loss is about
:<math>P_{\rm net} = 100 \ \mathrm{W}.</math>
The total energy radiated in one day is about 9 MJ ([[megajoule]]s), or 2000 kcal (food [[calorie]]s). [[Basal metabolic rate]] for a 40-year-old male is about 35 kcal/(m2·h),<ref name="Harris1918">{{cite journal|author = Harris J, Benedict F|title = A Biometric Study of Human Basal Metabolism.|journal = Proc Natl Acad Sci USA| volume = 4|issue = 12| pages = 370–3|year = 1918|pmid = 16576330|doi = 10.1073/pnas.4.12.370|pmc = 1091498
|bibcode = 1918PNAS....4..370H }}</ref> which is equivalent to 1700 kcal per day assuming the same 2 m2 area. However, the mean metabolic rate of sedentary adults is about 50% to 70% greater than their basal rate.<ref>{{cite journal|author=Levine, J|title=Nonexercise activity thermogenesis (NEAT): environment and biology|journal=Am J Physiol Endocrinol Metab|volume=286|year=2004|pages=E675–E685|url=http://ajpendo.physiology.org/cgi/content/full/286/5/E675|doi=10.1152/ajpendo.00562.2003|pmid=15102614|issue=5}}</ref>

There are other important thermal loss mechanisms, including [[convection]] and [[evaporation]]. Conduction is negligible – the [[Nusselt number]] is much greater than unity. Evaporation via [[perspiration]] is only required if radiation and convection are insufficient to maintain a steady state temperature (but evaporation from the lungs occurs regardless).{{citation needed|date=July 2012}} Free convection rates are comparable, albeit somewhat lower, than radiative rates.<ref>{{cite web
| author=DrPhysics.com
| title=Heat Transfer and the Human Body
| url=http://www.drphysics.com/convection/convection.html
| accessdate=2007-06-24}}</ref> Thus, radiation accounts for about two-thirds of thermal energy loss in cool, still air. Given the approximate nature of many of the assumptions, this can only be taken as a crude estimate. Ambient air motion, causing forced convection, or evaporation reduces the relative importance of radiation as a thermal loss mechanism.

Application of [[Wien's displacement law|Wien's Law]] to human body emission results in a peak wavelength of
:<math>\lambda_{\rm peak} = \frac{2.898\times 10^-3 \ \mathrm{K} \cdot \mathrm{nm}}{305 \ \mathrm{K}} = 9.50 \ \mu\mathrm{m}.</math>
For this reason, thermal imaging devices for human subjects are most sensitive in the 7–14 micron range.

==Temperature relation between a planet and its star==

The black-body law may be used to estimate the temperature of a planet orbiting the Sun.

[[Image:Erbe.gif|thumb|300px|Earth's longwave thermal [[Earth's energy budget#Outgoing energy|radiation]] intensity, from clouds, atmosphere and ground]]
The temperature of a planet depends on several factors:
*Incident radiation from its star
*Emitted radiation of the planet, e.g., [[Earth's energy budget#Outgoing energy|Earth's infrared glow]]
*The [[albedo]] effect causing a fraction of light to be reflected by the planet
*The [[greenhouse effect]] for planets with an atmosphere
*Energy generated internally by a planet itself due to [[radioactive decay]], [[tidal heating]], and [[Kelvin–Helmholtz mechanism|adiabatic contraction due by cooling]].

The analysis only considers the Sun's heat for a planet in a Solar System.

The [[Stefan–Boltzmann law]] gives the total [[power (physics)|power]] (energy/second) the Sun is emitting:

[[Image:Sun-Earth-Radiation.png|frame|The Earth only has an absorbing area equal to a two dimensional disk, rather than the surface of a sphere.]]
:<math>P_{\rm S\ emt} = 4 \pi R_{\rm S}^2 \sigma T_{\rm S}^4 \qquad \qquad (1)</math>
where
:<math>\sigma \,</math> is the [[Stefan–Boltzmann law|Stefan–Boltzmann constant]],
:<math>T_{\rm S} \,</math> is the effective temperature of the Sun, and
:<math>R_{\rm S} \,</math> is the radius of the Sun.

The Sun emits that power equally in all directions. Because of this, the planet is hit with only a tiny fraction of it. The power from the Sun that strikes the planet (at the top of the atmosphere) is:

:<math>P_{\rm SE} = P_{\rm S\ emt} \left( \frac{\pi R_{\rm E}^2}{4 \pi D^2} \right) \qquad \qquad (2)</math>
where
:<math>R_{\rm E} \,</math> is the radius of the planet and
:<math>D \,</math> is the [[astronomical unit]], the distance between the [[Sun]] and the planet.

Because of its high temperature, the Sun emits to a large extent in the ultraviolet and visible (UV-Vis) frequency range. In this frequency range, the planet reflects a fraction <math>\alpha</math> of this energy where <math>\alpha</math> is the [[albedo]] or reflectance of the planet in the UV-Vis range. In other words, the planet absorbs a fraction <math>1-\alpha</math> of the Sun's light, and reflects the rest. The power absorbed by the planet and its atmosphere is then:

:<math>P_{\rm abs} = (1-\alpha)\,P_{\rm SE} \qquad \qquad (3)</math>

Even though the planet only absorbs as a circular area <math>\pi R^2</math>, it emits equally in all directions as a sphere. If the planet were a perfect black body, it would emit according to the [[Stefan–Boltzmann law]]

:<math>P_{\rm emt\,bb} = 4 \pi R_{\rm E}^2 \sigma T_{\rm E}^4 \qquad \qquad (4)</math>

where <math>T_{\rm E} </math> is the temperature of the planet. This temperature, calculated for the case of the planet acting as a black body by setting <math>P_{\rm abs} = P_{\rm emt\,bb}</math>, is known as the [[effective temperature]]. The actual temperature of the planet will likely be different, depending on its surface and atmospheric properties. Ignoring the atmosphere and greenhouse effect, the planet, since it is at a much lower temperature than the Sun, emits mostly in the infrared (IR) portion of the spectrum. In this frequency range, it emits <math>\overline{\epsilon}</math> of the radiation that a black body would emit where <math>\overline{\epsilon}</math> is the average emissivity in the IR range. The power emitted by the planet is then:

:<math>P_{\rm emt} = \overline{\epsilon}\,P_{\rm emt\,bb} \qquad \qquad (5)</math>

For a body in [[Radiative equilibrium#Definitions of radiative equilbrium#radiative exchange equilibrium|radiative exchange equilibrium]] with its surroundings, the rate at which it emits radiant energy is equal to the rate at which it absorbs it:<ref name="Prevost 1791">Prevost, P. (1791). Mémoire sur l'equilibre du feu. ''Journal de Physique'' (Paris), vol 38 pp. 314-322.</ref><ref>Iribarne, J.V., Godson, W.L. (1981). ''Atmospheric Thermodynamics'', second edition, D. Reidel Publishing, Dordrecht, ISBN 90-277-1296-4, page 227.</ref>

:<math>P_{\rm abs}=P_{\rm emt} \qquad \qquad (6)</math>

Substituting the expressions for solar and planet power in equations 1–6 and simplifying yields the estimated temperature of the planet, ignoring greenhouse effect, ''T''P:

:<math>T_P = T_S\sqrt{\frac{R_S\sqrt{\frac{1-\alpha}{\overline{\varepsilon}}}}{2D}}</math>

In other words, given the assumptions made, the temperature of a planet depends only on the surface temperature of the Sun, the radius of the Sun, the distance between the planet and the Sun, the albedo and the IR emissivity of the planet.

===Temperature of Earth===

Substituting the measured values for the Sun and Earth yields:
:<math>T_{\rm S} = 5778 \ \mathrm{K},</math><ref name="NASA">[http://nssdc.gsfc.nasa.gov/planetary/factsheet/sunfact.html NASA Sun Fact Sheet]</ref>
:<math>R_{\rm S} = 6.96 \times 10^8 \ \mathrm{m},</math><ref name="NASA"/>
:<math>D = 1.496 \times 10^{11} \ \mathrm{m},</math><ref name="NASA"/>
:<math>\alpha = 0.306 \ </math><ref name="Cole">{{cite book|author=Cole, George H. A.; Woolfson, Michael M.
|title=Planetary Science: The Science of Planets Around Stars (1st ed.)
|publisher=Institute of Physics Publishing|year=2002|isbn=0-7503-0815-X|pages = 36–37, 380–382|url = http://books.google.com/?id=Bgsy66mJ5mYC&pg=RA3-PA382&dq=black-body+emissivity+greenhouse+intitle:Planetary-Science+inauthor:cole}}</ref>

With the average emissivity set to unity, the [[effective temperature]] of the Earth is:
:<math>T_{\rm E} = 254.356\ \mathrm{K}</math>

or −18.8 °C.

This is the temperature of the Earth if it radiated as a perfect black body in the infrared, ignoring greenhouse effects (which can raise the surface temperature of a body above what it would be if it were a perfect black body in all spectrums<ref>''Principles of Planetary Climate'' by Raymond T. Peirrehumbert, Cambridge University Press (2011), p. 146. From Chapter 3 which is available online [http://www-das.uwyo.edu/~deshler/Atsc4400_5400_Climate/PierreHumbert_Climate_Ch3.pdf here], p. 12 mentions that Venus' black-body temperature would be 330 K "in the zero albedo case", but that due to atmospheric warming, its actual surface temperature is 740 K.</ref>), and assuming an unchanging albedo. The Earth in fact radiates not quite as a perfect black body in the infrared which will raise the estimated temperature a few degrees above the effective temperature. If we wish to estimate what the temperature of the Earth would be if it had no atmosphere, then we could take the albedo and emissivity of the Moon as a good estimate. The albedo and emissivity of the Moon are about 0.1054<ref name="Saari">{{cite journal |last1=Saari |first1=J. M. |last2=Shorthill |first2= R. W.|year=1972 |title=The Sunlit Lunar Surface. I. Albedo Studies and Full Moon |journal=The Moon |volume=5 |issue=1-2 |pages=161–178 |bibcode=1972Moon....5..161S |doi=10.1007/BF00562111 }}</ref> and 0.95<ref>Lunar and Planetary Science XXXVII (2006) 2406</ref> respectively, yielding an estimated temperature of about 1.36 °C.

Estimates of the Earth's average albedo vary in the range 0.3–0.4, resulting in different estimated effective temperatures. Estimates are often based on the [[solar constant]] (total insolation power density) rather than the temperature, size, and distance of the Sun. For example, using 0.4 for albedo, and an insolation of 1400 W m−2, one obtains an effective temperature of about 245 K.<ref>{{cite book
|title = Space physics and space astronomy
|author = Michael D. Papagiannis
|publisher = Taylor & Francis
|year = 1972
|isbn = 978-0-677-04000-4
|pages = 10–11
|url = http://books.google.com/?id=SpgOAAAAQAAJ&pg=PA10}}</ref>
Similarly using albedo 0.3 and solar constant of 1372 W m−2, one obtains an effective temperature of 255 K.<ref>{{cite book
|title = Climate Change an Integrated Perspective
|author = Willem Jozef Meine Martens and Jan Rotmans
|publisher = Springer
|year = 1999
|isbn = 978-0-7923-5996-8
|pages = 52–55
|url = http://books.google.com/?id=o1SELkgK6PcC&pg=RA1-PA53&dq=Earth+effective-temperature+albedo+black-body+0.3
}}</ref><ref>{{cite book
|title = Astrobiology: Future Perspectives
|chapter = The Prebiotic Atmosphere of the Earth
|author = F. Selsis
|editor = Pascale Ehrenfreund et al.
|publisher = Springer
|year = 2004
|isbn = 978-1-4020-2587-7
|pages = 279–280
|url = http://books.google.com/?id=bA_uR3iwzQUC&pg=PA279&dq=Earth+effective-temperature+albedo+black-body+0.3
}}</ref><ref>Wallace, J.M., Hobbs, P.V. (2006). ''Atmospheric Science. An Introductory Survey'', second edition, Elsevier, Amsterdam, ISBN 978-0-12-732951-2, exercise 4.6, pages 119-120.</ref>

==Cosmology==

The [[cosmic microwave background]] radiation observed today is the most perfect black-body radiation ever observed in nature, with a temperature of about 2.7K.<ref name=White>White, M. (1999). "Anisotropies in the CMB". Proceedings of the Los Angeles Meeting, DPF 99. UCLA. http://arxiv.org/pdf/1106.2188v2.pdf.</ref> It is a "snapshot" of the radiation at the time of [[Decoupling (cosmology)|decoupling]] between matter and radiation in the early universe. Prior to this time, most matter in the universe was in the form of an ionized plasma in thermal equilibrium with radiation.

According to Kondepudi and Prigogine, at very high temperatures (above 1010K; such temperatures existed in the very early universe), where the thermal motion separates protons and neutrons in spite of the strong nuclear forces, electron-positron pairs appear and disappear spontanteously and are in thermal equilibrium with electromagnetic radiation. These particles form a part of the black body spectrum, in addition to the electromagnetic radiation.<ref>{{harvnb|Kondepudi|Prigogine|1998|pages = 227–228}}; also Section 11.6, pages 294–296.</ref>

==Doppler effect for a moving black body==

The [[relativistic Doppler effect]] causes a shift in the frequency ''f'' of light originating from a source that is moving in relation to the observer, so that the wave is observed to have frequency ''f''':
:<math>f' = f \frac{1 - \frac{v}{c} \cos \theta}{\sqrt{1-v^2/c^2}}, </math>
where ''v'' is the velocity of the source in the observer's rest frame, ''θ'' is the angle between the velocity vector and the observer-source direction measured in the reference frame of the source, and ''c'' is the [[speed of light]].<ref>The Doppler Effect, T. P. Gill, Logos Press, 1965</ref> This can be simplified for the special cases of objects moving directly towards (''θ'' = π) or away (''θ'' = 0) from the observer, and for speeds much less than ''c''.

Through Planck's law the temperature spectrum of a black body is proportionally related to the frequency of light and one may substitute the temperature (''T'') for the frequency in this equation.

For the case of a source moving directly towards or away from the observer, this reduces to
:<math>T' = T \sqrt{\frac{c-v}{c+v}}.</math>
Here ''v'' > 0 indicates a receding source, and ''v'' < 0 indicates an approaching source.

This is an important effect in astronomy, where the velocities of stars and galaxies can reach significant fractions of ''c''. An example is found in the [[cosmic microwave background radiation]], which exhibits a dipole anisotropy from the Earth's motion relative to this black-body radiation field.

==History==

===Balfour Stewart===

In 1858, Balfour Stewart described his experiments on the thermal radiative emissive and absorptive powers of polished plates of various substances, compared with the powers of lamp-black surfaces, at the same temperature.<ref name="Stewart 1858">{{harvnb|Stewart|1858}}</ref> Stewart chose lamp-black surfaces as his reference because of various previous experimental findings, especially those of [[Pierre Prevost]] and of [[John Leslie (physicist)|John Leslie]]. He wrote "Lamp-black, which absorbs all the rays that fall upon it, and therefore possesses the greatest possible absorbing power, will possess also the greatest possible radiating power." More an experimenter than a logician, Stewart failed to point out that his statement presupposed an abstract general principle, that there exist either ideally in theory or really in nature bodies or surfaces that respectively have one and the same unique universal greatest possible absorbing power, likewise for radiating power, for every wavelength and equilibrium temperature.

Stewart measured radiated power with a thermo-pile and sensitive galvanometer read with a microscope. He was concerned with selective thermal radiation, which he investigated with plates of substances that radiated and absorbed selectively for different qualities of radiation rather than maximally for all qualities of radiation. He discussed the experiments in terms of rays which could be reflected and refracted, and which obeyed the Stokes-[[Helmholtz reciprocity]] principle (though he did not use an eponym for it). He did not in this paper mention that the qualities of the rays might be described by their wavelengths, nor did he use spectrally resolving apparatus such as prisms or diffraction gratings. His work was quantitative within these constraints. He made his measurements in a room temperature environment, and quickly so as to catch his bodies in a condition near the thermal equilibrium in which they had been prepared by heating to equilibrium with boiling water. His measurements confirmed that substances that emit and absorb selectively respect the principle of selective equality of emission and absorption at thermal equilibrium.

Stewart offered a theoretical proof that this should be the case separately for every selected quality of thermal radiation, but his mathematics was not rigorously valid.<ref name="Siegel">{{harvnb|Siegel|1976}}</ref> He made no mention of thermodynamics in this paper, though he did refer to conservation of ''[[vis viva]]''. He proposed that his measurements implied that radiation was both absorbed and emitted by particles of matter throughout depths of the media in which it propagated. He applied the Helmholtz reciprocity principle to account for the material interface processes as distinct from the processes in the interior material. He did not postulate unrealizable perfectly black surfaces. He concluded that his experiments showed that in a cavity in thermal equilibrium, the heat radiated from any part of the interior bounding surface, no matter of what material it might be composed, was the same as would have been emitted from a surface of the same shape and position that would have been composed of lamp-black. He did not state explicitly that the lamp-black-coated bodies that he used as reference must have had a unique common spectral emittance function that depended on temperature in a unique way.

===Gustav Kirchhoff===

In 1859, not knowing of Stewart's work, [[Gustav Kirchhoff|Gustav Robert Kirchhoff]] reported the coincidence of the wavelengths of spectrally resolved lines of absorption and of emission of visible light. Importantly for thermal physics, he also observed that bright lines or dark lines were apparent depending on the temperature difference between emitter and absorber.<ref>{{harvnb|Kirchhoff|1860a}}</ref>

Kirchhoff then went on to consider bodies that emit and absorb heat radiation, in an opaque enclosure or cavity, in equilibrium at temperature {{math|''T''}}.

Here is used a notation different from Kirchhoff's. Here, the emitting power {{math|''E''(''T'', ''i'')}} denotes a dimensioned quantity, the total radiation emitted by a body labeled by index {{math|''i''}} at temperature {{math|''T''}}. The total absorption ratio {{math|''a''(''T'', ''i'')}} of that body is dimensionless, the ratio of absorbed to incident radiation in the cavity at temperature {{math|''T''}} . (In contrast with Balfour Stewart's, Kirchhoff's definition of his absorption ratio did not refer in particular to a lamp-black surface as the source of the incident radiation.) Thus the ratio {{math|''E''(''T'', ''i'') / ''a''(''T'', ''i'')}} of emitting power to absorption ratio is a dimensioned quantity, with the dimensions of emitting power, because {{math|''a''(''T'', ''i'')}} is dimensionless. Also here the wavelength-specific emitting power of the body at temperature {{math|''T''}} is denoted by {{math|''E''(''λ'', ''T'', ''i'')}} and the wavelength-specific absorption ratio by {{math|''a''(''λ'', ''T'', ''i'')}} . Again, the ratio {{math|''E''(''λ'', ''T'', ''i'') / ''a''(''λ'', ''T'', ''i'')}} of emitting power to absorption ratio is a dimensioned quantity, with the dimensions of emitting power.

In a second report made in 1859, Kirchhoff announced a new general principle or law for which he offered a theoretical and mathematical proof, though he did not offer quantitative measurements of radiation powers.<ref>{{harvnb|Kirchhoff|1860b}}</ref> His theoretical proof was and still is considered by some writers to be invalid.<ref name="Siegel"/><ref name="Schirrmacher 2001">{{harvnb|Schirrmacher|2001}}</ref> His principle, however, has endured: it was that for heat rays of the same wavelength, in equilibrium at a given temperature, the wavelength-specific ratio of emitting power to absorption ratio has one and the same common value for all bodies that emit and absorb at that wavelength. In symbols, the law stated that the wavelength-specific ratio {{math|''E''(''λ'', ''T'', ''i'') / ''a''(''λ'', ''T'', ''i'')}} has one and the same value for all bodies, that is for all values of index {{math|''i''}} . In this report there was no mention of black bodies.

In 1860, still not knowing of Stewart's measurements for selected qualities of radiation, Kirchhoff pointed out that it was long established experimentally that for total heat radiation, of unselected quality, emitted and absorbed by a body in equilibrium, the dimensioned total radiation ratio {{math|''E''(''T'', ''i'') / ''a''(''T'', ''i'')}}, has one and the same value common to all bodies, that is, for every value of the material index {{math|''i''}}.<ref name="Kirchhoff 1860c">{{harvnb|Kirchhoff|1860c}}</ref> Again without measurements of radiative powers or other new experimental data, Kirchhoff then offered a fresh theoretical proof of his new principle of the universality of the value of the wavelength-specific ratio {{math|''E''(''λ'', ''T'', ''i'') / ''a''(''λ'', ''T'', ''i'')}} at thermal equilibrium. His fresh theoretical proof was and still is considered by some writers to be invalid.<ref name="Siegel"/><ref name="Schirrmacher 2001"/>

But more importantly, it relied on a new theoretical postulate of "perfectly black bodies," which is the reason why one speaks of Kirchhoff's law. Such black bodies showed complete absorption in their infinitely thin most superficial surface. They correspond to Balfour Stewart's reference bodies, with internal radiation, coated with lamp-black. They were not the more realistic perfectly black bodies later considered by Planck. Planck's black bodies radiated and absorbed only by the material in their interiors; their interfaces with contiguous media were only mathematical surfaces, capable neither of absorption nor emission, but only of reflecting and transmitting with refraction.<ref>{{harvnb|Planck|1914|page=11}}</ref>

Kirchhoff's proof considered an arbitrary non-ideal body labeled {{math|''i''}} as well as various perfect black bodies labeled {{math|BB}} . It required that the bodies be kept in a cavity in thermal equilibrium at temperature {{math|''T''}} . His proof intended to show that the ratio {{math|''E''(''λ'', ''T'', ''i'') / ''a''(''λ'', ''T'', ''i'')}} was independent of the nature {{math|''i''}} of the non-ideal body, however partly transparent or partly reflective it was.

His proof first argued that for wavelength {{math|''λ''}} and at temperature {{math|''T''}}, at thermal equilibrium, all perfectly black bodies of the same size and shape have the one and the same common value of emissive power {{math|''E''(''λ'', ''T'', BB)}}, with the dimensions of power. His proof noted that the dimensionless wavelength-specific absorption ratio {{math|''a''(''λ'', ''T'', BB)}} of a perfectly black body is by definition exactly 1. Then for a perfectly black body, the wavelength-specific ratio of emissive power to absorption ratio {{math|''E''(''λ'', ''T'', BB) / ''a''(''λ'', ''T'', BB)}} is again just {{math|''E''(''λ'', ''T'', BB)}}, with the dimensions of power. Kirchhoff considered, successively, thermal equilibrium with the arbitrary non-ideal body, and with a perfectly black body of the same size and shape, in place in his cavity in equilibrium at temperature {{math|''T''}} . He argued that the flows of heat radiation must be the same in each case. Thus he argued that at thermal equilibrium the ratio {{math|''E''(''λ'', ''T'', ''i'') / ''a''(''λ'', ''T'', ''i'')}} was equal to {{math|''E''(''λ'', ''T'', BB)}}, which may now be denoted {{math|''B''''λ'' (''λ'', ''T'')}}, a continuous function, dependent only on {{math|''λ''}} at fixed temperature {{math|''T''}}, and an increasing function of {{math|''T''}} at fixed wavelength {{math|''λ''}}, at low temperatures vanishing for visible but not for longer wavelengths, with positive values for visible wavelengths at higher temperatures, which does not depend on the nature {{math|''i''}} of the arbitrary non-ideal body. (Geometrical factors, taken into detailed account by Kirchhoff, have been ignored in the foregoing.)

Thus [[Kirchhoff's law of thermal radiation]] can be stated: ''For any material at all, radiating and absorbing in thermodynamic equilibrium at any given temperature {{math|T}}, for every wavelength {{math|λ}}, the ratio of emissive power to absorptive ratio has one universal value, which is characteristic of a perfect black body, and is an emissive power which we here represent by {{math|Bλ (λ, T)}} .'' (For our notation {{math|''B''''λ'' (''λ'', ''T'')}}, Kirchhoff's original notation was simply {{math|''e''}}.)<ref name="Kirchhoff 1860c"/><ref>{{harvnb|Chandrasekhar|1950|p=8}}</ref><ref>{{harvnb|Milne|1930|page=80}}</ref><ref>{{harvnb|Rybicki|Lightman|1979|pages=16–17}}</ref><ref>{{harvnb|Mihalas|Weibel-Mihalas|1984|page=328}}</ref><ref>{{harvnb|Goody|Yung|1989|pages=27–28}}</ref>

Kirchhoff announced that the determination of the function {{math|''B''''λ'' (''λ'', ''T'')}} was a problem of the highest importance, though he recognized that there would be experimental difficulties to be overcome. He supposed that like other functions that do not depend on the properties of individual bodies, it would be a simple function. Occasionally by historians that function {{math|''B''''λ'' (''λ'', ''T'')}} has been called "Kirchhoff's (emission, universal) function,"<ref>[[Friedrich Paschen|Paschen, F.]] (1896), personal letter cited by {{harvnb|Hermann|1971|page=6}}</ref><ref>{{harvnb|Hermann|1971|page=7}}</ref><ref>{{harvnb|Kuhn|1978|pages=8, 29}}</ref><ref>{{harvnb|Mehra and Rechenberg|1982|pages=26, 28, 31, 39}}</ref> though its precise mathematical form would not be known for another forty years, till it was discovered by Planck in 1900. The theoretical proof for Kirchhoff's universality principle was worked on and debated by various physicists over the same time, and later.<ref name="Schirrmacher 2001"/> Kirchhoff stated later in 1860 that his theoretical proof was better than Balfour Stewart's, and in some respects it was so.<ref name="Siegel"/> Kirchhoff's 1860 paper did not mention the second law of thermodynamics, and of course did not mention the concept of entropy which had not at that time been established. In a more considered account in a book in 1862, Kirchhoff mentioned the connection of his law with [[Carnot's principle]], which is a form of the second law.<ref>{{harvnb|Kirchhoff|1862/1882|page=573}}</ref>

According to Helge Kragh, "Quantum theory owes its origin to the study of thermal radiation, in particular to the "black-body" radiation that Robert Kirchhoff had first defined in 1859–1860."<ref>{{harvnb|Kragh|1999|page=58}}</ref>

==See also==

{{colbegin|3}}
* [[Bolometer]]
* [[Color temperature]]
* [[Infrared thermometer]]
* [[Photon polarization]]
* [[Planck's law]]
* [[Pyrometry]]
* [[Rayleigh–Jeans law]]
* [[Thermography]]
* [[Sakuma–Hattori equation]]
{{colend}}

==References==
{{Reflist|2}}

=== Bibliography ===
{{refbegin}}
*{{cite book
|last1=Chandrasekhar |first1=S.
|author1-link=Subrahmanyan Chandrasekhar
|year=1950
|title=Radiative Transfer
|publisher=[[Oxford University Press]]
|ref=harv
}}
*{{cite book
|last1=Goody |first1=R. M.
|last2=Yung |first2=Y. L.
|author2-link=Yuk L. Yung
|year=1989
|title=Atmospheric Radiation: Theoretical Basis
|edition=2nd
|publisher=[[Oxford University Press]]
|isbn=978-0-19-510291-8
|ref=harv
}}
*{{Cite book
|last1=Hermann |first1=A.
|author1-link=Armin Hermann
|others=Nash, C.W. (transl.)
|year=1971
|title=The Genesis of Quantum Theory
|publisher=[[MIT Press]]
|isbn=0-262-08047-8
|ref=harv
}} a translation of ''Frühgeschichte der Quantentheorie (1899–1913)'', Physik Verlag, Mosbach/Baden.
*{{cite journal
|last1=Kirchhoff |first1=G.
|author1-link=Gustav Kirchhoff
|last2=[27 October 1859]
|year=1860a
|title=Über die Fraunhofer'schen Linien
|journal=[[Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin]]
|pages=662–665
|doi=
|ref=harv
}}
*{{cite journal
|last1=Kirchhoff |first1=G.
|author1-link=Gustav Kirchhoff
|last2=[11 December 1859]
|date= 1860b
|title=Über den Zusammenhang zwischen Emission und Absorption von Licht und Wärme
|journal=[[Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin]]
|pages=783–787
|doi=
|ref=harv
}}
*{{cite journal
|last1=Kirchhoff |first1=G.
|author1-link=Gustav Kirchhoff
|year=1860c
|title=Ueber das Verhältniss zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme and Licht
|journal=[[Annalen der Physik und Chemie]]
|volume=109
|pages=275–301
|doi=
|ref=harv
}} Translated by Guthrie, F. as {{cite journal
|last1=Kirchhoff |first1=G.
|year=1860
|title=On the relation between the radiating and absorbing powers of different bodies for light and heat
|journal=[[Philosophical Magazine]]
|volume=Series 4, volume 20 |pages=1–21
}}
*{{Citation
| last = Kirchhoff
| first = G.
| author-link = Gustav Kirchhoff
| title = Gessamelte Abhandlungen
| place = Leipzig
| publisher = Johann Ambrosius Barth
| series =
| volume =
| origyear = 1862
| year = 1882
| month=
| edition =
| chapter = Ueber das Verhältniss zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme und Licht
| chapterurl =
| page =
| pages = 571–598
| ref = harv
}}
*{{cite book
|last1=Kondepudi |first1=D.
|author1-link=Dilip Kondepudi
|last2=Prigogine |first2=I.
|author2-link=Ilya Prigogine
|year=1998
|title=Modern Thermodynamics. From Heat Engines to Dissipative Structures
|publisher=[[John Wiley & Sons]]
|isbn=0-471-97393-9
|ref=harv
}}
*{{cite book
|last1=Kragh |first1=H.
|author1-link=Helge Kragh
|year=1999
|title=Quantum Generations: a History of Physics in the Twentieth Century
|publisher=[[Princeton University Press]]
|isbn=0-691-01206-7
|ref=harv
}}
*{{cite book
|last1=Kuhn |first1=T. S.
|author1-link=Thomas Kuhn
|year=1978
|title=Black–Body Theory and the Quantum Discontinuity
|publisher=[[Oxford University Press]]
|isbn=0-19-502383-8
|ref=harv
}}
*{{cite book
|last1=Loudon |first1=R.
|author1-link=Rodney Loudon
|year=2000
|origyear=1973
|title=The Quantum Theory of Light
|publisher=[[Cambridge University Press]]
|edition=third
|isbn=0-19-850177-3
|ref=harv
}}
*{{cite book
|last1=Mandel |first1=L.
|author1-link=Leonard Mandel
|last2=Wolf |first2=E.
|author2-link=Emil Wolf
|year=1995
|title=Optical Coherence and Quantum Optics
|publisher=[[Cambridge University Press]]
|isbn=0-521-41711-2
|ref=harv
}}
*{{cite book
|last1=Mehra |first1=J.
|author1-link=Jagdish Mehra
|last2=Rechenberg |first2=H.
|author2-link=Hlemut Rechenberg
|year=1982
|title=The Historical Development of Quantum Theory
|volume=volume 1, part 1
|publisher=[[Springer-Verlag]]
|isbn=0-387-90642-8
|ref=harv
}}
*{{cite book
|last1=Mihalas |first1=D.
|author1-link=Dimitri Mihalas
|last2=Weibel-Mihalas |first2=B.
|year=1984
|title=Foundations of Radiation Hydrodynamics
|publisher=[[Oxford University Press]]
|isbn=0-19-503437-6
|ref=harv
}}
*{{cite journal
|last1=Milne |first1=E.A.
|author1-link=Edward Arthur Milne
|year=1930
|title=Thermodynamics of the Stars
|journal=[[Handbuch der Astrophysik]]
|volume=3, part 1
|pages=63–255
|doi=
|ref=harv
}}
*{{cite book
|last1=Partington |first1=J.R.
|author1-link=J.R. Partington
|year=1949
|title=An Advanced Treatise on Physical Chemistry. Volume 1. Fundamental Principles. The Properties of Gases
|publisher=[[Longman|Longmans, Green and Co.]]
|ref=harv
}}
*{{cite book
|last1=Planck |first1=M.
|author1-link=Max Planck
|others=translated by Masius, M.
|origyear= 1912
|year=1914
|title=The Theory of Heat Radiation
|publisher=[[P. Blakiston's Sons & Co.]]
|ref=harv
}}
*{{Cite book
|last1=Rybicki |first1=G. B.
|last2=Lightman |first2=A. P.
|author2-link=Alan Lightman
|year=1979
|title=Radiative Processes in Astrophysics
|url=http://books.google.com/books?id=LtdEjNABMlsC&printsec=frontcover
|publisher=[[John Wiley & Sons]]
|isbn=0-471-82759-2
|ref=harv
}}
*{{cite book
|last1=Schirrmacher |first1=A.
|author1-link=Arne Schirrmacher
|year=2001
|title=Experimenting theory: the proofs of Kirchhoff's radiation law before and after Planck
|publisher=[[Münchner Zentrum für Wissenschafts und Technikgeschichte]]
|ref=harv
}}
*{{cite journal
|last1=Siegel |first1=D.M.
|year=1976
|title=Balfour Stewart and Gustav Robert Kirchhoff: two independent approaches to "Kirchhoff's radiation law"
|journal=[[Isis]]
|volume=67 |pages=565–600
|doi=
|ref=harv
}}
*{{cite journal
|last1=Stewart |first1=B.
|author1-link=Balfour Stewart
|year=1858
|title=An account of some experiments on radiant heat
|journal=[[Transactions of the Royal Society of Edinburgh]]
|volume=22 |pages=1–20
|doi=
|ref=harv
}}
*{{cite journal
|last1=Wien |first1=W.
|author1-link=Wilhelm Wien
|year=1894
|title=Temperatur und Entropie der Strahlung
|journal=[[Annalen der Physik]]
|volume=288 |pages=132–165
|doi= 10.1002/andp.18942880511
|ref=harv
|bibcode = 1894AnP...288..132W }}
{{refend}}

==Further reading==

*{{cite book|author=Kroemer, Herbert; Kittel, Charles
|title=Thermal Physics |edition=2nd|publisher=W. H. Freeman Company|year=1980
|isbn=0-7167-1088-9}}
*{{cite book|author=Tipler, Paul; Llewellyn, Ralph
|title=Modern Physics |edition=4th|publisher=W. H. Freeman|year=2002|isbn=0-7167-4345-0}}

==External links==
*[http://www.spectralcalc.com/blackbody/blackbody.html Calculating Black-body Radiation] Interactive calculator with Doppler Effect. Includes most systems of units.
*[http://academo.org/demos/colour-temperature-relationship/ Color-to-Temperature demonstration] at Academo.org
*[http://hyperphysics.phy-astr.gsu.edu/Hbase/thermo/coobod.html#c1 Cooling Mechanisms for Human Body] – From Hyperphysics
*[http://www.x20.org/library/thermal/blackbody.htm Descriptions of radiation emitted by many different objects]
*[http://webphysics.davidson.edu/Applets/java11_Archive.html Black-Body Emission Applet]
*[http://demonstrations.wolfram.com/BlackbodySpectrum/ "Blackbody Spectrum"] by Jeff Bryant, [[Wolfram Demonstrations Project]], 2007.

{{DEFAULTSORT:Black Body}}
[[Category:Infrared]]
[[Category:Heat transfer]]
[[Category:Electromagnetic radiation]]
[[Category:Astrophysics]]

Schrödinger field

2013-07-16T11:01:13Z

130.83.36.104: /* Identical particles */

{{About|the structure in plasma physics||Double layer (disambiguation){{!}}Double layer}}

[[Image:Saturn-aurora.jpg|thumb|300px|Saturnian aurora whose reddish colour is characteristic of ionized hydrogen [[Plasma (physics)|plasma]].<ref>Astronomy Picture of the Day: [http://antwrp.gsfc.nasa.gov/apod/ap011223.html 23 December 2001] (NASA)</ref> Powered by the Saturnian equivalent of (filamentary) Birkeland currents, streams of charged particles from the [[interplanetary medium]] interact with the planet's magnetic field and funnel down to the poles.<ref>Isbell, J.; Dessler, A. J.; Waite, J. H. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1984JGR....8910715I&db_key=PHY&data_type=HTML&format=&high=42ca922c9c16745 Magnetospheric energization by interaction between planetary spin and the solar wind]" (1984) ''Journal of Geophysical Research'', Volume 89, Issue A12, pp. 10715–10722</ref> Double layers are associated with (filamentary) currents,<ref>Theisen, William L. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1994PhDT........19T&db_key=AST&data_type=HTML&format=&high=42ca922c9c05019 Langmuir Bursts and Filamentary Double Layers in Plasmas.]" (1994) Ph.D Thesis U. of Iowa, 1994</ref><ref>Deverapalli, C. M.; Singh, N.; Khazanov, I. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2005AGUFMSM41C1202D&db_key=AST&data_type=HTML&format=&high=42ca922c9c05019 Filamentary Structures in U-Shaped Double Layers]" (2005) American Geophysical Union, Fall Meeting 2005, abstract #SM41C-1202</ref> and their electric fields accelerate ions and [[electron]]s.<ref>Borovsky, Joseph E. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1992PhRvL..69.1054B&db_key=PHY&data_type=HTML&format=&high=42ca922c9c27309 Double layers do accelerate particles in the auroral zone]" (1992) ''Physical Review Letters'' (ISSN 0031-9007), vol. 69, no. 7, Aug. 17, 1992, pp. 1054–1056</ref>]]

A '''double layer''' is a [[structure]] in a [[Plasma (physics)|plasma]] and consists of two parallel layers with opposite electrical charge. The sheets of charge cause a strong electric field and a correspondingly sharp change in [[voltage]] ([[electrical potential]]) across the double layer. Ions and electrons which enter the double layer are accelerated, decelerated, or reflected by the electric field. In general, double layers (which may be curved rather than flat) separate regions of plasma with quite different characteristics. Double layers are found in a wide variety of plasmas, from discharge tubes to space plasmas to the [[Birkeland current]]s supplying the Earth's [[Aurora (astronomy)|aurora]], and are especially common in current-carrying plasmas. Compared to the sizes of the plasmas which contain them, double layers are very thin (typically ten [[Debye length]]s), with widths ranging from a few millimeters for laboratory plasmas to thousands of kilometres for astrophysical plasmas.

Other names for a double layer are electrostatic double layer, electric double layer, plasma double layers, electrostatic shock (a type of double layer which is oriented at an oblique angle to the magnetic field in such a way that the perpendicular electric field is much stronger than the parallel electric field),<ref>Temerin, M.; Mozer, F. S., "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1987dla..conf..295T&db_key=AST&data_type=HTML&format=&high=42ca922c9c17461 Double Layers Above the Aurora]" (1987) NASA Conference Publication, #2469</ref> space charge layer.<ref>Block, L. P. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1978Ap%26SS..55...59B&db_key=AST&data_type=HTML&format=&high=42ca922c9c32320 A double layer review]" (1978) ''Astrophysics and Space Science'', vol. 55, no. 1, May 1978, pp. 59–83</ref> In laser physics, a double layer is sometimes called an ambipolar electric field.<ref>Bulgakova, Nadezhda M. et al., "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2000PhRvE..62.5624B&db_key=PHY&data_type=HTML&format=&high=42ca922c9c30563 Double layer effects in laser-ablation plasma plumes]", ''Physical Review E'' (Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics), Volume 62, Issue 4, October 2000, pp. 5624–5635</ref> Double layers are conceptually related to the concept of a 'sheath' (''see'' [[Debye sheath]]).

The adopted electrical symbol for a double layer, when represented in an electrical circuit is
────'''DL'''────. If there is a net current present, then the DL is oriented with the base of the L in line with direction of current.<ref>''Double Layers in Astrophysics'', NASA Conference Publication 2469 (NASA CP-2469), (1987) Edited by Alton C. Williams and Tauna W. Moorhead</ref>

An overview of double layers in space, experiment and simulation is given in the introduction of ref.<ref>Meige, A. "[http://prl.anu.edu.au/Members/mdl112/thesis.pdf Numerical modeling of low-pressure plasmas: applications to electric double layers.]" Ph.D Thesis, The Australian National University, 2006</ref>

== Double layer classification ==
[[Image:HallThruster 2.jpg|thumb|300px|[[Hall effect thruster]]. The electric fields utilised in plasma thrusters (in particular the [[Helicon Double Layer Thruster]]) may be in the form of double layers.<ref>See "[http://www.esa.int/gsp/ACT/propulsion/helicon_double_layer.htm Helicon Double Layer Thruster study]", European Space Agency; "[http://www.esa.int/esaCP/SEM6HSVLWFE_index_0.html ESA accelerates towards a new space thruster]" (2005)</ref>]]

Double layers may be classified in the following ways:
*''Weak'' and ''strong'' double layers. The strength of a double layer is expressed as the ratio of the [[potential difference|potential drop]] in comparison with the plasma’s equivalent [[thermal energy]], or in comparison with the [[rest mass]] energy of the [[electron]]s. A double layer is said to be strong if the potential drop across the layer is greater than the equivalent thermal energy of the plasma’s components. This means that for strong double layers there are four different components to the plasma:
#The electrons entering at the low potential side of the double layer which are accelerated;
#The ions entering at the high potential side of the double layer which are accelerated;
#The electrons entering at the high potential side of the double layer which are decelerated and successively reflected; and
#The ions which enter the double layer at the low potential side of the double layer which are decelerated and reflected.
:Note that in the case of a weak double layer, the electrons and ions entering from the “wrong” side are decelerated by the electric field, however most will not be reflected, as the potential drop is not strong enough.
*''Relativistic'' or ''nonrelativistic'' double layers. A double layer is said to be relativistic if the potential drop over the layer is so large that the total gain in energy of the particles is larger than the rest mass energy of the electron. The charge distribution in a relativistic double layer is such that the charge density is located in two very thin layers, and inside the double layer the density is constant at and very low compared to the rest of the plasma. In this respect, the double layer is similar to the charge distribution in a [[capacitor]]. As a special case of a relativistic double layer one can consider the vacuum gap at the magnetic polar cap of a pulsar.
*''Current carrying'' and ''current-free'' double layers may both occur. ''Current carrying'' double layers may be generated by current-driven plasma instabilities which amplify variations of the plasma density. ''Current-free'' double layers form on the interface between two plasma regions with different characteristics, and their associated electric field maintains a balance between the penetration of electrons in either direction (so that the net current is low).

== Double layer formation ==
[[Image:Double layer formation.png|thumb|320px|Doubler layer formation. Hotter electrons moving into a cooler plasma region (Diagram 1, top) cause a charge imbalance, resulting in a double layer that is able to accelerate electrons across it (Diagram 2, bottom).]]

There are two different kinds of double layers, which are formed differently:

===Current carrying double layers===
'''Current carrying double layers''' may arise in plasmas carrying a current. Various [[Instability|instabilities]] can be responsible for the formation of these layers. One example is the [[Buneman instability]] which occurs when the streaming velocity of the electrons (basically the current density divided by the electron density) exceeds the electron [[thermal velocity]] of the plasma. Double layers (and other phase space structures) are often formed in the non-linear phase of the instability. One way of viewing the Buneman instability is to describe what happens when the current (in the form of a zero temperature electron beam) has to pass through a region of decreased ion density. In order to prevent charge from accumulating, the current in the system must be the same everywhere (in this 1D model). The electron density also has to be close to the ion density (quasineutrality), so there is also a dip in electron density. The electrons must therefore be accelerated into the density cavity, to maintain the same current density with a lower density of charge carriers. This implies that the density cavity is at a high electrical potential. As a consequence, the ions are accelerated out of the cavity, amplifying the density perturbation. Then there is the situation of a double-double layer, of which one side will most likely be convected away by the plasma, leaving a regular double layer. This is the process in which double layers are produced along planetary magnetic field lines in so-called [[Birkeland current]]s.

===Current-free double layers===
'''Current-free double layers''' occur at the boundary between plasma regions with different plasma properties. Consider a plasma divided into two regions by a plane, which has a higher electron temperature on one side than on the other (the same analysis can also be done for different densities). This means that the electrons on one side of the interface have a greater thermal velocity. The electrons may stream freely in either direction, and the flux of electrons from the hot plasma to the cold plasma will be greater than the flux of the electrons from the cold plasma to the hot plasma, because the electrons from the hot side have a greater average speed. Because many more electrons enter the cold plasma than exit it, part of the cold region becomes negatively charged. The hot plasma, conversely, becomes positively charged. Therefore, an electric field builds up, which starts to accelerate electrons towards the hot region, reducing the net flux. In the end, the electric field builds up until the fluxes of electrons in either direction are equal, and further charge build up in the two plasmas is prevented. The potential drop is in fact exactly equal to the difference in [[thermal energy]] between the two plasma regions in this case, so such a double layer is a marginally strong double layer.

[[Image:double-layer-formation-summary.png|thumb|center|400px|'''Double Layer Formation Summary'''. Double layers are formed in four main ways<ref>Singh, Nagendra; Thiemann, H.; Schunk, R. W., "[http://adsabs.harvard.edu/abs/1987dla..conf..183S Electric Fields and Double Layers in Plasmas] (1987) Double Layers in Astrophysics, Proceedings of a Workshop held in Huntsville, Ala., 17–19 Mar. 1986. Edited by Alton C. Williams and Tauna W. Moorehead. NASA Conference Publication, #2469"</ref>]]

===Double layer formation mechanisms===
[[Image:Moon-Mdf-2005.jpg|thumb|300px|The '''Moon.''' The prediction of a lunar double layer<ref>Borisov, N.; Mall, U. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2002JPlPh..67..277B&db_key=PHY&data_type=HTML&format=&high=42ca922c9c05280 The structure of the double layer behind the Moon]" (2002) ''Journal of Plasma Physics'', vol. 67, Issue 04, pp. 277–299</ref> was confirmed in 2003.<ref>Halekas, J. S.; Lin, R. P.; Mitchell, D. L. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2003GeoRL..30uPLA1H&db_key=AST&data_type=HTML&format=&high=42ca922c9c27734 Inferring the scale height of the lunar nightside double layer]" (2003) ''Geophysical Research Letters'', Volume 30, Issue 21, pp. PLA 1-1. ([http://sprg.ssl.berkeley.edu/adminstuff/webpubs/2003_grl_2117.pdf PDF])</ref> In the shadows, the Moon charges negatively in the interplanetary medium.<ref>Halekas, J. S ''et al.'' "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2002GeoRL..29j..77H&db_key=AST&data_type=HTML&format=&high=42ca922c9c05119 Evidence for negative charging of the lunar surface in shadow]" (2002) ''Geophysical Research Letters'', Volume 29, Issue 10, pp. 77–81</ref>]]
The details of the formation mechanism depend on the environment of the plasma (e.g. double layers in the laboratory, ionosphere, solar wind, fusion, etc.). Proposed mechanisms for their formation have included:

*1971: Between plasmas of different temperatures<ref>Hultqvist, Bengt, "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1971P%26SS...19..749H&db_key=AST&data_type=HTML&format=&high=42ca922c9c23669 On the production of a magnetic-field-aligned electric field by the interaction between the hot magnetospheric plasma and the cold ionosphere]" (1971) ''Planetary and Space Science'', Vol. 19, p.749. See also: Ishiguro, S.; Kamimura, T.; Sato, T., "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1985PhFl...28.2100I&db_key=PHY&data_type=HTML&format=&high=42ca922c9c26001 Double layer formation caused by contact between different temperature plasmas]" (1985) ''Physics of Fluids'' (ISSN 0031-9171), vol. 28, July 1985, p. 2100–2105.</ref>
*1976: In laboratory plasmas<ref>Torven, S. ""[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1979wisp.proc..109T&db_key=AST&data_type=HTML&format=&high=42ca922c9c26029 Formation of Double Layers in Laboratory Plasmas] (1976) ''Astrophysics and Space Science Library'', Vol. 74, p.109</ref>
*1982: Disruption of a neutral [[current sheet]]<ref>Stenzel, R. L., "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1982GeoRL...9..680S&db_key=AST&data_type=HTML&format=&high=42ca922c9c18563 Double layer formation during current sheet disruptions in a reconnection experiment]" (1982) ''Geophysical Research Letters'', vol. 9, June 1982, pp. 680–683.</ref>
*1983: Injection of non-neutral electron current into a cold plasma<ref>Thiemann, H.; Singh, N.; Schunk, R. W. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1983erbp.conf..269T&db_key=AST&data_type=HTML&format=&high=42ca922c9c18563 Formation of v-shaped potentials]" (1983) In ESA Sixth ESA Symp. on European Rocket and Balloon Programs and Related Res. pp 269–275</ref>
*1985: Increasing the current density in a plasma<ref>Yamamoto, Takashi; Kan, J. R. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1985P%26SS...33..853Y&db_key=AST&data_type=HTML&format=&high=42ca922c9c18563 Double layer formation due to current injection]" (1985) ''Planetary and Space Science'', Volume 33, Issue 7, pp. 853–861.</ref>
*1986: In the accretion column of a neutron star<ref>Williams, A. C. et al. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1986ApJ...305..759W&db_key=AST&data_type=HTML&format=&high=42ca922c9c18563 Accretion onto neutron stars with the presence of a double layer]" (1986) ''Astrophysical Journal'', Part 1 (ISSN 0004-637X), vol. 305, June 15, 1986, pp. 759–766.</ref>
*1986: By pinches in cosmic plasma regions<ref>Peratt, Anthony L. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1986ITPS...14..639P&db_key=AST&data_type=HTML&format=&high=42ca922c9c18563 Evolution of the plasma universe. I – Double radio galaxies, quasars, and extragalactic jets]" (1986) IEEE ''Transactions on Plasma Science'' (ISSN 0093-3813), vol. PS-14, Dec. 1986, pp. 639–660.</ref>
*1987: In a plasma constrained by a magnetic mirror<ref>Lennartsson, W. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1987dla..conf..275L&db_key=AST&data_type=HTML&format=&high=42ca922c9c26029 Some Aspects of Double Layer Formation in a Plasma Constrained by a Magnetic Mirror]" (1987) NASA Conference Publication, #2469. Washington, DC: NASA Scientific and Technical Information Branch., p.275</ref>
*1988: By an electrical discharge<ref>Lindberg, Lennart "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1988Ap%26SS.144....3L&db_key=AST&data_type=HTML&format=&high=42ca922c9c18563 Observations of propagating double layers in a high current discharge]" (1988) ''Astrophysics and Space Science'' (ISSN 0004-640X), vol. 144, no. 1–2, May 1988, pp. 3–13.</ref>
*1988: Current-driven instabilities (strong double layers)<ref>Raadu, Michael A.; Rasmussen, J. Juul "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1988Ap%26SS.144...43R&db_key=AST&data_type=HTML&format=&high=42ca922c9c18563 Dynamical aspects of electrostatic double layers]" (1988) ''Astrophysics and Space Science'' (ISSN 0004-640X), vol. 144, no. 1–2, May 1988, pp. 43–71</ref>
*1988: Spacecraft-ejected electron beams<ref>Singh, Nagendra; Hwang, K. S. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1988JGR....9310035S&db_key=AST&data_type=HTML&format=&high=42ca922c9c18563 Electric potential structures and propagation of electron beams injected from a spacecraft into a plasma]" (1988) ''Journal of Geophysical Research'' (ISSN 0148-0227), vol. 93, Sept. 1, 1988, pp. 10035–10040.</ref>
*1989: From shock waves in a plasma<ref>Lembege, B.; Dawson, J. M. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1989PhRvL..62.2683L&db_key=AST&data_type=HTML&format=&high=42ca922c9c18563 Formation of double layers within an oblique collisionless shock]" (1989) ''Physical Review Letters'' (ISSN 0031-9007), vol. 62, June 5, 1989, pp. 2683–2686</ref>
*2000: Laser radiation<ref>Bulgakova, Nadezhda M. et al. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2000PhRvE..62.5624B&db_key=PHY&data_type=HTML&format=&high=42ca922c9c30563 Double layer effects in laser-ablation plasma plumes]" (2000) ''Physical Review E'' (Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics), Volume 62, Issue 4, October 2000, pp. 5624–5635</ref>
*2002: When magnetic field-aligned currents encounter density cavities<ref>Singh, Nagendra "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2002GeoRL..29g..51S&db_key=AST&data_type=HTML&format=&high=42ca922c9c18563 Spontaneous formation of current-driven double layers in density depletions and its relevance to solitary Alfven waves]" (2002) ''Geophysical Research Letters'', Volume 29, Issue 7, pp. 51</ref>
*2003: By the incidence of plasma on the dark side of the Moon's surface<ref>Halekas, J. S.; Lin, R. P.; Mitchell, D. L. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2003GeoRL..30uPLA1H&db_key=AST&data_type=HTML&format=&high=42ca922c9c29136 Inferring the scale height of the lunar nightside double layer]" (2003) ''Geophysical Research Letters'', Volume 30, Issue 21, pp. PLA 1-1</ref>

== Features and characteristics of double layers ==
[[File:Double layer characteristics.png|thumb|300px|Double layer characteristics showing the potential (Φ), electric field (E) and space charge distribution (ρ) across the layer]]
{| class="wikitable" style="text-align:center; float:right; clear:right; margin-left:1em"
|+ Typical Double Layers
|-
! Location
! Typical Voltage drop
! Source
|-
| Ionosphere
| 102–104V
| Satellite
|-
| Solar
| 109–1011V
| Estimated<ref>Carlqvist, P. "[http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1982Ap%26SS..87...21C On the physics of relativistic double layers]" (1982) ''Astrophysics and Space Science'', vol. 87, no. 1–2, Oct. 1982, pp. 21–39</ref>
|-
| Neutron star
| 1015V
| Estimated<ref>Hamilton, R. J., Lamb, F. K., & Miller, M. C. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1994ApJS...90..837H Disk-accreting magnetic neutron stars as high-energy particle accelerators]" (1994) (p.939) ''Astrophysical Journal Supplement Series'' (ISSN 0067-0049), vol. 90, no. 2, pp. 837–839</ref>
|}
*'''Thickness''': The production of a double layer requires regions with a significant excess of positive or negative charge, that is, where [[Plasma (physics)#Potentials|quasi-neutrality]] is violated.<ref>Block, L. P. "[http://articles.adsabs.harvard.edu//full/seri/Ap%2BSS/0055//0000060.000.html A double layer review]" (1978) ''Astrophysics and Space Science'', vol. 55, no. 1, May 1978, p. 60.</ref><ref>Hasan, S. S.; Ter Haar, D. "[http://articles.adsabs.harvard.edu//full/seri/Ap%2BSS/0056//0000092.000.html The Alfven–Carlquist double-layer theory of solar flares]" (1978) ''Astrophysics and Space Science'', vol. 56, no. 1, June 1978, p. 92.</ref> In general, quasi-neutrality can only be violated on scales of the order of the [[Debye length]]. The thickness of a double layer is of the order of ten Debye lengths, which is a few centimeters in the [[ionosphere]], a few tens of meters in the [[interplanetary medium]], and tens of kilometers in the [[intergalactic medium]].
*'''Particle acceleration''': The potential drop across the double layer will accelerate electrons and positive ions in opposite directions. The magnitude of the potential drop determines the acceleration of the charged particles. In strong double layers, this will result in beams or jets of charged particles.
*'''Particle populations''': As described in the formation of double layers, there are four populations of charge particles inside a double layer. Note that in the case of weak double layers not all electrons and ions entering "from the wrong side" will be reflected, and therefore there will also be a population of decelerated electrons and ions.
*'''Particle flux''': For non-relativistic current carrying double layers the electrons comprise the main part of the particle flux. The Langmuir condition states that the ratio of the electron and the ion current across the layer is given by the square root of the mass ratio of the ions to the electrons.<ref>Block, L. P. "[http://articles.adsabs.harvard.edu//full/seri/Ap%2BSS/0055//0000060.000.html A double layer review]" (1978) ''Astrophysics and Space Science'', vol. 55, no. 1, May 1978, p. 65.</ref> For relativistic double layers the current ratio is 1; i.e. equal amounts of current are carried by the electrons and the ions.
*'''Energy supply''': In a certain limit, the voltage drop across a current-carrying double layer is proportional to the total current, and it might be thought of as a [[resistor|resistive]] element (or ''load'') which absorbs energy in an electric circuit. [[Anthony Peratt (physicist)|Anthony Peratt]] (1991) wrote: "Since the double layer acts as a load, there has to be an external source maintaining the potential difference and driving the current. In the laboratory this source is usually an electrical power supply, whereas in space it may be the magnetic energy stored in an extended current system, which responds to a change in current with an inductive voltage".<ref name="peratt1992">Peratt, Anthony L. [http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1992ppu..book.....P&db_key=AST&data_type=HTML&format=&high=42ca922c9c15428 ''Physics of the Plasma Universe''] (1992) Springer-Verlag</ref>
*'''Stability''': Double layers in laboratory plasmas may be stable or unstable depending on the parameter regime.<ref>Torven, S. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1982JPhD...15.1943T&db_key=PHY&data_type=HTML&format=&high=42ca922c9c12484 High-voltage double layers in a magnetised plasma column]" (1982) ''Journal of Physics D: Applied Physics'', Volume 15, Issue 10, pp. 1943–1949</ref> Various types of instabilities may occur, often arising due to the formation of [[Charged particle beam|beam]]s of ions and electrons. Unstable double layers are ''noisy'' in the sense that they produce oscillations across a wide frequency band. A lack of plasma stability may also lead to a dramatic change in configuration often referred to as an [[explosion]] (and hence ''exploding double layer''). In one example, the region enclosed in the double layer rapidly expands and evolves.<ref>B Song, N D Angelo and R L Merlino "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1992JPhD...25..938S&db_key=PHY&data_type=HTML&format=&high=42ca922c9c22502 Stability of a spherical double layer produced through ionization]" (1992) ''Journal of ''Physics D: Applied Physics'', Volume 25, Issue 6, pp. 938–941</ref> An [[explosion]] of this type was first discovered in [[mercury arc rectifier]]s used in high-power direct-current transmission lines, where the voltage drop across the device was seen to increase by several orders of magnitude. Double layers may also drift, usually in the direction of the emitted [[electron beam]], and in this respect are natural analogues to the smooth—bore [[magnetron]].<ref>Koenraad Mouthaan and Charles Süsskind, [http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=JAPIAU000037000007002598000001&idtype=cvips&gifs=yes Statistical Theory of Electron Transport in the Smooth-Bore Magnetron] (1966) ''Journal of Applied Physics'' June 1966, Volume 37, Issue 7, pp. 2598–2606</ref> (not to be confused with a unit of magnetic moment, the [[Bohr magneton]], which is created by the "classical circular motion" of an electron around a proton).
*'''Magnetised plasmas''': Double layers can both form in normal and magnetized plasmas.
*'''Cellular nature''': While double layers are relatively thin, they will spread over the entire cross surface of a laboratory container. Likewise where adjacent plasma regions have different properties, double layers will form and tend to cellularise the different regions.<ref>Alfven, H. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1982PhyS....2...10A&db_key=PHY&data_type=HTML&format=&high=42ca922c9c15283 Paradigm transition in cosmic plasma physics]" (1982) ''Physica Scripta'', vol. T2:1, 1982, pp. 10–19</ref>
*'''Energy transfer''': Double layers facilitate the transfer of electrical energy into kinetic energy, dW/dt=I·ΔV where I is the electric current dissipating energy into a double layer with a voltage drop of ΔV. Alfvén points out that the current may well consist exclusively of low-energy particles.<ref>Alfven, H.; Carlqvist, P. "[http://articles.adsabs.harvard.edu/cgi-bin/nph-iarticle_query?1978Ap%26SS..55..487A Interstellar clouds and the formation of stars]" (1978) ''Astrophysics and Space Science'', vol. 55, no. 2, May 1978, pp. 487–509</ref> Torvén ''et al.'' also report that plasma may spontaneously transfer magnetically stored energy into kinetic energy by electric double layers.<ref>S Torven, L Lindberg and R T Carpenter "[http://www.iop.org/EJ/abstract/0741-3335/27/2/005 Spontaneous transfer of magnetically stored energy to kinetic energy by electric double layers]" (1985) ''Plasma Phys. Control. Fusion'' 27 143–158</ref>
*'''Oblique double layer''': An oblique double layer has its electric field not parallel to the background magnetic field; i.e., it is not field-aligned.
*'''Simulation''': Double layers may be modelled using kinetic computer models like particle-in-cell (PIC) simulations. In some cases it is reasonable to treat the plasma as essentially one- or two-dimensional to reduce the computational cost of a simulation.
*'''Bohm Criterion''': A double layer cannot exist under all circumstances. In order to achieve that the electric field vanishes at the boundaries of the double layer, an existence criterion says that there is a maximum to the temperature of the ambient plasma. This is the so-called Bohm criterion.<ref>Raadu, Michael A.; Rasmussen, J. Juul "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1988Ap%26SS.144...43R&db_key=AST&data_type=HTML&format=&high=42ca922c9c10172 Dynamical aspects of electrostatic double layers]" (1988) ''Astrophysics and Space Science'' (ISSN 0004-640X), vol. 144, no. 1–2, May 1988, pp. 43–71.</ref> A mathematical description is given in the math section. In the theory of the [[Debye sheath]] there is a related but not identical condition also known as the [[Bohm criterion]].
*'''Bio-physical analogy''': A model of plasma double layers has been used to investigate their applicability to understanding ion transport across biological cell membranes.<ref>Gimmell, Jennifer ''et al.'' "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=2002APS..OSF.1P017G&db_key=PHY&data_type=HTML&format=&high=42ca922c9c29235 Bio-plasma physics: Measuring Ion Transport Across Cell membranes with Plasmas]" (2002) American Physical Society, Ohio Section Fall Meeting in conjunction with, abstract #1P.017</ref> Brazilian researchers have note that "Concepts like ''charge neutrality'', ''[[Debye length]]'', and ''double layer'' are very useful to explain the electrical properties of a [[cellular membrane]]."<ref>Mituo Uehara ''et al.'' "[http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000068000005000450000001&idtype=cvips&gifs=yes Physics and Biology: Bio-plasma physics]" ''American Journal of Physics'' May 2000 Volume 68, Issue 5, pp. 450–455</ref> Plasma physicist [[Hannes Alfvén]] also noted that association of double layers with cellular structure,<ref>Alfven, H. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1982STIN...8228234A&db_key=AST&data_type=HTML&format=&high=42ca922c9c17782 Cosmology in the plasma universe – an introductory exposition]" '''IEEE Transactions on Plasma Science''' (ISSN 0093-3813), vol. 18, Feb. 1990, pp. 5–10.</ref> as had [[Irving Langmuir]] before him, who coined the name "plasma" after its resemblance to blood cells.<ref>G. L. Rogoff, Ed., "Introduction", ''IEEE Transactions on Plasma Science'', vol. 19, p. 989, Dec. 1991. See extract on the [http://plasmacoalition.org/what.htm Plasma Coalition web site]</ref>

==History of double layers==
[[Image:Alfven-wave-double-layer.gif|thumb|300px|A cluster of double layers forming in an [[Alfvén wave]], about a sixth of the distance from the left. Click for more details]]
The research of these objects is a relatively young phenomenon. Although it was already known in the 1920s that a plasma has a limited capacity for current maintenance, [[Irving Langmuir]]<ref>Langmuir, Irving "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1929PhRv...33..954L&db_key=PHY&data_type=HTML&format=&high=42ca922c9c20559 The Interaction of Electron and Positive Ion Space Charges in Cathode Sheaths]" (1929) ''Physical Review'', vol. 33, Issue 6, pp. 954–989</ref> characterized double layers in the laboratory and called these structures double-sheaths. It was not until the 1950s that a thorough study of double layers started in the laboratory (e.g. Schönhuber, 1958).{{Citation needed|date=February 2007}} At the moment many groups are working on this topic theoretically, experimentally and numerically.
It was first proposed by [[Hannes Alfvén]] (the developer of magnetohydrodynamics) that the polar lights or Aurora Borealis are created by accelerated electrons in the magnetosphere of the Earth.<ref>Alfvén, H., "On the theory of magnetic storms and aurorae", ''Tellus'', 10, 104,. 1958.</ref> He supposed that the electrons were accelerated electrostatically by an electric field localized in a small volume bounded by two charged regions. This so-called double layer would accelerate electrons Earthwards. Many experiments with rockets and satellites have been performed to investigate the magnetosphere and acceleration regions. The first indication for the existence of electric field aligned along the magnetic field (or double layers) in the magnetosphere was by a rocket experiment by McIlwain (1960). Later, in 1977, [[Forrest Mozer]] reported that satellites had detected the signature of double layers (which he called electrostatic shocks) in the magnetosphere.<ref>Mozer, F. S. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1977PhRvL..38..292M&db_key=AST&data_type=HTML&format=&high=42ca922c9c26692 Observations of paired electrostatic shocks in the polar magnetosphere]" (1977) ''Physical Review Letters'', vol. 38, Feb. 7, 1977, pp. 292–295</ref>

The most definite proof of these structures was obtained by the Viking satellite,<ref>Boström, Rolf "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1992ITPS...20..756B&db_key=PHY&data_type=HTML&format=&high=42ca922c9c00530 Observations of weak double layers on auroral field lines]" (1992) IEEE ''Transactions on Plasma Science'' (ISSN 0093-3813), vol. 20, no. 6, pp. 756–763</ref> which measures the differential potential structures in the magnetosphere with probes mounted on 40m long booms. These probes can measure the local particle density and the potential difference between two points 80m apart. Asymmetric potential structures with respect to 0 V were measured, which means that the structure has a net potential and can be regarded as a double layer. The particle densities measured in such structures can be as low as 33% of the background density. The structures usually have an extent of 100 m (a few tens of Debye lengths). The filling factor of the lower magnetosphere with such solitary structures is about 10%. If one out of 5 such structures has a net potential drop of 1 V then the total potential drop over a region of 5000 km would be more than the 1 kV which is needed for the electrons to create the aurora. Magnetospheric double layers typically have a strength <math>e\phi_{DL}/k_B T_e \approx 0.1</math> (where the electron temperature is assumed to lie in the range (<math>2 eV \leq k_B T_e \leq 20 eV</math>) and are therefore weak.
The American [[Fast Auroral Snapshot Explorer|FAST]] spacecraft found strong double layers in the auroral acceleration region.<ref>Ergun, R. E., et al. "[http://adsabs.harvard.edu/abs/2002PhPl....9.3685E Parallel electric fields in the upward current region of the aurora: Indirect and direct observations]" (2002) ''Physics of Plasmas'', Volume 9, Issue 9, pp. 3685–3694</ref>
Strong Double layers were also found in the return current region by Andersson et al.<ref>Andersson, L., et al. "[http://adsabs.harvard.edu/abs/2002PhPl....9.3600A Characteristics of parallel electric fields in the downward current region of the aurora]" (2002) ''Physics of Plasmas'', Volume 9, Issue 8, pp. 3600–3609</ref> The return current region is where electrons move upward from the ionosphere to close the auroral current circuit.

In the laboratory, double layers can be created in different devices. They are investigated in double plasma machines, triple plasma machines, and [[Q-machine]]s. The stationary potential structures which can be measured in these machines agree very well with what one would expect theoretically. An example of a laboratory double layer can be seen in the figure below, taken from Torvén and Lindberg (1980), where we can see how well-defined and confined the potential drop of a double layer in a double plasma machine is.
One of the interesting things of the experiment by Torvén and Lindberg (1980)<ref>Torven, S.; Lindberg, L. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1980pfdl.rept.....T&db_key=PHY&data_type=HTML&format=&high=42ca922c9c17951 Properties of a fluctuating double layer in a magnetized plasma column]" (1982) ''Journal of Physics D: Applied Physics'', Volume 13, Issue 12, pp. 2285–2300</ref> is that not only did they measure the potential structure in the double plasma machine but they also found high-frequency fluctuating electric fields at the high-potential side of the double layer (also shown in the figure). These fluctuations are probably due to a beam-plasma interaction outside the double layer which excites plasma turbulence. Their observations are consistent with experiments on electromagnetic radiation emitted by double layers in a double plasma machine by Volwerk (1993),<ref>Volwerk, M. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1993JPhD...26.1192V&db_key=PHY&data_type=HTML&format=&high=42ca922c9c03565 Radiation from electrostatic double layers in laboratory plasmas]" (1993) ''Journal of Physics D: Applied Physics'', Volume 26, Issue 8, pp. 1192–1202</ref> who, however, also observed radiation from the double layer itself. The power of these fluctuations has a maximum around the plasma frequency of the ambient plasma.
It was later found that the electrostatic high-frequency fluctuations near the double layer can be concentrated in a narrow region, sometimes called the hf-spike,<ref>Gunell, H., et al. "[http://adsabs.harvard.edu/abs/1996JPhD...29..643G Bursts of high-frequency plasma waves at an electric double layer]" (1996) ''Journal of Physics D: Applied Physics'', Volume 29, Issue 3, pp. 643–654</ref> on the high potential side of the double layer.
Subsequently, both radio emissions, near the plasma frequency, and whistler waves at much lower frequencies were seen to emerge from this region.<ref>Brenning, N., et al. "[http://adsabs.harvard.edu/abs/2006JGRA..11111212B Radiation from an electron beam in a magnetized plasma: Whistler mode wave packets]" (2006) ''Journal of Geophysical Research, Volume 111'', Volume 111, Issue A11, CiteID A11212</ref>
Similar whistler wave structures were observed together with electron beams near Saturn's moon [[Enceladus (moon)|Enceladus]],<ref>Gurnett, D. A., et al. "[http://adsabs.harvard.edu/abs/2011GeoRL..3806102G Auroral hiss, electron beams and standing Alfvén wave currents near Saturn's moon Enceladus]" (2011) ''Geophysical Research Letters'', Volume 38, Issue 6, CiteID L06102</ref> suggesting the presence of a double layer at lower altitude.

A recent development in double layer experiments is the investigation of so-called stairstep double layers. It has been observed that a potential drop in a plasma column can be split up into different parts. Transitions from a single double layer into two-, three-, or greater-step double layers are strongly sensitive to the boundary conditions of the plasma (Hershkowitz, 1992).{{Citation needed|date=February 2007}} These experiments can give us information about the formation of the magnetospheric double layers and their possible role in creating the aurora.

Some scientists have subsequently suggested a role of double layers in solar flares.<ref>Hasan, S. S.; Ter Haar, D. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1978Ap%26SS..56...89H&db_key=AST&data_type=HTML&format=&high=42ca922c9c28323 The Alfven-Carlquist double-layer theory of solar flares]" (1978) ''Astrophysics and Space Science'' vol. 56, no. 1, June 1978, pp. 89–107</ref><ref>Khan, J. I. "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1989PASAu...8...29K&db_key=AST&data_type=HTML&format=&high=42ca922c9c28323 A model for solar flares invoking weak double layers]" (1989) ''Astronomical Society of Australia'', Proceedings (ISSN 0066-9997), vol. 8, no. 1, 1989, pp. 29–31</ref><ref>Volwerk, Martin; Kuijpers, Jan "[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?bibcode=1994ApJS...90..589V&db_key=AST&data_type=HTML&format=&high=42ca922c9c28323 Strong double layers, existence criteria, and annihilation: an application to solar flares]" (1994) ''Astrophysical Journal Supplement Series'' (ISSN 0067-0049), vol. 90, no. 2, pp. 589–593</ref>

== Mathematical description of a double layer ==
In this section we will take a closer look at the mathematics behind double layers. We first describe a semi-quantitative criterion for the formation of a density dip. We then describe a particularly simple kind of double layer. We then explain how to use the [[distribution function]] and the Vlasov-Poisson equation to model more-complex double layers.

=== Formation of a density dip ===
First we will take a look at the generation of a double layer in a current-carrying plasma. In 1968 Alfvén and Carlqvist showed that a density dip in a current carrying plasma can be favorable for the generation of a double layer. In this case we look at the plasma as a combination of two fluids, the moving electron fluid and the immobile ion fluid which acts as a neutralizing background. The electron fluid is treated as an essentially zero temperature beam and the ions are assumed to be collisional, and possess some finite temperature.

The density dip in the plasma (of both electrons and ions) will cause an electric field to be generated in order to keep the current density at the same level; i.e., electrons are accelerated in the decreasing part into the dip and decelerated in the increasing part out of the dip. However, this electric field will also have an influence on the first as immobile assumed ions. These ions will be driven out of the density dip, increasing it, and thereby increasing the electric field. When all ions are gone, the electric field has reached its maximum value over the dip. Note that we then have a double-double layer (increasing and decreasing electric field), and one side needs to be transported away.

We will use the quasi-static, non-relativistic description of this mechanism, which is governed by the [[continuity equation]] and the [[momentum]] equation:

:<math>\frac{\partial}{\partial x} (n_e v_e) = 0,</math>
:<math>m_e v_e \frac{\partial v_e}{\partial x} = - e E.</math>

Combining these two equations we get an expression for the electric field dependent on the electron density:
<math>E = - \frac{\partial}{\partial x} \left(\frac{m_e j_e^2}{2 n_e^2e^3}\right)</math>
where <math>j_e = -n_e e v_e</math> the electron current density. The ions will experience an outward force due to this electric field, with
<math>F = e n_i E = - \frac{\partial}{\partial x}(m_e n_e v_e^2)</math>.
Only when the force of the electric field can overcome the force by the ion [[pressure]] [[gradient]] can the evacuation of the density dip take place. Comparing the two forces (pressure and electric) assuming a quasi-neutral, [[thermal]] plasma shows after integration that this can only happen when <math>k_B T_i < m_e v_e^2</math>. This happens to be the Bohm criterion for the stability of a double layer (more below).

=== Current-carrying double layers formed by single, zero temperature beams ===
[[Image:Doublelayer.png|thumb|300px|right|The electron (red) and ion (blue) distributions in phase (velocity and position) space for the double layer described in this section. There are two beam-like (zero temperature) components for each species. The ion velocity is not to scale: it is exaggerated for the purposes of visualisation.]]
We consider how a single zero-temperature beam of ions and a single zero-temperature beam of electrons, together with a trapped, zero-velocity ion component, and a trapped, zero-velocity electron component, may form a particular class of double layer. The trapped components are referred to as the 'ambient plasma' and will later be allowed to have finite temperature.

[[Poisson's equation]] and the conservation of momentum and number density are used to analyse the structure of these double layers, in the 1D, time-independent limit. We are looking for double-layerlike solutions, where there is a well localised region with a potential gradient, outside of which the electric field is zero. The region can be divided into the interval inside the double layer, where there is only one ion component and one electron component, but there is a finite field, and the outside region, where the electric field is zero. For the moment, we need only consider the inside region and the densities and velocities associated with the beams inside the layer.

The electron beam component is streaming with positive velocity <math>v_{e}(x)</math> (to the right), and the ion beam is streaming with negative velocity <math>v_{i}(x)</math> (to the left). Here, the conservation of particle energy means that <math>v_{\alpha}^2(x)/2 m + q_{\alpha} \phi(x) </math> is a constant, and the conservation of particle number means that the current <math>j_{\alpha} = q n_{\alpha}(x) v_{\alpha}(x)</math> is also a constant.

:<math>|v_{\alpha}(\phi_{\alpha})| = \sqrt{v_{\alpha,0}^2 + \frac{2 e \phi_{\alpha}}{m_{\alpha}}},</math>
:<math>n_{\alpha}(\phi_{\alpha}) = n_0 v_{\alpha,0} v_{\alpha}^{-1}(\phi_{\alpha}),</math>

where <math>\phi_e = \phi</math> and <math>\phi_i = \phi_{DL}-\phi</math>. Here <math>n_0</math> and <math>v_{\alpha,0}</math> are respectively the electron (and ion) density and particle drift speed at the low (high) potential side of the double layer.

Now we use [[Poisson’s equation]] to obtain the maximal current through the double layer, as a function of the potential drop, the fraction of current carried by the ions as compared to the electrons and a temperature limit for the ''ambient plasma''. We chose <math>\phi(x=0) = 0</math> and <math>\phi(x=d) = \phi_{DL}</math>, with <math>d</math> the thickness of the double layer.

:<math>
\rho_e = \frac{j_e}{v_e},
\rho_i = \frac{j_i}{v_i}.
</math>

Thus we can write [[Poisson’s equation]] in the region inside the double layer as

:<math>- \frac{1}{4 \pi} \frac{\partial^2 \phi}{\partial x^2} = \frac{j_i}{ \sqrt{ v_{i,0}^2 + 2 e (\phi_{DL}-\phi)/m_i } } - \frac{j_e}{ \sqrt{v_{e,0}^2 + 2 e \phi / m_e } }.</math>

Introducing an integration factor <math>d\phi/dx</math> at both sides and integrating over <math>x</math> at the left hand side and over <math>\phi</math> on the right hand side the first integration leads to the square of the electric field <math>(d\phi/dx)^2</math>. The assumption that there is no electric field outside the double layer then leads to the ‘’Langmuir condition’’ for non-relativistic double layers:

:<math>\frac{j_e}{j_i} = \sqrt{\frac{m_i}{m_e}}.</math>

For this double layer (in a hydrogen plasma) the electron current dominates the ion current by a factor of <math>\sqrt{1836}</math>. (Note that for the same theory for ultra-relativistic double layers gives this fraction equal to 1). Further integration, as done by Raadu (1989), then leads to the ''Langmuir-Child'' relation:

:<math>j d^2 = (j_e + j_i) d^2 = \left( 1 + \sqrt{\frac{m_e}{m_i}}\right) \frac{4\varepsilon_0C_0}{9} \sqrt{\frac{2e}{m_e}} \phi_{DL}^{1.5},</math>

where <math>C_0</math> is expressed in terms of the elliptical integrals E and K:

:<math>C_0 = 2^{-1.5} (4 \sqrt{2} E ( \sin\pi/8) - (1 + 2\sqrt{2}K(\sin\pi/8))^2 \approx 1.86518.</math>

If we now allow the ambient plasma to be at finite temperature we have to take into account reflected particles more carefully and examine how far they can penetrate into the repulsive electric field. We describe the ambient plasma by a [[Boltzmann]] distribution over the double layer:

:<math>n_{\alpha,R} = n_{\alpha,0} e^{- \frac{e \phi_{\alpha}}{k_B T_{\alpha}}}.</math>

The densities of the reflected particles are now added to [[Poisson’s equation]]. In order that the particles in the 'ambient plasma' be truly trapped we require that their temperature be lower than the double layer potential. This can be seen in terms of the restriction that the potential and the electric field have to vanish at the boundaries of the double layer. The precise condition is known as the ‘’Bohm criterion’’:

:<math>m_e v_e^2 > k_B T_i.</math>

A double layer of this type cannot form if this criterion is not met. This is the same condition under which a double layer can be formed by an ion density dip (or equivalently, for instability to parallel wavemodes like the ion acoustic or Buneman instability) as discussed before.

=== The Vlasov–Poisson equation ===
In general the plasma distributions near a double layer are necessarily strongly non-[[Maxwell–Boltzmann distribution|Maxwellian]], and therefore inaccessible to fluid models. In order to analyse double layers in full generality, the plasma must be described using the particle [[distribution function]] <math>f_{\alpha}(\vec{x},t;\vec{v})</math>, which describes the number of particles of species <math>\alpha</math> having approximately the [[velocity]]<math>\vec{v}</math> near the place <math>\vec{x}</math> and time <math>t</math>.

The [[Vlasov equation#The Vlasov–Poisson equation|Vlasov-Poisson equations]] give the time-dependent interaction of a plasma (described using the particle distribution) with its self-consistent electric field. The Vlasov-Poisson equations are a combination of the [[Vlasov equation]] for each species <math>\alpha</math> (we take the nonrelativistic zero-magnetic field limit):

:<math>\frac{\partial}{\partial t} f_\alpha + \vec{v} \cdot \frac{\partial}{\partial \vec{x}} f_\alpha + \frac{q_\alpha \vec{E}}{m_\alpha} \cdot \frac{\partial}{\partial \vec{v}} f_\alpha = 0,</math>

and [[Poisson’s equation]] for self-consistent electric field:

:<math>\nabla \cdot \vec{E} = -\frac{\partial^2\phi}{\partial x^2} = 4 \pi \rho.</math>

Here <math>q_\alpha</math> is the particle’s electric charge, <math>m_\alpha</math> is the particle’s mass, <math>\vec{E}(\vec{x},t)</math> is the [[electric field]], <math>\phi(\vec{x}, t)</math> the [[electric potential]] and <math>\rho</math> is the [[electric charge]] density.

==See also==
*[[List of plasma (physics) articles]]

==Footnotes==

{{Reflist|colwidth=30em}}

== External links ==
*[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1978Ap&SS..55...59B A double layer review] (1978), L.P. Block
*[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1981Ap&SS..74..189R Electrostatic double layers and a plasma evacuation process] (1981), Raadu, M. A.; Carlqvist, P.
*[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1982Ap&SS..87...21C On the physics of relativistic double layers] (1982), Per Carlqvist
*[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1985IAUS..107..113S On the role of double layers in astrophysical plasmas] (1985), Smith, R. A.
*[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1988Ap&SS.144...43R Dynamical aspects of electrostatic double layers] (1988) M. A. Raadu, & J. J. Rasmussen
*[http://www.physics.uiowa.edu/~rmerlino/fildl.pdf Filamentary Double Layers] (PDF, 1993), W.L. Theisen, R.T.Carpenter, R.L.Merlino
*[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1994SSRv...68...29R Energy release in double layers] (1985), Raadu, M. A.
*[http://adsabs.harvard.edu/cgi-bin/nph-bib_query?1988Ap&SS.144..149H Parallel electric fields accelerating ions and electrons in the same direction] (1988), Hultqvist, Bengt; Lundin, Rickard
*[http://sprg.ssl.berkeley.edu/adminstuff/webpubs/2002_pop_3695.pdf Parallel electric fields in the upward current region of the aurora: Numerical solutions] (2002, PDF), R.W. Ergun, et al.
* [http://prl.anu.edu.au/Members/mdl112/thesis.pdf Numerical modeling of low-pressure plasmas: applications to electric double layers] (2006, PDF), A. Meige, PhD thesis

== References ==

*Alfvén, H., ''On the theory of magnetic storms and aurorae'', Tellus, 10, 104, 1958.
*Peratt, A., ''Physics of the Plasma Universe'', 1991
* Raadu, M.,A., ''The physics of double layers and their role in astrophysics'', Physics Reports, 178, 25–97, 1989.

[[Category:Plasma physics]]
[[Category:Space plasmas]]

[[ja:電気二重層]]

Minimum degree spanning tree

2013-03-08T16:40:44Z

130.83.2.27:

[[Image:Rounding & sphericity EN.svg|thumb|300px|Schematic representation of difference in grain shape. Two parameters are shown: sphericity (vertical) and [[rounding (sediment)|rounding]] (horizontal).]]
'''Sphericity''' is a measure of how spherical (round) an object is. As such, it is a specific example of a [[compactness measure of a shape]]. Defined by Wadell in 1935,<ref>{{cite journal |first=Hakon |last=Wadell |title=Volume, Shape and Roundness of Quartz Particles |journal=Journal of Geology |volume=43 |year=1935 |pages=250–280 |doi=10.1086/624298 |issue=3 }}</ref> the sphericity, <math>\Psi </math>, of a particle is: the ratio of the [[surface area]] of a [[sphere]] (with the same [[volume]] as the given particle) to the surface area of the particle:

:<math>\Psi = \frac{\pi^{\frac{1}{3}}(6V_p)^{\frac{2}{3}}}{A_p}
</math>

where <math>V_p</math> is volume of the particle and <math>A_p</math> is the surface area of the particle. The sphericity of a sphere is 1 and, by the [[isoperimetric inequality]], any particle which is not a sphere will have sphericity less than 1.

== Ellipsoidal objects ==
{{See_also|Earth radius}}
The sphericity, <math>\Psi </math>, of an [[oblate spheroid]] (similar to the shape of the planet [[Earth]]) is:

:<math>\Psi =
\frac{\pi^{\frac{1}{3}}(6V_p)^{\frac{2}{3}}}{A_p} =
\frac{2\sqrt[3]{ab^2}}{a+\frac{b^2}{\sqrt{a^2-b^2}}\ln{(\frac{a+\sqrt{a^2-b^2}}b)}},
</math> 

where ''a'' and ''b'' are the [[Semi-major axis|semi-major]] and [[semi-minor axis|semi-minor]] axes respectively.

== Derivation ==

Hakon Wadell defined sphericity as the surface area of a
sphere of the same volume as the particle divided by the actual surface area of the particle.

First we need to write surface area of the sphere, <math>A_s</math> in terms of the volume of the particle, <math>V_p</math>

:<math>A_{s}^3 = \left(4 \pi r^2\right)^3 = 4^3 \pi^3 r^6 = 4 \pi \left(4^2 \pi^2 r^6\right) = 4 \pi \cdot 3^2 \left(\frac{4^2 \pi^2}{3^2} r^6\right) = 36 \pi \left(\frac{4 \pi}{3} r^3\right)^2 = 36\,\pi V_{p}^2
</math>

therefore

:<math>A_{s} = \left(36\,\pi V_{p}^2\right)^{\frac{1}{3}} = 36^{\frac{1}{3}} \pi^{\frac{1}{3}} V_{p}^{\frac{2}{3}} = 6^{\frac{2}{3}} \pi^{\frac{1}{3}} V_{p}^{\frac{2}{3}} = \pi^{\frac{1}{3}} \left(6V_{p}\right)^{\frac{2}{3}}
</math>

hence we define <math>\Psi</math> as:

:<math>
\Psi = \frac{A_s}{A_p} = \frac{ \pi^{\frac{1}{3}} \left(6V_{p}\right)^{\frac{2}{3}} }{A_{p}}
</math>

== Sphericity of common objects ==
{| border="1" cellpadding="7" style="margin:0 auto; text-align:center; border-collapse: collapse;"
|-
!Name
!Picture
!Volume
!Area
!Sphericity
|-
| colspan=5 align=left|'''[[Platonic Solids]]'''
|-
|[[tetrahedron]]
| [[Image:tetrahedron.jpg|50px|Tetrahedron]]
| <math>\frac{\sqrt{2}}{12}\,s^3</math> || <math>\sqrt{3}\,s^2</math> || <math>\left(\frac{\pi}{6\sqrt{3}}\right)^{\frac{1}{3}} \approx 0.671</math>
|-
|[[cube]] (hexahedron)
|[[Image:hexahedron.jpg|50px|Hexahedron (cube)]]
| <math>\,s^3</math> || <math>6\,s^2</math> ||
<math>\left(
\frac{\pi}{6}
\right)^{\frac{1}{3}} \approx 0.806</math>
|-
|[[octahedron]]
|[[Image:octahedron.svg|50px|Octahedron]]
| <math> \frac{1}{3} \sqrt{2}\, s^3</math> || <math> 2 \sqrt{3}\, s^2</math> ||
<math>\left(
\frac{\pi}{3\sqrt{3}}
\right)^{\frac{1}{3}} \approx 0.846 </math>
|-
|[[dodecahedron]]
| [[Image:POV-Ray-Dodecahedron.svg|50px|Dodecahedron]]
| <math> \frac{1}{4} \left(15 + 7\sqrt{5}\right)\, s^3</math> || <math> 3 \sqrt{25 + 10\sqrt{5}}\, s^2</math> ||
<math>\left(
\frac{\left(15 + 7\sqrt{5}\right)^2 \pi}{12\left(25+10\sqrt{5}\right)^{\frac{3}{2}}}
\right)^{\frac{1}{3}} \approx 0.910</math>
|-
|[[icosahedron]]
| [[Image:icosahedron.jpg|50px|Icosahedron]]
| <math>\frac{5}{12}\left(3+\sqrt{5}\right)\, s^3</math> || <math>5\sqrt{3}\,s^2</math> || <math>\left(
\frac{ \left(3 + \sqrt{5} \right)^2 \pi}{60\sqrt{3}}
\right)^{\frac{1}{3}} \approx 0.939</math>
|-
| colspan=5 align=left|'''Round Shapes'''
|-
|ideal [[cone (geometry)|cone]] <math>(h=2\sqrt{2}r)</math>
|| || <math>\frac{1}{3} \pi\, r^2 h </math> 
<math>= \frac{2\sqrt{2}}{3} \pi\, r^3</math>
|| <math>\pi\, r (r + \sqrt{r^2 + h^2}) </math> 
<math>= 4 \pi\, r^2 </math>
|| <math>\left(
\frac{1}{2}
\right)^{\frac{1}{3}} \approx 0.794</math>
|-
|hemisphere (half sphere)
|
| <math>\frac{2}{3} \pi\, r^3</math> || <math>3 \pi\, r^2</math> ||
<math>\left(
\frac{16}{27}
\right)^{\frac{1}{3}} \approx 0.840</math>
|-
|ideal [[cylinder (geometry)|cylinder]] <math>(h=2\,r)</math>
|
| <math>\pi r^2 h = 2 \pi\,r^3</math> || <math>2 \pi r ( r + h ) = 6 \pi\,r^2</math> ||
<math>\left(
\frac{2}{3}
\right)^{\frac{1}{3}} \approx 0.874</math>
|-
|ideal [[torus]] <math>(R=r)</math>
|
| <math>2 \pi^2 R r^2 = 2 \pi^2 \,r^3</math> || <math>4 \pi^2 R r = 4 \pi^2\,r^2</math> ||
<math>\left(
\frac{9}{4 \pi}
\right)^{\frac{1}{3}} \approx 0.894</math>
|-
|[[sphere]]
|
| <math>\frac{4}{3} \pi r^3</math> || <math>4 \pi\,r^2</math> ||
<math>
1\,</math>
|}

==Sphericity in statistics==
In statistical analyses, sphericity relates to the equality of the [[variance]]s of the differences between levels of the repeated measures factor. Sphericity requires that the variances for each set of difference scores are equal. A sufficient (but not necessary) condition for sphericity is that the variances of the sets of scores are equal and the covariances of the sets of scores are equal. This is an assumption of an [[Analysis of variance|ANOVA]] with a repeated measures factor, where violations of this assumption can invalidate the analysis conclusions. [[Mauchly's sphericity test]] is one of the [[Statistics|statistical tests]] used to evaluate sphericity.

== See also ==
*[[Rounding (sediment)]]
*[[Sphericity scale]]

==References==
{{Reflist}}

== External links ==
* [http://www.howround.com/ How round is your circle?]
*[http://people.uncw.edu/dockal/gly312/grains/grains.htm Grain Morphology: Roundness, Surface Features, and Sphericity of Grains]

[[Category:Geometric measurement]]
[[Category:Spheres]]
[[Category:Metalworking terminology]]
[[Category:Metrology]]

Recursively inseparable sets

2012-07-19T15:25:11Z

130.83.2.200:

You need to read a newspaper regularly, you will see the how professional write their essay. People who work in custom writing industry are usually professional freelance writers with undergraduate or graduate degree and some years of experience in academic writing. There are many starters that miss the kind of ability it s essential submit good essays that could represent their own undertaking impressively and convincingly to trainers. Precision and depth are definitely the all-important elements here. s students are taught that how to write an effective essay but the problem is this that when they reach to the final stage I mean to say that when they are up to the age of writing an essay then the problem comes that they are not enough capable to write it more perfectly as they want to get the perfect marks. They subsequently look to to assist them with their writing assignments. We have been writing critical essays for a while and the experience has gotten our writers get used to writing these essays. If you practice this regularly, then one day surely you will improve yourself. s program is to show exact knowledge, then exact title is compulsory for choice and if they are eager to create in detail work with comprehensive overview, in this situation extensive theme is greatest selection for the title selection. Students use academic writing services because such services are extremely helpful to them in their studies. Therefore, we try to balance between quality and prices. Descriptive essays, as the name indicate are used to provide a vivid description of a person, place or thing. However, [stillneverthelesseven nownonetheless] the majority of students encounter the [problemsdifficultiesissues] in relation [for yourfor theto yourfor ones] essay writing. This is to ensure writers who provide you with services are qualified. Spelling is something that students tend to overlook in their rush to complete their essays. This way will assist as it is so complicated to detect mistakes in your work. Do they specify that the work needs to be double-spaced. There is much to consider what to write and when you have to write, making certain that your content is applicable, following correct style and using correct grammar, and finally making sure to submit it by the deadline. essay against online material and a database of previously submitted papers. It does not have to be a formal outline, but just sketch out the order of your essay in some method that you understand. If you liked this article and you would like to receive far more facts pertaining to [http://www.freshwidewallpapers.com/profile/elwid buy literary analysis essay] kindly pay a visit to the web-page. By the time [of thefrom thewith theon thein the] course [jobswork] accomplishment, the students [usuallygenerallytypicallynormallycommonly] are supposed [to getto obtainto haveto acquirefor getting] [requiredneedednecessary] essay writing skills, [as well asin addition toalong with] creative [methodtechniqueapproachprocedureprocess] [in thewithin theinside the] realm of academic writing. Academic essays are written from a personal standpoint. You cannot write long and long sentence giving no meaning at all, instead you can write one that explains everything, this will save a lot of time of your reader and your as well. This ensures that you know where the piece is going and do not deviate too far from the primary topic. That's enough words to make even the most attentive reader snore.

Lippmann–Schwinger equation

2012-06-13T12:10:46Z

130.83.36.104: /* Creating wavepackets */

The name of the author is Figures. California is exactly where her house is but she requirements to move simply because of her family. I am a meter reader. To collect coins is 1 of the issues I love most. Also visit my blog: [http://www.neweracinema.com/tube/blog/181510 home std test kit]

Gullstrand–Painlevé coordinates

2011-03-03T15:24:27Z

130.83.36.104: /* Free-float coordinates */

Hi there, I am Alyson Pomerleau and I believe it sounds fairly great when you say it. Her family members lives in Ohio. To perform lacross is the factor I love psychic readings ([http://kard.dk/?p=24252 http://kard.dk]) most of all. Distributing production has been his profession for some time.