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Black-body radiation
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(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.)- the content below is remote from Wikipedia
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missing image!
- PlanckianLocus.png -
The color (chromaticity) of black-body radiation depends on reverse the temperature of the black body; the locus of such colors, shown here in CIE 1931 x,y space, is known as the Planckian locus.
Black-body radiation is the thermal electromagnetic radiation within or surrounding a body in thermodynamic equilibrium with its environment, emitted by a black body (an idealized opaque, non-reflective body). It has a specific spectrum and reverse intensity that depends only on the body's temperature, which is assumed for the sake of calculations and theory to be uniform and constant.{{harvnb|Loudon|2000}}, Chapter 1.{{harvnb|Mandel|Wolf|1995}}, Chapter 13.{{harvnb|Kondepudi|Prigogine|1998}}, Chapter 11.{{harvnb|Landsberg|1990}}, Chapter 13.The thermal radiation spontaneously emitted by many ordinary objects can be approximated as black-body 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 in the infrared spectrum and cannot be perceived by the human eye. Since, by definition, the human eye cannot perceive light waves below the visible frequency, a black body, viewed in the dark at the lowest just faintly visible temperature, subjectively appears grey, even though its objective physical spectrum peak is in the infrared range.Partington, J.R. (1949), p. 466. When it becomes a little hotter, it appears dull red. As its temperature increases further it becomes yellow, white, and ultimately blue-white.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.BOOK, Introduction to Astronomy and Cosmology, Ian Morison,weblink 48, 0-470-03333-9, 2008, J Wiley & Sons, Black holes are near-perfect black bodies, in the sense that they absorb all the radiation that falls on them. It has been proposed that they emit black-body radiation (called Hawking radiation), with a temperature that depends on the mass of the black hole.BOOK, Modeling black hole evaporation,weblink Chapter 1: Introduction, Alessandro Fabbri, JosÃ© Navarro-Salas, 1-86094-527-9, 2005, Imperial College Press, The term black body was introduced by Gustav Kirchhoff in 1860.From (Kirchhoff, 1860) (Annalen der Physik und Chemie), p. 277: "Der Beweis, welcher fÃ¼r die ausgesprochene Behauptung hier gegeben werden soll, â€¦ vollkommen schwarze, oder kÃ¼rzer schwarze, nennen." (The proof, which shall be given here for the proposition stated [above], rests on the assumption that bodies are conceivable which in the case of infinitely small thicknesses, completely absorb all rays that fall on them, thus [they] neither reflect nor transmit rays. I will call such bodies "completely black [bodies]" or more briefly "black [bodies]".) See also (Kirchhoff, 1860) (Philosophical Magazine), p. 2. Black-body radiation is also called thermal radiation, cavity radiation, complete radiation or temperature radiation.- PlanckianLocus.png -
The color (chromaticity) of black-body radiation depends on reverse the temperature of the black body; the locus of such colors, shown here in CIE 1931 x,y space, is known as the Planckian locus.
Theory
Spectrum
Black-body radiation has a characteristic, continuous frequency spectrum that depends only on the body's temperature,BOOK,weblink 41, Â§2.3: Thermodynamic equilibrium and black-body radiation, The astrophysics of emission-line stars, Tomokazu Kogure, Kam-Ching Leung, 0-387-34500-0, 2007, Springer,
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.Wien, W. (1893). Eine neue Beziehung der Strahlung schwarzer KÃ¶rper zum zweiten Hauptsatz der WÃ¤rmetheorie, Sitzungberichte der KÃ¶niglich-PreuÃŸischen Akademie der Wissenschaften (Berlin), 1893, 1: 55â€“62.Lummer, O., Pringsheim, E. (1899). Die Vertheilung der Energie im Spectrum des schwarzen KÃ¶rpers, Verhandlungen der Deutschen Physikalischen Gessellschaft (Leipzig), 1899, 1: 23â€“41.{{harvnb|Planck|1914}} As the temperature increases past about 500 degrees Celsius, black bodies start to emit significant amounts of visible light. Viewed in the dark by the human eye, the first faint glow appears as a "ghostly" grey (the visible light is actually red, but low intensity light activates only the eye's grey-level sensors). 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.Draper, J.W. (1847). On the production of light by heat, London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, series 3, 30: 345â€“360. weblink{{harvnb|Partington|1949|pages = 466â€“467, 478}}. 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,{{harvnb|Goody|Yung|1989|pages=482, 484}} is an approximate 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.Black body
All normal (baryonic) matter emits electromagnetic radiation when it has a temperature above absolute zero. The radiation represents a conversion of a body's internal energy into electromagnetic energy, and is therefore called thermal radiation. It is a spontaneous process of radiative distribution of missing image!- Color temperature black body 800-12200K.svg">thumb|512px|Color of a black body from 800 K to 12200 K. This range of colors approximates the range of colors of stars of different temperatures, as seen or photographed in the night sky.Conversely all normal matter absorbs electromagnetic radiation to some degree. An object that absorbs all radiation falling on it, at all wavelengths, 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.{{harvnb|Planck|1914|page=42}} Graphite and 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. 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.{{harvnb|Wien|1894}}{{harvnb|Planck|1914|page=43}}Black-body radiation has the unique absolutely stable distribution of radiative intensity that can persist in thermodynamic equilibrium in a cavity. 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.BOOK,weblink 107, Â§4.2.2: Calculation of Planck's law, Passive infrared detection: theory and applications, Joseph Caniou, 0-7923-8532-2, 1999, Springer, For a black body (a perfect absorber) there is no reflected radiation, and so the spectral radiance is entirely due 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 {{val|798|u=K}}.BOOK, Radiation heat transfer: a statistical approach, J. R. Mahan, 3rd, Wiley-IEEE, 2002, 978-0-471-21270-6, 58,weblink At {{val|1000|u=K}}, a small opening in the wall of a large uniformly heated opaque-walled cavity (such as an oven), viewed from outside, looks red; at {{val|6000|u=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.Pahoehoe toe.jpg -
Further explanation
According to the Classical Theory of 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 would imply infinite heat capacity , as well as an nonphysical spectrum of emitted radiation that grows without bound with increasing frequency, a problem known as the ultraviolet catastrophe.In the longer wavelengths this deviation is not so noticeable, as h nu and nh nu are very small. In the shorter wavelengths of the ultraviolet range, however, classical theory predicts the energy emitted tends to infinity, hence the ultraviolet catastrophe. As all possible vibrational modes (including those whose energy less than h nu-The quantum of energy), the energy summed to infinity. The theory even predicted that all bodies would emit most of their energy in the ultraviolet range, clearly contradicted by the experimental data which showed a different peak wavelength at different temperatures. (see also wiens law) (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.)Instead, in the quantum treatment of this problem ,the numbers of the energy modes are quantized, attenuating the spectrum at high frequency in agreement with experimental observation and resolving the catastrophe. The modes that had more energy than the thermal energy of the substance itself were not considered , and by quantization - modes having infinitesimally little energy were excluded. Thus for shorter wavelengths very few modes(having energy more than h nu) were allowed, supporting the data that the energy emitted is reduced for wavelengths less than the wavelength of the observed peak of emission.Notice that there are two factors responsible for the shape of the graph. Firstly, longer wavelengths have a larger number of modes associated with them. Secondly, shorter wavelengths have more energy associated per mode. The two factors combined give the characteristic maximum wavelength .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.JOURNAL, Planck, Max, Max_Planck, Ueber das Gesetz der Energieverteilung im Normalspectrum, On the law of the distribution of energy in the normal spectrum, Annalen der Physik, 4th series, 4, 3, 553â€“563, 1901,weblink German, 10.1002/andp.19013090310, 1901AnP...309..553P, 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. 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 photons and the black-body cavity was thought of as containing a gas of photons. In addition, it led to the development of quantum probability distributions, called Fermiâ€“Dirac statistics and Boseâ€“Einstein statistics, each applicable to a different class of particles, fermions and bosons.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 Stefanâ€“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 invisibleâ€”indeed, the radiation of visible light increases monotonically with temperature.BOOK, Landau, L. D., E. M. Lifshitz, Statistical Physics, 3rd Edition Part 1, 1996, Butterworthâ€“Heinemann, Oxford, 0-521-65314-2, The Stefanâ€“Boltzmann law also says that the total radiant heat energy emitted from a surface is proportional to the fourth power of its absolute temperature. The law was formulated by Josef Stefan in 1879 and later derived by Ludwig Boltzmann. The formula {{nowrap|1=E = ÏƒT4}} is given, where E is the radiant heat emitted from a unit of area per unit time, T is the absolute temperature, and {{nowrap|1=Ïƒ = {{val|5.670367|e=-8|u=WÂ·mâˆ’2â‹…Kâˆ’4}}}} is the Stefanâ€“Boltzmann constant.ENCYCLOPEDIA, Stefan-Boltzmann law, EncyclopÃ¦dia Britannica,weblink 2019,Equations
Planck's law of black-body radiation
Planck's law states that{{harvnb|Rybicki|Lightman|1979|p=22}}
B_nu(nu, T) = frac{2hnu^3}{c^2}frac{1}{e^{hnu/kT} - 1},
where
BÎ½(T) is the spectral radiance (the power per unit solid angle and per unit of area normal to the propagation) density of frequency Î½ radiation per unit frequency at thermal equilibrium at temperature T.
h is the Planck constant;
c is the speed of light in a vacuum;
k is the Boltzmann constant;
Î½ is the frequency of the electromagnetic radiation;
T is the absolute temperature of the body.
For a black body surface the spectral radiance density (defined per unit of area normal to the propagation) is independent of the angle theta of emission with respect to the normal. However, this means that, following Lambert's cosine law, B_nu(T) cos theta is the radiance density per unit area of emitting surface as the surface area involved in generating the radiance is increased by a factor 1/cos theta with respect to an area normal to the propagation direction. At oblique angles, the solid angle spans involved do get smaller, resulting in lower aggregate intensities.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, lambda_max, is a function only of the temperature:
lambda_max = frac{b}{T},
where the constant b, known as Wien's displacement constant, is equal to {{val|2.897771955|e=-3|u=m K}}.WEB, Wien wavelength displacement law constant,weblink The NIST Reference on Constants, Units, and Uncertainty, NIST, February 8, 2019, Planck's law was also stated above as a function of frequency. The intensity maximum for this is given by
nu_max = T times 5.879 times 10^{10} mathrm{Hz}/mathrm{K}.
WEB
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, Dr. Rod
, Wien's Displacement Law and Other Ways to Characterize the Peak of Blackbody Radiation
, HyperPhysics
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Provides 5 variations of Wien's displacement law, Dr. Rod
, Wien's Displacement Law and Other Ways to Characterize the Peak of Blackbody Radiation
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Stefanâ€“Boltzmann law
By integrating B_nu(T) over the frequency the integrated radiance L is
L=frac{2pi^5}{15} frac{k^4 T^4}{c^2h^3} frac{1}{pi}=:sigma T^4 frac{1}{pi}
by using int_0^infty dx, frac{x^3}{e^x - 1}=frac{pi^4}{15} with x equiv frac{hnu}{k T} and with sigma equiv frac{2pi^5}{15} frac{k^4}{c^2h^3}=5.670373 times 10^{-8} frac{W}{m^2 K^4} being the Stefanâ€“Boltzmann constant. The radiance L is then
sigma T^4 frac{cos theta}{pi}
per unit of emitting surface.On a side note, at a distance d, the intensity dI per area dA of radiating surface is the useful expression
dI=sigma T^4 frac{costheta}{pi d^2}dA
when the receiving surface is perpendicular to the radiation.By subsequently integrating over the solid angle Omega (where theta- content above as imported from Wikipedia
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