**Black-body radiation** is the thermal 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). It has a specific spectrum and intensity that depends only on the body's temperature, which is assumed for the sake of calculations and theory to be uniform and constant.^{[1]}^{[2]}^{[3]}^{[4]}

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 infra-red and cannot be perceived by the human eye. Because the human eye cannot perceive light waves at lower frequencies, 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.^{[5]} 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.^{[6]} 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.^{[7]}

The term *black body* was introduced by Gustav Kirchhoff in 1860.^{[8]} Black-body radiation is also called thermal radiation, *cavity radiation*, *complete radiation* or *temperature radiation*.

Black-body radiation has a characteristic, continuous frequency spectrum that depends only on the body's temperature,^{[9]} 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.^{[10]}^{[11]}^{[12]} 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.^{[13]}^{[14]} 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,^{[15]} 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.

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 entropy.

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.^{[16]} 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.^{[16]} 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.^{[17]}^{[18]}

Black-body radiation has the unique absolutely stable distribution of radiative intensity that can persist in thermodynamic equilibrium in a cavity.^{[16]} 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.^{[19]} 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 798 K.^{[20]} 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.

Two bodies that are at the same temperature stay in mutual 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.^{[21]}^{[22]} 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*.^{[23]}

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.^{[24]} 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 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 temperature and the fact that the walls are opaque and at least partly absorptive, but not on the particular material of which they are built nor on the material in the cavity (compare with emission spectrum).

The radiance or observed intensity is not a function of direction. Therefore, a black body is a perfect 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.

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 local thermodynamic equilibrium still follow 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 stars 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 holes, 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.

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 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.

In the longer wavelengths this effect is not so noticeable (As hv is very small, allowing nhv to be almost infinitesimally small and thus a very large number of vibrational modes. But in the shorter wavelengths the classical theory predicted the energy emitted tended to Infinity (In the ultraviolet range; hence ultraviolet catastrophe). As all possible vibrational modes including those having energy less than hv were considered, the energy added up to infinity. It even predicted that all bodies would emit maximum energy in the ultraviolet range, clearly against the experimental data which showed a different peak wavelength at different temperatures.

Instead, in quantum theory the numbers of the modes are quantized, cutting off the spectrum at high frequency in agreement with experimental observation and resolving the catastrophe. The modes definitely cannot have more energy than the thermal energy of the substance itself, and by quantization infinitesimally small modes too were not allowed. Thus for shorter wavelengths very few modes were allowed, supporting the data that energy emitted reduces for wavelength shorter 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 larger number of modes associated with them. Secondly, shorter wavelengths have more energy associated per mode. The study of the laws of black bodies and the failure of classical physics to describe them helped establish the foundations of quantum mechanics.

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.^{[27]} 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 Fermi–Dirac statistics and Bose–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 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 monotonically with temperature.^{[28]} The Stefan–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 *E* = *σT*^{4} is given, where *E* is the radiant heat emitted from a unit of area per unit time, *T* is the absolute temperature, and *σ* = 5.670367×10^{−8} W·m^{−2}⋅K^{−4} is the Stefan–Boltzmann constant.^{[29]}

Planck's law states that^{[30]}

where

- B
(_{ν}*T*) is the spectral radiance (the power per unit solid angle and per unit of area normal to the propagation) density of frequency*ν*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;*ν*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 of emission with respect to the normal. However, this means that, following Lambert's cosine law, is the radiance density per unit area of emitting surface as the surface area involved in generating the radiance is increased by a factor 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 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, , is a function only of the temperature:

where the constant *b*, known as Wien's displacement constant, is equal to 2.8977729(17)×10^{−3} K m.^{[31]}

Planck's law was also stated above as a function of frequency. The intensity maximum for this is given by

- .
^{[32]}

By integrating over the frequency the integrated radiance is

by using with and with being the Stefan–Boltzmann constant. The radiance is then

per unit of emitting surface.

On a side note, at a distance d, the intensity per area of radiating surface is the useful expression

when the receiving surface is perpendicular to the radiation.

By subsequently integrating over the solid angle (where ) the Stefan–Boltzmann law is calculated, stating that the power *j** emitted per unit area of the surface of a black body is directly proportional to the fourth power of its absolute temperature:

by using

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 the darkness of the man's glasses. |

The human body radiates energy as infrared light. The net power radiated is the difference between the power emitted and the power absorbed:

Applying the Stefan–Boltzmann law,

where *A* and *T* are the body surface area and temperature, is the emissivity, and *T*_{0} is the ambient temperature.

The total surface area of an adult is about 2 m^{2}, and the mid- and far-infrared emissivity of skin and most clothing is near unity, as it is for most nonmetallic surfaces.^{[33]}^{[34]} Skin temperature is about 33 °C,^{[35]} but clothing reduces the surface temperature to about 28 °C when the ambient temperature is 20 °C.^{[36]} Hence, the net radiative heat loss is about

The total energy radiated in one day is about 8 MJ, or 2000 kcal (food calories). Basal metabolic rate for a 40-year-old male is about 35 kcal/(m^{2}·h),^{[37]} which is equivalent to 1700 kcal per day, assuming the same 2 m^{2} area. However, the mean metabolic rate of sedentary adults is about 50% to 70% greater than their basal rate.^{[38]}

There are other important thermal loss mechanisms, including convection and evaporation. Conduction is negligible – the Nusselt number is much greater than unity. Evaporation by perspiration is only required if radiation and convection are insufficient to maintain a steady-state temperature (but evaporation from the lungs occurs regardless). Free-convection rates are comparable, albeit somewhat lower, than radiative rates.^{[39]} 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 law to human-body emission results in a peak wavelength of

For this reason, thermal imaging devices for human subjects are most sensitive in the 7–14 micrometer range.

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

The temperature of a planet depends on several factors:

- Incident radiation from its star
- Emitted radiation of the planet, e.g., 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 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 (energy/second) the Sun is emitting:

where

- is the Stefan–Boltzmann constant,
- is the effective temperature of the Sun, and
- 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:

where

- is the radius of the planet and
- is 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 of this energy where is the albedo or reflectance of the planet in the UV-Vis range. In other words, the planet absorbs a fraction of the Sun's light, and reflects the rest. The power absorbed by the planet and its atmosphere is then:

Even though the planet only absorbs as a circular area , 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

where is the temperature of the planet. This temperature, calculated for the case of the planet acting as a black body by setting , 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 of the radiation that a black body would emit where is the average emissivity in the IR range. The power emitted by the planet is then:

For a body in radiative exchange equilibrium with its surroundings, the rate at which it emits radiant energy is equal to the rate at which it absorbs it:^{[40]}^{[41]}

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}:

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.

Notice that a gray (flat spectrum) ball where comes to the same temperature as a black body no matter how dark or light gray .

Substituting the measured values for the Sun and Earth yields:

^{[42]}^{[42]}^{[42]}^{[43]}

With the average emissivity set to unity, the effective temperature of the Earth is:

or −18.8 °C.

This is the temperature of the Earth if it radiated as a perfect black body in the infrared, assuming an unchanging albedo and 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^{[44]}). 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^{[45]} and 0.95^{[46]} 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.^{[47]}
Similarly using albedo 0.3 and solar constant of 1372 W m^{−2}, one obtains an effective temperature of 255 K.^{[48]}^{[49]}^{[50]}

The cosmic microwave background radiation observed today is the most perfect black-body radiation ever observed in nature, with a temperature of about 2.7 K.^{[51]} It is a "snapshot" of the radiation at the time of 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, though not full thermodynamic, equilibrium with radiation.

According to Kondepudi and Prigogine, at very high temperatures (above 10^{10} K; 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 spontaneously and are in thermal equilibrium with electromagnetic radiation. These particles form a part of the black body spectrum, in addition to the electromagnetic radiation.^{[52]}

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'*:

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.^{[53]} 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

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.

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.^{[23]} Stewart chose lamp-black surfaces as his reference because of various previous experimental findings, especially those of Pierre Prevost and of 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.^{[54]} 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.

In 1859, not knowing of Stewart's work, 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.^{[55]}

Kirchhoff then went on to consider some bodies that emit and absorb heat radiation, in an opaque enclosure or cavity, in equilibrium at temperature *T*.

Here is used a notation different from Kirchhoff's. Here, the emitting power *E*(*T*, *i*) denotes a dimensioned quantity, the total radiation emitted by a body labeled by index *i* at temperature *T*. The total absorption ratio *a*(*T*, *i*) of that body is dimensionless, the ratio of absorbed to incident radiation in the cavity at temperature *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 *E*(*T*, *i*) / *a*(*T*, *i*) of emitting power to absorption ratio is a dimensioned quantity, with the dimensions of emitting power, because *a*(*T*, *i*) is dimensionless. Also here the wavelength-specific emitting power of the body at temperature *T* is denoted by *E*(*λ*, *T*, *i*) and the wavelength-specific absorption ratio by *a*(*λ*, *T*, *i*) . Again, the ratio *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.^{[56]} His theoretical proof was and still is considered by some writers to be invalid.^{[54]}^{[57]} 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 *E*(*λ*, *T*, *i*) / *a*(*λ*, *T*, *i*) has one and the same value for all bodies, that is for all values of index *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 *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 *i*.^{[58]} 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 *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.^{[54]}^{[57]}

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.^{[59]}

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

His proof first argued that for wavelength *λ* and at temperature *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 *E*(*λ*, *T*, BB), with the dimensions of power. His proof noted that the dimensionless wavelength-specific absorption ratio *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 *E*(*λ*, *T*, BB) / *a*(*λ*, *T*, BB) is again just *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 *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 *E*(*λ*, *T*, *i*) / *a*(*λ*, *T*, *i*) was equal to *E*(*λ*, *T*, BB), which may now be denoted *B*_{λ} (*λ*, *T*), a continuous function, dependent only on *λ* at fixed temperature *T*, and an increasing function of *T* at fixed wavelength *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 *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 T, for every wavelength 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 B _{λ} (λ, T) .* (For our notation

Kirchhoff announced that the determination of the function *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 *B*_{λ} (*λ*, *T*) has been called "Kirchhoff's (emission, universal) function,"^{[65]}^{[66]}^{[67]}^{[68]} 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.^{[57]} Kirchhoff stated later in 1860 that his theoretical proof was better than Balfour Stewart's, and in some respects it was so.^{[54]} 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.^{[69]}

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."^{[70]}

- Bolometer
- Color temperature
- Infrared thermometer
- Photon polarization
- Planck's law
- Pyrometry
- Rayleigh–Jeans law
- Thermography
- Sakuma–Hattori equation
- Terahertz radiation

**^**Loudon 2000, Chapter 1.**^**Mandel & Wolf 1995, Chapter 13.**^**Kondepudi & Prigogine 1998, Chapter 11.**^**Landsberg 1990, Chapter 13.**^**Partington, J.R. (1949), p. 466.**^**Ian Morison (2008).*Introduction to Astronomy and Cosmology*. J Wiley & Sons. p. 48. ISBN 0-470-03333-9.**^**Alessandro Fabbri; José Navarro-Salas (2005). "Chapter 1: Introduction".*Modeling black hole evaporation*. Imperial College Press. ISBN 1-86094-527-9.**^**From (Kirchhoff, 1860) (*Annalen der Physik und Chemie*), p. 277:*"Der Beweis, welcher für die ausgesprochene Behauptung hier gegeben werden soll, …*vollkommen schwarze*, oder kürzer*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.**^**Tomokazu Kogure; Kam-Ching Leung (2007). "§2.3: Thermodynamic equilibrium and black-body radiation".*The astrophysics of emission-line stars*. Springer. p. 41. ISBN 0-387-34500-0.**^**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.**^**Lummer, O., Pringsheim, E. (1899). Die Vertheilung der Energie im Spectrum des schwarzen Körpers,*Verhandlungen der Deutschen Physikalischen Gessellschaft*(Leipzig), 1899,**1**: 23–41.**^**Planck 1914**^**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. [1]**^**Partington 1949, pp. 466–467, 478.**^**Goody & Yung 1989, pp. 482, 484- ^
^{a}^{b}^{c}Planck 1914, p. 42 **^**Wien 1894**^**Planck 1914, p. 43**^**Joseph Caniou (1999). "§4.2.2: Calculation of Planck's law".*Passive infrared detection: theory and applications*. Springer. p. 107. ISBN 0-7923-8532-2.**^**J. R. Mahan (2002).*Radiation heat transfer: a statistical approach*(3rd ed.). Wiley-IEEE. p. 58. ISBN 978-0-471-21270-6.**^**de Groot, SR., Mazur, P. (1962).*Non-equilibrium Thermodynamics*, North-Holland, Amsterdam.**^**Kondepudi & Prigogine 1998, Section 9.4.- ^
^{a}^{b}Stewart 1858 **^**Huang, Kerson (1967).*Statistical Mechanics*. New York: John Wiley & Sons. ISBN 0-471-81518-7.**^**Gannon, Megan (December 21, 2012). "New 'Baby Picture' of Universe Unveiled". Space.com. Retrieved December 21, 2012.**^**Bennett, C.L.; Larson, L.; Weiland, J.L.; Jarosk, N.; Hinshaw, N.; Odegard, N.; Smith, K.M.; Hill, R.S.; Gold, B.; Halpern, M.; Komatsu, E.; Nolta, M.R.; Page, L.; Spergel, D.N.; Wollack, E.; Dunkley, J.; Kogut, A.; Limon, M.; Meyer, S.S.; Tucker, G.S.; Wright, E.L. (December 20, 2012). "Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Final Maps and Results".**1212**: 5225. arXiv:1212.5225. Bibcode:2013ApJS..208...20B. doi:10.1088/0067-0049/208/2/20.**^**Planck, Max (1901). "Ueber das Gesetz der Energieverteilung im Normalspectrum" [On the law of the distribution of energy in the normal spectrum].*Annalen der Physik*. 4th series (in German).**4**(3): 553–563. Bibcode:1901AnP...309..553P. doi:10.1002/andp.19013090310.**^**Landau, L. D.; E. M. Lifshitz (1996).*Statistical Physics*(3rd Edition Part 1 ed.). Oxford: Butterworth–Heinemann. ISBN 0-521-65314-2.**^**"Stefan-Boltzmann law".*Encyclopædia Britannica*. 2019.**^**Rybicki & Lightman 1979, p. 22**^**"Wien wavelength displacement law constant".*The NIST Reference on Constants, Units, and Uncertainty*. NIST. Retrieved February 8, 2019.**^**Nave, Dr. Rod. "Wien's Displacement Law and Other Ways to Characterize the Peak of Blackbody Radiation".*HyperPhysics*. Provides 5 variations of Wien's displacement law**^**Infrared Services. "Emissivity Values for Common Materials". Retrieved 2007-06-24.**^**Omega Engineering. "Emissivity of Common Materials". Retrieved 2007-06-24.**^**Farzana, Abanty (2001). "Temperature of a Healthy Human (Skin Temperature)".*The Physics Factbook*. Retrieved 2007-06-24.**^**Lee, B. "Theoretical Prediction and Measurement of the Fabric Surface Apparent Temperature in a Simulated Man/Fabric/Environment System" (PDF). Archived from the original (PDF) on 2006-09-02. Retrieved 2007-06-24.**^**Harris J, Benedict F; Benedict (1918). "A Biometric Study of Human Basal Metabolism".*Proc Natl Acad Sci USA*.**4**(12): 370–3. Bibcode:1918PNAS....4..370H. doi:10.1073/pnas.4.12.370. PMC 1091498. PMID 16576330.**^**Levine, J (2004). "Nonexercise activity thermogenesis (NEAT): environment and biology".*Am J Physiol Endocrinol Metab*.**286**(5): E675–E685. doi:10.1152/ajpendo.00562.2003. PMID 15102614.**^**DrPhysics.com. "Heat Transfer and the Human Body". Retrieved 2007-06-24.**^**Prevost, P. (1791). Mémoire sur l'equilibre du feu.*Journal de Physique*(Paris), vol 38 pp. 314-322.**^**Iribarne, J.V., Godson, W.L. (1981).*Atmospheric Thermodynamics*, second edition, D. Reidel Publishing, Dordrecht, ISBN 90-277-1296-4, page 227.- ^
^{a}^{b}^{c}NASA Sun Fact Sheet **^**Cole, George H. A.; Woolfson, Michael M. (2002).*Planetary Science: The Science of Planets Around Stars (1st ed.)*. Institute of Physics Publishing. pp. 36–37, 380–382. ISBN 0-7503-0815-X.**^***Principles of Planetary Climate*by Raymond T. Peirrehumbert, Cambridge University Press (2011), p. 146. From Chapter 3 which is available online here Archived March 28, 2012, at the Wayback Machine, 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.**^**Saari, J. M.; Shorthill, R. W. (1972). "The Sunlit Lunar Surface. I. Albedo Studies and Full Moon".*The Moon*.**5**(1–2): 161–178. Bibcode:1972Moon....5..161S. doi:10.1007/BF00562111.**^**Lunar and Planetary Science XXXVII (2006) 2406**^**Michael D. Papagiannis (1972).*Space physics and space astronomy*. Taylor & Francis. pp. 10–11. ISBN 978-0-677-04000-4.**^**Willem Jozef Meine Martens & Jan Rotmans (1999).*Climate Change an Integrated Perspective*. Springer. pp. 52–55. ISBN 978-0-7923-5996-8.**^**F. Selsis (2004). "The Prebiotic Atmosphere of the Earth". In Pascale Ehrenfreund; et al.*Astrobiology: Future Perspectives*. Springer. pp. 279–280. ISBN 978-1-4020-2587-7.**^**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.**^**White, M. (1999). "Anisotropies in the CMB". arXiv:astro-ph/9903232. Bibcode:1999dpf..conf.....W.**^**Kondepudi & Prigogine 1998, pp. 227–228; also Section 11.6, pages 294–296.**^**The Doppler Effect, T. P. Gill, Logos Press, 1965- ^
^{a}^{b}^{c}^{d}Siegel 1976 **^**Kirchhoff 1860a**^**Kirchhoff 1860b- ^
^{a}^{b}^{c}Schirrmacher 2001 - ^
^{a}^{b}Kirchhoff 1860c **^**Planck 1914, p. 11**^**Chandrasekhar 1950, p. 8**^**Milne 1930, p. 80**^**Rybicki & Lightman 1979, pp. 16–17**^**Mihalas & Weibel-Mihalas 1984, p. 328**^**Goody & Yung 1989, pp. 27–28**^**Paschen, F. (1896), personal letter cited by Hermann 1971, p. 6**^**Hermann 1971, p. 7**^**Kuhn 1978, pp. 8, 29**^**Mehra and Rechenberg 1982, pp. 26, 28, 31, 39**^**Kirchhoff & 1862/1882, p. 573**^**Kragh 1999, p. 58

- Chandrasekhar, S. (1950).
*Radiative Transfer*. Oxford University Press. - Goody, R. M.; Yung, Y. L. (1989).
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*The Genesis of Quantum Theory*. Nash, C.W. (transl.). MIT Press. ISBN 0-262-08047-8. a translation of*Frühgeschichte der Quantentheorie (1899–1913)*, Physik Verlag, Mosbach/Baden. - Kirchhoff, G.; [27 October 1859] (1860a). "Über die Fraunhofer'schen Linien" [On Fraunhofer's lines].
*Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin*: 662–665. - Kirchhoff, G.; [11 December 1859] (1860b). "Über den Zusammenhang zwischen Emission und Absorption von Licht und Wärme" [On the relation between emission and absorption of light and heat].
*Monatsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin*: 783–787. - Kirchhoff, G. (1860c). "Ueber das Verhältniss zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme and Licht" [On the relation between bodies' emission capacity and absorption capacity for heat and light].
*Annalen der Physik und Chemie*.**109**(2): 275–301. Bibcode:1860AnP...185..275K. doi:10.1002/andp.18601850205. Translated by Guthrie, F. as Kirchhoff, G. (1860). "On the relation between the radiating and absorbing powers of different bodies for light and heat".*Philosophical Magazine*. Series 4, volume 20: 1–21. - Kirchhoff, G. (1882) [1862], "Ueber das Verhältniss zwischen dem Emissionsvermögen und dem Absorptionsvermögen der Körper für Wärme und Licht",
*Gessamelte Abhandlungen*, Leipzig: Johann Ambrosius Barth, pp. 571–598 - Kondepudi, D.; Prigogine, I. (1998).
*Modern Thermodynamics. From Heat Engines to Dissipative Structures*. John Wiley & Sons. ISBN 0-471-97393-9. - Kragh, H. (1999).
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*Black–Body Theory and the Quantum Discontinuity*. Oxford University Press. ISBN 0-19-502383-8. - Landsberg, P. T. (1990).
*Thermodynamics and statistical mechanics*(Reprint ed.). Courier Dover Publications. ISBN 0-486-66493-7. - Lavenda, Bernard Howard (1991).
*Statistical Physics: A Probabilistic Approach*. John Wiley & Sons. pp. 41–42. ISBN 978-0-471-54607-8. - Loudon, R. (2000) [1973].
*The Quantum Theory of Light*(third ed.). Cambridge University Press. ISBN 0-19-850177-3. - Mandel, L.; Wolf, E. (1995).
*Optical Coherence and Quantum Optics*. Cambridge University Press. ISBN 0-521-41711-2. - Mehra, J.; Rechenberg, H. (1982).
*The Historical Development of Quantum Theory*. volume 1, part 1. Springer-Verlag. ISBN 0-387-90642-8. - Mihalas, D.; Weibel-Mihalas, B. (1984).
*Foundations of Radiation Hydrodynamics*. Oxford University Press. ISBN 0-19-503437-6. - Milne, E.A. (1930). "Thermodynamics of the Stars".
*Handbuch der Astrophysik*. 3, part 1: 63–255. - Partington, J.R. (1949).
*An Advanced Treatise on Physical Chemistry. Volume 1. Fundamental Principles. The Properties of Gases*. Longmans, Green and Co. - Planck, M. (1914) [1912].
*The Theory of Heat Radiation*. translated by Masius, M. P. Blakiston's Sons & Co. - Rybicki, G. B.; Lightman, A. P. (1979).
*Radiative Processes in Astrophysics*. John Wiley & Sons. ISBN 0-471-82759-2. - Schirrmacher, A. (2001).
*Experimenting theory: the proofs of Kirchhoff's radiation law before and after Planck*. Münchner Zentrum für Wissenschafts und Technikgeschichte. - Siegel, D.M. (1976). "Balfour Stewart and Gustav Robert Kirchhoff: two independent approaches to "Kirchhoff's radiation law"".
*Isis*.**67**(4): 565–600. doi:10.1086/351669. - Stewart, B. (1858). "An account of some experiments on radiant heat".
*Transactions of the Royal Society of Edinburgh*.**22**: 1–20. - Wien, W. (1894). "Temperatur und Entropie der Strahlung" [Temperature and entropy of radiation].
*Annalen der Physik*.**288**(5): 132–165. Bibcode:1894AnP...288..132W. doi:10.1002/andp.18942880511.

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*Thermal Physics*(2nd ed.). W. H. Freeman Company. ISBN 0-7167-1088-9. - Tipler, Paul; Llewellyn, Ralph (2002).
*Modern Physics*(4th ed.). W. H. Freeman. ISBN 0-7167-4345-0.

- Black-body radiation JavaScript Interactives Black-body radiation by Fu-Kwun Hwang and Loo Kang Wee
- Calculating Black-body Radiation Interactive calculator with Doppler Effect. Includes most systems of units.
- Color-to-Temperature demonstration at Academo.org
- Cooling Mechanisms for Human Body – From Hyperphysics
- Descriptions of radiation emitted by many different objects
- Black-Body Emission Applet
- "Blackbody Spectrum" by Jeff Bryant, Wolfram Demonstrations Project, 2007.

A black body or blackbody is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. A white body is one with a "rough surface [that] reflects all incident rays completely and uniformly in all directions."A black body in thermal equilibrium (that is, at a constant temperature) emits electromagnetic radiation called black-body radiation. The radiation is emitted according to Planck's law, meaning that it has a spectrum that is determined by the temperature alone (see figure at right), not by the body's shape or composition.

An ideal black body in thermal equilibrium has two notable properties:

It is an ideal emitter: at every frequency, it emits as much or more thermal radiative energy as any other body at the same temperature.

It is a diffuse emitter: the energy is radiated isotropically, independent of direction.An approximate realization of a black surface is a hole in the wall of a large insulated enclosure (an oven). Any light entering the hole is reflected or absorbed at the internal surfaces of the body and is unlikely to re-emerge, making the hole a nearly perfect absorber. When the radiation confined in such an enclosure is in thermal equilibrium, the radiation emitted from the hole will be as great as from any body at that equilibrium temperature.

Real materials emit energy at a fraction—called the emissivity—of black-body energy levels. By definition, a black body in thermal equilibrium has an emissivity of ε = 1.0. A source with lower emissivity independent of frequency often is referred to as a gray body.

Construction of black bodies with emissivity as close to one as possible remains a topic of current interest.In astronomy, the radiation from stars and planets is sometimes characterized in terms of an effective temperature, the temperature of a black body that would emit the same total flux of electromagnetic energy.

Boltzmann constant

The **Boltzmann constant** (*k*_{B} or k) is a physical constant relating the average relative kinetic energy of particles in a gas with the temperature of the gas and occurs in Planck's law of black-body radiation and in Boltzmann's entropy formula. It was introduced by Max Planck, but named after Ludwig Boltzmann.

It is the gas constant R divided by the Avogadro constant *N*_{A}:

The Boltzmann constant has the dimension energy divided by temperature, the same as entropy.
As of 2017, its value in SI units is a measured quantity. The recommended value (as of 2015, with standard uncertainty in parentheses) is 64852(79)×10^{−23} J/K. 1.380

Current measurements of the Boltzmann constant depend on the definition of the kelvin in terms of the triple point of water. In the redefinition of SI base units adopted at the 26th General Conference on Weights and Measures (CGPM) on 16 November 2018, the definition of the kelvin was changed to one based on a fixed, exact numerical value of the Boltzmann constant, similar to the way that the speed of light was given an exact numerical value at the 17th CGPM in 1983. The final value (based on the 2017 CODATA adjusted value of 64903(51)×10^{−23} J/K) is 1.380649×10^{−23} J/K. 1.380

In quantum mechanics, the results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number of molecules which do not interact with each other except for instantaneous thermalizing collisions. This simple model can be used to describe the classical ideal gas as well as the various quantum ideal gases such as the ideal massive Fermi gas, the ideal massive Bose gas as well as black body radiation (photon gas) which may be treated as a massless Bose gas, in which thermalization is usually assumed to be facilitated by the interaction of the photons with an equilibrated mass.

Using the results from either Maxwell–Boltzmann statistics, Bose–Einstein statistics or Fermi–Dirac statistics, and considering the limit of a very large box, the Thomas-Fermi approximation (named after Enrico Fermi and Llewellyn Thomas) is used to express the degeneracy of the energy states as a differential, and summations over states as integrals. This enables thermodynamic properties of the gas to be calculated with the use of the partition function or the grand partition function. These results will be applied to both massive and massless particles. More complete calculations will be left to separate articles, but some simple examples will be given in this article.

Gaussian noise**Gaussian noise**, named after Carl Friedrich Gauss, is statistical noise having a probability density function (PDF) equal to that of the normal distribution, which is also known as the Gaussian distribution. In other words, the values that the noise can take on are Gaussian-distributed.

The probability density function of a Gaussian random variable is given by:

where represents the grey level, the mean value and the standard deviation.

A special case is *white Gaussian noise*, in which the values at any pair of times are identically distributed and statistically independent (and hence uncorrelated). In communication channel testing and modelling, Gaussian noise is used as additive white noise to generate additive white Gaussian noise.

In telecommunications and computer networking, communication channels can be affected by wideband Gaussian noise coming from many natural sources, such as the thermal vibrations of atoms in conductors (referred to as thermal noise or Johnson-Nyquist noise), shot noise, black body radiation from the earth and other warm objects, and from celestial sources such as the Sun.

Principal sources of Gaussian noise in digital images arise during acquisition e.g. sensor noise caused by poor illumination and/or high temperature, and/or transmission e.g. electronic circuit noise. In digital image processing Gaussian noise can be reduced using a spatial filter, though when smoothing an image, an undesirable outcome may result in the blurring of fine-scaled image edges and details because they also correspond to blocked high frequencies. Conventional spatial filtering techniques for noise removal include: mean (convolution) filtering, median filtering and Gaussian smoothing.

Gustav KirchhoffGustav Robert Kirchhoff (German: [ˈkɪʁçhɔf]; 12 March 1824 – 17 October 1887) was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects.

He coined the term black-body radiation in 1862, and at least two different sets of concepts are named "Kirchhoff's laws" after him. The Bunsen–Kirchhoff Award for spectroscopy is named after him and his colleague, Robert Bunsen.

Heinrich RubensHeinrich Rubens (30 March 1865, Wiesbaden, Nassau, Germany – 17 July 1922, Berlin, Germany) was a German physicist. He is known for his measurements of the energy of black-body radiation which led Max Planck to the discovery of his radiation law. This was the genesis of quantum theory.

After having attended realgymnasium Wöhlerschule in Frankfurt am Main, he started in 1884 to study electrical engineering at the technical universities in Darmstadt and Berlin. The following year he switched to physics at the University of Berlin which was more to his liking. After just one semester there he transferred to Strasbourg. There he benefited much from the lectures by August Kundt who in 1888 took over the vacant position of Hermann Helmholtz at the University of Berlin. Rubens followed after and got his doctors degree there the same year. In the period 1890–1896 he was employed as an assistant at the physics institute and made his habilitation in 1892. He was then a privatdozent and was allowed to teach. Already then he was praised for his experimental investigations of infrared radiation.

Rubens got a permanent position in 1896 as docent at the Technical University of Berlin in Berlin-Charlottenburg. He could continue his experimental research at the nearby Physikalisch-Technische Reichsanstalt. It was there he in 1900 did his important measurements of black-body radiation which made him world-famous. He was promoted to professor the same year.

After Paul Drude retired in 1906 from his professorship at the University in Berlin, the position was given to Rubens. He was at the same time appointed director of the physics institute. In this way he could influence and lead a large group of colleagues and students. The year after he was elected to the Prussian Academy of Sciences and became in 1908 a corresponding member Göttingen Academy of Sciences and Humanities. He participated at the two first Solvay conferences after having received the Rumford Medal in 1910 "on the ground of his researches on radiation, especially of long wave length.".

Heinrich Rubens died in 1922 after a longer illness. At a memorial meeting in the science academy the following year Max Planck said about him:

Without the intervention of Rubens the formulation of the radiation law and thereby the foundation of quantum theory would perhaps have arisen in quite a different manner, or perhaps not have developed in Germany at all.

He is buried at the Alter St.-Matthäus-Kirchhof in Berlin-Schöneberg with his wife Marie. She took her life in 1941 for fear of being deported and killed by the Nazis. The burial place is near that of Gustav Kirchhoff, who founded spectroscopy and formulated the first laws of black-body radiation.

History of quantum mechanicsThe **history of quantum mechanics** is a fundamental part of the history of modern physics. Quantum mechanics' history, as it interlaces with the history of quantum chemistry, began essentially with a number of different scientific discoveries: the 1838 discovery of cathode rays by Michael Faraday; the 1859–60 winter statement of the black-body radiation problem by Gustav Kirchhoff; the 1877 suggestion by Ludwig Boltzmann that the energy states of a physical system could be *discrete*; the discovery of the photoelectric effect by Heinrich Hertz in 1887; and the 1900 quantum hypothesis by Max Planck that any energy-radiating atomic system can theoretically be divided into a number of discrete "energy elements" *ε* (epsilon) such that each of these energy elements is proportional to the frequency *ν* with which each of them individually radiate energy, as defined by the following formula:

where *h* is a numerical value called Planck's constant.

Then, Albert Einstein in 1905, in order to explain the photoelectric effect previously reported by Heinrich Hertz in 1887, postulated consistently with Max Planck's quantum hypothesis that light itself is made of individual quantum particles, which in 1926 came to be called photons by Gilbert N. Lewis. The photoelectric effect was observed upon shining light of particular wavelengths on certain materials, such as metals, which caused electrons to be ejected from those materials only if the light quantum energy was greater than the work function of the metal's surface.

The phrase "quantum mechanics" was coined (in German, *Quantenmechanik*) by the group of physicists including Max Born, Werner Heisenberg, and Wolfgang Pauli, at the University of Göttingen in the early 1920s, and was first used in Born's 1924 paper *"Zur Quantenmechanik"*. In the years to follow, this theoretical basis slowly began to be applied to chemical structure, reactivity, and bonding.

In radiation thermodynamics, a hohlraum (a non-specific German word for a "hollow space" or "cavity") is a cavity whose walls are in radiative equilibrium with the radiant energy within the cavity. This idealized cavity can be approximated in practice by making a small perforation in the wall of a hollow container of any opaque material. The radiation escaping through such a perforation will be a good approximation to black-body radiation at the temperature of the interior of the container.

Max PlanckMax Karl Ernst Ludwig Planck, ForMemRS (German: [ˈplaŋk]; English: ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918.Planck made many contributions to theoretical physics, but his fame as a physicist rests primarily on his role as the originator of quantum theory, which revolutionized human understanding of atomic and subatomic processes. In 1948 the German scientific institution the Kaiser Wilhelm Society (of which Planck was twice president) was renamed the Max Planck Society (MPS). The MPS now includes 83 institutions representing a wide range of scientific directions.

Non-ionizing radiationNon-ionizing (or non-ionising) radiation refers to any type of electromagnetic radiation that does not carry enough energy per quantum (photon energy) to ionize atoms or molecules—that is, to completely remove an electron from an atom or molecule. Instead of producing charged

ions when passing through matter, non-ionizing electromagnetic radiation has sufficient energy only for excitation, the movement of an electron to a higher energy state. Ionizing radiation which has a higher frequency and shorter wavelength than nonionizing radiation, has many uses but can be a health hazard; exposure to it can cause burns, radiation sickness, cancer, and genetic damage. Using ionizing radiation requires elaborate radiological protection measures which in general are not required with nonionizing radiation.

The region at which radiation becomes considered as "ionizing" is not well defined, since different molecules and atoms ionize at different energies. The usual definitions have suggested that radiation with particle or photon energies less than 10 electronvolts (eV) be considered non-ionizing. Another suggested threshold is 33 electronvolts, which is the energy needed to ionize water molecules. The light from the Sun that reaches the earth is largely composed of non-ionizing radiation, since the ionizing far-ultraviolet rays have been filtered out by the gases in the atmosphere, particularly oxygen. The remaining ultraviolet radiation from the Sun causes molecular damage (for example, sunburn) by photochemical and free-radical-producing means.Different biological effects are observed for different types of non-ionizing radiation. A difficulty is that there is no controversy that the upper frequencies of non-ionizing radiation near these energies (much of the spectrum of UV light and some visible light) is capable of non-thermal biological damage, similar to ionizing radiation. Health debate therefore centers on the non-thermal effects of radiation of much lower frequencies (microwave, millimeter and radiowave radiation). The International Agency for Research on Cancer recently stated that there could be some risk from non-ionizing radiation to humans. But a subsequent study reported that the basis of the IARC evaluation was not consistent with observed incidence trends. This and other reports suggest that there is virtually no way that results on which the IARC based its conclusions are correct. The Bioinitiative Report 2012 makes the claim that there are significant health risk associated with low frequency non-ionizing electromagnetic radiation. This report claims that statistically significant increases in cancer among those exposed to even low power levels, low frequency, non-ionizing radiation. There is considerable debate on this matter. Currently regulatory bodies around the world have not seen the need to change current safety standards.

PhotonThe photon is a type of elementary particle, the quantum of the electromagnetic field including electromagnetic radiation such as light, and the force carrier for the electromagnetic force (even when static via virtual particles). The photon has zero rest mass and always moves at the speed of light within a vacuum.

Like all elementary particles, photons are currently best explained by quantum mechanics and exhibit wave–particle duality, exhibiting properties of both waves and particles. For example, a single photon may be refracted by a lens and exhibit wave interference with itself, and it can behave as a particle with definite and finite measurable position or momentum, though not both at the same time as per the Heisenberg's uncertainty principle. The photon's wave and quantum qualities are two observable aspects of a single phenomenon—they cannot be described by any mechanical model; a representation of this dual property of light that assumes certain points on the wavefront to be the seat of the energy is not possible. The quanta in a light wave are not spatially localized.

The modern concept of the photon was developed gradually by Albert Einstein in the early 20th century to explain experimental observations that did not fit the classical wave model of light. The benefit of the photon model is that it accounts for the frequency dependence of light's energy, and explains the ability of matter and electromagnetic radiation to be in thermal equilibrium. The photon model accounts for anomalous observations, including the properties of black-body radiation, that others (notably Max Planck) had tried to explain using semiclassical models. In that model, light is described by Maxwell's equations, but material objects emit and absorb light in quantized amounts (i.e., they change energy only by certain particular discrete amounts). Although these semiclassical models contributed to the development of quantum mechanics, many further experiments beginning with the phenomenon of Compton scattering of single photons by electrons, validated Einstein's hypothesis that light itself is quantized. In 1926 the optical physicist Frithiof Wolfers and the chemist Gilbert N. Lewis coined the name "photon" for these particles. After Arthur H. Compton won the Nobel Prize in 1927 for his scattering studies, most scientists accepted that light quanta have an independent existence, and the term "photon" was accepted.

In the Standard Model of particle physics, photons and other elementary particles are described as a necessary consequence of physical laws having a certain symmetry at every point in spacetime. The intrinsic properties of particles, such as charge, mass, and spin, are determined by this gauge symmetry. The photon concept has led to momentous advances in experimental and theoretical physics, including lasers, Bose–Einstein condensation, quantum field theory, and the probabilistic interpretation of quantum mechanics. It has been applied to photochemistry, high-resolution microscopy, and measurements of molecular distances. Recently, photons have been studied as elements of quantum computers, and for applications in optical imaging and optical communication such as quantum cryptography.

Photon gasIn physics, a photon gas is a gas-like collection of photons, which has many of the same properties of a conventional gas like hydrogen or neon – including pressure, temperature, and entropy. The most common example of a photon gas in equilibrium is black body radiation.

Photons are part of family of particles known as bosons, particles that follow Bose-Einstein statistics and with integer spin. A gas of bosons with only one type of particle is uniquely described by three state functions such as the temperature, volume, and the number of particles. However, for a black body, the energy distribution is established by the interaction of the photons with matter, usually the walls of the container. In this interaction, the number of photons is not conserved. As a result, the chemical potential of the black body photon gas is zero. The number of state variables needed to describe a black body state is thus reduced from three to two (e.g. temperature and volume).

Planck's lawPlanck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature T, when there is no net flow of matter or energy between the body and its environment.At the end of the 19th-century, physicists were unable to explain why the observed spectrum of black body radiation, which by then had been accurately measured, diverged significantly at higher frequencies from that predicted by existing theories. In 1900, Max Planck empirically derived a formula for the observed spectrum by assuming that a hypothetical electrically charged oscillator in a cavity that contained black body radiation could only change its energy in a minimal increment, E, that was proportional to the frequency of its associated electromagnetic wave. This resolved the problem of the ultraviolet catastrophe predicted by classical physics.

It was a pioneering insight of modern physics and is of fundamental importance to quantum theory.

Planck constantThe Planck constant (denoted h, also called Planck's constant) is a physical constant that is the quantum of electromagnetic action, which relates the energy carried by a photon to its frequency. A photon's energy is equal to its frequency multiplied by the Planck constant. The Planck constant is of fundamental importance in quantum mechanics, and in metrology it is the basis for the definition of the kilogram.

At the end of the 19th century, physicists were unable to explain why the observed spectrum of black body radiation, which by then had been accurately measured, diverged significantly at higher frequencies from that predicted by existing theories. In 1900, Max Planck empirically derived a formula for the observed spectrum by assuming that a hypothetical electrically charged oscillator in a cavity that contained black body radiation could only change its energy in a minimal increment, E, that was proportional to the frequency of its associated electromagnetic wave. He was able to calculate the proportionality constant, h, from the experimental measurements, and that constant is named in his honor. In 1905, the value E was associated by Albert Einstein with a "quantum" or minimal element of the energy of the electromagnetic wave itself. The light quantum behaved in some respects as an electrically neutral particle, as opposed to an electromagnetic wave. It was eventually called a photon.

Since energy and mass are equivalent, the Planck constant also relates mass to frequency. By 2017, the Planck constant had been measured with sufficient accuracy in terms of the SI base units, that it was central to replacing the metal cylinder, called the International Prototype of the Kilogram (IPK), that had defined the kilogram since 1889. The new definition was unanimously approved at the General Conference on Weights and Measures (CGPM) on 16 November 2018 as part of the 2019 redefinition of SI base units. For this new definition of the kilogram, the Planck constant, as defined by the ISO standard, was set to 6.62607015×10−34 J⋅s exactly. The kilogram was the last SI base unit to be re-defined by a fundamental physical property to replace a physical artefact.

RadiationIn physics, radiation is the emission or transmission of energy in the form of waves or particles through space or through a material medium. This includes:

electromagnetic radiation, such as radio waves, microwaves, infrared, visible light, ultraviolet, x-rays, and gamma radiation (γ)

particle radiation, such as alpha radiation (α), beta radiation (β), and neutron radiation (particles of non-zero rest energy)

acoustic radiation, such as ultrasound, sound, and seismic waves (dependent on a physical transmission medium)

gravitational radiation, radiation that takes the form of gravitational waves, or ripples in the curvature of spacetime.Radiation is often categorized as either ionizing or non-ionizing depending on the energy of the radiated particles. Ionizing radiation carries more than 10 eV, which is enough to ionize atoms and molecules, and break chemical bonds. This is an important distinction due to the large difference in harmfulness to living organisms. A common source of ionizing radiation is radioactive materials that emit α, β, or γ radiation, consisting of helium nuclei, electrons or positrons, and photons, respectively. Other sources include X-rays from medical radiography examinations and muons, mesons, positrons, neutrons and other particles that constitute the secondary cosmic rays that are produced after primary cosmic rays interact with Earth's atmosphere.

Gamma rays, X-rays and the higher energy range of ultraviolet light constitute the ionizing part of the electromagnetic spectrum. The word "ionize" refers to the breaking of one or more electrons away from an atom, an action that requires the relatively high energies that these electromagnetic waves supply. Further down the spectrum, the non-ionizing lower energies of the lower ultraviolet spectrum cannot ionize atoms, but can disrupt the inter-atomic bonds which form molecules, thereby breaking down molecules rather than atoms; a good example of this is sunburn caused by long-wavelength solar ultraviolet. The waves of longer wavelength than UV in visible light, infrared and microwave frequencies cannot break bonds but can cause vibrations in the bonds which are sensed as heat. Radio wavelengths and below generally are not regarded as harmful to biological systems. These are not sharp delineations of the energies; there is some overlap in the effects of specific frequencies.The word radiation arises from the phenomenon of waves radiating (i.e., traveling outward in all directions) from a source. This aspect leads to a system of measurements and physical units that are applicable to all types of radiation. Because such radiation expands as it passes through space, and as its energy is conserved (in vacuum), the intensity of all types of radiation from a point source follows an inverse-square law in relation to the distance from its source. Like any ideal law, the inverse-square law approximates a measured radiation intensity to the extent that the source approximates a geometric point.

Radiation pressureRadiation pressure is the pressure exerted upon any surface due to the exchange of momentum between the object and the electromagnetic field. This includes the momentum of light or electromagnetic radiation of any wavelength which is absorbed, reflected, or otherwise emitted (e.g. black body radiation) by matter on any scale (from macroscopic objects to dust particles to gas molecules).The forces generated by radiation pressure are generally too small to be noticed under everyday circumstances; however, they are important in some physical processes. This particularly includes objects in outer space where it is usually the main force acting on objects besides gravity, and where the net effect of a tiny force may have a large cumulative effect over long periods of time. For example, had the effects of the sun's radiation pressure on the spacecraft of the Viking program been ignored, the spacecraft would have missed Mars orbit by about 15,000 km (9,300 mi). Radiation pressure from starlight is crucial in a number of astrophysical processes as well. The significance of radiation pressure increases rapidly at extremely high temperatures, and can sometimes dwarf the usual gas pressure, for instance in stellar interiors and thermonuclear weapons.

Radiation pressure can equally well be accounted for by considering the momentum of a classical electromagnetic field or in terms of the momenta of photons, particles of light. The interaction of electromagnetic waves or photons with matter may involve an exchange of momentum. Due to the law of conservation of momentum, any change in the total momentum of the waves or photons must involve an equal and opposite change in the momentum of the matter it interacted with (Newton's third law of motion), as is illustrated in the accompanying figure for the case of light being perfectly reflected by a surface. This transfer of momentum is the general explanation for what we term radiation pressure.

Thermodynamic temperature**Thermodynamic temperature** is the absolute measure of temperature and is one of the principal parameters of thermodynamics.

Thermodynamic temperature is defined by the third law of thermodynamics in which the theoretically lowest temperature is the null or zero point. At this point, absolute zero, the particle constituents of matter have minimal motion and can become no colder. In the quantum-mechanical description, matter at absolute zero is in its ground state, which is its state of lowest energy. Thermodynamic temperature is often also called **absolute temperature**, for two reasons: one, proposed by Kelvin, that it does not depend on the properties of a particular material; two that it refers to an absolute zero according to the properties of the ideal gas.

The International System of Units specifies a particular scale for thermodynamic temperature. It uses the kelvin scale for measurement and selects the triple point of water at 273.16 K as the fundamental fixing point. Other scales have been in use historically. The Rankine scale, using the degree Fahrenheit as its unit interval, is still in use as part of the English Engineering Units in the United States in some engineering fields. ITS-90 gives a practical means of estimating the thermodynamic temperature to a very high degree of accuracy.

Roughly, the temperature of a body at rest is a measure of the mean of the energy of the translational, vibrational and rotational motions of matter's particle constituents, such as molecules, atoms, and subatomic particles. The full variety of these kinetic motions, along with potential energies of particles, and also occasionally certain other types of particle energy in equilibrium with these, make up the total internal energy of a substance. Internal energy is loosely called the heat energy or thermal energy in conditions when no work is done upon the substance by its surroundings, or by the substance upon the surroundings. Internal energy may be stored in a number of ways within a substance, each way constituting a "degree of freedom". At equilibrium, each degree of freedom will have on average the same energy: where is the Boltzmann constant, unless that degree of freedom is in the quantum regime. The internal degrees of freedom (rotation, vibration, etc.) may be in the quantum regime at room temperature, but the translational degrees of freedom will be in the classical regime except at extremely low temperatures (fractions of kelvins) and it may be said that, for most situations, the thermodynamic temperature is specified by the average translational kinetic energy of the particles.

ThermoluminescenceThermoluminescence is a form of luminescence that is exhibited by certain crystalline materials, such as some minerals, when previously absorbed energy from electromagnetic radiation or other ionizing radiation is re-emitted as light upon heating of the material. The phenomenon is distinct from that of black-body radiation.

Wien's displacement law*Not to be confused with Wien distribution law.*

**Wien's displacement law** states that the black body radiation curve for different temperature peaks at a wavelength is inversely proportional to the temperature. The shift of that peak is a direct consequence of the Planck radiation law, which describes the spectral brightness of black body radiation as a function of wavelength at any given temperature. However, it had been discovered by Wilhelm Wien several years before Max Planck developed that more general equation, and describes the entire shift of the spectrum of black body radiation toward shorter wavelengths as temperature increases.

Formally, Wien's displacement law states that the spectral radiance of black body radiation per unit wavelength, peaks at the wavelength *λ*_{max} given by:

where *T* is the absolute temperature in kelvins. *b* is a constant of proportionality called *Wien's displacement constant*, equal to 7729(17)×10^{−3} m⋅K, or to obtain wavelength in micrometers, 2.897*b* ≈ 2900 μm⋅K. If one is considering the peak of black body emission per unit frequency or per proportional bandwidth, one must use a different proportionality constant. However, the form of the law remains the same: the peak wavelength is inversely proportional to temperature, and the peak frequency is directly proportional to temperature.

Wien's displacement law may be referred to as "Wien's law", a term which is also used for the Wien approximation.

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