Black body

As the temperature decreases, the peak of the blackbody radiation curve moves to lower intensities and longer wavelengths. The blackbody radiation graph is also compared with the classical model of Rayleigh and Jeans.

The color (chromaticity) of blackbody radiation depends on 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.

A black body is an idealized physical body that absorbs all incident electromagnetic radiation. Because of this perfect absorptivity at all wavelengths, a black body is also the best possible emitter of thermal radiation, which it radiates incandescently in a characteristic, continuous spectrum that depends on the body's temperature. At Earth-ambient temperatures this emission is in the infrared region of the electromagnetic spectrum and is not visible. The object appears black, since it does not reflect or emit any visible light.

The thermal radiation from a black body is energy converted electrodynamically from the body's pool of internal thermal energy at any temperature greater than absolute zero. It is called blackbody radiation and has a frequency distribution with a characteristic frequency of maximum radiative power that shifts to higher frequencies with increasing temperature. As the temperature increases past a few hundred degrees Celsius, black bodies start to emit visible wavelengths, appearing red, orange, yellow, white, and blue with increasing temperature. When an object is visually white, it is emitting a substantial fraction as ultraviolet radiation.

Blackbody emission provides insight into the thermodynamic equilibrium state of cavity radiation. If each Fourier mode of the absolutely stable equilibrium radiation in a cavity with perfectly reflective walls were considered as a degree of freedom, and if all those degrees of freedom could freely exchange energy, then, according to the equipartition theorem in classical physics, each degree of freedom would have one and the same quantity of energy. This approach led to the paradox known as the ultraviolet catastrophe, that there would be an infinite amount of energy in any continuous field. The study of the laws of black bodies helped to establish the foundations of quantum mechanics.

The term black body was introduced by Gustav Kirchhoff in 1860. When used as a compound adjective, the term is typically written as one word in English, e.g. in blackbody radiation, but sometimes also hyphenated, as in black-body radiation.

Explanation

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

Conversely all 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 blackbody radiation.

The concept of the black body is an idealization, because the only perfect black bodies in nature are black holes. Graphite and lamp black, with emissivities greater than 0.95, however, are good approximations to a black material. Experimentally, blackbody 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 blackbody radiation emanating from the opening.

Blackbody radiation has the unique absolutely stable distribution of radiative intensity that can persist in thermodynamic equilibrium in a cavity.

Blackbody 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. At 1000 K, the opening in the oven 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 such that almost all light entering is absorbed, it will be a good approximation to a blackbody, so 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 blackbody curve. Different curves are obtained by varying the temperature.

The temperature of a Pāhoehoe lava flow can be estimated by observing its color. The result agrees well with measured temperatures of lava flows at about 1000 to 1200 °C.

Two things that are at the same temperature stay in 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 there are no strange correlations between the process of emission and absorption: the process of emission is not affected by the absorption, but only by the thermal state of the emitting body. This means that the total light emitted by a body at temperature T, black or not, is always equal to the total light that the body would absorb were it to be surrounded by light at temperature T.

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 blackbody curve. This means that the blackbody 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 blackbody 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. 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, blackbody 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 opacity and partial reflectivity of the walls, but not on the particular material of which they are built nor on the material in the cavity (compare with emission spectrum).

Calculating the blackbody 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 blackbody radiation. 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 blackbody 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.

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.

The distribution of the cosmic microwave background radiation across the universe as measured by the Wilkinson Microwave Anisotropy Probe. It is the most precisely known thermal emission spectrum and corresponds to a temperature of 2.725 K and an emission peak frequency of 160.2 GHz.

With non-black surfaces, the deviations from ideal blackbody 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 blackbody spectrum is exhibited by the cosmic microwave background radiation. Hawking radiation is the hypothetical blackbody radiation emitted by black holes.

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.

Blackbody simulators

File:Extended Source Black Body.JPG

A typical industrial extended-source-plate type blackbody device.

A black body is an idealized object, and by definition has an emissivity of e = 1.0. In practice, common applications define all sources of infrared radiation as a black body when the object approaches that emissivity and is greater than approximately 0.99. A source with lower emissivity is often referred to as a gray body.

An example of a nearly perfect black body is super black, produced from a nickel-phosphorus alloy. In 2009, a team of Japanese scientists created a material even closer to an ideal black body, based on vertically aligned single-walled carbon nanotubes. This absorbs between 98% and 99% of the incoming light in the spectral range from the ultra-violet to the far-infrared regions.

Equations governing black bodies

Planck's law of blackbody radiation

Main article: Planck's law

Planck's law states that

$I(\nu ,T)={\frac {2h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\frac {h\nu }{kT}}-1}}.$

where

I(ν,T) is the energy per unit time (or the power) radiated per unit area of emitting surface in the normal direction per unit solid angle per unit frequency by a black body 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; and

T is the temperature of the body in kelvins.

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.

A consequence of Wien's displacement law is that the wavelength at which the intensity of the radiation produced by a black body is at a maximum, $\lambda _{\max }$, it 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 2.8977685(51)×10⁻³ K m.

Note that the peak intensity can be expressed in terms of intensity per unit wavelength or in terms of intensity per unit frequency. The expression for the peak wavelength given above refers to the intensity per unit wavelength; meanwhile the Planck's Law section above was in terms of intensity per unit frequency. The frequency at which the power per unit frequency is maximised is given by

$\nu _{\max }=T\times 59\ \mathrm {GHz} \mathrm {K} ^{-1}$.

Stefan–Boltzmann law

The Stefan–Boltzmann law states that the power emitted per unit area of the surface of a black body is directly proportional to the fourth power of its absolute temperature:

$j^{\star }=\sigma T^{4},$

where j*is the total power radiated per unit area, T is the absolute temperature and σ = 5.67×10⁻⁸ W m⁻² K⁻⁴ is the Stefan–Boltzmann constant.

Human body emission

Much of a person's energy is radiated away in the form of infrared light. Some materials are transparent in the infrared, while opaque to visible light, as is the plastic bag in this infrared image (bottom). Other materials are transparent to visible light, while opaque or reflective in the infrared, noticeable by darkness of the man's glasses.

As all matter, the human body radiates some of a person's energy away as infrared light.

The net power radiated is the difference between the power emitted and the power absorbed:

$P_{\mathrm {net} }=P_{\mathrm {emit} }-P_{\mathrm {absorb} }.\,$

Applying the Stefan–Boltzmann law,

$P_{\rm {net}}=A\sigma \varepsilon \left(T^{4}-T_{0}^{4}\right).$

The total surface area of an adult is about 2 m², and the mid- and far-infrared emissivity of skin and most clothing is near unity, as it is for most nonmetallic surfaces. Skin temperature is about 33 °C, but clothing reduces the surface temperature to about 28 °C when the ambient temperature is 20 °C. Hence, the net radiative heat loss is about

$P_{\rm {net}}=100\ \mathrm {W} .$

The total energy radiated in one day is about 9 MJ (megajoules), or 2000 kcal (food calories). Basal metabolic rate for a 40-year-old male is about 35 kcal/(m²·h), which is equivalent to 1700 kcal per day assuming the same 2 m² area. However, the mean metabolic rate of sedentary adults is about 50% to 70% greater than their basal rate.

There are other important thermal loss mechanisms, including convection and evaporation. Conduction is negligible – the Nusselt number is much greater than unity. Evaporation via perspiration is only required if radiation and convection are insufficient to maintain a steady state temperature (but evaporation from the lungs occurs regardless). Free convection rates are comparable, albeit somewhat lower, than radiative rates. 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

$\lambda _{\rm {peak}}={\frac {2.898\times 10^{6}\ \mathrm {K} \cdot \mathrm {nm} }{305\ \mathrm {K} }}=9500\ \mathrm {nm} .$

For this reason, thermal imaging devices for human subjects are most sensitive in the 7000–14000 nanometer range.

Temperature relation between a planet and its star

The blackbody law may be used to estimate the temperature of a planet orbiting the Sun.

Earth's longwave thermal radiation intensity, from clouds, atmosphere and ground

The temperature of a planet depends on several factors:

• Incident radiation from its sun
• 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.

This example is concerned with the balance of incident and emitted radiation, which is the most important impact for the inner planets in the Solar System.

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

The Earth only has an absorbing area equal to a two dimensional disk, rather than the surface of a sphere.

$P_{\rm {S\ emt}}=4\pi R_{\rm {S}}^{2}\sigma T_{\rm {S}}^{4}\qquad \qquad (1)$

where

$\sigma \,$ is the Stefan–Boltzmann constant,

$T_{\rm {S}}\,$ is the surface temperature of the Sun, and

$R_{\rm {S}}\,$ 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:

$P_{\rm {SE}}=P_{\rm {S\ emt}}\left({\frac {\pi R_{\rm {E}}^{2}}{4\pi D^{2}}}\right)\qquad \qquad (2)$

where

$R_{\rm {E}}\,$ is the radius of the planet and

$D\,$ is the astronomical unit, the distance between the Sun and the planet.

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

$P_{\rm {abs}}=(1-\alpha )\,P_{\rm {SE}}\qquad \qquad (3)$

Even though the planet only absorbs as a circular area $\pi R^{2}$, 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

$P_{\rm {emt\,bb}}=4\pi R_{\rm {E}}^{2}\sigma T_{\rm {E}}^{4}\qquad \qquad (4)$

where $T_{\rm {E}}$ is the temperature of the planet. 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 ${\overline {\epsilon }}$ of the radiation that a black body would emit where ${\overline {\epsilon }}$ is the average emissivity in the IR range. The power emitted by the planet and its atmosphere is then:

$P_{\rm {emt}}={\overline {\epsilon }}\,P_{\rm {emt\,bb}}\qquad \qquad (5)$

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:

$P_{\rm {abs}}=P_{\rm {emt}}\qquad \qquad (6)$

Substituting the expressions for solar and planet power in equations 1–6 and simplifying yields:

$T_{E}=T_{S}{\sqrt {\frac {R_{S}{\sqrt {\frac {1-\alpha }{\overline {\varepsilon }}}}}{2D}}}$

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

Temperature of Earth

Substituting the measured values for the Sun and Earth yields:

$T_{\rm {S}}=5778\ \mathrm {K} ,$

$R_{\rm {S}}=6.96\times 10^{8}\ \mathrm {m} ,$

$D=1.496\times 10^{11}\ \mathrm {m} ,$

$\alpha =0.306\$

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

$T_{\rm {E}}=254.356\ \mathrm {K}$

or −18.8 °C.

This is the temperature of the Earth if it radiated as a perfect black body in the infrared, ignoring greenhouse effects, and assuming an unchanging albedo. The Earth in fact radiates not quite as a perfect black body in the infrared which will raise the estimated temperature a few degrees above the effective temperature. If we wish to estimate what the temperature of the Earth would be if it had no atmosphere, then we could take the albedo and emissivity of the moon as a good estimate. The albedo and emissivity of the moon are about 0.1054 and 0.95 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. Similarly using albedo 0.3 and solar constant of 1372 W m⁻², one obtains an effective temperature of 255 K.

Doppler effect for a moving black body

The relativistic Doppler effect causes a shift in the frequency f of light originating from a source that is moving in relation to the observer, so that the wave is observed to have frequency f':

$f'=f{\frac {1-{\frac {v}{c}}\cos \theta }{\sqrt {1-v^{2}/c^{2}}}},$

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. 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

$T'=T{\sqrt {\frac {c-v}{c+v}}}.$

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.

See also

• Bolometer
• Color temperature
• Effective temperature
• Emissivity
• Infrared thermometer
• Photon polarization
• Pyrometry
• Rayleigh–Jeans law
• Super black
• Thermal radiation
• Thermography
• Ultraviolet catastrophe
• Sakuma–Hattori equation

Further reading

• Kroemer, Herbert; Kittel, Charles (1980). Thermal Physics (2nd ed.). W. H. Freeman Company. ISBN 0716710889.{{cite book}}: CS1 maint: multiple names: authors list (link)
• Tipler, Paul; Llewellyn, Ralph (2002). Modern Physics (4th ed.). W. H. Freeman. ISBN 0716743450.{{cite book}}: CS1 maint: multiple names: authors list (link)

External links

• Dimensionless Constants and Blackbody Radiation Laws EJTP, 8, No. 25 (2011)
• Calculating Blackbody Radiation Interactive calculator with Doppler Effect. Includes most systems of units.
• Cooling Mechanisms for Human Body – From Hyperphysics
• Descriptions of radiation emitted by many different objects
• BlackBody Emission Applet
• "Blackbody Spectrum" by Jeff Bryant, Wolfram Demonstrations Project, 2007.
• Carbon Nanotube Black Body (AIST nano tech 2009)