Kirchhoff's law of thermal radiation

Kirchhoff's law describes the relationship between radiation absorption and emission of a real object in the thermal equilibrium. It states that radiation absorption and emission correspond to each other: A black surface heats up in sunlight on lighter than a white (like white washed houses in warm countries ).

The German physicist Gustav Robert Kirchhoff formulated the radiation law in 1859 while he developed the method of spectroscopy. It formed the foundation for the study of thermal radiation and thus also of Max Planck's quantum hypothesis.

Terms

• The spectral radiance (unit: W m- 2 Hz -1 SR-1) a body temperature indicates which emits radiation power of the body at a frequency in represented by the polar angle and the azimuth angle direction per unit area, per frequency interval and per unit solid angle. The spectral radiance of a black body is independent of direction and is given by Planck's radiation law.

• The spectral irradiance (unit: W m- 2 Hz -1 SR-1) is the radiant power, which meets at the frequency of the given by the polar angle and the azimuth angle direction per unit area, for each frequency interval, and per unit of solid angle on the body. The spectral irradiance is always equal to the spectral radiation density of the surrounding radiation field. If the body is especially surrounded by black body radiation, its spectral radiance and thus the spectral irradiance are given by Planck's radiation law.

• The directional spectral absorptance indicates what fraction of the body is absorbed in the temperature and the frequency of the direction from the () coming spectral irradiance.

• The directional spectral emissivity is the ratio of the temperature of a body at a frequency in the direction () emitted to the spectral radiance emitted by a black body of the same temperature spectral radiance:

Derivation

The body is viewed with blackbody radiation of temperature in thermal equilibrium. The body will absorb a portion of the incident radiation in accordance with its degree of absorption. So that the balance is retained, but it has to emit the amount of energy absorbed again in order to replace the energy extracted from the cavity in each case at the same frequencies in the same directions.

For the frequency and the direction ( ), the radiation absorbed power is given by

The emitted radiation power is determined by the spectral radiance of the body

In thermal equilibrium absorbed and emitted radiation power must be equal:

Rearranging yields

In this form was the Kirchhoff's law in the 19th century known ( Gustav Robert Kirchhoff, 1859). On the left are size, depending upon the particular characteristics of the subject body as was already known due to thermodynamic arguments related to the black-body radiation, that the function on the right side is independent of the body characteristics universal feature alone of the wavelength and the temperature must be ( " Kirchhoff function"). This function could be specified explicitly later by Max Planck and is now known as Planck's radiation law.

This formulation can also be seen immediately that the spectral radiance of a body whose absorption coefficient for all directions and frequencies takes the value 1, with the given by Planck's radiation law spectral radiance matches: a black body is a Planck radiator.

Because the spectral radiance of the body must proportionally increase the absorption coefficient in order to ensure constancy of the right side of the absorption coefficient, but a value of 1 can not exceed the spectral radiation density of the body can not exceed the spectral radiance of the blackbody addition: no body can more radiation emit as a black body of the same temperature.

Plotting the apparent meaning thereby become of the black body radiation as a reference calculation by referring the spectral radiance of a body by introducing its emissivity on the spectral radiance of the black body,

So provides equating the absorbed and emitted spectral radiance:

In thermal equilibrium, the directional spectral absorptance and the directional spectral emissivity are the same for the same frequencies and directions:

Good absorbers are good emitters.

Kirchhoff's radiation law is initially in thermal equilibrium, ie when the radiation balance between the radiating body and interacting with him radiation bath is balanced. It is also usually a very good approximation for the body, which are not in thermal equilibrium with the environment, as long as do not change their directional spectral absorption and emission levels under these conditions.

Limitations

Integrated radiation quantities

The equality of absorption and emissivity applies in full generality only for the directional spectral absorptance and the directional spectral emissivity. These quantities, which describe explicitly the direction and frequency dependence of the absorption and emission processes, however, are often not available. It is known for a material usually only the over all directions in the hemisphere integrated hemispherical spectral emissivity, or integrated over all frequencies directional total emissivity or even just the over all directions of the half-space and integrated over all frequencies hemispherical total emissivity. Here the equality holds with the corresponding integrated absorption coefficients only in special cases, especially as the integrated absorption coefficients also depend on the direction and frequency distribution of the incident radiation, ie, in contrast to the emission levels of purely material properties.

The most important cases in which Kirchhoff's law of radiation still remains valid, are the following:

• For diffuse ( ie with direction -independent emissivity ) radiating surfaces is the hemispherical spectral absorption coefficient equal to the hemispherical spectral and the directional spectral emissivity:
• For gray ( ie with frequency- independent emissivity ) radiating surfaces of the directional total absorption coefficient is equal to the directional total emissivity and the directional spectral emissivity:
• For diffuse and gray radiating surfaces of the hemispherical total absorption coefficient is equal to the hemispherical total emissivity and the directional spectral emissivity:

Real bodies are often a good approximation diffuse emitters. The demand for gray radiant surface is usually poorly met, but can often be regarded approximately as a given when absorbed and emitted radiation only in the frequency ranges have appreciable intensities, where the emissivity is approximately constant.

• Non-metals (ie, electrical insulators, dielectrics ) behave generally in a good approximation as a diffuse reflector. In addition, their directional spectral emissivity approximately constant in many cases for wavelengths greater than about 1 to 3 microns. For the radiation exchange in the long-wavelength region (especially thermal radiation at not too high temperatures) dielectrics may therefore be often approximately treated as a diffuse gray bodies and it is.
• For metals (ie electrical conductors ), however, the directional dependence of the emissivity does not allow approximation by a diffuse emitters usually. In addition, their spectral emissivity at long wavelengths also not constant, so that they also provide no gray light bulb; It is therefore usually. Oxide or contamination can the radiation properties of metals, however, greatly changed and which approach of dielectrics.

Also dielectrics can no longer be treated as Gray bodies when the radiation exchange to be considered shorter-wavelength spectral forms with, so if in particular the absorption of solar radiation is considered. Dielectrics typically for wavelengths below 1-3 microns relatively low, about relatively high spectral absorption and emission levels. The solar radiation in the range low absorbance, ie is integrated over all of the wavelengths absorbed rather low. The thermal radiation is in the range of high emission levels, is thus integrated over all wavelengths, emitted very effective. The same is true for metals, where, however, the spectral emissivity at short wavelengths is usually somewhat higher than at longer wavelengths. In these cases, the total degrees of absorption and total emissivity can assume very different values ​​under certain circumstances.

The following table compares the total hemispherical absorptance of solar radiation and the hemispherical total emissivity at = 300 K for some materials:

White painted surfaces can thus in the solar radiation remain relatively cool ( low radiation absorption, high heat emission). On the other hand, metal foils with special selective coatings in the solar radiation strongly heat ( radiation absorption coefficient to 0.95, thermal emittance < 0.05, use in solar panels as " heat traps" ). White coated radiators can in daylight (ie in the solar spectrum ) appear bright friendly ( low absorption ), while they radiate heat well in the long wavelength range (high emission). Snow from sun radiation slowly melted (solar radiation is in the range of low absorption) by the heat radiation of the wall, however, much quicker: heat radiation is in the range of high emission, including high absorbency.

Outside the thermal equilibrium

The equality of absorption and emissivity must be maintained in thermal equilibrium in each case for all directions and at all frequencies. In the non-equilibrium deviations thereof may occur:

• Beugungsseffekte on the surface can deflect incident beam in a different direction so given that in that direction a total of more radiated power than even for a black body would be allowed (). However, this does not violate the conservation of energy, since the excess energy is simply redistributed and elsewhere is missing. In the sum over all angles, the energy conservation is maintained.
• An optically non-linear (e.g., fluorescent ) body can absorb radiation at one frequency and radiate a different frequency. Again, it involves only a redistribution: The conservation of energy is not given for a particular frequency, but well integrated over all frequencies.

Application Examples

• Good reflective body absorb little radiation, so are themselves poor radiator. Thus, emergency blankets are often made of reflective material to lose as little heat radiation. Thermos flasks are both durable mirrored the one hand to reflect the heat radiation of content and kept warm on the other hand leave as little own heat radiation to a kaltzuhaltenden content.

• A furnace kept heated and become in thermal equilibrium. Then no structures are visible inside the oven: Objects in the oven, which absorb well the radiation are also good emitters. Objects that absorb bad, either transparent ( gas) or it may reflect some of the radiation, they do not radiate themselves. All the elements in the furnace thus have the same irradiance and therefore can not be distinguished on the basis of the radiation.

General: If a body of any kind is related to the thermal radiation in the vacuum in thermal equilibrium, its emitted and reflected radiation total is always equal to the black-body radiation. ( This fact is sometimes referred to as the second Kirchhoff 's law).

• A transparent body appearing in the visible spectral range is no radiation absorbed, therefore it can emit no radiation in that region. Since the Earth's atmosphere is transparent, it can emit at visible wavelengths no thermally excited light. Light that comes from the atmosphere, is either scattered at impurities or air molecules sunlight ( diffuse radiation ) or arises in the higher layers by recombination of ionized air molecules ( airglow ) or collisional excitation ( aurora ). In other selective wavelength ranges, however, trace gases contained (water vapor, carbon dioxide, ozone) absorb some very intense in the air, then on the same wavelengths as well as intense thermal radiation emit (greenhouse gases ). Were the eye in these areas sensitive, seemed to him the atmosphere, because at the same time emitting and absorbing, as luminous mist.

• The Fraunhofer lines in the solar spectrum caused by the fact that gases absorb certain wavelengths of light emitted from deeper photospheric layers light into cooler regions of the photosphere or in the atmosphere. Observing such a gas under conditions in which it emits light itself, then this light is composed of spectral lines, which occur at exactly the same wavelengths as those caused by this gas between Fraunhofer lines. Thus, the gas emits particularly well to those wavelengths, which are also well absorbed.

• Hot gas flame emits little light. The bluish light is produced from non-thermal radiation excitations of the gas molecules ( see figure). In furnaces, the heat transfer happens but mainly by flame radiation, which therefore needs to be kept as intensively as possible by selecting suitable combustion conditions or additives. With a reduced supply of oxygen is formed due to incomplete combustion of black soot, which shines like a black body (see candle). The soot can also be controlled by the addition of carbon-rich hydrocarbons or carbon dust ( carburization ). Only to the infrared emission lines located in the combustion products of carbon dioxide and water vapor (greenhouse gases ) are the flame without soot radiation.

Examples, to the Kirchhoff's law of radiation can not be used:

• A cold illuminant ( eg LED, fluorescent light) emitted at each wavelength significantly more radiation energy than an equally warm blackbody. Kirchhoff's law can emissivities greater than one for thermal emitters not to. It is here, however, does not apply since these bulbs which light is produced by thermally but not other types of excitation (see Luminescence ).