Kirchhoff's law of radiation

The kirchhoff radiation law describes the connection between absorption and emission of a real body in thermal equilibrium . It states that radiation absorption and emission correspond to each other at a given wavelength: A body that absorbs well also radiates well.

The German physicist Gustav Robert Kirchhoff formulated the radiation law in 1859 while he was developing the method of spectroscopy . It formed the cornerstone of the investigation of thermal radiation and thus also of Max Planck's quantum hypothesis .

Kirchhoff's law of radiation: (a) A body that absorbs well also radiates well. (b) Reversed case.

Terms

• The spectral radiance (unit: W m ⁻² Hz ⁻¹ sr ⁻¹ ) of a body of temperature indicates which radiant power the body emits at the frequency in the direction given by the polar angle and the azimuth angle per unit area, per frequency interval and per solid angle unit . The spectral radiance of a black body is independent of direction and is given by Planck's law of radiation . ${\displaystyle L_{\Omega \nu }(\beta ,\varphi ,\nu ,T)}$ ${\displaystyle T}$ ${\displaystyle \nu }$ ${\displaystyle \beta }$ ${\displaystyle \varphi }$ ${\displaystyle L_{\Omega \nu }^{o}(\nu ,T)}$

• The spectral irradiance (unit: W m ⁻² Hz ⁻¹ sr ⁻¹ ) is the radiant power that hits the body at the frequency from the direction given by the polar angle and the azimuth angle per unit area, per frequency interval and per solid angle unit. The spectral radiation density is always the same as the spectral radiation density of the surrounding radiation field. If the body is surrounded by cavity radiation , its spectral radiation density and thus also the spectral radiation density are given by Planck's law of radiation. ${\displaystyle K_{\Omega \nu }(\beta ,\varphi ,\nu )}$ ${\displaystyle \nu }$ ${\displaystyle \beta }$ ${\displaystyle \varphi }$

• The directed spectral absorption indicates which fraction the body absorbs at the temperature and the frequency of the spectral irradiance coming from the direction ( ). ${\displaystyle a_{\nu }^{\prime }(\beta ,\varphi ,\nu ,T)}$ ${\displaystyle T}$ ${\displaystyle \nu }$ ${\displaystyle \beta ,\varphi }$

• The directional spectral emissivity is the ratio of the spectral radiance emitted by a body of temperature at the frequency in the direction to the spectral radiance emitted by a blackbody of the same temperature: ${\displaystyle \varepsilon _{\nu }^{\prime }(\beta ,\varphi ,\nu ,T)}$ ${\displaystyle T}$ ${\displaystyle \nu }$ ${\displaystyle (\beta ,\varphi )}$

${\displaystyle \varepsilon _{\nu }^{\prime }(\beta ,\varphi ,\nu ,T)={\frac {L_{\Omega \nu }(\beta ,\varphi ,\nu ,T)}{L_{\Omega \nu }^{o}(\nu ,T)}}}$.

Derivation

Let the body under consideration be in thermal equilibrium with cavity radiation of temperature . The body is a portion of the incident radiation in accordance with its degree of absorption absorb . In order to maintain the equilibrium, it must re-radiate the absorbed amount of energy at the same frequencies in the same directions in order to replace the energy taken from the cavity. ${\displaystyle T}$

For the frequency and the direction ( ) the absorbed radiant power is given by ${\displaystyle \nu }$ ${\displaystyle \beta ,\varphi }$

${\displaystyle a_{\nu }^{\prime }(\beta ,\varphi ,\nu ,T)\cdot K_{\Omega \nu }(\beta ,\varphi ,\nu )=a_{\nu }^{\prime }(\beta ,\varphi ,\nu ,T)\cdot L_{\Omega \nu }^{o}(\nu ,T)}$.

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

${\displaystyle L_{\Omega \nu }(\beta ,\varphi ,\nu ,T)}$.

In thermal equilibrium, absorbed and emitted radiation power must be the same:

${\displaystyle L_{\Omega \nu }(\beta ,\varphi ,\nu ,T)=a_{\nu }^{\prime }(\beta ,\varphi ,\nu ,T)\cdot L_{\Omega \nu }^{o}(\nu ,T)}$.

Rearranging results

${\displaystyle {\frac {L_{\Omega \nu }(\beta ,\varphi ,\nu ,T)}{a_{\nu }^{\prime }(\beta ,\varphi ,\nu ,T)}}=L_{\Omega \nu }^{o}(\nu ,T)}$.

The kirchhoff law was already known in this form in the 19th century ( Gustav Robert Kirchhoff , 1859). On the left side there are quantities that depend on the special properties of the body under consideration, while due to thermodynamic arguments in connection with the cavity radiation it was already known that the function on the right side is a universal function independent of the body properties solely of wavelength and temperature must be (“kirchhoff's function”). This function could later be explicitly stated by Max Planck and is known today as Planck's radiation law .

This formulation also shows that the spectral radiance of a body, whose degree of absorption assumes the value 1 for all directions and frequencies, corresponds to the spectral radiance given by Planck's law of radiation: A black body is a Planckian radiator .

Since the spectral radiance of the body must increase proportionally to the degree of absorption in order to ensure constancy of the right-hand side, but the degree of absorption cannot exceed the value 1, the spectral radiance of the body cannot rise above the spectral radiance of the blackbody: No body can emit more radiation than a black body of the same temperature.

The black body is therefore used as a reference. If one relates the spectral radiance of a body by introducing its emissivity to the spectral radiance of the blackbody

${\displaystyle L_{\Omega \nu }(\beta ,\varphi ,\nu ,T)=\varepsilon _{\nu }^{\prime }(\beta ,\varphi ,\nu ,T)\cdot L_{\Omega \nu }^{o}(\nu ,T)}$,

equating the absorbed and emitted spectral radiance yields:

${\displaystyle a_{\nu }^{\prime }(\beta ,\varphi ,\nu ,T)\cdot L_{\Omega \nu }^{o}(\nu ,T)=\varepsilon _{\nu }^{\prime }(\beta ,\varphi ,\nu ,T)\cdot L_{\Omega \nu }^{o}(\nu ,T)\Leftrightarrow a_{\nu }^{\prime }(\beta ,\varphi ,\nu ,T)=\varepsilon _{\nu }^{\prime }(\beta ,\varphi ,\nu ,T)}$.

In thermal equilibrium, the directed spectral absorption coefficient and the directed spectral emissivity are the same for the same frequencies and directions:

${\displaystyle a_{\nu }^{\prime }=\varepsilon _{\nu }^{\prime }}$

Good absorbers are good emitters.

Kirchhoff's law of radiation is initially valid in thermal equilibrium, i.e. when the radiation balance between the radiating body and the radiation bath interacting with it is balanced. As a rule, it also applies to a very good approximation for bodies that are not in thermal equilibrium with their surroundings, as long as their directed spectral absorption and emissivity levels do not change under these conditions.

restrictions

Integrated radiation quantities

The equality of absorption and emissivity applies in full generality only to the directional spectral absorption and the directed spectral emissivity . However, these quantities, which describe the explicit direction and frequency dependence of the absorption and emission processes, are often not available. For a material, mostly only the hemispherical spectral emissivity integrated over all directions of the half-space or the directed total emissivity integrated over all frequencies or the hemispheric total emissivity integrated over all directions of the half-space and over all frequencies is known . Here the equality with the corresponding integrated degrees of absorption only applies in special cases, especially since the integrated degrees of absorption also depend on the direction and frequency distribution of the incident radiation, i.e., in contrast to the emissivities, they are not pure material properties. ${\displaystyle \varepsilon _{\nu }(\nu ,T)}$ ${\displaystyle \varepsilon ^{\prime }(\beta ,\varphi ,T)}$ ${\displaystyle \varepsilon (T)}$

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

• For diffuse (i.e. with direction-independent emissivity) radiating surfaces, the hemispherical spectral absorption coefficient is equal to the hemispherical spectral and the directional spectral emissivity:

${\displaystyle a_{\nu }(\nu ,T)=\varepsilon _{\nu }(\nu ,T)=\varepsilon _{\nu }^{\prime }(\nu ,T)}$

• For gray (i.e. with frequency-independent emissivity) radiating surfaces, the directed total absorption factor is equal to the directed total emissivity and the directed spectral emissivity:

${\displaystyle a^{\prime }(\beta ,\varphi ,T)=\varepsilon ^{\prime }(\beta ,\varphi ,T)=\varepsilon _{\nu }^{\prime }(\beta ,\varphi ,T)}$

• For diffuse and gray surfaces the total hemispherical absorption coefficient is equal to the total hemispherical emissivity and the directed spectral emissivity:

${\displaystyle a(T)=\varepsilon (T)=\varepsilon _{\nu }^{\prime }(T)}$

Real bodies are often diffuse emitters to a good approximation . The requirement for a gray surface is usually poorly fulfilled, but can be regarded as given if absorbed and emitted radiation only show noticeable intensities in the frequency ranges in which the emissivity is approximately constant.

• Non-metals (i.e. electrical non-conductors, dielectrics) usually behave as diffuse emitters to a good approximation. In addition, their directed spectral emissivity is in many cases approximately constant for wavelengths over approx. 1 to 3 μm. For the radiation exchange in the long-wave range ( thermal radiation at temperatures that are not too high), dielectrics can therefore often be treated approximately as diffuse gray emitters and it is . ${\displaystyle a(T)\approx \varepsilon (T)}$
• In the case of metals (i.e. electrical conductors), on the other hand, the directional dependency of the emissivity generally does not allow an approximation by a diffuse radiator. In addition, their spectral emissivity is not constant even at long wavelengths, so that they do not represent gray emitters; it is therefore usually . Oxide layers or contamination can change the radiation properties of metals and approach those of dielectrics. ${\displaystyle a(T)\neq \varepsilon (T)}$

Even dielectrics can no longer be treated as gray emitters if the radiation exchange to be considered includes shorter-wave spectral ranges, i.e. if in particular the absorption of solar radiation is to be considered. Dielectrics typically have relatively low levels of spectral absorption and emissivity for wavelengths below 1 to 3 μm and relatively high levels above that. The solar radiation is in the range of low levels of absorption, so, integrated over all wavelengths, is slightly absorbed. The thermal radiation is in the range of high emissivities, so it is effectively emitted when integrated over all wavelengths. The same applies to metals in which the spectral emissivity is higher at short wavelengths than at longer wavelengths. In these cases, the total absorption coefficient and the total emission coefficient can assume different values.

The following table compares the hemispherical total absorption coefficient for solar radiation and the hemispherical total emissivity at = 300 K for some materials: ${\displaystyle a}$ ${\displaystyle \varepsilon }$ ${\displaystyle T}$

material	${\displaystyle a}$	${\displaystyle \varepsilon }$
Roofing felt, black	0.82	0.91
Brick, red	0.75	0.93
Zinc white	0.22	0.92
Snow, clean	0.20 ... 0.35	0.95
Polished chrome	0.40	0.07
Gold, polished	0.29	0.026
Polished copper	0.18	0.03
Copper, oxidized	0.70	0.45

Areas painted white can remain relatively cool when exposed to solar radiation (low radiation absorption, high heat emission). On the other hand, metal foils with special selective coatings can heat up strongly in solar radiation (radiation absorption level up to 0.95, heat emission level <0.05, use in solar collectors as "heat traps"). White painted radiators can appear friendly bright in daylight (i.e. in the solar spectrum) (low absorption), while in the long-wave range they radiate heat well (high emission). Snow is only melted slowly by solar radiation (solar radiation is in the range of low absorption), but much faster by the heat radiation of a wall: heat radiation is in the range of high emission, i.e. also high absorption.

Outside of thermal equilibrium

The equality of the degree of absorption and emissivity must be maintained in thermal equilibrium for all directions and for all frequencies. Deviations from this can occur in the non-equilibrium:

• Diffraction effects on the surface can deflect incident radiation in another direction, so that overall more radiation power is emitted in that direction than would be permissible even for a black body ( ) ${\displaystyle \varepsilon >1}$ . However, this does not mean a violation of the conservation of energy, since the excess energy has only been redistributed and is missing elsewhere. In the sum over all angles the energy conservation is preserved.
• An optically non-linear (e.g. fluorescent ) body can absorb radiation of one frequency and emit it at another frequency. Again, it is just a matter of redistribution: the conservation of energy is not given for a certain frequency, but is integrated over all frequencies.

Application examples

• Well-reflective bodies absorb little radiation and are therefore also bad emitters themselves. Thus, emergency blankets often made of reflective material to minimize heat about losing radiation. Thermos flasks are internally mirrored in order on the one hand to reflect the heat radiation of a content to be kept warm and on the other hand to give off as little heat radiation as possible to a content to be kept cold.
• A kiln is heated and kept in thermal equilibrium. Then no structures can be seen inside the furnace: Objects in the furnace that absorb the radiation well are also good emitters. Objects that absorb poorly are either transparent (gases) or they reflect the part of the radiation that they do not emit themselves. All elements in the furnace therefore have the same radiation density and can therefore not be differentiated based on the radiation.

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

• A body that appears transparent does not absorb any radiation in the visible spectral range , so it cannot emit any radiation in this range. Since the earth's atmosphere is transparent, it cannot radiate thermally excited light at visible wavelengths. Light that comes from the atmosphere is either sunlight scattered by impurities or the air molecules ( diffuse radiation ) or is created in the higher layers through recombination of ionized air molecules ( airglow ) or shock excitation ( polar light ). In other selective wavelength ranges, on the other hand, trace gases contained in the air (water vapor, carbon dioxide, ozone) are sometimes absorbed very intensively, which then also emit thermal radiation just as intensely at the same wavelengths ( greenhouse gases ). If the eye were sensitive in these areas, the atmosphere, because it is emitting and absorbing at the same time, would appear to be a shining mist.
• The Fraunhofer's lines in the solar spectrum arise from the fact that gases in cooler areas of the photosphere or in the earth's atmosphere absorb certain wavelengths of the light emitted by deeper photosphere layers. If one observes such a gas under conditions in which it emits light itself, this light is composed of spectral lines which occur at exactly the same wavelengths as the Fraunhofer's lines caused by this gas. The gas emits particularly well at those wavelengths at which it also absorbs well.

Blue glowing spirit flame and its line spectrum.

• Hot gas flames give off little light. The bluish light arises from radiation excitations of the gas molecules (see picture). In furnaces, the heat is mainly transferred by flame radiation, which must therefore be kept as intensive as possible by choosing suitable combustion conditions or using additives. If the oxygen supply is reduced, black soot forms due to incomplete combustion , which glows like a black body (see also candle ). Soot production can also be controlled by adding carbon-rich hydrocarbons or coal dust (carburization). Only on the emission lines of the combustion products water vapor and carbon dioxide ( greenhouse gases ) located in the infrared does the flame emit radiation even without soot particles.

Examples to which the kirchhoff radiation law is not applicable:

• A cold lamp (e.g. light-emitting diode , fluorescent lamp ) emits significantly more radiant energy at individual wavelengths than a black body at the same temperature. Kirchhoff's law does not allow emissivities greater than one for thermal radiators. However, it cannot be used here, as the light in these illuminants is not generated thermally but by other types of excitation (see luminescence ).

literature

• HD Baehr, K. Stephan: Heat and mass transfer. 4th edition. Springer-Verlag, Berlin 2004, ISBN 3-540-40130-X ; Cape. 5: thermal radiation