Wien's law of displacement

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A dimmed incandescent filament glows red at approx. 700 ° C, and orange to yellowish-white at higher temperatures

The Viennese law of displacement , named after Wilhelm Wien , states that the wavelength at which a black body at the absolute temperature T emits the most intense radiation is inversely proportional to the temperature. For example, if the temperature of the radiator doubles, the wavelength at which its radiation maximum lies is halved. The general experience that the glowing colors of a glowing body are initially reddish and shift to bluer, i.e. shorter wavelengths with increasing temperature, is based on this connection.

In addition to this particularly frequently used formulation, which considers the wavelength of the most intense radiation, other formulations of the law that describe the frequency of the most intense radiation or the wavelength or frequency of the highest photon rate are more useful for some applications.

Nowadays the law of displacement is most simply derived from Planck's law of radiation , which describes the energy distribution in the radiation of a black body and allows the temperature-dependent position of the radiation maximum to be easily calculated. However, due to thermodynamic considerations, a few years before Planck's discovery, Vienna was able to derive general laws to which the (at that time still unknown) radiation law of the black body had to be subject, and which also include the law of displacement.

General

Planck radiation spectra for different temperatures

The thermal radiation given off by a black body is a mixture of electromagnetic waves from a wide range of wavelengths. The distribution of the radiation intensity over the individual wavelengths is described by Planck's law of radiation . It shows a clear maximum, the position of which can easily be calculated using Wien's displacement law.

The higher the temperature of a body, the shorter the wavelengths the maximum of the distribution. This is why steel, for example, emits invisible infrared radiation (“thermal radiation”) at room temperature, while warm, glowing steel shines dark red. Hot liquid steel glows almost white because, in addition to shifting the maximum into a shorter-wave, bluish area, the intensity of all wavelengths in the spectrum is increased (white light consists of several wavelengths of the visible spectrum ).

Maximum radiation intensity

The most common formulation of the law of displacement describes the wavelength at which the maximum radiation intensity lies. It is:

${\displaystyle \lambda _{\mathrm {max} }={\frac {2897{,}8\,\mathrm {\mu m\ K} }{T}}}$

With

${\displaystyle \lambda _{\mathrm {max} }}$	: Wavelength at which the intensity is maximum (in μm )
${\displaystyle T}$	: absolute temperature of the black body (in K )

Occasionally, instead of the wavelength, the frequency at which the intensity maximum is located is of interest. This frequency is:

${\displaystyle \nu _{\mathrm {max} }=5{,}8789\cdot 10^{10}\,\mathrm {Hz\,K^{-1}} \cdot T}$

This frequency is not the frequency that would correspond to the maximum wavelength according to the conversion formula that applies to all waves , but rather a temperature-independent factor of approx. Smaller. The position of the maximum is therefore different depending on whether the radiation distribution is viewed as a function of the wavelength or the frequency. This fact, which at first appears paradoxical, is explained in more detail in the next section. ${\displaystyle \textstyle \nu =c/\lambda }$ ${\displaystyle \lambda _{\mathrm {max} }}$ ${\displaystyle 0{,}6}$

Maximum of the photon rate

For some processes such as photosynthesis , instead of the incident radiation intensity, the incident photon rate is decisive. The wavelength at which the maximum of the photon rate lies is

${\displaystyle \lambda _{\mathrm {max,Ph} }={\frac {3669{,}7\,\mathrm {\mu m\ K} }{T}}}$

The frequency at which the maximum of the photon rate lies is

${\displaystyle \nu _{\mathrm {max,Ph} }=3{,}3206\cdot 10^{10}\,\mathrm {Hz\,K^{-1}} \cdot T}$

Here, too, the frequency of the maximum is not simply obtained by conversion from the wavelength of the maximum.

Different maxima in wavelength and frequency representation

The fact that the position of the intensity maximum is different, depending on whether the radiation distribution is viewed as a function of the wavelength or the frequency - that there is no objective position of the maximum - is based on the fact that the radiation distribution is a density distribution. In the form of Planck's curve, the wavelengths at the two intensity maxima differ by a factor of approx . Independently of the temperature . In sunlight this means z. B. that the intensity maximum with regard to the wavelength is 500 nm (green), but with regard to the frequency it is approx. 830 nm, i.e. in the near infrared which is invisible to humans . ${\displaystyle 0{,}6}$

In the case of a radiation spectrum, it is namely not possible to specify an associated radiation intensity for a given individual wavelength. Since the radiated power emitted contains a finite number of watts in each wavelength interval, but the interval consists of an infinite number of wavelengths, there are zero watts for each individual wavelength.

Therefore, one does not consider a single wavelength , but a small wavelength interval surrounding the respective wavelength, sets the (finite) radiation power contained in this interval in relation to the (finite) interval width and allows the interval to shrink to zero. Although the power contained in the interval and the width of the interval both approach zero, the ratio of the two tends towards a finite limit value, the spectral power density at the wavelength under consideration ${\displaystyle \lambda }$ ${\displaystyle \Delta P(\lambda )}$ ${\displaystyle \Delta \lambda }$ ${\displaystyle S_{\lambda }}$ ${\displaystyle \lambda }$

${\displaystyle {\frac {\Delta P(\lambda )}{\Delta \lambda }}\to S_{\lambda }(\lambda )}$,

which is measured, for example, in watts per micrometer. Diagrams showing the spectrum of the radiated power show this quantity as a curve plotted against the wavelength. The concept of power spectral density is the same as that underlying mass density , for example : the mass contained in a given point of an object is zero because a point has no volume. But if you consider the mass that is contained in a small volume surrounding the point and calculate its ratio to the volume, you also get a finite numerical value for a volume that shrinks towards zero: the mass density at this point.

If a spectral power density given as a function of the wavelength is to be converted into the frequency-dependent representation , the numerical value for follows from the condition that the radiated power contained in a wavelength interval must be the same as in the frequency interval , the limits of which are obtained by converting the limits of the wavelength interval . ${\displaystyle S_{\lambda }(\lambda )}$ ${\displaystyle S_{\nu }(\nu )}$ ${\displaystyle S_{\nu }(\nu )}$ ${\displaystyle \Delta \lambda }$ ${\displaystyle \Delta P=S_{\lambda }(\lambda )\ \Delta \lambda }$ ${\displaystyle \Delta \nu }$

So consider the interval between the wavelengths and - in the case of solar radiation, these limit wavelengths could be marked, for example, by Fraunhofer lines . The conversion of the interval width into the frequency-dependent representation results ${\displaystyle \lambda _{1}}$ ${\displaystyle \lambda _{2}}$

${\displaystyle \Delta \lambda =\lambda _{2}-\lambda _{1}={\frac {c}{\nu _{2}}}-{\frac {c}{\nu _{1}}}={\frac {c(\nu _{1}-\nu _{2})}{\nu _{1}\nu _{2}}}=(-){\frac {c}{\nu _{1}\nu _{2}}}\Delta \nu }$,

In the following the minus sign is ignored, since only the amounts of the interval widths are of interest. (The minus sign only reflects the fact that the frequency increases when the wavelength decreases.) Infinitesimally small intervals are required to convert the spectra. To do this, let go of the above expression or simply form the derivative ${\displaystyle \nu _{1}\to \nu _{2}}$

${\displaystyle {\frac {\mathrm {d} \lambda }{\mathrm {d} \nu }}={\frac {\mathrm {d} (c/\nu )}{\mathrm {d} \nu }}=(-){\frac {c}{\nu ^{2}}}}$,

From which follows

${\displaystyle \mathrm {d} \lambda ={\frac {c}{\nu ^{2}}}\mathrm {d} \nu }$.

If, for example, the wavelength axis is subdivided into equally large wavelength intervals, the associated frequency intervals become increasingly wider for larger frequencies. ${\displaystyle \mathrm {d} \lambda }$ ${\displaystyle \mathrm {d} \nu }$

Since the radiation power contained in the respective interval considered must be the same regardless of the selected variables: ${\displaystyle \mathrm {d} P}$

${\displaystyle \mathrm {d} P=S_{\nu }(\nu )\ \mathrm {d} \nu }$,

and at the same time

${\displaystyle \mathrm {d} P=S_{\lambda }(\lambda )\ \mathrm {d} \lambda }$

follows for the spectral power density

${\displaystyle S_{\nu }(\nu )\ \mathrm {d} \nu =S_{\lambda }(\lambda )\ \mathrm {d} \lambda =S_{\lambda }(\lambda )\ {\frac {c}{\nu ^{2}}}\ \mathrm {d} \nu }$

and thus

${\displaystyle S_{\nu }(\nu )={\frac {c}{\nu ^{2}}}\ S_{\lambda }(\lambda )}$

The numerical value of the spectral power density in the frequency representation must therefore decrease with increasing frequency by the same factor by which the width of the frequency intervals increases.

For example, if the radiation source under consideration has a constant spectral power density ( ) in the wavelength representation , the spectral power density in the frequency representation decreases as the square of the frequency, i.e. it is in particular not constant: ${\displaystyle S_{\lambda }(\lambda )=const}$

${\displaystyle S_{\nu }(\nu )={\frac {const}{\nu ^{2}}}}$.

If the radiation source has a maximum at a certain wavelength in the wavelength representation , this wavelength is constant in an infinitesimal environment. Then, according to the above explanation, at this wavelength it cannot be constant at this wavelength , i.e. it cannot have a maximum there either . ${\displaystyle S_{\lambda }}$ ${\displaystyle S_{\lambda }}$ ${\displaystyle S_{\nu }}$

Wavelength-dependent quantities that are not density distributions are converted from the wavelength representation into the frequency representation by assigning the quantity assigned to the wavelength to the frequency . Examples are the wavelength-dependent transmittance of a filter or the wavelength-dependent sensitivity curve of the eye . ${\displaystyle \lambda }$ ${\displaystyle \nu ={\tfrac {c}{\lambda }}}$

Derivations

Maximum radiation power in the wavelength display

The spectral specific radiation of a black body of temperature is described by Planck's law of radiation and reads in the wavelength representation: ${\displaystyle T}$

${\displaystyle M_{\lambda }^{o}(\lambda ,T)={\frac {2\pi hc^{2}}{\lambda ^{5}}}\cdot {\frac {1}{e^{\frac {hc}{\lambda kT}}-1}}}$

${\displaystyle M_{\lambda }^{o}(\lambda ,T)}$	:	spectral specific radiation in W · m −2 m −1
${\displaystyle h\,}$	:	Planck's quantum of action in Js
${\displaystyle c\,}$	:	Speed ​​of light in m · s −1
${\displaystyle k\,}$	:	Boltzmann's constant in J · K −1
${\displaystyle T\,}$	:	absolute temperature of the heater surface in K
${\displaystyle \lambda \,}$	:	considered wavelength in m

We are looking for the wavelength at which this function assumes the maximum. Setting the derivative to zero yields: ${\displaystyle \lambda _{\mathrm {max} }}$ ${\displaystyle \lambda }$

${\displaystyle {\frac {hc}{\lambda kT}}\cdot {\frac {e^{\frac {hc}{\lambda kT}}}{e^{\frac {hc}{\lambda kT}}-1}}-5={\frac {hc}{\lambda kT}}\cdot {\frac {1}{1-e^{-{\frac {hc}{\lambda kT}}}}}-5=0}$.

The substitution simplifies the expression to: ${\displaystyle x:={\frac {hc}{\lambda kT}}}$

${\displaystyle {\frac {x}{1-e^{-x}}}-5=0}$.

The numerical solution gives

${\displaystyle x=4{,}965\,114\,231\,7\dots }$,

and the back substitution leads to the similar law of displacement in the wavelength representation:

${\displaystyle \lambda _{\mathrm {max} }={\frac {hc}{xkT}}=:{\frac {b}{T}}={\frac {2897{,}8\,\mathrm {\mu m\,K} }{T}}}$

The wavelength of maximum radiation power shifts so at a temperature change simply inversely proportional to the absolute temperature of the black body: Doubles the temperature of the radiator, the largest radiation power occurs at half the wavelength.

The constant is also known as the shift constant . Since the quantum of action, the speed of light and Boltzmann's constant have exact values since the SI units were redefined in 2019, the constant of displacement has also been exact since then. Their exact value is: ${\displaystyle b}$

${\displaystyle b=2\,897{,}771\,955\ldots \,\mathrm {\mu m\,K} \ }$.

The spectral specific emission of the maximum is proportional to : ${\displaystyle T^{5}}$

${\displaystyle M_{\lambda }^{o}(\lambda _{\mathrm {max} },T)={\frac {2\pi \,x^{5}k^{5}}{h^{4}c^{3}}}{\frac {1}{e^{x}-1}}\cdot T^{5}}$.

Maximum radiated power in the frequency representation

In the frequency representation, the specific spectral emission is given by

${\displaystyle M_{\nu }^{o}(\nu ,T)={\frac {2\pi h\nu ^{3}}{c^{2}}}{\frac {1}{e^{\left({\frac {h\nu }{kT}}\right)}-1}}}$.

Setting the derivative with respect to the frequency to zero gives: ${\displaystyle \nu }$

${\displaystyle 3-{\frac {h\nu }{kT}}{\frac {1}{1-e^{\left(-{\frac {h\nu }{kT}}\right)}}}=0}$.

The substitution simplifies the expression to . ${\displaystyle {\tilde {x}}:={\frac {h\nu }{kT}}}$ ${\displaystyle 3-{\frac {\tilde {x}}{1-e^{-{\tilde {x}}}}}=0}$

The numerical solution gives

${\displaystyle {\tilde {x}}=2{,}8214393721\dots }$,

and back substitution leads to the similar law of displacement in the frequency representation:

${\displaystyle \nu _{\mathrm {max} }={\frac {{\tilde {x}}kT}{h}}=:b'T=5{,}879\cdot 10^{10}\,\mathrm {Hz\,K^{-1}} \cdot T}$

The frequency of maximum radiant power is shifted proportionally to the absolute temperature of the radiator. The exact value of the Viennese constant b ' in the frequency representation is:

${\displaystyle b'=5{,}878\,925\,757\ldots \,\cdot \,10^{10}\,\mathrm {Hz/K} }$.

The spectral specific emission of the maximum is proportional to : ${\displaystyle T^{3}}$

${\displaystyle M_{\nu }^{o}(\nu _{\mathrm {max} },T)={\frac {2\pi \,{\tilde {x}}^{3}k^{3}}{h^{2}c^{2}}}{\frac {1}{e^{\tilde {x}}-1}}\cdot T^{3}}$.

Maximum photon rate in the wavelength display

The spectral specific emission, expressed by the emission rate of the photons, is given in the wavelength representation by

${\displaystyle {\tilde {M}}_{\lambda }^{o}(\lambda ,T)={\frac {2\pi c}{\lambda ^{4}}}{\frac {1}{e^{\frac {hc}{\lambda kT}}-1}}}$.

Setting the derivative to zero yields: ${\displaystyle \lambda }$

${\displaystyle {\frac {hc}{\lambda kT}}\cdot {\frac {1}{1-e^{-{\frac {hc}{\lambda kT}}}}}-4=0}$.

The substitution simplifies the expression to . ${\displaystyle x:={\frac {hc}{\lambda kT}}}$ ${\displaystyle {\frac {x}{1-e^{-x}}}-4=0}$

The numerical solution gives

${\displaystyle {\hat {x}}=3{,}9206903948\dots }$,

and back substitution leads to the similar law of displacement for the photon rate in the wavelength representation:

${\displaystyle \lambda _{\rm {max}}={\frac {hc}{{\hat {x}}kT}}={\frac {3669{,}7\,\mathrm {\mu m\,K} }{T}}}$

The spectral photon rate of the maximum is proportional to . ${\displaystyle T^{4}}$

Maximum photon rate in the frequency representation

In the frequency representation, the spectral specific emission, expressed by the emission rate of the photons, is given by

${\displaystyle {\tilde {M}}_{\nu }^{o}(\nu ,T)={\frac {2\pi \nu ^{2}}{c^{2}}}{\frac {1}{e^{\left({\frac {h\nu }{kT}}\right)}-1}}}$.

${\displaystyle 2-{\frac {h\nu }{kT}}\,{\frac {1}{1-e^{-~{\frac {h\nu }{kT}}}}}=0}$.

The substitution simplifies the expression to . ${\displaystyle {\check {x}}:={\frac {h\nu }{kT}}}$ ${\displaystyle 2-{\frac {\check {x}}{1-e^{-{\check {x}}}}}=0}$

The numerical solution gives

${\displaystyle {\check {x}}=1{,}5936242600\dots }$,

and back substitution leads to the similar law of displacement for the photon rate in the frequency representation:

${\displaystyle \nu _{\rm {max}}={\frac {{\check {x}}kT}{h}}=3{,}320578\cdot 10^{10}\,\mathrm {Hz\,K^{-1}} \cdot T}$

The spectral photon rate of the maximum is proportional to . ${\displaystyle T^{2}}$

Application examples

Taking the sun λ _max  ≈ 500  nm and considers them approximately as a black body , it follows according to the Wien's displacement law its surface temperature to about 5800  K . The temperature determined in this way is called what temperature . Compare it with the effective temperature of 5777 K determined using the Stefan-Boltzmann law . The difference arises from the fact that the assumption on which the two calculations are based, that the sun is a black body, is fulfilled to a good approximation, but not perfectly .

Glow colors provide information about the temperature of hot (above approx. 500 ° C), glowing materials.

Other examples are the radiant surface of the earth and greenhouse gases. At temperatures in the range of 0 ° C, the radiation maximum is in the infrared range around 10 μm. In the case of greenhouse gases, there is also the fact that they are only partially (selective) black bodies.

history

The version of the law of displacement originally drawn up by Vienna described the change in the entire energy distribution curve of a black body with a change in temperature, not just the shift in the radiation maximum.

Based on the experimental investigations by Josef Stefan and the thermodynamic derivation by Ludwig Boltzmann , it was known that the radiant power thermally emitted by a black body with absolute temperature increases with the fourth power of temperature ( main article : Stefan-Boltzmann's law ). The distribution of the radiation energy over the various emitted wavelengths was still unknown. ${\displaystyle T}$

On the basis of thermodynamic considerations, Vienna was able to derive a "law of displacement" which established a connection between the wavelength distributions at different temperatures. If the shape of the energy distribution for a given temperature had been known, one could have obtained the entire curve for any other temperature by shifting and changing the shape of the curve appropriately: ${\displaystyle \varphi (\lambda )}$

“ If the distribution of the energy as a function of the wavelength is given for any temperature , it can now be derived for any other temperature . Let us again think of those as abscissas and those plotted as ordinates. The area between the curve and the abscissa axis is the total energy . You now have to change each one so that it remains constant. One cuts at the site of the original a narrow piece of the width and the content of such Diess piece is located at the position after the change have shifted, from the width is made. Since the energy quantum must remain constant, so is ${\displaystyle \vartheta _{0}}$ ${\displaystyle \vartheta }$ ${\displaystyle \lambda }$ ${\displaystyle \varphi (\lambda )}$ ${\displaystyle \psi }$ ${\displaystyle \lambda }$ ${\displaystyle \lambda \vartheta }$ ${\displaystyle \lambda _{0}}$ ${\displaystyle d\lambda _{0}}$ ${\displaystyle \varphi _{0}\ d\lambda _{0}}$ ${\displaystyle \lambda }$ ${\displaystyle d\lambda _{0}}$ ${\displaystyle d\lambda ={\frac {\vartheta _{0}}{\vartheta }}d\lambda _{0}}$ ${\displaystyle \varphi _{0}\ d\lambda _{0}}$

${\displaystyle \varphi \ d\lambda =\varphi _{0}\ d\lambda _{0},\quad \varphi =\varphi _{0}\ {\frac {d\lambda _{0}}{d\lambda }}=\varphi _{0}\ {\frac {\vartheta }{\vartheta _{0}}}}$.

Now, in addition, each changes proportionally with the temperature according to Stefan's law , so it will be the new ordinate ${\displaystyle \varphi }$ ${\displaystyle {\frac {\vartheta ^{4}}{\vartheta _{0}^{4}}}}$

${\displaystyle \varphi =\varphi _{0}\ {\frac {\vartheta ^{5}}{\vartheta _{0}^{5}}}}$.

In this way you get all the points of the new energy curve. "

The real wavelength distribution of blackbody radiation was still unknown, but an additional condition was found which it had to be subject to when the temperature changed. With the help of some additional assumptions, Vienna was able to derive a radiation law which, in the event of temperature changes, actually behaves as required by the displacement law. However, the comparison with the experiment showed that this Viennese radiation law in the long-wave range delivers values that are too low.

Max Planck was finally able to derive Planck's law of radiation through a clever interpolation between the Rayleigh-Jeans law (correct for large wavelengths) and Wien's radiation law (correct for small wavelengths), which correctly reproduces the emitted radiation in all wavelength ranges.

Nowadays Wien's law of displacement no longer plays a role in its original version, because Planck's law of radiation correctly describes the spectral distribution at any temperature and therefore no “shifts” to a desired temperature are necessary. Only the temperature-related shift of the radiation maximum, which can already be derived from the original version of the shift law , has survived under the name of Wien's shift law .

Web links

Wikibooks: Collection of formulas for Planck's radiation law  - learning and teaching materials

• Hilke Stümpel: Learning module "Physics of Thermal Radiation". Radiation budget. In: WEBGEO basics / climatology. Institute for Physical Geography (IPG) at the University of Freiburg , accessed on August 8, 2017 . Flash player  required .
• Michael Gaedtke: Vienna's law of displacement - derivation step by step. Retrieved on August 8, 2017 (derivation of the law of displacement from Planck's radiation formula). 

Remarks

1. ↑ In addition to the items in the simplicity of using total emitted by the emitter spectral power density also, for example, such a curve spectral radiance , spectral emissivity or volume-based spectral energy density present. The explanations regarding the position of the maxima apply equally in all these cases.

Individual evidence

1. ↑ ^{a b} Helmut Kraus: The Earth's Atmosphere: An Introduction to Meteorology . Springer, 2004, ISBN 978-3-540-20656-9 , pp. 101 ( limited preview in Google Book search).
2. ↑ ^{a b c d} see: JB Tatum: Stellar Atmospheres. Chapter2: Blackbody Radiation. In: On-line lecture notes. S. 5 PDF 217 KB, accessed June 12, 2007.
3. ↑ CODATA Recommended Values. National Institute of Standards and Technology, accessed June 4, 2019 .  Value for ${\displaystyle b}$
4. ↑ ^{a b} J. B. Tatum: Stellar Atmospheres. Chapter2: Blackbody Radiation. In: On-line lecture notes. P. 6 PDF 217 KB, accessed June 12, 2007.
5. ↑ CODATA Recommended Values. National Institute of Standards and Technology, accessed June 4, 2019 .  Value for ${\displaystyle b'}$
6. ↑ Willy Wien: A new relationship between black body radiation and the second law of heat theory. Meeting reports of the Royal Prussian Academy of Sciences in Berlin, publ. D. Kgl. Akad. D. Wiss., Berlin 1893, first half volume 1893, p. 55 ( digitized version )