The Wien's law of radiation was an empirical test of Wilhelm Wien , of a black body radiation emitted thermal radiation , depending on the wavelength to be described. It reproduces Vienna's law of displacement qualitatively correctly.

## history

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. ${\ displaystyle T}$

Based on thermodynamic considerations, Vienna was able to derive its law of displacement, which established a connection between the wavelength distributions at different temperatures:

" If you imagine [...] the energy plotted at a temperature as a function of the wavelength, this curve would remain unchanged at a changed temperature if the scale of the drawing were changed so that the ordinates were reduced in the ratio 1 / θ 4 and the abscissas would be increased in the ratio θ. "

The wavelength distribution of the radiation was thus still unknown, but an additional condition was found which the real wavelength distribution had to be subject to in the event of a temperature change. Nowadays, this general form of the law of displacement no longer plays a role, because Planck's law of radiation describes the spectral displacement in the event of a change in temperature very specifically. Only the temperature-related shift of the radiation maximum, which already follows from the law of displacement, has survived under the name of Wien's law of displacement .

With the help of some additional assumptions, Vienna was able to derive a radiation law that behaves as required by the displacement law in the case of temperature changes.

## definition

Comparison of the Viennese and Planck's radiation law

The Viennese radiation law reads in the form given by Wilhelm Wien in 1896:

${\ displaystyle \ phi (\ lambda) = {\ frac {C} {\ lambda ^ {5}}} \ cdot {\ frac {1} {\ mathrm {e} ^ {\ left ({\ frac {c} {\ lambda T}} \ right)}}}}$

With

• ${\ displaystyle \ phi (\ lambda)}$: specific spectral emission
• ${\ displaystyle \ lambda}$: Wavelength
• ${\ displaystyle T}$: absolute temperature
• ${\ displaystyle C}$: First radiation constant (see below)
• ${\ displaystyle c}$: Second radiation constant.

As expected, it has a radiation maximum, but delivers values ​​that are too low in the long-wave range, see picture.

## Connection with Planck's law of radiation

Max Planck corrected the above Deficiency in 1900 through a clever interpolation between Wien's radiation law (correct for small wavelengths) and Rayleigh-Jeans law (correct for large wavelengths). He found

${\ displaystyle \ phi (\ lambda) = {\ frac {C} {\ lambda ^ {5}}} \ cdot {\ frac {1} {\ mathrm {e} ^ {\ left ({\ frac {c} {\ lambda T}} \ right)} - ​​1}}}$

and from it developed Planck's law of radiation within a few weeks , which is also considered to be the birth of quantum physics .

For small wavelengths or small temperatures (in general: for small products ) the exponential term in the denominator of Planck's formula becomes large against one: ${\ displaystyle \ lambda}$${\ displaystyle T}$${\ displaystyle \ lambda \ cdot T}$

${\ displaystyle \ lambda \ cdot T \ ll 1 \ Rightarrow \ mathrm {e} ^ {\ left ({\ frac {c} {\ lambda T}} \ right)} \ gg 1}$

In these cases, the one can be neglected compared to the larger term:

${\ displaystyle \ Rightarrow e ^ {\ left ({\ frac {c} {\ lambda T}} \ right)} - ​​1 \ approx e ^ {\ left ({\ frac {c} {\ lambda T}} \ right)} \ gg 0}$

and Planck's formula goes over into Wien’s formula, which in this sense can be regarded as the limit case of Planck’s law of radiation.

## Constants

It is remarkable that the constants assumed by Wien and by Planck were expressed by the natural constants Boltzmann constant , the speed of light and the new constant : ${\ displaystyle C}$${\ displaystyle c}$ ${\ displaystyle h}$

${\ displaystyle C = 2 \ cdot \ pi \ cdot h \ cdot c_ {0} ^ {2}}$
${\ displaystyle c = {\ frac {h \ cdot c_ {0}} {k _ {\ rm {B}}}}}$

With

The "auxiliary constant" was later named Planck's quantum of action in honor of Planck . ${\ displaystyle h}$

## literature

• Willy Wien: About the energy distribution in the emission spectrum of a black body. In: Annals of Physics . No. 294, 1896. pp. 662-669 ( doi : 10.1002 / andp.18962940803 , PDF file ; 317 kB).
• Max Planck: About an improvement of Wien's spectral equation. In: Negotiations of the German physical society. 2, No. 13, 1900, pp. 202–204 ( PDF file ; 88 kB)