Wien's radiation law

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

• ${\displaystyle h}$: Planck's quantum of action
• ${\displaystyle c_{0}}$: Speed ​​of light in a vacuum
• ${\displaystyle k_{\rm {B}}}$: Boltzmann constant .

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)

Web links

• Michael Komma: Planck's radiation formula. Accessed August 8, 2017 (comparison of the radiation laws of Planck, Vienna and Rayleigh / Jeans with Maple). 

Individual evidence

1. ↑ W. Wien: About the energy distribution in the emission spectrum of a black body. Annalen der Physik, Volume 294, No. 8, pp. 662-669 (1896); here: p. 666 PDF