Rayleigh Jeans Law

Comparing the Rayleigh-Jeans law (blue) to the Wien's (red) and the Planck's radiation law (black), for a blackbody of temperature 1000 K.

The Rayleigh-Jeans law describes the dependence of the specific radiation of a black body on the light wavelength , which is theoretically to be expected at a given temperature in the context of classical electrodynamics and statistical thermodynamics . It was derived for the first time in 1900 by the English physicist John William Strutt, 3rd Baron Rayleigh , but his formula still had an incorrect prefactor. The correct formula was published five years later by the English physicist, astronomer and mathematician Sir James Jeans . ${\displaystyle M}$ ${\displaystyle \lambda }$

The Rayleigh-Jeans law only agrees with the measurements at long wavelengths (see picture, the measured values ​​correspond to the Planck curve). In the case of small wavelengths, on the other hand, it delivers values ​​that are far too large, which make the energy of the total radiation, the spectral radiation integrated over the entire wavelength range, strive towards infinity at any temperature . This behavior marks a failure of classical physics and is therefore referred to as an ultraviolet catastrophe . ${\displaystyle T>0\;\mathrm {K} }$

The Rayleigh Jeans Law reads:

${\displaystyle M(\lambda )\,\mathrm {d} \lambda ={\frac {2\,c\,\pi }{\lambda ^{4}}}\,k_{\mathrm {B} }\,T\,\mathrm {d} \lambda }$

With

• ${\displaystyle M(\lambda )}$: Spectral specific radiation
• ${\displaystyle c}$: Speed ​​of light
• ${\displaystyle k_{\mathrm {B} }}$: Boltzmann constant
• ${\displaystyle T}$: absolute temperature of the blackbody.

The behavior at small wavelengths, i.e. high frequencies (and thus correspondingly high energy of the quanta), is correctly described by Wien's radiation law of 1896, which, however, cannot be explained with classical physics. In 1900 Max Planck found the radiation law named after him , which agrees with the measurements at all temperatures in the entire wavelength range, and whose first successful theoretical interpretation is regarded as the beginning of quantum physics . Planck's law of radiation is an interpolation formula for the other two laws and contains them as a limiting case of large and small wavelengths.

Derivation

The starting point for deriving the Rayleigh-Jeans law is cavity radiation . Standing light waves are formed in a cavity, which, due to the boundary conditions of having to form wave nodes on the walls of the cavity, can only assume discrete wave numbers . For the sake of simplicity, a cube with the edge length is assumed below . Then the condition on the wave number is ${\displaystyle a}$ ${\displaystyle k}$

${\displaystyle k={\frac {\pi }{a}}{\sqrt {n_{x}^{2}+n_{y}^{2}+n_{z}^{2}}}}$

with three whole numbers . The number of modes per volume unit thus results from ${\displaystyle n_{x},n_{y},n_{z}}$ ${\displaystyle n}$

${\displaystyle n={\frac {8\pi \nu ^{3}}{3c^{3}}}}$

with frequency . ${\displaystyle \textstyle \nu ={\frac {k}{2\pi }}c}$

The spectral modes dense, so the number of modes per frequency interval is: ${\displaystyle \mathrm {d} \nu }$

${\displaystyle {\frac {\mathrm {d} n}{\mathrm {d} \nu }}\mathrm {d} \nu ={\frac {8\pi \nu ^{2}}{c^{3}}}\mathrm {d} \nu }$

and the spectral energy density is the product of the spectral mode density with the mean energy per mode : ${\displaystyle w(\nu )}$ ${\displaystyle {\bar {E}}_{\nu }}$

${\displaystyle w(\nu )\mathrm {d} \nu ={\frac {\mathrm {d} n}{\mathrm {d} \nu }}{\bar {E}}_{\nu }\mathrm {d} \nu }$

The Rayleigh-Jeans law assumes the validity of the uniform distribution theorem for the mean energy . This says

${\displaystyle {\bar {E}}_{\nu }=k_{\mathrm {B} }T}$

and thus

${\displaystyle w(\nu )\,\mathrm {d} \nu ={\frac {8\,\pi \,\nu ^{2}}{c^{3}}}\,k_{\mathrm {B} }\,T\,\mathrm {d} \nu }$

To rewrite this in terms of wavelength, the relationship and therefore applies ${\displaystyle \nu =c/\lambda }$

${\displaystyle w(\lambda )\,\mathrm {d} \lambda ={\frac {8\,\pi }{c\,\lambda ^{2}}}\,k_{\mathrm {B} }\,T\,\left|{\frac {\mathrm {d} \nu }{\mathrm {d} \lambda }}\right|\,\mathrm {d} \lambda ={\frac {8\,\pi }{\lambda ^{4}}}\,k_{\mathrm {B} }\,T\,\mathrm {d} \lambda }$.

The Rayleigh-Jeans law follows from the relationship between radiation density and energy density . ${\displaystyle \textstyle M={\frac {c}{4\pi }}w}$

Rayleigh-Jeans law as an approximation of Planck's law of radiation

Planck's law of radiation reads

${\displaystyle M(\lambda )\,\mathrm {d} \lambda ={\frac {2\pi hc^{2}}{\lambda ^{5}}}{\frac {1}{e^{\left({\frac {hc}{\lambda k_{\mathrm {B} }T}}\right)}-1}}\,\mathrm {d} \lambda }$

• ${\displaystyle h}$is Planck's quantum of action

In the long wavelength range

${\displaystyle {\begin{alignedat}{2}&\lambda &&\gg {\frac {hc}{k_{\mathrm {B} }T}}\Leftrightarrow \\&{\frac {hc}{\lambda k_{\mathrm {B} }T}}&&\ll 1,\end{alignedat}}}$

the approximation can be used ( ) and the Rayleigh-Jeans law is obtained immediately. ${\displaystyle e^{x}-1\approx x}$ ${\displaystyle x=(hc)/(\lambda k_{\mathrm {B} }T)}$

The adjacent diagrams show a comparison of the three radiation formulas according to Planck, Vienna and Rayleigh-Jeans (above in linear, below in double logarithmic representation). The Rayleigh-Jeans and Planck predictions are in good agreement for long wavelengths; Rayleigh-Jeans deviates increasingly upwards towards smaller wavelengths. Vienna, on the other hand, describes the borderline case of small wavelengths (here μm) very well, but is clearly too low for larger wavelengths. ${\displaystyle \lambda <5}$

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. ^ L. Rayleigh: Remarks upon the Law of Complete Radiation . In: Phil. Mag. Band 49 , 1900, pp. 539-540 .
2. ↑ JH Jeans: On the partition of energy between matter and Aether . In: Phil. Mag. Band 10 , 1905, pp. 91-98 .
3. ↑ This physically nonsensical divergence of the Rayleigh-Jeans law at small wavelengths (high radiation frequencies) was first described (independently of one another) in 1905 by Einstein, Rayleigh and Jeans. The term ultraviolet catastrophe was first used by Paul Ehrenfest in 1911:
   P. Ehrenfest : Which features of the light quantum hypothesis play an essential role in the theory of thermal radiation? In: Annals of Physics . tape 341 , no. 11 , 1911, pp. 91-118 .
4. ↑ Wolfgang Demtröder: Experimentalphysik 2: Electricity and optics . 3. Edition. Springer, Berlin Heidelberg New York 2004, ISBN 3-540-20210-2 , pp. 201-203 .
5. ↑ Wolfgang Demtröder: Experimental Physics 3: Atoms, Molecules and Solids . 3. Edition. Springer, Berlin Heidelberg New York 2005, ISBN 3-540-21473-9 , pp. 76 .