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Ultraviolet catastrophe

## Summary

The ultraviolet catastrophe is the error at short wavelengths in the Rayleigh–Jeans law (depicted as "classical theory" in the graph) for the energy emitted by an ideal black body. The error, much more pronounced for short wavelengths, is the difference between the black curve (as classically predicted by the Rayleigh–Jeans law) and the blue curve (the measured curve as predicted by Planck's law).

The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was the prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium would emit unbound quantity of energy as wavelength decreases, corresponding to ultraviolet light.[1]

The term "ultraviolet catastrophe" was first used in 1911 by Paul Ehrenfest,[2] but the concept originated with the 1900 statistical derivation of the Rayleigh–Jeans law. The phrase refers to the fact that the Rayleigh–Jeans law accurately predicts experimental results at radiative frequencies below 105 GHz, but begins to diverge with empirical observations as these frequencies reach the ultraviolet region of the electromagnetic spectrum.[3]

Since the first use of this term, it has also been used for other predictions of a similar nature, as in quantum electrodynamics and such cases as ultraviolet divergence.

## Problem

The Rayleigh-Jeans law is an approximation to the spectral radiance of electromagnetic radiation as a function of wavelength from a black body at a given temperature through classical arguments. For wavelength ${\displaystyle \lambda }$, it is:

${\displaystyle B_{\lambda }(T)={\frac {2ck_{\mathrm {B} }T}{\lambda ^{4}}},}$

where ${\displaystyle B_{\lambda }}$ is the spectral radiance, the power emitted per unit emitting area, per steradian, per unit wavelength; ${\displaystyle c}$ is the speed of light; ${\displaystyle k_{\mathrm {B} }}$ is the Boltzmann constant; and ${\displaystyle T}$ is the temperature in kelvins. For frequency ${\displaystyle \nu }$, the expression is instead

${\displaystyle B_{\nu }(T)={\frac {2\nu ^{2}k_{\mathrm {B} }T}{c^{2}}}.}$

This formula is obtained from the equipartition theorem of classical statistical mechanics which states that all harmonic oscillator modes (degrees of freedom) of a system at equilibrium have an average energy of ${\displaystyle k_{\rm {B}}T}$.

The "ultraviolet catastrophe" is the expression of the fact that the formula misbehaves at higher frequencies, i.e. ${\displaystyle B_{\nu }(T)\to \infty }$ as ${\displaystyle \nu \to \infty }$.

An example, from Mason's A History of the Sciences,[4] illustrates multi-mode vibration via a piece of string. As a natural vibrator, the string will oscillate with specific modes (the standing waves of a string in harmonic resonance), dependent on the length of the string. In classical physics, a radiator of energy will act as a natural vibrator. And, since each mode will have the same energy, most of the energy in a natural vibrator will be in the smaller wavelengths and higher frequencies, where most of the modes are.

According to classical electromagnetism, the number of electromagnetic modes in a 3-dimensional cavity, per unit frequency, is proportional to the square of the frequency. This therefore implies that the radiated power per unit frequency should be proportional to frequency squared. Thus, both the power at a given frequency and the total radiated power is unlimited as higher and higher frequencies are considered: this is clearly unphysical as the total radiated power of a cavity is not observed to be infinite, a point that was made independently by Einstein and by Lord Rayleigh and Sir James Jeans in 1905.

## Solution

In 1900, Max Planck derived the correct form for the intensity spectral distribution function by making some strange (for the time) assumptions. In particular, Planck assumed that electromagnetic radiation can be emitted or absorbed only in discrete packets, called quanta, of energy: ${\textstyle E_{\text{quanta}}=h\nu =h{\frac {c}{\lambda }}}$, where h is Planck's constant, v is the frequency of light, c is the speed of light and λ is the wavelength of light. Planck's assumptions led to the correct form of the spectral distribution functions:

${\displaystyle B_{\lambda }(\lambda ,T)={\frac {2hc^{2}}{\lambda ^{5}}}{\frac {1}{e^{hc/(\lambda k_{\mathrm {B} }T)}-1}}}$

Albert Einstein (in 1905) and Satyendra Nath Bose (in 1924) solved the problem by postulating that Planck's quanta were real physical particles – what we now call photons, not just a mathematical fiction. They modified statistical mechanics in the style of Boltzmann to an ensemble of photons. Einstein's photon had an energy proportional to its frequency and also explained an unpublished law of Stokes and the photoelectric effect.[5] This published postulate was specifically cited by the Nobel Prize in Physics committee in their decision to award the prize for 1921 to Einstein.[6]

## References

1. ^ Ultraviolet Radiation in the Solar System, M. Vázquez, Arnold Hanslmeier, year 2005, pages 6-7
2. ^ Ehrenfest 1911
3. ^ McQuarrie, Donald A.; Simon, John D. (1997). Physical chemistry: a molecular approach (rev. ed.). Sausalito, Calif.: Univ. Science Books. ISBN 978-0-935702-99-6.
4. ^ Mason, Stephen F. (1962). A History of the Sciences. Collier Books. p. 550.
5. ^ Stone, A. Douglas (2013). Einstein and the Quantum. Princeton University Press.
6. ^ "The Nobel Prize in Physics: 1921". Nobelprize.org. Nobel Media AB. 2017. Retrieved December 13, 2017. For his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect.

### Bibliography

• Ehrenfest, P. (1911). "Welche Z{\"u}ge der Lichtquantenhypothese spielen in der Theorie der W{\"a}rmestrahlung eine wesentliche Rolle?" [In which features of the light quantum hypothesis does thermal radiation play an essential role?]. Annalen der Physik. 341 (11): 91–118. Bibcode:1911AnP...341...91E. doi:10.1002/andp.19113411106.