Ultraviolet catastrophe

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 will emit radiation in all frequency ranges, emitting more energy as the frequency increases. By calculating the total amount of radiated energy (i.e., the sum of emissions in all frequency ranges), it can be shown that a black body is likely to release an arbitrarily high amount of energy. This would cause all matter to instantaneously radiate all of its energy until it is near absolute zero – indicating that a new model for the behaviour of black bodies was needed.

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 term "ultraviolet catastrophe" was first used in 1911 by Paul Ehrenfest, 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.[1] Since the first appearance of the 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 ultraviolet catastrophe results 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 $kT$ .

An example, from Mason's A History of the Sciences,[2] 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: $E_{\text{quanta}}=h\nu =h{\frac {c}{\lambda }}$ , where h is Planck's constant. Planck's assumptions led to the correct form of the spectral distribution functions: $B_{\lambda }(\lambda ,T)={\frac {2hc^{2}}{\lambda ^{5}}}{\frac {1}{e^{hc/(\lambda k_{\mathrm {B} }T)}-1}}$ . 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.[3] This published postulate was specifically cited by the Nobel Prize in Physics committee in their decision to award the prize for 1921 to Einstein.[4]

References

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.
Mason, Stephen F. (1962). A History of the Sciences. Collier Books. p. 550.
Stone, A. Douglas (2013). Einstein and the Quantum. Princeton University Press.
"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.

Ultraviolet catastrophe

Problem

Solution

See also

References

Further reading