Black Body Radiation And The Ultraviolet Catastrophe Essay Writing

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 blackbody would release an infinite amount of energy, contradicting the principles of conservation of energy and indicating that a new model for the behaviour of blackbodies was needed.

The term "ultraviolet catastrophe" was first used in 1911 by Paul Ehrenfest, but the concept originated with the 1900 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[edit]

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 .

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 follow the Rayleigh–Jeans law, and 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[edit]

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 only be emitted or absorbed in discrete packets, called quanta, of energy: , where h is Planck's constant. Planck's assumptions led to the correct form of the spectral distribution functions: . Albert Einstein solved the problem by postulating that Planck's quanta were real physical particles—what we now call photons, not just a mathematical fiction. He 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]

See also[edit]

References[edit]

Further reading[edit]

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).
  1. ^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. 
  2. ^Mason, Stephen F. (1962). A History of the Sciences. Collier Books. p. 550. 
  3. ^Stone, A. Douglas (2013). Einstein and the Quantum. Princeton University Press. 
  4. ^"The Nobel Prize in Physics: 1921". Nobelprize.org. Nobel Media AB. 2017. Retrieved December 13, 2017.  

On Dec. 14, 1900, in a lecture to the German Physics Society in Berlin, Max Planck presented a mathematical derivation that introduced quantum mechanics principles to the field of classical physics. In his paper, “On the Theory of the Law of Energy Distribution in the Normal Spectrum,” Planck suggested in mathematical terms that energy could be emitted in discrete packets called quanta.

Prior to Planck’s work, physicists had relied on the classical wave theory of light to explain the behavior of light and the radiation of electromagnetic energy. However, classical theory could not explain sufficiently the absorption or emission of light, such as when metal glows following exposure to high temperatures. In 1860, German physicist Gustav Robert Kirchoff conceived of the blackbody—a hypothetical ideal body or surface that absorbs and reemits all radiant energy falling on it. The blackbody became central to efforts to explain energy radiation.

In the 1890s, German physicist Wilhelm Wien determined in that the maximum wavelength reached by radiation is inversely proportional to the absolute temperature of the emitting body. The accuracy of Wien’s law, however, was lost on longer wavelengths. A formula developed by English physicist John William Strutt (Lord Rayleigh) was similarly problematic at short wavelengths. Rayleigh’s formula actually led to the “ultraviolet catastrophe”—the incorrect notion that radiation emission from a blackbody could be of unlimited intensity.

Planck’s radiation law. E = energy; λ = wavelength; h = Planck’s constant; c = the speed of light; k = the Boltzmann constant; and T = absolute temperature. (Encyclopædia Britannica, Inc.)

Planck’s derivation became known as Planck’s radiation law. And it is now understood that a blackbody heated to several hundred degrees emits primarily infrared radiation, but at higher temperatures, as radiation energy increases, the emitted spectrum shifts to shorter wavelengths that fall within the visible portion of the electromagnetic spectrum. This is why, for example, several moments after heating a fire poker, its tip becomes visibly red. Planck received the 1918 Nobel Prize for Physics for his contributions to the development of quantum theory.

Photo credits: Encyclopædia Britannica, Inc.

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