Black Body Radiation And The Ultraviolet Catastrophe Essay

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.

You are referring to the so-called 'ultraviolet catastrophe', in which classical physics predicted that the electromagnetic spectrum emitted by warm objects should show an increasing intensity at higher and higher frequencies. This impossible outcome of course disagreed with experimental evidence, and was eventually explained by postulating that energy could only be emitted in discrete packets of definite size. Just as you can only pay cash in increments of 1 cent (U.S.), nature has a smallest unit of energy which is not infinitely divisible.

Assuming that energy could not exist at arbitrary levels, but instead exists in packages of discrete sizes, the spectrum calculation changed to one that agrees with observations. As temperature increases, radiation emission reaches a peak at higher and higher frequencies but decreases at frequencies beyond that peak.

It was the attempt to explain the 'ultraviolet catastrophe' that lead to the development of quantum physics and understanding of the statistical rather than absolute outcomes of matter/energy interactions.

Classical mechanics could not accurately predict the spectrum of radiation emitted by a heated body. In fact, it predicted that a hot body should radiate an infinite amount of energy.

This means that, according to classical theory, if we switch on an oven, the amount of energy contained in radiation waves inside of it will be infinite. And that's bad news for the chips we put inside...

The electromagnetic radiation we are considering is wave-like in character and the waves that are produced in the oven must fit perfectly. They have a whole number of peaks and troughs in the same way that vibrations on a plucked string fit the length of the string nicely. Even with this restriction, there are still an infinite number of waves that can fit inside the oven - a wave with one oscillation, a wave with two oscillations, a wave with three, etc. Here's the trouble - classical mechanics predicts that each of these waves will exist and that each will contribute the same amount of energy to the total in the oven. That's a total of .... um, .... lots.

It was Max Planck that solved this problem in 1900. He suggested that each wave has an intrinsic, associated energy (totally independent of the temperature of the oven). Waves with smaller wavelengths (and therefore higher frequencies) have higher energies. He gave the following equation to determine the energy of a wave:

Energy of a wave = (Planck's constant) * (Frequency of the wave)

This idea is familiar to us today, we know that x-rays have more energy than rays of light, which in turn are more powerful than radio waves. These are all examples of electromagnetic radiation - but with different frequencies, and therefore, according to Planck's idea, different energies.

This solved the problem of blackbody radiation in the following way. Because each wave has a particular energy, there are only a finite number of waves that have an energy that is less than that associated with the oven. Not all waves that fit in the oven actually exist - there are some who require more energy than is available. If a particular wavelength has too high an energy threshold, it doesn't contribute anything - it doesn't 'wave' at all! This means that the total number of waves (and therefore the total energy) in the oven is finite, and even more exiting, it fits the spectrum discovered experimentally.

It was Einstein who realised what Planck was really doing here. Beneath the concept that each wave has an associated energy is a deeper one - that electromagnetic waves can be 'quantisized'. The idea is that waves can only carry particular values of energy because they actually arrive as particles, each with a specific energy value. It isn't possible for half a particle to arrive ! Solving the blackbody dilemma was therefore one of the first steps to quantum theory - this is ultimately what scientists had to do about the blackbody conflict between theory and experimental evidence - they embarked upon a new theory altogether !

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