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The Radiation Spectra of a Blackbody
as a Function of Absolute Temperature

During the 19th century there was an ongoing quest to understand the nature of light. Isaac Newton believed that it was a radiation of particles, or as he called them corpuscles. This belief prevailed during the 18th century. At the beginning of the 19th century Fresnel and others put forth evidence that light was a wave phenomenon. The wave theory prevailed throughout the 19th century culminated in the work of James Clerk Maxwell demonstrating that light was just a special case of electromagnetic waves. The existence of other electromagnetic radiation was established. The wave theory of radiation seemed unassailable.

After the invention of the spectroscope it was possible to establish the spectrum or wavelength profile of light from various sources such as the Sun or heated objects. This prompted a subsidiary quest to understand and explain the spectrum of radiation from heated objects.

The character of radiation can be expressed by its frequency ν or its wavelength λ. These two measures are inversely related, their product being equal to velocity c of the radiation; i.e.,

νλ = c

The frequency of radiation is expressed in cycles per second but a more convenient representation for spectroscopic analysis is the reciprocal of the wavelength, a quantity called the wave number of the radiation. It is usually expressed in inverse centimeters, cm-1.

One major breakthrough in this quest came from the work of Kirchhoff. Kirchhoff established that materials radiated in the manner as they absorbed radiation. It had been found that light passing through transparent materials such as gases lost radiation near specific wavelengths. This was termed the absorption spectrum of the material. Also the change in radiation after reflection off of a solid material demonstrated an absorption spectrum. That the radiation spectrum of a material and its absorption spectrum were the same was a surprise. Now it is realized that absorption and radiation take place only near the frequencies which are characteristic of the structure of molecules in the materials. A molecule has a variety of vibration modes. When radiation of the same frequency as one of these modes hits the molecule, the molecule vibrates and then reradiates that energy but in a random direction. Radiation which is not of a frequency nearly equal to one of the vibration modes of the molecule does not interact with the molecules. If the impinging radiation had to have exactly the wavelength of the discrete spectral lines there would not be much interaction between the radiation an the molecules.

The spectrum is modified by the motion of the molecules. The Doppler effect is the modification of the perceived frequency of radiation due to the motion of the molecule. If the molecule is traveling in opposite direction from the incoming radiation the perceived frequency of the radiation is greater. Thus if radiation were slightly lower frequency than a vibration frequency of a molecule the Doppler effect could bring about a coincidence with the vibration frequency of the molecule. If a molecule were traveling in the same direction as incoming radiation the Doppler effect lowers its perceived frequency and thus could result in the absorption of radiation of a slightly higher frequency.

In effect the lines of the absorption spectrum are broadened by the Doppler effect. They are also broadened by collision frequency within a gas. For more on this broadening of the spectral lines see Spectral Line Shape.

With the insights of Kirchhoff physicists gave special attention to materials that absorb all of the radiation that falls upon them. These are called black bodies. Nothing is a perfect blackbody but there are some close approximations.

When blackbodies are heated they give off radiation with a particular profile or spectrum as shown below.

Once the spectra of blackbody radiation was established there was a concerted effort by the scientific community to explain its nature. One important step was Wien's Law; i.e., that the wavelength of maximum intensity λmax was inversely related to the temperature T of the blackbody.

λmax = γ/T

But the rest of the details of the spectrum of blackbody radiation eluded researchers until at the turn of the 20th century Max Planck discovered that the spectrum could be explained by radiation taking place in discrete increments, called quanta. However he first discovered the formula for the spectra of blackbodies by intuition and trial and error. That formula was

Eλ = C1/[exp(hc/(λkT))−1)λ5]

where C1, h and k are constants. The constant c is the speed of light. The constant h became known as Planck's constant and k is Boltzmann's constant. It was later determined that the constant C1 is equal to 2hc². In terms of frequency the formula is

Bν = C2ν²/[exp(hν/(kT))−1]

where the constant C2 was later determined to be equal to 2h/c².

The important point is that the empirical formula came first. Planck initially announced it as merely an improvement of Wien's radiation law. The empirical fit was impressive enough for Planck to know he had the right formula. The next step for him was to find a derivation of it. Planck was a classical physicist and only resorted to formulating quantum theory after finding that there was no other explanation for the formula. So it is completely wrong to think that Planck hypothesized the quantization of radiation and then derived the formula for the spectra of blackbodies as a function of temperature. Planck revolutionized radiation physics only after finding there was no other choice. He was of a conservative nature and not given to making radical speculations. But when confronted with the radical nature of the physical truth he accepted it.

Max Planck

For more on quantum theory see Planck's Derivation.

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