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Wien's Law was an early attempt to describe the blackbody radiation spectrum, proposed by the German scientist Wilhelm Wien in 1896. The Wien Law closely approximated the true shape of the blackbody spectrum at short wavelengths, but ultimately failed because it relied solely on classical physics. It was superseded by Planck's radiation law which correctly describes the blackbody spectrum at all wavelengths. However, Wien's Law is still used as a convenient approximation to the blackbody spectrum at short wavelengths.

The mystery of the blackbody

The nature of blackbody radiation was believed to be one of the last great unsolved problems in physics at the close of the nineteenth century. Although much work had been done on both thermodynamics and electromagnetic radiation, no one had yet combined both to explain the spectrum of light emitted by hot objects, first described by Kirchhoff in 1859.

Kirchhoff showed that blackbody radiation should be some function of the wavelength and temperature, as in

Eλ = constant × f(&lambda,T)

a relation based upon proven thermodynamic principles. The problem was that no one knew the exact form of that function, f.

Wien's Law

In 1893, Wilhelm Wien first derived his Displacement Law, performing a gedankenexperiment with a hypothetical blackbody -- a cavity, filled with radiation, that could be expanded and contracted at will. His work showed that the wavelength at which most radiation is emitted by a blackbody is inversely proportional to its temperature. This was an important step, as it suggested how the blackbody spectrum was related to the temperature, but it still didn't fully uncover the form of Kirchhoff's function.

In his 1896 paper (Ann. Physik, v.58, 622), Wien combined his Displacement Law with the Stefan-Boltzmann Law, and suggested that the blackbody spectrum could be fit to a function of the form

Eλ = (c1 / λ5) / exp(c2/λT)

where c1 and c2 are empirically defined constants. This relation is known as Wien's Law of blackbody radiation.

Wien's assumptions on exactly why the blackbody spectrum should have this form aren't entirely correct, but they matched observations very well. For a while, Wien's Law was believed to be reasonably correct. However, experiments to measure the blackbody spectrum in the far infrared (out to 60 microns) showed that Wien's Law diverged badly from reality. Thus, the Wien Law was incorrect, and the mystery of the blackbody continued.

Planck gets it right

In 1900 Max Planck derived his relation for a blackbody, which was found to be in perfect agreement with measurements at all wavelengths and all temperatures -- the problem of the blackbody was solved. Planck's Law appears nearly identical to Wien's except for the constants and a slight difference in the exponential term:

Eλ = (2hc2/ λ5) / (exp(hc/ λk T) - 1)

However, that "slight" difference masks a radical change in physical assumptions, particularly in Planck's treatment of light as individual "quanta", and the use of statistical mechanics to describe the emitting blackbody as a collection of quantum oscillators. Planck's relation is also radically different in that the constants are derived from physical principles, not from experimental fitting. The correct blackbody relation is generally considered one of the most important nails in the coffin of classical physics, and the birth of quantum mechanics.

Although Wien's law was proven incorrect, it was an important step in Planck's eventual derivation of the correct blackbody relation. Wien received the 1911 Nobel Prize in Physics mainly for his derivation of the Displacement Law and for his work on using cavities as experimental blackbodies, but his attempt at a law of blackbody radiation certainly contributed to his award.

Like many quantum mechanical and relativistic relations, Planck's Law reduces to the classical laws of both Wien and Rayleigh and Jeans in the short- and long-wavelength approximations. For the Wien Law, this happens when hc/λ is much larger than kT, making the exponential term much larger than -1.

Rybicki, G. and Lightman, A., Radiative Processes In Astrophysics, J. Wiley and Sons, 1976
Richtmeyer, F.K., Introduction to Modern Physics, McGraw-Hill, 1934

blackbody radiation
Rayleigh-Jeans Law -- Planck's radiation law -- Wien's Law

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