Wien's displacement law

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Wien's displacement law states that the wavelength distribution of thermal radiation from a black body at any temperature has essentially the same shape as the distribution at any other temperature, except that each wavelength is displaced on the graph. Apart from an overall T3 multiplicative factor, the average thermal energy in each mode with frequency ν only depends on the ratio ν/T. Restated in terms of the wavelength λ = c/ν, the distributions at corresponding wavelengths are related, where corresponding wavelengths are at locations proportional to 1/T.

From this general law, it follows that there is an inverse relationship between the wavelength of the peak of the emission of a black body and its temperature when expressed as a function of wavelength, and this less powerful consequence is often also called Wien's displacement law in many textbooks.

where λmax is the peak wavelength, T is the absolute temperature of the black body, and b is a constant of proportionality called Wien's displacement constant, equal to 2.8977685(51)×10−3
(2002 CODATA recommended value)

For wavelengths near the visible spectrum, it is often more convenient to use the nanometer in place of the meter as the unit of measure. In this case, b = 2,897,768.5(51) nm·K.

In the field of plasma physics temperatures are often measured in units of electron volts and the proportionality constant becomes b = 249.71066 nm·eV.


Explanation and familiar approximate applications

The law is named for Wilhelm Wien, who derived it in 1893 based on a thermodynamic argument. Wien considered adiabatic, or slow, expansion of a cavity containing waves of light in thermal equilibrium. He showed that under slow expansion or contraction, the energy of light reflecting off the walls changes in exactly the same way as the frequency. A general principle of thermodynamics is that a thermal equilibrium state, when expanded very slowly stays in thermal equilibrium. The adiabatic principle allowed Wien to conclude that for each mode, the adiabatic invariant energy/frequency is only a function of the other adiabatic invariant, the frequency/temperature.

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