Wien's displacement law

In physics, Wien's displacement law states that the black-body radiation curve for different temperatures will peak at different wavelengths that are inversely proportional to the temperature.

The shift of that peak is a direct consequence of the Planck radiation law, which describes the spectral brightness or intensity of black-body radiation as a function of wavelength at any given temperature.

However, it had been discovered by German physicist Wilhelm Wien several years before Max Planck developed that more general equation, and describes the entire shift of the spectrum of black-body radiation toward shorter wavelengths as temperature increases.

So the higher the temperature, the shorter or smaller the wavelength of the thermal radiation.

The lower the temperature, the longer or larger the wavelength of the thermal radiation.

However, the form of the law remains the same: the peak wavelength is inversely proportional to temperature, and the peak frequency is directly proportional to temperature.

There are other formulations of Wien's displacement law, which are parameterized relative to other quantities.

For these alternate formulations, the form of the relationship is similar, but the proportionality constant, b, differs.

Wien's displacement law is relevant to some everyday experiences: The law is named for Wilhelm Wien, who derived it in 1893 based on a thermodynamic argument.

[7] Wien considered adiabatic expansion of a cavity containing waves of light in thermal equilibrium.

Using Doppler's principle, 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.

Wien himself deduced this law theoretically in 1893, following Boltzmann's thermodynamic reasoning.

It had previously been observed, at least semi-quantitatively, by an American astronomer, Langley.

causes the color to change to orange then yellow, and finally blue at very high temperatures (10,000 K or more) for which the peak in radiation intensity has moved beyond the visible into the ultraviolet.

From this, he derived the "strong version" of Wien's displacement law: the statement that the blackbody spectral radiance is proportional to

A modern variant of Wien's derivation can be found in the textbook by Wannier[9] and in a paper by E. Buckingham[10] The consequence is that the shape of the black-body radiation function (which was not yet understood) would shift proportionally in frequency (or inversely proportionally in wavelength) with temperature.

When Max Planck later formulated the correct black-body radiation function it did not explicitly include Wien's constant

[11] Only 25 percent of the energy in the black-body spectrum is associated with wavelengths shorter than the value given by the peak-wavelength version of Wien's law.

The density function has different shapes for different parameterizations, depending on relative stretching or compression of the abscissa, which measures the change in probability density relative to a linear change in a given parameter.

The total radiance is the integral of the distribution over all positive values, and that is invariant for a given temperature under any parameterization.

Additionally, for a given temperature the radiance consisting of all photons between two wavelengths must be the same regardless of which distribution you use.

[13] However, the distribution shape depends on the parameterization, and for a different parameterization the distribution will typically have a different peak density, as these calculations demonstrate.

(in hertz), Wien's displacement law describes a peak emission at the optical frequency

Planck's law for the spectrum of black-body radiation predicts the Wien displacement law and may be used to numerically evaluate the constant relating temperature and the peak parameter value for any particular parameterization.

Commonly a wavelength parameterization is used and in that case the black body spectral radiance (power per emitting area per solid angle) is:

in millimetres, and using kelvins for the temperature yields:[17][2] Another common parameterization is by frequency.

The derivation yielding peak parameter value is similar, but starts with the form of Planck's law as a function of frequency

Marr and Wilkin (2012) contend that the widespread teaching of Wien's displacement law in introductory courses is undesirable, and it would be better replaced by alternate material.

They argue that teaching the law is problematic because: They suggest that the average photon energy be presented in place of Wien's displacement law, as being a more physically meaningful indicator of changes that occur with changing temperature.

They recommend that the Planck spectrum be plotted as a "spectral energy density per fractional bandwidth distribution," using a logarithmic scale for the wavelength or frequency.

Black-body radiation as a function of wavelength for various temperatures. Each temperature curve peaks at a different wavelength and Wien's law describes the shift of that peak.
There are a variety of ways of associating a characteristic wavelength or frequency with the Planck black-body emission spectrum. Each of these metrics scales similarly with temperature, a principle referred to as Wien's displacement law. For different versions of the law, the proportionality constant differs—so, for a given temperature, there is no unique characteristic wavelength or frequency.
Blacksmiths work iron when it is hot enough to emit plainly visible thermal radiation .
The color of a star is determined by its temperature, according to Wien's law. In the constellation of Orion , one can compare Betelgeuse ( T ≈ 3800 K, upper left), Rigel ( T = 12100 K, bottom right), Bellatrix ( T = 22000 K, upper right), and Mintaka ( T = 31800 K, rightmost of the 3 "belt stars" in the middle).
Planck blackbody spectrum parameterized by wavelength, fractional bandwidth (log wavelength or log frequency), and frequency, for a temperature of 6000 K