Wien's displacement law

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The wavelength corresponding to the peak emission in various black body spectra as a function of temperature

Wien's displacement law is a law of physics that states that there is an inverse relationship between the wavelength of the peak of the emission of a black body and its temperature.

$\lambda_{max} = \frac{b}{T}$

where

$\lambda_{max} \,$ is the peak wavelength in meters,

$T \,$ is the temperature of the blackbody in kelvins (K), and

b is a constant of proportionality, called Wien's displacement constant and equals 2.897 7685(51) × 10^–3 m K (2002 CODATA recommended value)

The two digits between the parentheses denotes the uncertainty (the standard deviation at 68.27% confidence level) in the two least significant digits of the mantissa.

For optical wavelengths, 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 7685(51) × 10⁶ nm K.

1 Explanation
2 Frequency form
3 Derivation
4 External links

[edit] Explanation

Wien's law states that the hotter an object is, the shorter the wavelength at which it will emit most of its radiation. For example, the surface temperature of the Sun is 5778 K. Using Wien's law, this temperature corresponds to a peak emission at a wavelength of 502 nm. This wavelength is fairly in the middle of the visual spectrum (see for example the article color), because of the spread resulting in white light. Due to the Rayleigh scattering of blue light by the atmosphere this white light is separated somewhat, resulting in a blue sky and a yellow sun.

A lightbulb has a glowing wire with a somewhat lower temperature, resulting in yellow light, and something that is "red hot" is again a little less hot.

The law is named for Wilhelm Wien, who formulated the relationship in 1893 based on empirical data.

[edit] Frequency form

In terms of frequency f (in hertz), Wien's displacement law becomes

$f_{max} = { \alpha k \over h} T \approx (5.879 \times 10^{10} \ \mathrm{Hz/K}) \cdot T$

where

$\alpha \approx 2.821439...$ is a constant resulting from the numerical solution of the maximization equation,

k is Boltzmann's constant,

h is Planck's constant, and

T is temperature (in kelvin).

Because the spectrum resulting from Planck's law of black body radiation takes a different shape in the frequency domain from that of the wavelength domain, the frequency location of the peak emission does not correspond to the peak wavelength using the simple relationship between frequency, wavelength, and the speed of light.

[edit] Derivation

Wilhelm Wien formulated this law, in 1893, based entirely on empirical observations, prior to the development of Planck's law of black body radiation. With the benefit of hindsight, however, it is now possible to derive Wien's law as a direct consequence of Planck's more general expression.

From Planck's law, we know that the spectrum of black body radiation is

$u(\lambda) = {8\pi h c\over \lambda^5}{1\over e^{h c/\lambda kT}-1}$

The value of $λ$ for which this function is maximized is sought. To find it, we differentiate $u (λ)$ with respect to $λ$ and set it equal to zero

${ \partial u \over \partial \lambda } = 8\pi h c\left( {hc\over kT \lambda^7}{e^{h c/\lambda kT}\over \left(e^{h c/\lambda kT}-1\right)^2} - {1\over\lambda^6}{5\over e^{h c/\lambda kT}-1}\right)=0$

${hc\over\lambda kT }{1\over 1-e^{-h c/\lambda kT}}-5=0$

If we define

$x\equiv{hc\over\lambda kT }$

then

${x\over 1-e^{-x}}-5=0$

This equation cannot be solved in terms of elementary functions. It can be solved in terms of Lambert's Product Log function but an exact solution is not important in this derivation. One can easily find the numerical value of $x$

$x = 4.965114231744276\ldots$ (dimensionless)

Solving for the wavelength $λ$ in units of nanometers, and using units of kelvins for the temperature yields:

$\lambda_{max} = {hc\over kx }{1\over T} = {2.89776829\ldots \times 10^6 \ \mathrm{nm \cdot K} \over T}$ .

The frequency form of Wien's displacement law is derived using similar methods, but starting with Planck's law in terms of frequency instead of wavelength.