Definitions

# Wien's displacement law

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.

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

where

$lambda_\left\{max\right\} ,$ 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 2.897 768 5(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 = 2.897768 5(51) × 106 nm K.

## Explanation and familiar approximate applications

The law is named for Wilhelm Wien, who formulated the relationship in 1893 based on a thermodynamic argument. Wien considered adiabatic expansion of a cavity containing waves of light in thermal equilibrium. He showed that under adiabatic expansion or contraction, the energy of light changes in the exact same way as the frequency. This means that the peak frequency should change with temperature as the energy goes. Wien did not interpret his constant b as a new fundamental constant of nature. This was done by Planck.

Wien's displacement law states that the hotter an object is, the shorter the wavelength at which it will emit most of its radiation, and further that the frequency for maximal or peak radiation power is found by dividing Wien's constant by the temperature in kelvins.

Examples:

• Light from the Sun and Moon. The surface temperature (or more correctly, the effective temperature) of the Sun is 5778 K. Using Wien's law, this temperature corresponds to a peak emission at a wavelength of 2.89777 million nm K/ 5778 K = 502 nm = about 5000 Å. This wavelength is (not incidentally) fairly in the middle of the most sensitive part of land animal visual spectrum acuity. Even nocturnal and twilight-hunting animals must sense light from the waning day and from the moon, which is reflected sunlight with this same wavelength distribution. Also, the average wavelength of starlight maximal power is in this region, due to the sun being in the middle of a common temperature range of stars.

[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].

Wien's constant may be used in different units, and many examples to calculate familiar situation types of radiation required use of only one or two significant figures:

• Light from incandescent bulbs and fires. 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. It is easy to calculate that a wood fire at 1500 K puts out peak radiation at 3 million nm K /1500 K = 2000 nm = 20,000 Å. This is far more energy in the infrared than in the visible band, which ends about 7500 Å.
• Radiation from mammals and the living human body. Mammals at roughly 300 K emit peak radiation at 3 thousand μm K / 300 K = 10 μm, in the far infrared. This is therefore the range of infrared wavelengths that pit viper snakes and passive IR cameras must sense.
• The wavelength of radiation from the Big Bang. A typical application of Wien's law would also be to the blackbody radiation resulting from the Big Bang. Remembering that Wien's displacement constant is about 3 mm K, and the temperature of the Big Bang background radiation is about 3 K (actually 2.7 K), it is apparent that the microwave background of the sky peaks in power at 2.9 mm K / 2.7 K = just over 1 mm wavelength in the microwave spectrum. This provides a convenient rule of thumb for why microwave equipment must be sensitive on both sides of this frequency band, in order to do effective research on the cosmic microwave background.

## Frequency form

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

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

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.

## Derivation

Wilhelm Wien first derived this law in 1893 by applying the laws of thermodynamics to electromagnetic radiation. As is typically the case with thermodynamic arguments, Wien's derivation determines the functional form of the relationship but does not specify the values of the constants b (in the temperature form) or $alpha$ (in the frequency form.) A modern variant of Wien's derivation can be found in the textbook by Wannier . Today, the usual practice is to derive the relationship from Planck's law of black body radiation, as this procedure also yields expressions for the constants b and $alpha$ in terms of fundamental constants.

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

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

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

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

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

If we define

$xequiv\left\{hcoverlambda kT \right\}$

then

$\left\{xover 1-e^\left\{-x\right\}\right\}-5=0$

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 $x$

$x = 4.965114231744276ldots$     (dimensionless)

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

$lambda_\left\{max\right\} = \left\{hcover x \right\}\left\{1over kT\right\} = \left\{2.89776829ldots times 10^6 mathrm\left\{nm cdot K\right\} over T\right\}$.

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. The upshot is simply to substitute 3 for 5 in the equation for the peak wavelength, solved with x = 2.82143937212…. Using 4 instead yields the appropriate wavelength-frequency-neutral peak at x = 3.92069039487….