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.
Formally, the wavelength version of Wien's displacement law states that the spectral radiance of black-body radiation per unit wavelength, peaks at the wavelength given by:
where is the absolute temperature and is a constant of proportionality called Wien's displacement constant, equal to or.
This is an inverse relationship between wavelength and temperature. 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. For visible radiation, hot objects emit bluer light than cool objects. If one is considering the peak of black body emission per unit frequency or per proportional bandwidth, one must use a different proportionality constant. 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,, differs.
Wien's displacement law may be referred to as "Wien's law", a term which is also used for the Wien approximation.
In "Wien's displacement law", the word displacement refers to how the intensity-wavelength graphs appear shifted for different temperatures.
Examples
Wien's displacement law is relevant to some everyday experiences:- A piece of metal heated by a blow torch first becomes "red hot" as the very longest visible wavelengths appear red, then becomes more orange-red as the temperature is increased, and at very high temperatures would be described as "white hot" as shorter and shorter wavelengths come to predominate the black body emission spectrum. Before it had even reached the red hot temperature, the thermal emission was mainly at longer infrared wavelengths, which are not visible; nevertheless, that radiation could be felt as it warms one's nearby skin.
- One easily observes changes in the color of an incandescent light bulb as the temperature of its filament is varied by a light dimmer. As the light is dimmed and the filament temperature decreases, the distribution of color shifts toward longer wavelengths and the light appears redder, as well as dimmer.
- A wood fire at 1500 K puts out peak radiation at about 2000 nanometers. 98% of its radiation is at wavelengths longer than 1000 nm, and only a tiny proportion at visible wavelengths. Consequently, a campfire can keep one warm but is a poor source of visible light.
- The effective temperature of the Sun is 5778 kelvin. Using Wien's law, one finds a peak emission per nanometer at a wavelength of about 500 nm, in the green portion of the spectrum near the peak sensitivity of the human eye. On the other hand, in terms of power per unit optical frequency, the Sun's peak emission is at 343 THz or a wavelength of 883 nm in the near infrared. In terms of power per percentage bandwidth, the peak is at about 635 nm, a red wavelength. About half of the Sun's radiation is at wavelengths shorter than 710 nm, about the limit of the human vision. Of that, about 12% is at wavelengths shorter than 400 nm, ultraviolet wavelengths, which is invisible to an unaided human eye. A large amount of the Sun's radiation falls in the fairly small visible spectrum and passes through the atmosphere.
- The preponderance of emission in the visible range, however, is not the case in most stars. The hot supergiant Rigel emits 60% of its light in the ultraviolet, while the cool supergiant Betelgeuse emits 85% of its light at infrared wavelengths. With both stars prominent in the constellation of Orion, one can easily appreciate the color difference between the blue-white Rigel and the red Betelgeuse. While few stars are as hot as Rigel, stars cooler than the Sun or even as cool as Betelgeuse are very commonplace.
- Mammals with a skin temperature of about 300 K emit peak radiation at around 10 μm in the far infrared. This is therefore the range of infrared wavelengths that pit viper snakes and passive IR cameras must sense.
- When comparing the apparent color of lighting sources, it is customary to cite the color temperature. Although the spectra of such lights are not accurately described by the black-body radiation curve, a color temperature is quoted for which black-body radiation would most closely match the subjective color of that source. For instance, the blue-white fluorescent light sometimes used in an office may have a color temperature of 6500 K, whereas the reddish tint of a dimmed incandescent light may have a color temperature of 2000 K. Note that the informal description of the former color as "cool" and the latter as "warm" is exactly opposite the actual temperature change involved in black-body radiation.
Discovery
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. This upward shift in with is familiar to everyone—when an iron is heated in a fire, the first visible radiation is deep red, the lowest frequency visible light. Further increase in causes the color to change to orange then yellow, and finally blue at very high temperatures for which the peak in radiation intensity has moved beyond the visible into the ultraviolet.
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. From this, he derived the "strong version" of Wien's displacement law: the statement that the blackbody spectral radiance is proportional to for some function of a single variable. A modern variant of Wien's derivation can be found in the textbook by Wannier and in a paper by E. Buckingham
The consequence is that the shape of the black-body radiation function would shift proportionally in frequency with temperature. When Max Planck later formulated the correct black-body radiation function it did not explicitly include Wien's constant. Rather, the Planck constant was created and introduced into his new formula. From the Planck constant and the Boltzmann constant, Wien's constant can be obtained.
Peak differs according to parameterization
| Parameterized by | x | b |
| Mean photon energy | 5327 | |
| 10% percentile | 2195 | |
| 25% percentile | 2898 | |
| 50% percentile | 4107 | |
| 70% percentile | 5590 | |
| 90% percentile | 9376 |
The results in the tables above summarize results from other sections of this article. Percentiles are percentiles of the Planck blackbody spectrum. 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.
Notice that for a given temperature, different parameterizations imply different maximal wavelengths. In particular, the curve of intensity per unit frequency peaks at a different wavelength than the curve of intensity per unit wavelength.
For example, using and parameterization by wavelength, the wavelength for maximal spectral radiance is with corresponding frequency. For the same temperature, but parameterizing by frequency, the frequency for maximal spectral radiance is with corresponding wavelength.
These functions are radiance density functions, which are probability density functions scaled to give units of radiance. 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. Since wavelength and frequency have a reciprocal relation, they represent significantly non-linear shifts in probability density relative to one another.
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. That is to say, integrating the wavelength distribution from to will result in the same value as integrating the frequency distribution between the two frequencies that correspond to and, namely from to. 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.
The important point of Wien's law, however, is that any such wavelength marker, including the median wavelength is proportional to the reciprocal of temperature. That is, the shape of the distribution for a given parameterization scales with and translates according to temperature, and can be calculated once for a canonical temperature, then appropriately shifted and scaled to obtain the distribution for another temperature. This is a consequence of the strong statement of Wien's law.