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Wien's displacement law states that the
black-body radiation Black-body radiation is the thermal electromagnetic radiation within, or surrounding, a body in thermodynamic equilibrium with its environment, emitted by a black body (an idealized opaque, non-reflective body). It has a specific, continuous spec ...
curve for different temperatures will peak at different
wavelengths In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
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
Wilhelm Wien Wilhelm Carl Werner Otto Fritz Franz Wien (; 13 January 1864 – 30 August 1928) was a German physicist who, in 1893, used theories about heat and electromagnetism to deduce Wien's displacement law, which calculates the emission of a blackbody ...
several years before
Max Planck Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial contributions to theoretical ...
developed that more general equation, and describes the entire shift of the spectrum of black-body radiation toward shorter wavelengths as temperature increases. Formally, Wien's displacement law states that the
spectral radiance In radiometry, spectral radiance or specific intensity is the radiance of a surface per unit frequency or wavelength, depending on whether the spectrum is taken as a function of frequency or of wavelength. The SI unit of spectral radiance in freque ...
of black-body radiation per unit wavelength, peaks at the wavelength ''λ''peak given by: :\lambda_\text = \frac where ''T'' is the absolute temperature. ''b'' is a
constant of proportionality In mathematics, two sequences of numbers, often experimental data, are proportional or directly proportional if their corresponding elements have a constant ratio, which is called the coefficient of proportionality or proportionality constant ...
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. Wien's displacement law may be referred to as "Wien's law", a term which is also used for the
Wien approximation Wien's approximation (also sometimes called Wien's law or the Wien distribution law) is a law of physics used to describe the spectrum of thermal radiation (frequently called the blackbody function). This law was first derived by Wilhelm Wien in ...
.


Examples

Wien's displacement law is relevant to some everyday experiences: *A piece of metal heated by a
blow torch A blowtorch, also referred to as a blowlamp, is an ambient air fuel-burning gas lamp used for applying flame and heat to various applications, usually metalworking. Early blowtorches used liquid fuel, carried in a refillable reservoir attach ...
first becomes "red hot" as the very longest
visible wavelength The visible spectrum is the portion of the electromagnetic spectrum that is visible to the human eye. Electromagnetic radiation in this range of wavelengths is called ''visible light'' or simply light. A typical human eye will respond to wavele ...
s 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 Infrared (IR), sometimes called infrared light, is electromagnetic radiation (EMR) with wavelengths longer than those of visible light. It is therefore invisible to the human eye. IR is generally understood to encompass wavelengths from arou ...
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 An incandescent light bulb, incandescent lamp or incandescent light globe is an electric light with a wire filament heated until it glows. The filament is enclosed in a glass bulb with a vacuum or inert gas to protect the filament from oxida ...
(which produces light through thermal radiation) 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 (390–700 nanometers). Consequently, a campfire can keep one warm but is a poor source of visible light. *The effective temperature of the
Sun The Sun is the star at the center of the Solar System. It is a nearly perfect ball of hot plasma, heated to incandescence by nuclear fusion reactions in its core. The Sun radiates this energy mainly as light, ultraviolet, and infrared radi ...
is 5778 Kelvin. Using Wien's law, one finds a peak emission per nanometer (of wavelength) 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. Regardless of how one wants to plot the spectrum, 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. It can be appreciated that a rather large amount of the Sun's radiation falls in the fairly small
visible spectrum The visible spectrum is the portion of the electromagnetic spectrum that is visible to the human eye. Electromagnetic radiation in this range of wavelengths is called ''visible light'' or simply light. A typical human eye will respond to wavele ...
. *The preponderance of emission in the visible range, however, is not the case in most
stars A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Earth ...
. The hot supergiant
Rigel Rigel is a blue supergiant star in the constellation of Orion. It has the Bayer designation β Orionis, which is Latinized to Beta Orionis and abbreviated Beta Ori or β Ori. Rigel is the brightest and most massive componentan ...
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 (''T'' = 12100 K) and the red Betelgeuse (''T'' ≈ 3300 K). While few stars are as hot as Rigel, stars cooler than the sun or even as cool as Betelgeuse are very commonplace. *
Mammals Mammals () are a group of vertebrate animals constituting the class Mammalia (), characterized by the presence of mammary glands which in females produce milk for feeding (nursing) their young, a neocortex (a region of the brain), f ...
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 The Crotalinae, commonly known as pit vipers,Mehrtens JM (1987). ''Living Snakes of the World in Color''. New York: Sterling Publishers. 480 pp. . crotaline snakes (from grc, κρόταλον ''krotalon'' castanet), or pit adders, are a subfa ...
snakes and passive IR cameras must sense. *When comparing the apparent color of lighting sources (including
fluorescent lights A fluorescent lamp, or fluorescent tube, is a low-pressure mercury-vapor gas-discharge lamp that uses fluorescence to produce visible light. An electric current in the gas excites mercury vapor, which produces short-wave ultraviolet ligh ...
,
LED lighting An LED lamp or LED light bulb is an electric light that produces light using light-emitting diodes (LEDs). LED lamps are significantly more energy-efficient than equivalent incandescent lamps and can be significantly more efficient than ...
,
computer monitor A computer monitor is an output device that displays information in pictorial or textual form. A discrete monitor comprises a visual display, support electronics, power supply, housing, electrical connectors, and external user controls. The d ...
s, and
photoflash A flash is a device used in photography that produces a brief burst of light (typically lasting 1/1000 to 1/200 of a second) at a color temperature of about 5500  K to help illuminate a scene. A major purpose of a flash is to illuminate a ...
), it is customary to cite the
color temperature Color temperature is the color of light emitted by an idealized opaque, non-reflective body at a particular temperature measured in kelvins. The color temperature scale is used to categorize the color of light emitted by other light sources ...
. Although the spectra of such lights are not accurately described by the black-body radiation curve, a color temperature (the
correlated color temperature Color temperature is the color of light emitted by an idealized opaque, non-reflective body at a particular temperature measured in kelvins. The color temperature scale is used to categorize the color of light emitted by other light sources ...
) 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 (and an actual filament temperature) of 2000 K. Note that the informal description of the former (bluish) color as "cool" and the latter (reddish) as "warm" is exactly opposite the actual temperature change involved in black-body radiation.


Discovery

The law is named for
Wilhelm Wien Wilhelm Carl Werner Otto Fritz Franz Wien (; 13 January 1864 – 30 August 1928) was a German physicist who, in 1893, used theories about heat and electromagnetism to deduce Wien's displacement law, which calculates the emission of a blackbody ...
, who derived it in 1893 based on a thermodynamic argument. 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. A general principle of thermodynamics is that a thermal equilibrium state, when expanded very slowly, stays in thermal equilibrium. 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 Langley may refer to: People * Langley (surname), a common English surname, including a list of notable people with the name * Dawn Langley Simmons (1922–2000), English author and biographer * Elizabeth Langley (born 1933), Canadian perfor ...
. This upward shift in \nu_ with T is familiar to everyone—when an iron is heated in a fire, the first visible radiation (at around 900 K) is deep red, the lowest frequency visible light. Further increase in T 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. 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 \nu^3 F(\nu/T) for some function ''F'' 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 (which was not yet understood) would shift proportionally in frequency (or inversely proportionally in wavelength) with temperature. When
Max Planck Max Karl Ernst Ludwig Planck (, ; 23 April 1858 – 4 October 1947) was a German theoretical physicist whose discovery of energy quanta won him the Nobel Prize in Physics in 1918. Planck made many substantial contributions to theoretical ...
later formulated the correct black-body radiation function it did not explicitly include Wien's constant ''b''. Rather, the
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
''h'' was created and introduced into his new formula. From the Planck constant ''h'' and the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
''k'', Wien's constant ''b'' can be obtained.


Frequency-dependent formulation

For spectral flux considered per unit
frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz The hertz ...
d\nu (in
hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that one he ...
), Wien's displacement law describes a peak emission at the optical frequency \nu_\text given by: :\nu_\text = kT \approx (5.879 \times 10^ \ \mathrm) \cdot T or equivalently :h \nu_\text = \alpha k T \approx (2.431 \times 10^ \ \mathrm) \cdot T where is a constant resulting from the maximization equation, ''k'' is the
Boltzmann constant The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constan ...
, ''h'' is the
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivale ...
, and ''T'' is the temperature (in
kelvin The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and ph ...
s). With the emission now considered per unit frequency, this peak now corresponds to a wavelength about 76% longer than the peak considered per unit wavelength. The relevant math is detailed in the next section.


Derivation from Planck's law

Planck's law In physics, Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature , when there is no net flow of matter or energy between the body and its environment. A ...
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: :u_(\lambda,T) = . Differentiating ''u''(λ,''T'') with respect to λ and setting the derivative equal to zero gives: : = 2 h c^2\left( - \right)=0, which can be simplified to give: : - 5 = 0. By defining: :x\equiv, the equation becomes one in the single variable ''x'': :-5=0. which is equivalent to: :x = 5(1-e^)\,. This equation is solved by : x = 5+W_0(-5e^) where W_0 is the principal branch of the Lambert ''W'' function, and gives . Solving for the wavelength ''λ'' in millimetres, and using kelvins for the temperature yields: :


Parameterization by frequency

Another common parameterization is by ''frequency''. The derivation yielding peak parameter value is similar, but starts with the form of
Planck's law In physics, Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature , when there is no net flow of matter or energy between the body and its environment. A ...
as a function of frequency ''ν'': :u_(\nu,T) = . The preceding process using this equation yields: :- + 3 = 0. The net result is: :x = 3(1-e^)\,. This is similarly solved with the Lambert ''W'' function: : x = 3+W_0(-3e^) giving . Solving for ''ν'' produces: :


Maxima differ according to parameterization

Notice that for a given temperature, parameterization by frequency implies a different maximal wavelength than parameterization by 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 ''λ''1 to ''λ''2 will result in the same value as integrating the frequency distribution between the two frequencies that correspond to ''λ''1 and λ2, namely from ''c''/''λ''2 to ''c''/''λ''1. 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. Using the implicit equation x = 4(1-e^) yields the peak in the spectral radiance density function expressed in the parameter radiance ''per proportional bandwidth''. (That is, the density of irradiance per frequency bandwidth proportional to the frequency itself, which can be calculated by considering infinitesimal intervals of (or equivalently ) rather of frequency itself.) This is perhaps a more intuitive way of presenting "wavelength of peak emission". That yields . The important point of Wien's law, however, is that ''any'' such wavelength marker, including the median wavelength (or, alternatively, the wavelength below which ''any'' specified percentage of the emission occurs) 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.


See also

*
Wien approximation Wien's approximation (also sometimes called Wien's law or the Wien distribution law) is a law of physics used to describe the spectrum of thermal radiation (frequently called the blackbody function). This law was first derived by Wilhelm Wien in ...
*
Emissivity The emissivity of the surface of a material is its effectiveness in emitting energy as thermal radiation. Thermal radiation is electromagnetic radiation that most commonly includes both visible radiation (light) and infrared radiation, which is ...
* Sakuma–Hattori equation *
Stefan–Boltzmann law The Stefan–Boltzmann law describes the power radiated from a black body in terms of its temperature. Specifically, the Stefan–Boltzmann law states that the total energy radiated per unit surface area of a black body across all wavelengths ...
*
Thermometer A thermometer is a device that measures temperature or a temperature gradient (the degree of hotness or coldness of an object). A thermometer has two important elements: (1) a temperature sensor (e.g. the bulb of a mercury-in-glass thermomete ...
*
Ultraviolet catastrophe The ultraviolet catastrophe, also called the Rayleigh–Jeans catastrophe, was the prediction of late 19th century/early 20th century classical physics that an ideal black body at thermal equilibrium would emit an unbounded quantity of energ ...


References


Further reading

* *


External links


Eric Weisstein's World of Physics
{{blackbody radiation laws Statistical mechanics Foundational quantum physics Light 1893 in science 1893 in Germany