Wien's displacement law
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 of black-body radiation as a function of wavelength at any given temperature. However, it had been discovered by 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, Wien 's displacement law states that the spectral radiance of black-body radiation per unit wavelength, peaks at the wavelength "λ" given by: where "T" is the absolute temperature. "b" 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. Wien's displacement law may be referred to as "Wien's law", a term which is also used for the Wien approximation. Examples. Wien's displacement law is relevant to some everyday experiences: Discovery. The law is named for Wilhelm Wien, who derived it 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 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. This upward shift in νmax 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 formula_2 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 later formulated the correct black-body radiation function it did not explicitly include Wien's constant "b". Rather, Planck's constant "h" was created and introduced into his new formula. From Planck's constant "h" and  the Boltzmann constant "k", Wien's constant "b" can be obtained. Frequency-dependent formulation. For spectral flux considered per unit frequency formula_3 (in hertz), Wien's displacement law describes a peak emission at the optical frequency formula_4 given by: or equivalently where is a constant resulting from the maximization equation, "k" is the Boltzmann constant, "h" is the Planck constant, and "T" is the temperature (in kelvins). 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 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: Differentiating "u"(λ,"T") with respect to λ and setting the derivative equal to zero gives: which can be simplified to give: By defining: the equation becomes one in the single variable "x": which is equivalent to: This  equation is solved by where formula_14 is the principal branch of the Lambert W function, and gives to double precision floating point accuracy. 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 as a function of frequency ν: The preceding process using this equation yields: The net result is: This is similarly solved with Lambert's W function: giving to double precision floating point accuracy. 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" to "λ" will result in the same value as integrating the frequency  distribution between the two frequencies that correspond to "λ" and λ, namely from "c"/"λ" to "c"/"λ". 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 value 4 to solve the implicit equation 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 taking considering infinitesimal intervals of formula_19 rather of frequency itself.) This is perhaps  a more intuitive way of presenting "wavelength of peak emission". That yields to double precision floating point accuracy. 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.