Blackbody Peak Wavelength Calculator

Wien's displacement law falls out of the Planck spectrum: differentiate the spectral radiance with respect to wavelength, set the derivative to zero, and the transcendental equation that results has the root x = 4.965114, giving a peak at lambda_max = hc / (4.965114 k T). Collecting the constants gives Wien's displacement constant b = 2.897771955 x 10^-3 m K, a value now exact because h, c and k were fixed by the 2019 SI redefinition. The inverse relation is steep in practical terms: the Sun's photosphere at 5,772 K peaks near 502 nm in the green, human skin near 305 K peaks at about 9.5 micrometers, and the cosmic microwave background at 2.7255 K peaks near 1.06 mm. One subtlety trips up most first-time users. The location of the peak depends on how the spectrum is parameterized. Expressed per unit frequency instead of per unit wavelength, the maximum sits at a different place, given by nu_max = 5.879 x 10^10 Hz per kelvin, which for the Sun corresponds to about 850 nm rather than 502 nm. Neither answer is wrong; they are peaks of different functions. The other caveat is that real emitters are gray bodies with emissivity below 1 and often strong spectral structure. A stellar spectrum is riddled with absorption lines and molecular bands, so the temperature quoted for a star is an effective temperature fitted to its total output rather than a literal blackbody reading. Thermal imagers exploit the same physics, which is why longwave infrared cameras work in the 8 to 14 micrometer window.

Gustav Kirchhoff introduced the idea of a perfectly black body at Heidelberg in 1859 and 1860, and proved that the ratio of emissive to absorptive power is a universal function of temperature and wavelength alone. Finding that function became one of the central problems of nineteenth-century physics. Josef Stefan inferred the fourth-power total-emission law empirically in 1879, and Ludwig Boltzmann derived it thermodynamically in 1884. Wilhelm Wien, working at the Physikalisch-Technische Reichsanstalt in Berlin, published the displacement law in 1893 and received the 1911 Nobel Prize in Physics for his work on heat radiation. Wien's 1896 distribution fitted short wavelengths but failed in the infrared, while the Rayleigh-Jeans expression developed between 1900 and 1905 diverged at short wavelengths. The decisive data came from the same Berlin institute, where Otto Lummer and Ernst Pringsheim, and separately Heinrich Rubens and Ferdinand Kurlbaum, measured cavity radiation far into the infrared. Rubens brought his results to Max Planck in October 1900; Planck presented an interpolation formula that matched them on 19 October 1900 and, on 14 December 1900, the derivation requiring energy to be exchanged in discrete quanta of size h times frequency. That constant became Planck's constant and the starting point of quantum theory. In 1990 the FIRAS instrument on NASA's COBE satellite measured the cosmic microwave background as the closest blackbody spectrum ever recorded.