Wien approximation

Wien derived his law from thermodynamic arguments, several years before Planck introduced the quantization of radiation. The law may be written as I(\nu, T) = \frac{2 h \nu^3}{c^2} e^{-\frac{h \nu}{k_\text{B}T}}, (note the simple exponential frequency dependence of this approximation) or, by introducing natural Planck units, I(\nu, x) = 2 \nu^3 e^{-x}, where: This equation may also be written as I(\lambda, T) = \frac{2hc^2}{\lambda^5} e^{-\frac{hc}{\lambda k_\text{B} T}}, where I(\lambda, T) is the amount of energy per unit surface area per unit time per unit solid angle per unit wavelength emitted at a wavelength λ. Wien acknowledges Friedrich Paschen in his original paper as having supplied him with the same formula based on Paschen's experimental observations. occurs at a wavelength \lambda_\text{max} = \frac{hc}{5k_\text{B}T} \approx \frac{\mathrm{0.2878 ~ cm \cdot K}}{T}, and frequency \nu_\text{max} = \frac{3k_\text{B}T}{h} \approx \mathrm{6.25 \times 10^{10}~\frac{Hz}{K}} \cdot T. ==Relation to Planck's law==

The Wien approximation was originally proposed as a description of the complete spectrum of thermal radiation, although it failed to accurately describe long-wavelength (low-frequency) emission. However, it was soon superseded by Planck's law, which accurately describes the full spectrum, derived by treating the radiation as a photon gas and accordingly applying Bose–Einstein in place of Maxwell–Boltzmann statistics. Planck's law may be given as I(\nu, T) = \frac{2h\nu^3}{c^2} \frac{1}{e^{\frac{h\nu}{kT}} - 1}. The Wien approximation may be derived from Planck's law by assuming h\nu \gg kT. When this is true, then \frac{1}{e^{\frac{h\nu}{kT}} - 1} \approx \frac{1}{e^{\frac{h\nu}{kT}}} = e^{-\frac{h\nu}{kT}}, and so the Wien approximation gets ever closer to Planck's law as the frequency increases. ==Other approximations of thermal radiation==

The Rayleigh–Jeans law developed by Lord Rayleigh may be used to accurately describe the long wavelength spectrum of thermal radiation but fails to describe the short wavelength spectrum of thermal emission. ==See also==