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 1896.[1][2][3] The equation does accurately describe the short-
wavelength (high-
frequency) spectrum of thermal emission from objects, but it fails to accurately fit the experimental data for long-wavelength (low-frequency) emission.[3]
Details
Wien derived his law from thermodynamic arguments, several years before Planck introduced the quantization of radiation.[1]
Wien's original paper did not contain the Planck constant.[1] In this paper, Wien took the wavelength of
black-body radiation and combined it with the
Maxwell–Boltzmann energy distribution for atoms. The exponential curve was created by the use of
Euler's numbere raised to the power of the temperature multiplied by a constant. Fundamental constants were later introduced by
Max Planck.[4]
where 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.[1]
The peak value of this curve, as determined by setting the
derivative of the equation equal to zero and solving,[7] occurs at
a wavelength
and frequency
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 However, it was soon superseded by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[5]
The Wien approximation may be derived from Planck's law by assuming . When this is true, then[5]
and so Planck's law approximately equals the Wien approximation at high frequencies.
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.[3][5]