Wien approximation

Wien derived his law from thermodynamic arguments, several years before Planck introduced the quantization of radiation.

Wien's original paper did not contain the Planck constant. 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 number e raised to the power of the temperature multiplied by a constant. Fundamental constants were later introduced by Max Planck.{{cite book

|last=Crepeau |first=J.

|year=2009

|chapter=A Brief History of the T⁴ Radiation Law

|title=ASME 2009 Heat Transfer Summer Conference

|publisher=ASME

|volume=1 |pages=59–65

|doi=10.1115/HT2009-88060

|isbn=978-0-7918-4356-7

}}

The law may be written as{{cite book

|last1=Rybicki |first1=G. B.

|last2=Lightman |first2=A. P.

|year=1979

|title=Radiative Processes in Astrophysics

|publisher=John Wiley & Sons

|isbn=978-0-471-82759-7

}}

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:

{{unbulleted list | style = padding-left: 1.5em

| $I(\backslash nu,\; T)$ is the amount of energy per unit surface area per unit time per unit solid angle per unit frequency emitted at a frequency ν, the so called spectral radiance,

| $T$ is the temperature of the black body,

| $x$ is the ratio of frequency over temperature,

| $h$ is the Planck constant,

| $c$ is the speed of light,

| $k\_\backslash text\{B\}$ is the Boltzmann constant.

}}

This equation may also be written as{{cite book

|last=Modest |first=M. F.

|year=2013

|title=Radiative Heat Transfer

|publisher=Academic Press

|isbn=978-0-12-386944-9

|pages=9, 15

}}

I(\lambda, T) = \frac{2hc^2}{\lambda^5} e^{-\frac{hc}{\lambda k_\text{B} T}},

where $I(\backslash lambda,\; T)$ 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.

The peak value of this curve, as determined by setting the derivative of the equation equal to zero and solving,{{cite book

|last1=Irwin |first1=J. A.

|year=2007

|title=Astrophysics: Decoding the Cosmos

|url=https://books.google.com/books?id=7SbN3iyoNf0C&pg=PA130

|page=130

|publisher=John Wiley & Sons

|isbn=978-0-470-01306-9

}} 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.

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 $h\backslash nu\; \backslash gg\; kT$. When this is true, then

\frac{1}{e^{\frac{h\nu}{kT}} - 1} \approx e^{-\frac{h\nu}{kT}},

and so the Wien approximation gets ever closer to Planck's law as the frequency increases.

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.

• ASTM Subcommittee E20.02 on Radiation Thermometry
• Sakuma–Hattori equation
• Ultraviolet catastrophe
• Wien's displacement law

{{reflist}}

Category:Statistical mechanics

Category:Electromagnetic radiation

Category:1896 in science

Category:1896 in Germany