Wien approximation

Details

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.

The law may be written as $${\displaystyle I(\nu ,T)=\frac{2h{\nu }^3}{c^2}{e}^{-\frac{h\nu }{k_{\text{B}}T}},}$$ (note the simple exponential frequency dependence of this approximation) or, by introducing natural Planck units, $${\displaystyle I(\nu ,x)=2{\nu }^3{e}^{-x},}$$ where:

• ${\displaystyle I(\nu ,T)}$ 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,
• ${\displaystyle T}$ is the temperature of the black body,
• ${\displaystyle x}$ is proportional to the ratio of frequency over temperature,
• ${\displaystyle h}$ is the Planck constant,
• ${\displaystyle c}$ is the speed of light,
• ${\displaystyle k_{\text{B}}}$ is the Boltzmann constant.

This equation may also be written as $${\displaystyle I(\lambda ,T)=\frac{2h{c}^2}{{\lambda }^5}{e}^{-\frac{hc}{\lambda k_{\text{B}}T}},}$$ where ${\displaystyle I(\lambda ,T)}$ 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.

The peak value of this curve, as determined by setting the derivative of the equation equal to zero and solving, occurs at a wavelength $${\displaystyle {\lambda }_{\text{max}}=\frac{hc}{5k_{\text{B}}T}\approx \frac{0.2878\mathrm{c}\mathrm{m}\cdot \mathrm{K}}{T},}$$ and frequency $${\displaystyle {\nu }_{\text{max}}=\frac{3k_{\text{B}}T}{h}\approx 6.25\times {10}^{10}\frac{\mathrm{H}\mathrm{z}}{\mathrm{K}}\cdot T.}$$