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The Planck Blackbody Formula in Units of Wavelength

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Units of Wavelength

(click on equations to view enlarged)

For many applications, particularly when dealing in the infrared region of the spectrum, the preferred spectral unit is wavelength in μm, λ = 10c/ν.  We can deduce the spectral radiance per μm, Lλ, from (1) by noting that

With this, and substituting ν = 10c/λ into (1), the spectral radiance per μm is:

(7)

To find the wavelength of the peak, we set the derivative to zero:

Letting x = 106hc/λkT

, we arrive at the transcendental equation 5(1 - e-x) = x, whose numerical solution, x = a5 ≈ 4.96511423174  provides

(8)

The peak value, found by substituting (8) into (7), is

(9)

As we did above with spectral units of Hz, we can derive these radiometric quantities in terms of photons per second.   Dividing (7) by the energy of a photon, 10hc/λ, gives the spectral photon radiance,

.                   (10)

The wavelength where this peaks is found by differentiating:

.                                           (11)

The peak spectral photon radiance is

.                       (12)

Lλ and LλP are shown in Fig. 2.  Note that as with units of Hz, the spectral radiance and spectral photon radiance have different behaviors, and distinctly different temperature dependences.

Fig. 2:  Spectral radiance, Lλ, (top) and the spectral photon radiance, LλP, (bottom) as a function of wavelength, λ, for various temperatures.  The small black dots indicate the wavelength and value of the peak, at 10 K temperature intervals.  Note that Lλ and  LλP have different wavelength dependences.  Although the peak wavelength is inversely proportional to T for both quantities, LλP peaks at a longer wavelength than Lλ.  Furthermore, the peak value of Lλ increases as 5, whereas the peak value of LλP increases as 4.

 Calculation of a Blackbody Radiance Units of Frequency Units of Wavelength Units of Wavenumbers Radiance: Integrating the Planck Equation In-band Radiance: Integrating the Planck Equation over a Finite Range Appendix A: Algorithms for Computing In-band Radiance Appendix B: The Doppler Effect Appendix C: Summary of Formulas References Blackbody Calculator Print Version

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Transmittance: ratio of received radiation intensity, I, to incident light intensity, I0

Transmittance: ratio of received radiation intensity, I, to incident light intensity, I0

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light with wavenumber between σ and σ + dσ

light with wavenumber between σ and σ + dσ
Isotopes are forms of an element whose nuclei have the same atomic number, the number of protons in the nucleus,but different atomic masses because they contain different numbers of neutrons.
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Wavenumber cm-1: the number of
wavelengths of light per centimeter

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VMR: volume mixing ratio. The fractional number of molecules of a species in a volume.

Individual vmrs and their sum must be between 0 and 1.

If the vmrs sum to less than 1, the rest of the gas in the cell is assumed transparent.(Lineshapes for molecules with vmr less than 1 are air-broadened.)
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Temperature Offset: The model atmosphere (US_Standard, Tropical, etc.) determines the temperature, pressure and gas concentrations at each height in the atmosphere. To adjust the temperature from the model value, enter a temperature offset (from -50 to 50 K). The Atmosphere Browser tool displays the temperature profiles for the model atmospheres.
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