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The Planck Blackbody Formula in Units of Wavenumbers

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Units of Wavenumbers

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Yet a third spectral unit, commonly used in spectroscopy, is wavenumber, the number of waves per cm: σ = ν /100c  cm-1.  Converting (1) to these units gives

.

(13)

Again, the peak is where the derivative with respect to wavenumber vanishes:

so

.                                            (14)

The peak value is

.                    (15)

The spectral photon radiance is found by dividing Lσ by the energy of a photon, 100hcσ :

.      (16)

We next find the wavenumber at the peak of the spectral photon radiance:

and                                                .                                            (17)

The peak spectral photon radiance is

.                    (18)

Fig. 3 shows plots of Lσ and LσP for various temperatures.  Note again the important difference between the spectral radiance and spectral photon radiance.

Fig. 3 - Spectral radiance, Lσ , (top) and the spectral photon radiance, LσP, (bottom) as a function of wavenumber, σ, for various temperatures.  The small black dots indicate the wavenumber and value of the peak, at 10 K temperature intervals.  Note that Lσ and  LσP have different wavenumber dependences.  Although the peak wavenumber is proportional to T for both quantities, Lσ peaks at a higher wavenumber than LσP.  Furthermore, the peak value of Lσ increases as 3, whereas the peak value of LσP increases as 2.

 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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