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← Revision 162 as of 2019-09-21 06:23:24 ⇥
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| Scattering is due to the polarization of species. The polarization can be summed from the behavior of individual resonances and damping factors (related to resonance bandwidth), which I have not yet been able to find. For mostly-isolated atoms in high vacuum, [[ | Beers line broadening ]] will not be relevant; the bandwidth $ \gamma $ is related to damping time, TBD . note: perhaps I can get the relevant numbers from [[ | HITRAN ]], but many of the resonances (especially for tightly bound molecules and deep atomic orbitals) will be far into the ultraviolet, where HITRAN may not go. TBD |
Scattering is due to the polarization of species. The polarization can be summed from the behavior of individual resonances and damping factors (related to natural linewidth and spontaneous emission rates), which I am still learning about. I |
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| The scattering from a single resonator at frequency $ \omega $ is (from Feynman Lectures on Physics Chapter X page X) proportional to | ----- |
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| $ \integral { \omega^4 \over { ( \omega^2 - {\omega_0}^2 )^2 - \gamma^2 omega^2 } ~ = ~ $ | The scattering cross section from a single resonator at frequency $ ~ \large \omega ~ $ is derived from [[ http://www.feynmanlectures.caltech.edu/I_32.html | Feynman Lectures on Physics, volume one, chapter 32, equation 32.15 ]], and is proportional to: |
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| $ \left( \left( \left( 2 {\omega_0}^4 - {\omega_0}^2 \gamma \left( \sqrt{ 4 {\omega_0}^2 + \gamma^2 } - 2 \gamma \right) + \gamma^3 \left( \gamma - \sqrt{ 4 {\omega_0}^2 + \gamma^2 } \right) \right $ |
$ { \huge \int } { \Large { \omega^4 \over { ( \omega^2 ~-~ \omega_0^2 )^2 ~+~ \gamma^2 \omega^2 } } ~ } { d \omega }$ |
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| $ atan( x \over \sqrt{ { 0.5 \gamma \left( \sqrt{ 4 {\omega_0}^2 + \gamma^2 } - \gamma \right) - {\omega_0}^2 } \right) $ | That can be integrated using [[ https://www.wolframalpha.com/input/?i=integral+of+x%5E4%2F%28%28x%5E2-a%5E2%29%5E2+%2B+x%5E2*b%5E2%29 | Wolfram alpha ]] for the integration, but that spews frightening number of radicals containing negative values, given that $ \omega_0 > \gamma > 0 $. And I don't have a list of resonances ( $\omega_0$ ) and resonance bandwidths ($\gamma$) anyway; the integration would be highly sensitive to resonance bandwidth if the resonances occur in the middle of the Rayleigh scattering maximum of the solar spectrum. |
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| $ \left( { \sqrt{2} \gamma \sqrt{ 4 {\omega_0}^2 + \gamma^2 } \sqrt{ \gamma \left{ \sqrt{ 4 {\omega_0}^2 + \gamma^2 } - 2 \gamma \right) - 2 {\omega_0}^2 } right) } right) $ | So, until I find accurate information, I'll just assume the low frequency polarization. Since the solar spectrum drops off exponentially in the UV, Rayleigh scattering seems to peak in the near-UV, around 300 nm wavelength. {{ attachment:spectrumgifKL.png }} |
Rayleigh Scattering of Isolated Species
( Species == ions, atoms, molecules )
- See Polarization for the low wavenumber (frequency / speed of light ) approximation used for Rayleigh scattering.
Scattering is due to the polarization of species. The polarization can be summed from the behavior of individual resonances and damping factors (related to natural linewidth and spontaneous emission rates), which I am still learning about. I
The scattering cross section from a single resonator at frequency ~ \large \omega ~ is derived from Feynman Lectures on Physics, volume one, chapter 32, equation 32.15, and is proportional to:
{ \huge \int } { \Large { \omega^4 \over { ( \omega^2 ~-~ \omega_0^2 )^2 ~+~ \gamma^2 \omega^2 } } ~ } { d \omega }
That can be integrated using Wolfram alpha for the integration, but that spews frightening number of radicals containing negative values, given that \omega_0 > \gamma > 0 . And I don't have a list of resonances ( \omega_0 ) and resonance bandwidths (\gamma) anyway; the integration would be highly sensitive to resonance bandwidth if the resonances occur in the middle of the Rayleigh scattering maximum of the solar spectrum.
So, until I find accurate information, I'll just assume the low frequency polarization.
Since the solar spectrum drops off exponentially in the UV, Rayleigh scattering seems to peak in the near-UV, around 300 nm wavelength.
