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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 natural linewidth and spontaneous emission rates), which I am still learning about. | 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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| $ { \huge \int } { \Large { \omega^4 \over { ( \omega^2 ~-~ \omega_0^2 )^2 ~+~ \gamma^2 \omega^2 } } ~ } { d \omega } ~=~~~~ { \Large \omega ~~ + } $ $ { { \large \left( 2 \omega_0^4 ~+~ 2 \omega_0^2 \gamma \left( \sqrt{ \gamma^2 ~-~ 4 \omega_0^2 } ~-~ 2 \gamma \right) ~+~ \gamma^3 \left( \gamma - \sqrt{ \gamma^2 ~-~ 4 \omega_0^2 } \right) \right) ~~ \tan^{-1} \left( { \Large { { \huge \omega } \over { \sqrt{ { \Large { \gamma \over 2 } } \left( \gamma ~-~ \sqrt{ \gamma^2 ~-~ 4 \omega_0^2 } \right) ~-~ \omega_0^2 } } } } \right) } \over { \large \sqrt{ 2 } ~ \gamma ~ \sqrt{ \gamma^2 ~-~ 4 \omega_0^2 } ~ \sqrt{ \gamma ~ \left( \gamma ~-~ \sqrt{ \gamma^2 ~-~ 4 \omega_0^2 } \right) ~-~ 2 \omega_0^2 } } } ~~ {\Large - } $ $ { { \large \left( 2 \omega_0^4 ~-~ 2 \omega_0^2 \gamma \left( \sqrt{ \gamma^2 ~-~ 4 \omega_0^2 } ~+~ 2 \gamma \right) ~+~ \gamma^3 \left( \gamma + \sqrt{ \gamma^2 ~-~ 4 \omega_0^2 } \right) \right) ~~ \tan^{-1} \left( { \Large { { \huge \omega } \over { \sqrt{ { \Large { \gamma \over 2 } } \left( \gamma ~+~ \sqrt{ \gamma^2 ~-~ 4 \omega_0^2 } \right) ~-~ \omega_0^2 } } } } \right) } \over { \large \sqrt{ 2 } ~ \gamma ~ \sqrt{ \gamma^2 ~-~ 4 \omega_0^2 } ~ \sqrt{ \gamma ~ \left( \gamma ~+~ \sqrt{ \gamma^2 ~-~ 4 \omega_0^2 } \right) ~-~ 2 \omega_0^2 } } } $ |
$ { \huge \int } { \Large { \omega^4 \over { ( \omega^2 ~-~ \omega_0^2 )^2 ~+~ \gamma^2 \omega^2 } } ~ } { d \omega }$ |
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| 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, and slightly reordered for visual symmetry. | 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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| A frightening number of radicals that could contain negative values, depending on the values of $ \omega_0 $ and $ \gamma $; we do know that $ \omega_0 > \gamma > 0 $, which suggests a heap of complex numbers. ---- |
So, until I find accurate information, I'll just assume the low frequency polarization. |
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| So, let's cheat. The solar irradiance data I have is actually in terms of wavelength; nanometers, not Terahertz. The wavelength $ \lambda = 2 \pi c / \omega $; let's derive the cross section in terms of wavelength. Integrating wavelength over small bins should yield similar results | Since the solar spectrum drops off exponentially in the UV, Rayleigh scattering seems to peak in the near-UV, around 300 nm wavelength. |
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| The actual equation will be a long series of many such terms, one per resonance. And it will really be computed with a C program, which iterates over the resonances, then over the frequency bins and values for the average vacuum solar spectrum. Much information missing, many opportunities for mistakes, and a serious lack of empirical data to compare it with. |
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.