Rayleigh Scattering of Isolated Species

( Species == ions, atoms, molecules )

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.

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 } ~=~~~~ { \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 } } }

Using Wolfram alpha for the integration, and slightly reordered for visual symmetry.

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, 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

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.