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Originally Posted by George
I assume the photon transfers its energy to an electron, raising it to the next, or higher, energy shell. Then, the reverse happens allowing the resurrected photon, if you will, to take off in any direction, I assume. Is this a fair picture? If so, wouldn't this greatly slow down the propogation rate of light through a medium. Is this random walking, or just random thinking? 
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Yup, on all counts. Except for the random thinking.
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Originally Posted by George
I don't understand the true use of the term. I simply read it somewhere and it stuck. Further help would be appreciated.
[BTW, I want to observe blues, not give them to others.  ]
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Well, when it looks like a white haze, it's Mie, but when it looks blue, it's Rayleigh (because Rayleigh favors blue, since the scatterers are so small they oscillate faster for higher frequency light, but Mie scattering has scatterers on the same size as the wavelength, so short wavelength light is less favored since the scatterer does not respond entirely coherently. Coherent enough?)