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I've read that during Rayleigh Scattering, a photon passes through a molecule and the photon's electric field creates a dipole in the molecules. But what I don't understand is what happens next. How do the molecules physically "scatter" the light? For example, the sky is blue because blue light gets scattered by the gas molecules in the atmosphere. But what physical process occurs that molecules can change the path of photons?

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Raman scattering is an inelastic scattering of light where as Rayleigh is an elastic scattering.

When EM waves interacts with the lattice molecules, the electrons are perturbed periodically with the same frequency as the incident wave. The oscillation or perturbation of electron cloud results in a periodic separation of charge with in the molecules, which is called induced dipole moment. The induced dipole moment becomes a source of EM radiation, resulting in the scattered light. The scattering has no preferential direction, it happens in all direction except the direction of the induced dipole moment.This scattering will be having the same frequency as the incoming wave and is called Rayleigh scattering.

In case of Raman scattering, the induced oscillating dipole moment will be modulated by some internal motions in the molecule like vibrational or rotational motion resulting in additional frequencies like Stokes and Antistokes.

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Raleigh scattering is the elastic scattering of light on particles/molecules much smaller than the wave length. The electric field of incident light excites dipole oscillations which emit (like a dipole antenna) electromagnetic radiation (a photon) with the same frequency but, in general, a different (random) direction. The intensity of Raleigh scattering is proportional to $f^4$, the 4th power of the light frequency, just like the electromagnetic wave emission intensity of an oscillating electric dipole.

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Actually the particles gain and re emit photons with a different frequency which causes scattering.

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