12 03 2008

# Rayleigh scattering

Rayleigh scattering causes the blue hue of the daytime sky and the reddening of the sun at sunset

Rayleigh scattering (named after Lord Rayleigh) is the scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of the light. It can occur when light travels in transparent solids and liquids, but is most prominently seen in gases. Rayleigh scattering of sunlight in clear atmosphere is the main reason why the sky is blue. Rayleigh and cloud-mediated scattering contribute to diffuse light (direct light being sunrays).

For scattering by particles similar to or larger than a wavelength, see Mie theory or discrete dipole approximation (they apply to the Rayleigh regime as well).

## Small size parameter approximation

Figure showing the more intense scattering of blue light by the atmosphere relative to red light.

The size of a scattering particle is parametrized by the ratio x of its characteristic dimension r and wavelength λ:

$x = \frac{2 \pi r} {\lambda}$.

Rayleigh scattering can be defined as scattering in small size parameter regime $x \ll 1$. The amount of Rayleigh scattering that occurs to a beam of light is dependent upon the size of the particles and the wavelength of the light; in particular, the scattering coefficient, and hence the intensity of the scattered light, varies for small size parameter inversely with the fourth power of the wavelength. Scattering from larger spherical particles is explained by the Mie theory for arbitrary size parameter x including small size parameter – in this case Mie theory reduces to Rayleigh approximation.

The intensity I of light scattered by a single small particle from a beam of unpolarized light of wavelength λ and intensity I0 is given by:

$I = I_0 \frac{ 1+\cos^2 \theta }{2 R^2} \left( \frac{ 2 \pi }{ \lambda } \right)^4 \left( \frac{ n^2-1}{ n^2+2 } \right)^2 \left( \frac{d}{2} \right)^6,$

where R is the distance to the particle, θ is the scattering angle, n is the refractive index of the particle, and d is the diameter of the particle.

The angular distribution of Rayleigh scattering, governed by the (1+cos2θ) term, is symmetric in the plane normal to the incident direction of the light, and so the forward scatter equals the backwards scatter. Integrating over the sphere surrounding the particle gives the Rayleigh scattering cross section σs:

$\sigma_s = \frac{ 2 \pi^5}{3} \frac{d^6}{\lambda^4} \left( \frac{ n^2-1}{ n^2+2 } \right)^2 .$

The Rayleigh scattering coefficient for a group of scattering particles is the number of particles per unit volume N times the cross-section. As with all wave effects, in incoherent scattering the scattered powers add arithmetically, while in coherent scattering, such as if the particles are very near each-other, the fields add arithmetically and the sum must be squared to obtain the total scattered power.

The strong wavelength dependence of the scattering (~λ-4) means that blue light is scattered much more than red light. In the atmosphere, this results in blue wavelength being scattered to a greater extent than longer wavelengths, and so one sees blue light coming from all regions of the sky. Direct radiation (from definition) is coming directly from the Sun. Rayleigh scattering is a good approximation to the manner in which light scattering occurs within various media for which scattering particles have small size parameter.

## Why is the sky blue?

When one looks at the sky during the day, rather than seeing the black of space, one sees light from Rayleigh scattering off the air. Rayleigh scattering is inversely proportional to the fourth power of wavelength, which means that the shorter wavelength of blue light will scatter more than the longer wavelengths of green and red light. This gives the sky a blue appearance. Conversely, when one looks towards the sun at sunset, one sees the colors that were not scattered away — the longer wavelength, red light.

While Rayleigh scattering explains the blue color, there wouldn’t be any light at all without some particles to do the scattering. Some of the scattering is from aerosols of sulfate particles. For years following large Plinian eruptions, the blue cast of the sky is notably brightened due to the persistent sulfate load of the stratospheric eruptive gases. Another source of scattering is from microscopic density fluctuations, resulting from the random motion of the air molecules. A region of higher or lower density has a slightly different refractive index than the surrounding medium, and therefore it acts like a short-lived particle that can scatter light.

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6 04 2011

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