# Rayleigh scattering

Rayleigh scattering causes the blue hue of the daytime sky and the reddening of the sun at sunset.
Rayleigh scattering is more evident after sunset. This picture was taken about one hour after sunset at 500m altitude, looking at the horizon where the sun had set.
The beam of a 5 mW green laser pointer is visible at night partly because of Rayleigh scattering on various particles and molecules present in air.

Rayleigh scattering, named after the British physicist Lord Rayleigh,[1] is the elastic scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of the light. The particles may be individual atoms or molecules. It can occur when light travels through transparent solids and liquids, but is most prominently seen in gases. Rayleigh scattering results from the electric polarizability of the particles. The oscillating electric field of a light wave acts on the charges within a particle, causing them to move at the same frequency. The particle therefore becomes a small radiating dipole whose radiation we see as scattered light.

Rayleigh scattering of sunlight in the atmosphere causes diffuse sky radiation, which is the reason for the blue color of the sky and the yellow tone of the sun itself.

Scattering by particles similar to or larger than the wavelength of light is typically treated by the Mie theory, the discrete dipole approximation and other computational techniques. Rayleigh scattering applies to particles that are small with respect to wavelengths of light, and that are optically "soft" (i.e. with a refractive index close to 1). On the other hand, Anomalous Diffraction Theory applies to optically soft but larger particles.

## Small size parameter approximation

The size of a scattering particle is parameterized 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 the small size parameter regime x ≪ 1. Scattering from larger spherical particles is explained by the Mie theory for an arbitrary size parameter x. For small x the Mie theory reduces to the Rayleigh approximation.

The amount of Rayleigh scattering that occurs for a beam of light depends upon the size of the particles and the wavelength of the light. Specifically, the intensity of the scattered light varies as the sixth power of the particle size, and varies inversely with the fourth power of the wavelength.[2]

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:

$I = I_0 \frac{ 1+\cos^2 \theta }{2 R^2} \left( \frac{ 2 \pi }{ \lambda } \right)^4 \left( \frac{ n^2-1}{ n^2+2 } \right)^2 \left( \frac{d}{2} \right)^6$[3]

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 Rayleigh scattering cross-section is given by

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

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, for incoherent scattering the scattered powers add arithmetically, while for 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.

## From molecules

Figure showing the greater proportion of blue light scattered by the atmosphere relative to red light.

Rayleigh scattering also occurs from individual molecules. Here the scattering is due to the molecular polarizability α, which describes how much the electrical charges on the molecule will move in an electric field. In this case, the Rayleigh scattering intensity for a single particle is given by[4]

$I = I_0 \frac{8\pi^4\alpha^2}{\lambda^4 R^2}(1+\cos^2\theta).$

The amount of Rayleigh scattering from a single particle can also be expressed as a cross section σ. For example, the major constituent of the atmosphere, nitrogen, has a Rayleigh cross section of 5.1×10−31 m2 at a wavelength of 532 nm (green light).[5] This means that at atmospheric pressure, about a fraction 10−5 of light will be scattered for every meter of travel.

The strong wavelength dependence of the scattering (~λ−4) means that shorter (blue) wavelengths are scattered more strongly than longer (red) wavelengths. This results in the indirect blue light coming from all regions of the sky. Rayleigh scattering is a good approximation of the manner in which light scattering occurs within various media for which scattering particles have a small size parameter.

## Reason for the blue color of the sky

Scattered blue light is polarized. The picture on the right is shot through a polarizing filter which removes light that is linearly polarized in a specific direction.

A portion of the beam of light coming from the sun scatters off molecules of gas and other small particles in the atmosphere. Here, Rayleigh scattering primarily occurs through sunlight's interaction with randomly located air molecules. Exactly equivalently, but from a purely macroscopic point of view, the scattering comes from the microscopic density fluctuations which result from the random distribution of molecules in the air. A region of higher or lower density has a slightly different refractive index from the surrounding medium, and therefore it acts like a short-lived scattering particle. It is this scattered light that gives the surrounding sky its brightness and its color. As previously stated, Rayleigh scattering is inversely proportional to the fourth power of wavelength, so that shorter wavelength violet and blue light will scatter more than the longer wavelengths (yellow and especially red light). However, the Sun, like any star has its own spectrum and so I0 in the scattering formula above is not constant but falls away in the violet. In addition the oxygen in the Earth's atmosphere absorbs wavelengths at the edge of the ultra-violet region of the spectrum. The resulting color, which appears like a pale blue, actually is a mixture of all the scattered colors, mainly blue and green. Conversely, glancing toward the sun, the colors that were not scattered away — the longer wavelengths such as red and yellow light — are directly visible, giving the sun itself a slightly yellowish hue. Viewed from space, however, the sky is black and the sun is white.

The reddening of sunlight is intensified when the sun is near the horizon, because the volume of air through which sunlight must pass is significantly greater than when the sun is high in the sky. The Rayleigh scattering effect is thus increased, removing virtually all blue light from the direct path to the observer. The remaining unscattered light is mostly of a longer wavelength, and therefore appears to be orange.

Some of the scattering can also be from sulfate particles. For years after large Plinian eruptions, the blue cast of the sky is notably brightened by the persistent sulfate load of the stratospheric gases. Some works of the artist J. M. W. Turner may owe their vivid red colours to the eruption of Mount Tambora in his lifetime.

In locations with little light pollution, the moonlit night sky is also blue, because moonlight is reflected sunlight, with a slightly lower color temperature due to the brownish color of the moon. The moonlit sky is not perceived as blue, however, because at low light levels human vision comes mainly from rod cells that do not produce any color perception (Purkinje effect).[citation needed]

## In optical fibers

Rayleigh scattering is an important component of the scattering of optical signals in optical fibers. Silica fibers are disordered materials, thus their density varies on a microscopic scale. The density fluctuations give rise to energy loss due to the scattered light, with the following coefficient:[6]

$\alpha_\text{scat} = \frac{8 \pi^3}{3 \lambda^4} n^8 p^2 k T_\text{f} \beta$

where n is the refraction index, p is the photoelastic coefficient of the glass, k is the Boltzmann constant, and β is the isothermal compressibility. Tf is a fictive temperature, representing the temperature at which the density fluctuations are "frozen" in the material.

## In porous materials

Rayleigh scattering in opalescent glass: it appears blue from the side, but orange light shines through.[7]

λ−4 Rayleigh-type scattering can also be exhibited by porous materials. An example is the strong optical scattering by nanoporous materials.[8] The strong contrast in refractive index between pores and solid parts of sintered alumina results in very strong scattering, with light completely changing direction each 5 micrometers on average. The λ−4-type scattering is caused by the nanoporous structure (a narrow pore size distribution around ~70 nm) obtained by sintering monodispersive alumina powder.

## References

1. ^ Lord Rayleigh (John Strutt) refined his theory of scattering in a series of papers that were issued over a period of decades. Here is a partial list of those papers:
1. John Strutt (1871) "On the light from the sky, its polarization and colour," Philosophical Magazine, series 4, vol.41, pages 107-120, 274-279.
2. John Strutt (1871) "On the scattering of light by small particles," Philosophical Magazine, series 4, vol. 41, pages 447-454.
3. John Strutt (1881) "On the electromagnetic theory of light," Philosophical Magazine, series 5, vol. 12, pages 81-101.
4. John Strutt (1899) "On the transmission of light through an atmosphere containing small particles in suspension, and on the origin of the blue of the sky," Philosophical Magazine, series 5, vol. 47, pages 375-394.
2. ^ Barnett, C.E. (1942). "Some application of wavelength turbidimetry in the infrared". J.Phys.Chem 46 (1): 69–75.
3. ^ Seinfeld and Pandis, Atmospheric Chemistry and Physics, 2nd Edition, John Wiley and Sons, New Jersey 2006, Chapter 15.1.1
4. ^ Rayleigh scattering at Hyperphysics
5. ^ Maarten Sneep and Wim Ubachs, Direct measurement of the Rayleigh scattering cross section in various gases. Journal of Quantitative Spectroscopy and Radiative Transfer, 92, 293 (2005).
6. ^ K.Rajagopal, Textbook on Engineering Physics, PHI, New Delhi 2008, part I, Chapt. 3
7. ^ Blue & red | Causes of Color
8. ^ T. Svensson & Z. Shen, "Laser spectroscopy of gas confined in nanoporous materials", Applied Physics Letters 96, 021107 (2010). [1]
• C.F. Bohren, D. Huffman, Absorption and scattering of light by small particles, John Wiley, New York 1983. Contains a good description of the asymptotic behavior of Mie theory for small size parameter (Rayleigh approximation).
• Ditchburn, R.W. (1963). Light (2nd ed.). London: Blackie & Sons. pp. 582–585. ISBN 0-12-218101-8.
• Chakraborti, Sayan (September 2007). "Verification of the Rayleigh scattering cross section". American Journal of Physics 75 (9): 824−826. arXiv:physics/0702101. Bibcode:2007AmJPh..75..824C. doi:10.1119/1.2752825.
• Ahrens, C. Donald (1994). Meteorology Today: an introduction to weather, climate, and the environment (5th ed.). St. Paul MN: West Publishing Company. pp. 88–89. ISBN 0-314-02779-3.