Rayleigh scattering

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

Rayleigh scattering (pronounced /ˈreɪli/ RAY-lee), named after the British physicist Lord Rayleigh (John William Strutt),^[1] is the (dominantly) elastic scattering of light or other electromagnetic radiation by particles much smaller than the wavelength of the radiation. Rayleigh scattering does not change the state of material, hence it is a parametric process. 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.

Rayleigh scattering of molecular nitrogen and oxygen in the atmosphere includes elastic scattering as well as the inelastic contribution from rotational Raman scattering in air, since the changes in wavenumber of the scattered photon are typically smaller than 50 cm⁻¹.^[2] This can lead to changes in the rotational state of the molecules. Furthermore, the inelastic contribution has the same wavelengths dependency as the elastic part.

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 often parameterized by the ratio

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

where r is its characteristic length (radius) and λ is the wavelength of the light. The amplitude of light scattered from within any transparent dielectric is proportional to the inverse square of its wavelength and to the volume of material, that is to the cube of its characteristic length. The wavelength dependence is characteristic of dipole scattering^[3] and the volume dependence will apply to any scattering mechanism. Objects with x ≫ 1 act as geometric shapes, scattering light according to their projected area. At the intermediate x ≃ 1 of Mie scattering, interference effects develop through phase variations over the object's surface. Rayleigh scattering applies to the case when the scattering particle is very small (x ≪ 1, with a particle size < 1 /10 wavelength^[4]) and the whole surface re-radiates with the same phase. Because the particles are randomly positioned, the scattered light arrives at a particular point with a random collection of phases; it is incoherent and the resulting intensity is just the sum of the squares of the amplitudes from each particle and therefore proportional to the inverse fourth power of the wavelength and the sixth power of its size.^[3][5] In detail, the intensity I of light scattered by any one of the small spheres of diameter d and refractive index n from a beam of unpolarized light of wavelength λ and intensity I₀ is given by

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

where R is the distance to the particle and θ is the scattering angle. Averaging this over all angles gives the Rayleigh scattering cross-section^[7]

$\sigma_\text{s} = \frac{ 2 \pi^5}{3} \frac{d^6}{\lambda^4} \left( \frac{ n^2-1}{ n^2+2 } \right)^2$.^[8]

The fraction of light scattered by a group of scattering particles is the number of particles per unit volume N times the cross-section. 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).^[9] This means that at atmospheric pressure, where there are about 2×10²⁵ molecules per cubic meter, about a fraction 10⁻⁵ of the 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.

From molecules

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

The expression above can also be written in terms of individual molecules by expressing the dependence on refractive index in terms of the molecular polarizability α, proportional to the dipole moment induced by the electric field of the light. In this case, the Rayleigh scattering intensity for a single particle is given in CGS-units by^[10]

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

Reason for the blue color of the sky

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

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.

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. 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 density of air and particles near the earth's surface 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).

In optical fibers

Rayleigh scattering is an important component of the scattering of optical signals in optical fibers. Silica fibers are glasses, disordered materials with microscopic variations of density and refractive index. These give rise to energy losses due to the scattered light, with the following coefficient:^[11]

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

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. T_f 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.^[12]

λ⁻⁴ Rayleigh-type scattering can also be exhibited by porous materials. An example is the strong optical scattering by nanoporous materials.^[13] 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.

See also

• Rayleigh sky model
• Rayleigh fading
• Ricean fading
• Raman scattering
• Optical phenomenon
• Dynamic light scattering
• Tyndall effect
• Critical opalescence
• Marian Smoluchowski
• Rayleigh Criterion
• Aerial perspective
• Parametric Process
• Bragg's law

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. A. T. Young, "Rayleigh scattering," Appl. Opt. 20, 522–535, 1981
3. "Cornell lectures" (PDF). Retrieved 2 April 2014.
4. http://hyperphysics.phy-astr.gsu.edu/hbase/atmos/blusky.html
5. Barnett, C.E. (1942). "Some application of wavelength turbidimetry in the infrared". J.Phys.Chem 46 (1): 69–75. doi:10.1021/j150415a009.
6. Seinfeld and Pandis, Atmospheric Chemistry and Physics, 2nd Edition, John Wiley and Sons, New Jersey 2006, Chapter 15.1.1
7. Cox, A.J. (2002). "An experiment to measure Mie and Rayleigh total scattering cross sections". American Journal of Physics 70: 620. Bibcode:2002AmJPh..70..620C. doi:10.1119/1.1466815.
8. Siegel, R., Howell, J.R., (2002). Thermal radiation heat transfer. p. 480. New York, NY: Taylor & Francis.
9. Sneep, Maarten; Ubachs, Wim (2005). "Direct measurement of the Rayleigh scattering cross section in various gases". Journal of Quantitative Spectroscopy and Radiative Transfer 92: 293. Bibcode:2005JQSRT..92..293S. doi:10.1016/j.jqsrt.2004.07.025.
10. Rayleigh scattering at Hyperphysics
11. K.Rajagopal, Textbook on Engineering Physics, PHI, New Delhi 2008, part I, Chapt. 3
12. Blue & red | Causes of Color
13. T. Svensson & Z. Shen, "Laser spectroscopy of gas confined in nanoporous materials", Applied Physics Letters 96, 021107 (2010).

• 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.

Further reading

• Pedro Lilienfeld, "A Blue Sky History." (2004). Optics and Photonics News. Vol. 15, Issue 6, pp. 32–39. doi:10.1364/OPN.15.6.000032. Gives a brief history of theories of why the sky is blue leading up to Rayleigh's discovery, and a brief description of Rayleigh scattering.

External links

• HyperPhysics description of Rayleigh scattering
• Full physical explanation of sky color, in simple terms